CINXE.COM
GitHub - kroitor/gjk.c: Gilbert-Johnson-Keerthi (GJK) collision detection algorithm in 200 lines of clean plain C
<!DOCTYPE html> <html lang="en" data-color-mode="auto" data-light-theme="light" data-dark-theme="dark" data-a11y-animated-images="system" data-a11y-link-underlines="true" > <head> <meta charset="utf-8"> <link rel="dns-prefetch" href="https://github.githubassets.com"> <link rel="dns-prefetch" href="https://avatars.githubusercontent.com"> <link rel="dns-prefetch" href="https://github-cloud.s3.amazonaws.com"> <link rel="dns-prefetch" href="https://user-images.githubusercontent.com/"> <link rel="preconnect" href="https://github.githubassets.com" crossorigin> <link rel="preconnect" href="https://avatars.githubusercontent.com"> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/light-74231a1f3bbb.css" /><link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/dark-8a995f0bacd4.css" /><link data-color-theme="dark_dimmed" crossorigin="anonymous" media="all" rel="stylesheet" data-href="https://github.githubassets.com/assets/dark_dimmed-f37fb7684b1f.css" /><link data-color-theme="dark_high_contrast" crossorigin="anonymous" media="all" rel="stylesheet" data-href="https://github.githubassets.com/assets/dark_high_contrast-9ac301c3ebe5.css" /><link data-color-theme="dark_colorblind" crossorigin="anonymous" media="all" rel="stylesheet" data-href="https://github.githubassets.com/assets/dark_colorblind-cd826e8636dc.css" /><link data-color-theme="light_colorblind" crossorigin="anonymous" media="all" rel="stylesheet" data-href="https://github.githubassets.com/assets/light_colorblind-f91b0f603451.css" /><link data-color-theme="light_high_contrast" crossorigin="anonymous" media="all" rel="stylesheet" data-href="https://github.githubassets.com/assets/light_high_contrast-83beb16e0ecf.css" /><link data-color-theme="light_tritanopia" crossorigin="anonymous" media="all" rel="stylesheet" data-href="https://github.githubassets.com/assets/light_tritanopia-6e122dab64fc.css" /><link data-color-theme="dark_tritanopia" crossorigin="anonymous" media="all" rel="stylesheet" data-href="https://github.githubassets.com/assets/dark_tritanopia-18119e682df0.css" /> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/primer-primitives-225433424a87.css" /> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/primer-cba26849680f.css" /> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/global-b6cb3703b934.css" /> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/github-ea73c9cb5377.css" /> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/repository-4fce88777fa8.css" /> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/code-0210be90f4d3.css" /> <script type="application/json" id="client-env">{"locale":"en","featureFlags":["contentful_lp_flex_features_actions","contentful_lp_flex_features_codespaces","contentful_lp_flex_features_code_review","contentful_lp_flex_features_code_search","contentful_lp_flex_features_discussions","contentful_lp_flex_features_issues","copilot_immersive_issue_preview","copilot_new_references_ui","copilot_chat_repo_custom_instructions_preview","copilot_chat_show_model_picker_on_retry","copilot_no_floating_button","copilot_topics_as_references","copilot_read_shared_conversation","copilot_duplicate_thread","dotcom_chat_client_side_skills","experimentation_azure_variant_endpoint","failbot_handle_non_errors","geojson_azure_maps","ghost_pilot_confidence_truncation_25","ghost_pilot_confidence_truncation_40","github_models_gateway_parse_params","github_models_o3_mini_streaming","insert_before_patch","issues_advanced_search_has_filter","issues_react_remove_placeholders","issues_react_blur_item_picker_on_close","issues_advanced_search_nested_ownership_filters","issues_dashboard_no_redirects","marketing_pages_search_explore_provider","primer_react_css_modules_ga","react_data_router_pull_requests","remove_child_patch","report_hydro_web_vitals","sample_network_conn_type","swp_enterprise_contact_form","site_copilot_pro_plus","site_proxima_australia_update","viewscreen_sandbox","issues_react_create_milestone","issues_react_cache_fix_workaround","lifecycle_label_name_updates","copilot_task_oriented_assistive_prompts","issue_types_prevent_private_type_creation","refresh_image_video_src","codespaces_prebuild_region_target_update","copilot_code_review_sign_up_closed"]}</script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/wp-runtime-2efce53a2203.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_oddbird_popover-polyfill_dist_popover_js-9da652f58479.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_arianotify-polyfill_ariaNotify-polyfill_js-node_modules_github_mi-3abb8f-46b9f4874d95.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/ui_packages_failbot_failbot_ts-952d624642a1.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/environment-f04cb2a9fc8c.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_primer_behaviors_dist_esm_index_mjs-0dbb79f97f8f.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_selector-observer_dist_index_esm_js-f690fd9ae3d5.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_relative-time-element_dist_index_js-62d275b7ddd9.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_text-expander-element_dist_index_js-78748950cb0c.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_auto-complete-element_dist_index_js-node_modules_github_catalyst_-8e9f78-a90ac05d2469.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_filter-input-element_dist_index_js-node_modules_github_remote-inp-b5f1d7-a1760ffda83d.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_markdown-toolbar-element_dist_index_js-ceef33f593fa.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_file-attachment-element_dist_index_js-node_modules_primer_view-co-c44a69-efa32db3a345.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/github-elements-394f8eb34f19.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/element-registry-0b7798be0424.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_braintree_browser-detection_dist_browser-detection_js-node_modules_githu-2906d7-2a07a295af40.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_lit-html_lit-html_js-be8cb88f481b.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_mini-throttle_dist_index_js-node_modules_morphdom_dist_morphdom-e-7c534c-a4a1922eb55f.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_turbo_dist_turbo_es2017-esm_js-a03ee12d659a.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_remote-form_dist_index_js-node_modules_delegated-events_dist_inde-893f9f-b6294cf703b7.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_color-convert_index_js-e3180fe3bcb3.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_quote-selection_dist_index_js-node_modules_github_session-resume_-947061-e7a6c4a19f98.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/ui_packages_updatable-content_updatable-content_ts-62f3e9c52ece.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/app_assets_modules_github_behaviors_task-list_ts-app_assets_modules_github_sso_ts-ui_packages-900dde-768abe60b1f8.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/app_assets_modules_github_sticky-scroll-into-view_ts-3e000c5d31a9.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/app_assets_modules_github_behaviors_ajax-error_ts-app_assets_modules_github_behaviors_include-d0d0a6-e7f74ee74d91.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/app_assets_modules_github_behaviors_commenting_edit_ts-app_assets_modules_github_behaviors_ht-83c235-4bcbbbfbe1d4.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/behaviors-4414ad8b510b.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_delegated-events_dist_index_js-node_modules_github_catalyst_lib_index_js-f6223d90c7ba.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/notifications-global-01e85cd1be94.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_virtualized-list_es_index_js-node_modules_github_template-parts_lib_index_js-94dc7a2157c1.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_remote-form_dist_index_js-node_modules_delegated-events_dist_inde-70450e-4b93df70b903.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/app_assets_modules_github_ref-selector_ts-52913063a0b9.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/codespaces-b419a25ee02f.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_filter-input-element_dist_index_js-node_modules_github_remote-inp-3eebbd-0763620ad7bf.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_mini-throttle_dist_decorators_js-node_modules_delegated-events_di-e161aa-9d41fb1b6c9e.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_file-attachment-element_dist_index_js-node_modules_github_remote--3c9c82-b71ef90fbdc7.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/repositories-10217e4e5a53.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_mini-throttle_dist_index_js-node_modules_github_catalyst_lib_inde-dbbea9-26cce2010167.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/code-menu-d6d3c94ee97e.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/primer-react-350730ea92ff.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/react-core-a8203875c6f9.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/react-lib-1622bd1e542f.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/octicons-react-cf2f2ab8dab4.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_emotion_is-prop-valid_dist_emotion-is-prop-valid_esm_js-node_modules_emo-41b1a8-555bc0cf9cab.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_mini-throttle_dist_index_js-node_modules_stacktrace-parser_dist_s-e7dcdd-9a233856b02c.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_oddbird_popover-polyfill_dist_popover-fn_js-55fea94174bf.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/notifications-subscriptions-menu-53aa08c61b34.js"></script> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/primer-react.a490b7c9fa319e5cb069.module.css" /> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/notifications-subscriptions-menu.1bcff9205c241e99cff2.module.css" /> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/primer-react.a490b7c9fa319e5cb069.module.css" /> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/notifications-subscriptions-menu.1bcff9205c241e99cff2.module.css" /> <title>GitHub - kroitor/gjk.c: Gilbert-Johnson-Keerthi (GJK) collision detection algorithm in 200 lines of clean plain C</title> <meta name="route-pattern" content="/:user_id/:repository" data-turbo-transient> <meta name="route-controller" content="files" data-turbo-transient> <meta name="route-action" content="disambiguate" data-turbo-transient> <meta name="current-catalog-service-hash" content="f3abb0cc802f3d7b95fc8762b94bdcb13bf39634c40c357301c4aa1d67a256fb"> <meta name="request-id" content="98B2:3B7070:2A4A:31EC:67F6918E" data-pjax-transient="true"/><meta name="html-safe-nonce" content="086fc181d31bb39040aac39ea1446eeac8a38fa12c07e01179d9082f2c026cb3" data-pjax-transient="true"/><meta name="visitor-payload" content="eyJyZWZlcnJlciI6IiIsInJlcXVlc3RfaWQiOiI5OEIyOjNCNzA3MDoyQTRBOjMxRUM6NjdGNjkxOEUiLCJ2aXNpdG9yX2lkIjoiNDU1NDgyMjkxMjMyNTM1Nzk2NiIsInJlZ2lvbl9lZGdlIjoic291dGhlYXN0YXNpYSIsInJlZ2lvbl9yZW5kZXIiOiJzb3V0aGVhc3Rhc2lhIn0=" data-pjax-transient="true"/><meta name="visitor-hmac" content="0979dd1ab2d1c64b60739dac1c469ca67e2e16a9c87b74d3fdcab3c6e31d76d1" data-pjax-transient="true"/> <meta name="hovercard-subject-tag" content="repository:48733330" data-turbo-transient> <meta name="github-keyboard-shortcuts" content="repository,copilot" data-turbo-transient="true" /> <meta name="selected-link" value="repo_source" data-turbo-transient> <link rel="assets" href="https://github.githubassets.com/"> <meta name="google-site-verification" content="Apib7-x98H0j5cPqHWwSMm6dNU4GmODRoqxLiDzdx9I"> <meta name="octolytics-url" content="https://collector.github.com/github/collect" /> <meta name="analytics-location" content="/<user-name>/<repo-name>" data-turbo-transient="true" /> <meta name="user-login" content=""> <meta name="viewport" content="width=device-width"> <meta name="description" content="Gilbert-Johnson-Keerthi (GJK) collision detection algorithm in 200 lines of clean plain C - kroitor/gjk.c"> <link rel="search" type="application/opensearchdescription+xml" href="/opensearch.xml" title="GitHub"> <link rel="fluid-icon" href="https://github.com/fluidicon.png" title="GitHub"> <meta property="fb:app_id" content="1401488693436528"> <meta name="apple-itunes-app" content="app-id=1477376905, app-argument=https://github.com/kroitor/gjk.c" /> <meta name="twitter:image" content="https://opengraph.githubassets.com/66c3dee164ac86ef3ac384849d2959c9cfd638e5c22c6d5ee7d83803cef66bbd/kroitor/gjk.c" /><meta name="twitter:site" content="@github" /><meta name="twitter:card" content="summary_large_image" /><meta name="twitter:title" content="GitHub - kroitor/gjk.c: Gilbert-Johnson-Keerthi (GJK) collision detection algorithm in 200 lines of clean plain C" /><meta name="twitter:description" content="Gilbert-Johnson-Keerthi (GJK) collision detection algorithm in 200 lines of clean plain C - kroitor/gjk.c" /> <meta property="og:image" content="https://opengraph.githubassets.com/66c3dee164ac86ef3ac384849d2959c9cfd638e5c22c6d5ee7d83803cef66bbd/kroitor/gjk.c" /><meta property="og:image:alt" content="Gilbert-Johnson-Keerthi (GJK) collision detection algorithm in 200 lines of clean plain C - kroitor/gjk.c" /><meta property="og:image:width" content="1200" /><meta property="og:image:height" content="600" /><meta property="og:site_name" content="GitHub" /><meta property="og:type" content="object" /><meta property="og:title" content="GitHub - kroitor/gjk.c: Gilbert-Johnson-Keerthi (GJK) collision detection algorithm in 200 lines of clean plain C" /><meta property="og:url" content="https://github.com/kroitor/gjk.c" /><meta property="og:description" content="Gilbert-Johnson-Keerthi (GJK) collision detection algorithm in 200 lines of clean plain C - kroitor/gjk.c" /> <meta name="hostname" content="github.com"> <meta name="expected-hostname" content="github.com"> <meta http-equiv="x-pjax-version" content="c61aa93e644a864ea23b6dce812b9c50ba97f801a1e73a52746d9df15f3b80ae" data-turbo-track="reload"> <meta http-equiv="x-pjax-csp-version" content="e26f9f0ba624ee85cc7ac057d8faa8618a4f25a85eab052c33d018ac0f6b1a46" data-turbo-track="reload"> <meta http-equiv="x-pjax-css-version" content="205838381d6e5f35c535dbb12458f905bc43e0b186c86bf75aabbd0c0f36537c" data-turbo-track="reload"> <meta http-equiv="x-pjax-js-version" content="2402d76e6eeebf5456b7ad77a50a8c5b9a1fe9fc9d4345abe8b1423242287bc5" data-turbo-track="reload"> <meta name="turbo-cache-control" content="no-preview" data-turbo-transient=""> <meta data-hydrostats="publish"> <meta name="go-import" content="github.com/kroitor/gjk.c git https://github.com/kroitor/gjk.c.git"> <meta name="octolytics-dimension-user_id" content="1294454" /><meta name="octolytics-dimension-user_login" content="kroitor" /><meta name="octolytics-dimension-repository_id" content="48733330" /><meta name="octolytics-dimension-repository_nwo" content="kroitor/gjk.c" /><meta name="octolytics-dimension-repository_public" content="true" /><meta name="octolytics-dimension-repository_is_fork" content="false" /><meta name="octolytics-dimension-repository_network_root_id" content="48733330" /><meta name="octolytics-dimension-repository_network_root_nwo" content="kroitor/gjk.c" /> <link rel="canonical" href="https://github.com/kroitor/gjk.c" data-turbo-transient> <meta name="turbo-body-classes" content="logged-out env-production page-responsive"> <meta name="browser-stats-url" content="https://api.github.com/_private/browser/stats"> <meta name="browser-errors-url" content="https://api.github.com/_private/browser/errors"> <meta name="release" content="6e6b03118d0c671eea4b816f3732461dd6c0bac9"> <link rel="mask-icon" href="https://github.githubassets.com/assets/pinned-octocat-093da3e6fa40.svg" color="#000000"> <link rel="alternate icon" class="js-site-favicon" type="image/png" href="https://github.githubassets.com/favicons/favicon.png"> <link rel="icon" class="js-site-favicon" type="image/svg+xml" href="https://github.githubassets.com/favicons/favicon.svg" data-base-href="https://github.githubassets.com/favicons/favicon"> <meta name="theme-color" content="#1e2327"> <meta name="color-scheme" content="light dark" /> <link rel="manifest" href="/manifest.json" crossOrigin="use-credentials"> </head> <body class="logged-out env-production page-responsive" style="word-wrap: break-word;"> <div data-turbo-body class="logged-out env-production page-responsive" style="word-wrap: break-word;"> <div class="position-relative header-wrapper js-header-wrapper "> <a href="#start-of-content" data-skip-target-assigned="false" class="px-2 py-4 color-bg-accent-emphasis color-fg-on-emphasis show-on-focus js-skip-to-content">Skip to content</a> <span data-view-component="true" class="progress-pjax-loader Progress position-fixed width-full"> <span style="width: 0%;" data-view-component="true" class="Progress-item progress-pjax-loader-bar left-0 top-0 color-bg-accent-emphasis"></span> </span> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/ui_packages_ui-commands_ui-commands_ts-2d52c8e72e64.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/keyboard-shortcuts-dialog-2560f573c7ca.js"></script> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/primer-react.a490b7c9fa319e5cb069.module.css" /> <react-partial partial-name="keyboard-shortcuts-dialog" data-ssr="false" data-attempted-ssr="false" > <script type="application/json" data-target="react-partial.embeddedData">{"props":{"docsUrl":"https://docs.github.com/get-started/accessibility/keyboard-shortcuts"}}</script> <div data-target="react-partial.reactRoot"></div> </react-partial> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_github_remote-form_dist_index_js-node_modules_delegated-events_dist_inde-94fd67-4898d1bf4b51.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/sessions-45d6658f8b6b.js"></script> <header class="HeaderMktg header-logged-out js-details-container js-header Details f4 py-3" role="banner" data-is-top="true" data-color-mode=light data-light-theme=light data-dark-theme=dark> <h2 class="sr-only">Navigation Menu</h2> <button type="button" class="HeaderMktg-backdrop d-lg-none border-0 position-fixed top-0 left-0 width-full height-full js-details-target" aria-label="Toggle navigation"> <span class="d-none">Toggle navigation</span> </button> <div class="d-flex flex-column flex-lg-row flex-items-center px-3 px-md-4 px-lg-5 height-full position-relative z-1"> <div class="d-flex flex-justify-between flex-items-center width-full width-lg-auto"> <div class="flex-1"> <button aria-label="Toggle navigation" aria-expanded="false" type="button" data-view-component="true" class="js-details-target js-nav-padding-recalculate js-header-menu-toggle Button--link Button--medium Button d-lg-none color-fg-inherit p-1"> <span class="Button-content"> <span class="Button-label"><div class="HeaderMenu-toggle-bar rounded my-1"></div> <div class="HeaderMenu-toggle-bar rounded my-1"></div> <div class="HeaderMenu-toggle-bar rounded my-1"></div></span> </span> </button> </div> <a class="mr-lg-3 color-fg-inherit flex-order-2 js-prevent-focus-on-mobile-nav" href="/" aria-label="Homepage" data-analytics-event="{"category":"Marketing nav","action":"click to go to homepage","label":"ref_page:Marketing;ref_cta:Logomark;ref_loc:Header"}"> <svg height="32" aria-hidden="true" viewBox="0 0 24 24" version="1.1" width="32" data-view-component="true" class="octicon octicon-mark-github"> <path d="M12 1C5.9225 1 1 5.9225 1 12C1 16.8675 4.14875 20.9787 8.52125 22.4362C9.07125 22.5325 9.2775 22.2025 9.2775 21.9137C9.2775 21.6525 9.26375 20.7862 9.26375 19.865C6.5 20.3737 5.785 19.1912 5.565 18.5725C5.44125 18.2562 4.905 17.28 4.4375 17.0187C4.0525 16.8125 3.5025 16.3037 4.42375 16.29C5.29 16.2762 5.90875 17.0875 6.115 17.4175C7.105 19.0812 8.68625 18.6137 9.31875 18.325C9.415 17.61 9.70375 17.1287 10.02 16.8537C7.5725 16.5787 5.015 15.63 5.015 11.4225C5.015 10.2262 5.44125 9.23625 6.1425 8.46625C6.0325 8.19125 5.6475 7.06375 6.2525 5.55125C6.2525 5.55125 7.17375 5.2625 9.2775 6.67875C10.1575 6.43125 11.0925 6.3075 12.0275 6.3075C12.9625 6.3075 13.8975 6.43125 14.7775 6.67875C16.8813 5.24875 17.8025 5.55125 17.8025 5.55125C18.4075 7.06375 18.0225 8.19125 17.9125 8.46625C18.6138 9.23625 19.04 10.2125 19.04 11.4225C19.04 15.6437 16.4688 16.5787 14.0213 16.8537C14.42 17.1975 14.7638 17.8575 14.7638 18.8887C14.7638 20.36 14.75 21.5425 14.75 21.9137C14.75 22.2025 14.9563 22.5462 15.5063 22.4362C19.8513 20.9787 23 16.8537 23 12C23 5.9225 18.0775 1 12 1Z"></path> </svg> </a> <div class="flex-1 flex-order-2 text-right"> <a href="/login?return_to=https%3A%2F%2Fgithub.com%2Fkroitor%2Fgjk.c" class="HeaderMenu-link HeaderMenu-button d-inline-flex d-lg-none flex-order-1 f5 no-underline border color-border-default rounded-2 px-2 py-1 color-fg-inherit js-prevent-focus-on-mobile-nav" data-hydro-click="{"event_type":"authentication.click","payload":{"location_in_page":"site header menu","repository_id":null,"auth_type":"SIGN_UP","originating_url":"https://github.com/kroitor/gjk.c","user_id":null}}" data-hydro-click-hmac="2ac8eb781f277273bc83a3a02d65530e3b132e22dc514c809d62270b9616962f" data-analytics-event="{"category":"Marketing nav","action":"click to Sign in","label":"ref_page:Marketing;ref_cta:Sign in;ref_loc:Header"}" > Sign in </a> </div> </div> <div class="HeaderMenu js-header-menu height-fit position-lg-relative d-lg-flex flex-column flex-auto top-0"> <div class="HeaderMenu-wrapper d-flex flex-column flex-self-start flex-lg-row flex-auto rounded rounded-lg-0"> <nav class="HeaderMenu-nav" aria-label="Global"> <ul class="d-lg-flex list-style-none"> <li class="HeaderMenu-item position-relative flex-wrap flex-justify-between flex-items-center d-block d-lg-flex flex-lg-nowrap flex-lg-items-center js-details-container js-header-menu-item"> <button type="button" class="HeaderMenu-link border-0 width-full width-lg-auto px-0 px-lg-2 py-lg-2 no-wrap d-flex flex-items-center flex-justify-between js-details-target" aria-expanded="false"> Product <svg opacity="0.5" aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-chevron-down HeaderMenu-icon ml-1"> <path d="M12.78 5.22a.749.749 0 0 1 0 1.06l-4.25 4.25a.749.749 0 0 1-1.06 0L3.22 6.28a.749.749 0 1 1 1.06-1.06L8 8.939l3.72-3.719a.749.749 0 0 1 1.06 0Z"></path> </svg> </button> <div class="HeaderMenu-dropdown dropdown-menu rounded m-0 p-0 pt-2 pt-lg-4 position-relative position-lg-absolute left-0 left-lg-n3 pb-2 pb-lg-4 d-lg-flex flex-wrap dropdown-menu-wide"> <div class="HeaderMenu-column px-lg-4 border-lg-right mb-4 mb-lg-0 pr-lg-7"> <div class="border-bottom pb-3 pb-lg-0 border-lg-bottom-0"> <ul class="list-style-none f5" > <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description pb-lg-3" data-analytics-event="{"location":"navbar","action":"github_copilot","context":"product","tag":"link","label":"github_copilot_link_product_navbar"}" href="https://github.com/features/copilot"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-copilot color-fg-subtle mr-3"> <path d="M23.922 16.992c-.861 1.495-5.859 5.023-11.922 5.023-6.063 0-11.061-3.528-11.922-5.023A.641.641 0 0 1 0 16.736v-2.869a.841.841 0 0 1 .053-.22c.372-.935 1.347-2.292 2.605-2.656.167-.429.414-1.055.644-1.517a10.195 10.195 0 0 1-.052-1.086c0-1.331.282-2.499 1.132-3.368.397-.406.89-.717 1.474-.952 1.399-1.136 3.392-2.093 6.122-2.093 2.731 0 4.767.957 6.166 2.093.584.235 1.077.546 1.474.952.85.869 1.132 2.037 1.132 3.368 0 .368-.014.733-.052 1.086.23.462.477 1.088.644 1.517 1.258.364 2.233 1.721 2.605 2.656a.832.832 0 0 1 .053.22v2.869a.641.641 0 0 1-.078.256ZM12.172 11h-.344a4.323 4.323 0 0 1-.355.508C10.703 12.455 9.555 13 7.965 13c-1.725 0-2.989-.359-3.782-1.259a2.005 2.005 0 0 1-.085-.104L4 11.741v6.585c1.435.779 4.514 2.179 8 2.179 3.486 0 6.565-1.4 8-2.179v-6.585l-.098-.104s-.033.045-.085.104c-.793.9-2.057 1.259-3.782 1.259-1.59 0-2.738-.545-3.508-1.492a4.323 4.323 0 0 1-.355-.508h-.016.016Zm.641-2.935c.136 1.057.403 1.913.878 2.497.442.544 1.134.938 2.344.938 1.573 0 2.292-.337 2.657-.751.384-.435.558-1.15.558-2.361 0-1.14-.243-1.847-.705-2.319-.477-.488-1.319-.862-2.824-1.025-1.487-.161-2.192.138-2.533.529-.269.307-.437.808-.438 1.578v.021c0 .265.021.562.063.893Zm-1.626 0c.042-.331.063-.628.063-.894v-.02c-.001-.77-.169-1.271-.438-1.578-.341-.391-1.046-.69-2.533-.529-1.505.163-2.347.537-2.824 1.025-.462.472-.705 1.179-.705 2.319 0 1.211.175 1.926.558 2.361.365.414 1.084.751 2.657.751 1.21 0 1.902-.394 2.344-.938.475-.584.742-1.44.878-2.497Z"></path><path d="M14.5 14.25a1 1 0 0 1 1 1v2a1 1 0 0 1-2 0v-2a1 1 0 0 1 1-1Zm-5 0a1 1 0 0 1 1 1v2a1 1 0 0 1-2 0v-2a1 1 0 0 1 1-1Z"></path> </svg> <div> <div class="color-fg-default h4">GitHub Copilot</div> Write better code with AI </div> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description pb-lg-3" data-analytics-event="{"location":"navbar","action":"github_advanced_security","context":"product","tag":"link","label":"github_advanced_security_link_product_navbar"}" href="https://github.com/security/advanced-security"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-shield-check color-fg-subtle mr-3"> <path d="M16.53 9.78a.75.75 0 0 0-1.06-1.06L11 13.19l-1.97-1.97a.75.75 0 0 0-1.06 1.06l2.5 2.5a.75.75 0 0 0 1.06 0l5-5Z"></path><path d="m12.54.637 8.25 2.675A1.75 1.75 0 0 1 22 4.976V10c0 6.19-3.771 10.704-9.401 12.83a1.704 1.704 0 0 1-1.198 0C5.77 20.705 2 16.19 2 10V4.976c0-.758.489-1.43 1.21-1.664L11.46.637a1.748 1.748 0 0 1 1.08 0Zm-.617 1.426-8.25 2.676a.249.249 0 0 0-.173.237V10c0 5.46 3.28 9.483 8.43 11.426a.199.199 0 0 0 .14 0C17.22 19.483 20.5 15.461 20.5 10V4.976a.25.25 0 0 0-.173-.237l-8.25-2.676a.253.253 0 0 0-.154 0Z"></path> </svg> <div> <div class="color-fg-default h4">GitHub Advanced Security</div> Find and fix vulnerabilities </div> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description pb-lg-3" data-analytics-event="{"location":"navbar","action":"actions","context":"product","tag":"link","label":"actions_link_product_navbar"}" href="https://github.com/features/actions"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-workflow color-fg-subtle mr-3"> <path d="M1 3a2 2 0 0 1 2-2h6.5a2 2 0 0 1 2 2v6.5a2 2 0 0 1-2 2H7v4.063C7 16.355 7.644 17 8.438 17H12.5v-2.5a2 2 0 0 1 2-2H21a2 2 0 0 1 2 2V21a2 2 0 0 1-2 2h-6.5a2 2 0 0 1-2-2v-2.5H8.437A2.939 2.939 0 0 1 5.5 15.562V11.5H3a2 2 0 0 1-2-2Zm2-.5a.5.5 0 0 0-.5.5v6.5a.5.5 0 0 0 .5.5h6.5a.5.5 0 0 0 .5-.5V3a.5.5 0 0 0-.5-.5ZM14.5 14a.5.5 0 0 0-.5.5V21a.5.5 0 0 0 .5.5H21a.5.5 0 0 0 .5-.5v-6.5a.5.5 0 0 0-.5-.5Z"></path> </svg> <div> <div class="color-fg-default h4">Actions</div> Automate any workflow </div> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description pb-lg-3" data-analytics-event="{"location":"navbar","action":"codespaces","context":"product","tag":"link","label":"codespaces_link_product_navbar"}" href="https://github.com/features/codespaces"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-codespaces color-fg-subtle mr-3"> <path d="M3.5 3.75C3.5 2.784 4.284 2 5.25 2h13.5c.966 0 1.75.784 1.75 1.75v7.5A1.75 1.75 0 0 1 18.75 13H5.25a1.75 1.75 0 0 1-1.75-1.75Zm-2 12c0-.966.784-1.75 1.75-1.75h17.5c.966 0 1.75.784 1.75 1.75v4a1.75 1.75 0 0 1-1.75 1.75H3.25a1.75 1.75 0 0 1-1.75-1.75ZM5.25 3.5a.25.25 0 0 0-.25.25v7.5c0 .138.112.25.25.25h13.5a.25.25 0 0 0 .25-.25v-7.5a.25.25 0 0 0-.25-.25Zm-2 12a.25.25 0 0 0-.25.25v4c0 .138.112.25.25.25h17.5a.25.25 0 0 0 .25-.25v-4a.25.25 0 0 0-.25-.25Z"></path><path d="M10 17.75a.75.75 0 0 1 .75-.75h6.5a.75.75 0 0 1 0 1.5h-6.5a.75.75 0 0 1-.75-.75Zm-4 0a.75.75 0 0 1 .75-.75h.5a.75.75 0 0 1 0 1.5h-.5a.75.75 0 0 1-.75-.75Z"></path> </svg> <div> <div class="color-fg-default h4">Codespaces</div> Instant dev environments </div> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description pb-lg-3" data-analytics-event="{"location":"navbar","action":"issues","context":"product","tag":"link","label":"issues_link_product_navbar"}" href="https://github.com/features/issues"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-issue-opened color-fg-subtle mr-3"> <path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1ZM2.5 12a9.5 9.5 0 0 0 9.5 9.5 9.5 9.5 0 0 0 9.5-9.5A9.5 9.5 0 0 0 12 2.5 9.5 9.5 0 0 0 2.5 12Zm9.5 2a2 2 0 1 1-.001-3.999A2 2 0 0 1 12 14Z"></path> </svg> <div> <div class="color-fg-default h4">Issues</div> Plan and track work </div> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description pb-lg-3" data-analytics-event="{"location":"navbar","action":"code_review","context":"product","tag":"link","label":"code_review_link_product_navbar"}" href="https://github.com/features/code-review"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-code-review color-fg-subtle mr-3"> <path d="M10.3 6.74a.75.75 0 0 1-.04 1.06l-2.908 2.7 2.908 2.7a.75.75 0 1 1-1.02 1.1l-3.5-3.25a.75.75 0 0 1 0-1.1l3.5-3.25a.75.75 0 0 1 1.06.04Zm3.44 1.06a.75.75 0 1 1 1.02-1.1l3.5 3.25a.75.75 0 0 1 0 1.1l-3.5 3.25a.75.75 0 1 1-1.02-1.1l2.908-2.7-2.908-2.7Z"></path><path d="M1.5 4.25c0-.966.784-1.75 1.75-1.75h17.5c.966 0 1.75.784 1.75 1.75v12.5a1.75 1.75 0 0 1-1.75 1.75h-9.69l-3.573 3.573A1.458 1.458 0 0 1 5 21.043V18.5H3.25a1.75 1.75 0 0 1-1.75-1.75ZM3.25 4a.25.25 0 0 0-.25.25v12.5c0 .138.112.25.25.25h2.5a.75.75 0 0 1 .75.75v3.19l3.72-3.72a.749.749 0 0 1 .53-.22h10a.25.25 0 0 0 .25-.25V4.25a.25.25 0 0 0-.25-.25Z"></path> </svg> <div> <div class="color-fg-default h4">Code Review</div> Manage code changes </div> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description pb-lg-3" data-analytics-event="{"location":"navbar","action":"discussions","context":"product","tag":"link","label":"discussions_link_product_navbar"}" href="https://github.com/features/discussions"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-comment-discussion color-fg-subtle mr-3"> <path d="M1.75 1h12.5c.966 0 1.75.784 1.75 1.75v9.5A1.75 1.75 0 0 1 14.25 14H8.061l-2.574 2.573A1.458 1.458 0 0 1 3 15.543V14H1.75A1.75 1.75 0 0 1 0 12.25v-9.5C0 1.784.784 1 1.75 1ZM1.5 2.75v9.5c0 .138.112.25.25.25h2a.75.75 0 0 1 .75.75v2.19l2.72-2.72a.749.749 0 0 1 .53-.22h6.5a.25.25 0 0 0 .25-.25v-9.5a.25.25 0 0 0-.25-.25H1.75a.25.25 0 0 0-.25.25Z"></path><path d="M22.5 8.75a.25.25 0 0 0-.25-.25h-3.5a.75.75 0 0 1 0-1.5h3.5c.966 0 1.75.784 1.75 1.75v9.5A1.75 1.75 0 0 1 22.25 20H21v1.543a1.457 1.457 0 0 1-2.487 1.03L15.939 20H10.75A1.75 1.75 0 0 1 9 18.25v-1.465a.75.75 0 0 1 1.5 0v1.465c0 .138.112.25.25.25h5.5a.75.75 0 0 1 .53.22l2.72 2.72v-2.19a.75.75 0 0 1 .75-.75h2a.25.25 0 0 0 .25-.25v-9.5Z"></path> </svg> <div> <div class="color-fg-default h4">Discussions</div> Collaborate outside of code </div> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description" data-analytics-event="{"location":"navbar","action":"code_search","context":"product","tag":"link","label":"code_search_link_product_navbar"}" href="https://github.com/features/code-search"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-code-square color-fg-subtle mr-3"> <path d="M10.3 8.24a.75.75 0 0 1-.04 1.06L7.352 12l2.908 2.7a.75.75 0 1 1-1.02 1.1l-3.5-3.25a.75.75 0 0 1 0-1.1l3.5-3.25a.75.75 0 0 1 1.06.04Zm3.44 1.06a.75.75 0 1 1 1.02-1.1l3.5 3.25a.75.75 0 0 1 0 1.1l-3.5 3.25a.75.75 0 1 1-1.02-1.1l2.908-2.7-2.908-2.7Z"></path><path d="M2 3.75C2 2.784 2.784 2 3.75 2h16.5c.966 0 1.75.784 1.75 1.75v16.5A1.75 1.75 0 0 1 20.25 22H3.75A1.75 1.75 0 0 1 2 20.25Zm1.75-.25a.25.25 0 0 0-.25.25v16.5c0 .138.112.25.25.25h16.5a.25.25 0 0 0 .25-.25V3.75a.25.25 0 0 0-.25-.25Z"></path> </svg> <div> <div class="color-fg-default h4">Code Search</div> Find more, search less </div> </a></li> </ul> </div> </div> <div class="HeaderMenu-column px-lg-4"> <div class="border-bottom pb-3 pb-lg-0 border-lg-bottom-0 border-bottom-0"> <span class="d-block h4 color-fg-default my-1" id="product-explore-heading">Explore</span> <ul class="list-style-none f5" aria-labelledby="product-explore-heading"> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"all_features","context":"product","tag":"link","label":"all_features_link_product_navbar"}" href="https://github.com/features"> All features </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary Link--external" target="_blank" data-analytics-event="{"location":"navbar","action":"documentation","context":"product","tag":"link","label":"documentation_link_product_navbar"}" href="https://docs.github.com"> Documentation <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-link-external HeaderMenu-external-icon color-fg-subtle"> <path d="M3.75 2h3.5a.75.75 0 0 1 0 1.5h-3.5a.25.25 0 0 0-.25.25v8.5c0 .138.112.25.25.25h8.5a.25.25 0 0 0 .25-.25v-3.5a.75.75 0 0 1 1.5 0v3.5A1.75 1.75 0 0 1 12.25 14h-8.5A1.75 1.75 0 0 1 2 12.25v-8.5C2 2.784 2.784 2 3.75 2Zm6.854-1h4.146a.25.25 0 0 1 .25.25v4.146a.25.25 0 0 1-.427.177L13.03 4.03 9.28 7.78a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042l3.75-3.75-1.543-1.543A.25.25 0 0 1 10.604 1Z"></path> </svg> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary Link--external" target="_blank" data-analytics-event="{"location":"navbar","action":"github_skills","context":"product","tag":"link","label":"github_skills_link_product_navbar"}" href="https://skills.github.com"> GitHub Skills <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-link-external HeaderMenu-external-icon color-fg-subtle"> <path d="M3.75 2h3.5a.75.75 0 0 1 0 1.5h-3.5a.25.25 0 0 0-.25.25v8.5c0 .138.112.25.25.25h8.5a.25.25 0 0 0 .25-.25v-3.5a.75.75 0 0 1 1.5 0v3.5A1.75 1.75 0 0 1 12.25 14h-8.5A1.75 1.75 0 0 1 2 12.25v-8.5C2 2.784 2.784 2 3.75 2Zm6.854-1h4.146a.25.25 0 0 1 .25.25v4.146a.25.25 0 0 1-.427.177L13.03 4.03 9.28 7.78a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042l3.75-3.75-1.543-1.543A.25.25 0 0 1 10.604 1Z"></path> </svg> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary Link--external" target="_blank" data-analytics-event="{"location":"navbar","action":"blog","context":"product","tag":"link","label":"blog_link_product_navbar"}" href="https://github.blog"> Blog <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-link-external HeaderMenu-external-icon color-fg-subtle"> <path d="M3.75 2h3.5a.75.75 0 0 1 0 1.5h-3.5a.25.25 0 0 0-.25.25v8.5c0 .138.112.25.25.25h8.5a.25.25 0 0 0 .25-.25v-3.5a.75.75 0 0 1 1.5 0v3.5A1.75 1.75 0 0 1 12.25 14h-8.5A1.75 1.75 0 0 1 2 12.25v-8.5C2 2.784 2.784 2 3.75 2Zm6.854-1h4.146a.25.25 0 0 1 .25.25v4.146a.25.25 0 0 1-.427.177L13.03 4.03 9.28 7.78a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042l3.75-3.75-1.543-1.543A.25.25 0 0 1 10.604 1Z"></path> </svg> </a></li> </ul> </div> </div> </div> </li> <li class="HeaderMenu-item position-relative flex-wrap flex-justify-between flex-items-center d-block d-lg-flex flex-lg-nowrap flex-lg-items-center js-details-container js-header-menu-item"> <button type="button" class="HeaderMenu-link border-0 width-full width-lg-auto px-0 px-lg-2 py-lg-2 no-wrap d-flex flex-items-center flex-justify-between js-details-target" aria-expanded="false"> Solutions <svg opacity="0.5" aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-chevron-down HeaderMenu-icon ml-1"> <path d="M12.78 5.22a.749.749 0 0 1 0 1.06l-4.25 4.25a.749.749 0 0 1-1.06 0L3.22 6.28a.749.749 0 1 1 1.06-1.06L8 8.939l3.72-3.719a.749.749 0 0 1 1.06 0Z"></path> </svg> </button> <div class="HeaderMenu-dropdown dropdown-menu rounded m-0 p-0 pt-2 pt-lg-4 position-relative position-lg-absolute left-0 left-lg-n3 d-lg-flex flex-wrap dropdown-menu-wide"> <div class="HeaderMenu-column px-lg-4 border-lg-right mb-4 mb-lg-0 pr-lg-7"> <div class="border-bottom pb-3 pb-lg-0 border-lg-bottom-0 pb-lg-3 mb-3 mb-lg-0"> <span class="d-block h4 color-fg-default my-1" id="solutions-by-company-size-heading">By company size</span> <ul class="list-style-none f5" aria-labelledby="solutions-by-company-size-heading"> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"enterprises","context":"solutions","tag":"link","label":"enterprises_link_solutions_navbar"}" href="https://github.com/enterprise"> Enterprises </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"small_and_medium_teams","context":"solutions","tag":"link","label":"small_and_medium_teams_link_solutions_navbar"}" href="https://github.com/team"> Small and medium teams </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"startups","context":"solutions","tag":"link","label":"startups_link_solutions_navbar"}" href="https://github.com/enterprise/startups"> Startups </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"nonprofits","context":"solutions","tag":"link","label":"nonprofits_link_solutions_navbar"}" href="/solutions/industry/nonprofits"> Nonprofits </a></li> </ul> </div> <div class="border-bottom pb-3 pb-lg-0 border-lg-bottom-0"> <span class="d-block h4 color-fg-default my-1" id="solutions-by-use-case-heading">By use case</span> <ul class="list-style-none f5" aria-labelledby="solutions-by-use-case-heading"> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"devsecops","context":"solutions","tag":"link","label":"devsecops_link_solutions_navbar"}" href="/solutions/use-case/devsecops"> DevSecOps </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"devops","context":"solutions","tag":"link","label":"devops_link_solutions_navbar"}" href="/solutions/use-case/devops"> DevOps </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"ci_cd","context":"solutions","tag":"link","label":"ci_cd_link_solutions_navbar"}" href="/solutions/use-case/ci-cd"> CI/CD </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"view_all_use_cases","context":"solutions","tag":"link","label":"view_all_use_cases_link_solutions_navbar"}" href="/solutions/use-case"> View all use cases </a></li> </ul> </div> </div> <div class="HeaderMenu-column px-lg-4"> <div class="border-bottom pb-3 pb-lg-0 border-lg-bottom-0"> <span class="d-block h4 color-fg-default my-1" id="solutions-by-industry-heading">By industry</span> <ul class="list-style-none f5" aria-labelledby="solutions-by-industry-heading"> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"healthcare","context":"solutions","tag":"link","label":"healthcare_link_solutions_navbar"}" href="/solutions/industry/healthcare"> Healthcare </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"financial_services","context":"solutions","tag":"link","label":"financial_services_link_solutions_navbar"}" href="/solutions/industry/financial-services"> Financial services </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"manufacturing","context":"solutions","tag":"link","label":"manufacturing_link_solutions_navbar"}" href="/solutions/industry/manufacturing"> Manufacturing </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"government","context":"solutions","tag":"link","label":"government_link_solutions_navbar"}" href="/solutions/industry/government"> Government </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"view_all_industries","context":"solutions","tag":"link","label":"view_all_industries_link_solutions_navbar"}" href="/solutions/industry"> View all industries </a></li> </ul> </div> </div> <div class="HeaderMenu-trailing-link rounded-bottom-2 flex-shrink-0 mt-lg-4 px-lg-4 py-4 py-lg-3 f5 text-semibold"> <a href="/solutions"> View all solutions <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-chevron-right HeaderMenu-trailing-link-icon"> <path d="M6.22 3.22a.75.75 0 0 1 1.06 0l4.25 4.25a.75.75 0 0 1 0 1.06l-4.25 4.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042L9.94 8 6.22 4.28a.75.75 0 0 1 0-1.06Z"></path> </svg> </a> </div> </div> </li> <li class="HeaderMenu-item position-relative flex-wrap flex-justify-between flex-items-center d-block d-lg-flex flex-lg-nowrap flex-lg-items-center js-details-container js-header-menu-item"> <button type="button" class="HeaderMenu-link border-0 width-full width-lg-auto px-0 px-lg-2 py-lg-2 no-wrap d-flex flex-items-center flex-justify-between js-details-target" aria-expanded="false"> Resources <svg opacity="0.5" aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-chevron-down HeaderMenu-icon ml-1"> <path d="M12.78 5.22a.749.749 0 0 1 0 1.06l-4.25 4.25a.749.749 0 0 1-1.06 0L3.22 6.28a.749.749 0 1 1 1.06-1.06L8 8.939l3.72-3.719a.749.749 0 0 1 1.06 0Z"></path> </svg> </button> <div class="HeaderMenu-dropdown dropdown-menu rounded m-0 p-0 pt-2 pt-lg-4 position-relative position-lg-absolute left-0 left-lg-n3 pb-2 pb-lg-4 d-lg-flex flex-wrap dropdown-menu-wide"> <div class="HeaderMenu-column px-lg-4 border-lg-right mb-4 mb-lg-0 pr-lg-7"> <div class="border-bottom pb-3 pb-lg-0 border-lg-bottom-0"> <span class="d-block h4 color-fg-default my-1" id="resources-topics-heading">Topics</span> <ul class="list-style-none f5" aria-labelledby="resources-topics-heading"> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"ai","context":"resources","tag":"link","label":"ai_link_resources_navbar"}" href="/resources/articles/ai"> AI </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"devops","context":"resources","tag":"link","label":"devops_link_resources_navbar"}" href="/resources/articles/devops"> DevOps </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"security","context":"resources","tag":"link","label":"security_link_resources_navbar"}" href="/resources/articles/security"> Security </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"software_development","context":"resources","tag":"link","label":"software_development_link_resources_navbar"}" href="/resources/articles/software-development"> Software Development </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"view_all","context":"resources","tag":"link","label":"view_all_link_resources_navbar"}" href="/resources/articles"> View all </a></li> </ul> </div> </div> <div class="HeaderMenu-column px-lg-4"> <div class="border-bottom pb-3 pb-lg-0 border-lg-bottom-0 border-bottom-0"> <span class="d-block h4 color-fg-default my-1" id="resources-explore-heading">Explore</span> <ul class="list-style-none f5" aria-labelledby="resources-explore-heading"> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary Link--external" target="_blank" data-analytics-event="{"location":"navbar","action":"learning_pathways","context":"resources","tag":"link","label":"learning_pathways_link_resources_navbar"}" href="https://resources.github.com/learn/pathways"> Learning Pathways <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-link-external HeaderMenu-external-icon color-fg-subtle"> <path d="M3.75 2h3.5a.75.75 0 0 1 0 1.5h-3.5a.25.25 0 0 0-.25.25v8.5c0 .138.112.25.25.25h8.5a.25.25 0 0 0 .25-.25v-3.5a.75.75 0 0 1 1.5 0v3.5A1.75 1.75 0 0 1 12.25 14h-8.5A1.75 1.75 0 0 1 2 12.25v-8.5C2 2.784 2.784 2 3.75 2Zm6.854-1h4.146a.25.25 0 0 1 .25.25v4.146a.25.25 0 0 1-.427.177L13.03 4.03 9.28 7.78a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042l3.75-3.75-1.543-1.543A.25.25 0 0 1 10.604 1Z"></path> </svg> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary Link--external" target="_blank" data-analytics-event="{"location":"navbar","action":"events_amp_webinars","context":"resources","tag":"link","label":"events_amp_webinars_link_resources_navbar"}" href="https://resources.github.com"> Events & Webinars <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-link-external HeaderMenu-external-icon color-fg-subtle"> <path d="M3.75 2h3.5a.75.75 0 0 1 0 1.5h-3.5a.25.25 0 0 0-.25.25v8.5c0 .138.112.25.25.25h8.5a.25.25 0 0 0 .25-.25v-3.5a.75.75 0 0 1 1.5 0v3.5A1.75 1.75 0 0 1 12.25 14h-8.5A1.75 1.75 0 0 1 2 12.25v-8.5C2 2.784 2.784 2 3.75 2Zm6.854-1h4.146a.25.25 0 0 1 .25.25v4.146a.25.25 0 0 1-.427.177L13.03 4.03 9.28 7.78a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042l3.75-3.75-1.543-1.543A.25.25 0 0 1 10.604 1Z"></path> </svg> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"ebooks_amp_whitepapers","context":"resources","tag":"link","label":"ebooks_amp_whitepapers_link_resources_navbar"}" href="https://github.com/resources/whitepapers"> Ebooks & Whitepapers </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"customer_stories","context":"resources","tag":"link","label":"customer_stories_link_resources_navbar"}" href="https://github.com/customer-stories"> Customer Stories </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary Link--external" target="_blank" data-analytics-event="{"location":"navbar","action":"partners","context":"resources","tag":"link","label":"partners_link_resources_navbar"}" href="https://partner.github.com"> Partners <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-link-external HeaderMenu-external-icon color-fg-subtle"> <path d="M3.75 2h3.5a.75.75 0 0 1 0 1.5h-3.5a.25.25 0 0 0-.25.25v8.5c0 .138.112.25.25.25h8.5a.25.25 0 0 0 .25-.25v-3.5a.75.75 0 0 1 1.5 0v3.5A1.75 1.75 0 0 1 12.25 14h-8.5A1.75 1.75 0 0 1 2 12.25v-8.5C2 2.784 2.784 2 3.75 2Zm6.854-1h4.146a.25.25 0 0 1 .25.25v4.146a.25.25 0 0 1-.427.177L13.03 4.03 9.28 7.78a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042l3.75-3.75-1.543-1.543A.25.25 0 0 1 10.604 1Z"></path> </svg> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"executive_insights","context":"resources","tag":"link","label":"executive_insights_link_resources_navbar"}" href="https://github.com/solutions/executive-insights"> Executive Insights </a></li> </ul> </div> </div> </div> </li> <li class="HeaderMenu-item position-relative flex-wrap flex-justify-between flex-items-center d-block d-lg-flex flex-lg-nowrap flex-lg-items-center js-details-container js-header-menu-item"> <button type="button" class="HeaderMenu-link border-0 width-full width-lg-auto px-0 px-lg-2 py-lg-2 no-wrap d-flex flex-items-center flex-justify-between js-details-target" aria-expanded="false"> Open Source <svg opacity="0.5" aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-chevron-down HeaderMenu-icon ml-1"> <path d="M12.78 5.22a.749.749 0 0 1 0 1.06l-4.25 4.25a.749.749 0 0 1-1.06 0L3.22 6.28a.749.749 0 1 1 1.06-1.06L8 8.939l3.72-3.719a.749.749 0 0 1 1.06 0Z"></path> </svg> </button> <div class="HeaderMenu-dropdown dropdown-menu rounded m-0 p-0 pt-2 pt-lg-4 position-relative position-lg-absolute left-0 left-lg-n3 pb-2 pb-lg-4 px-lg-4"> <div class="HeaderMenu-column"> <div class="border-bottom pb-3 pb-lg-0 pb-lg-3 mb-3 mb-lg-0 mb-lg-3"> <ul class="list-style-none f5" > <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description" data-analytics-event="{"location":"navbar","action":"github_sponsors","context":"open_source","tag":"link","label":"github_sponsors_link_open_source_navbar"}" href="/sponsors"> <div> <div class="color-fg-default h4">GitHub Sponsors</div> Fund open source developers </div> </a></li> </ul> </div> <div class="border-bottom pb-3 pb-lg-0 pb-lg-3 mb-3 mb-lg-0 mb-lg-3"> <ul class="list-style-none f5" > <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description" data-analytics-event="{"location":"navbar","action":"the_readme_project","context":"open_source","tag":"link","label":"the_readme_project_link_open_source_navbar"}" href="https://github.com/readme"> <div> <div class="color-fg-default h4">The ReadME Project</div> GitHub community articles </div> </a></li> </ul> </div> <div class="border-bottom pb-3 pb-lg-0 border-bottom-0"> <span class="d-block h4 color-fg-default my-1" id="open-source-repositories-heading">Repositories</span> <ul class="list-style-none f5" aria-labelledby="open-source-repositories-heading"> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"topics","context":"open_source","tag":"link","label":"topics_link_open_source_navbar"}" href="https://github.com/topics"> Topics </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"trending","context":"open_source","tag":"link","label":"trending_link_open_source_navbar"}" href="https://github.com/trending"> Trending </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary" data-analytics-event="{"location":"navbar","action":"collections","context":"open_source","tag":"link","label":"collections_link_open_source_navbar"}" href="https://github.com/collections"> Collections </a></li> </ul> </div> </div> </div> </li> <li class="HeaderMenu-item position-relative flex-wrap flex-justify-between flex-items-center d-block d-lg-flex flex-lg-nowrap flex-lg-items-center js-details-container js-header-menu-item"> <button type="button" class="HeaderMenu-link border-0 width-full width-lg-auto px-0 px-lg-2 py-lg-2 no-wrap d-flex flex-items-center flex-justify-between js-details-target" aria-expanded="false"> Enterprise <svg opacity="0.5" aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-chevron-down HeaderMenu-icon ml-1"> <path d="M12.78 5.22a.749.749 0 0 1 0 1.06l-4.25 4.25a.749.749 0 0 1-1.06 0L3.22 6.28a.749.749 0 1 1 1.06-1.06L8 8.939l3.72-3.719a.749.749 0 0 1 1.06 0Z"></path> </svg> </button> <div class="HeaderMenu-dropdown dropdown-menu rounded m-0 p-0 pt-2 pt-lg-4 position-relative position-lg-absolute left-0 left-lg-n3 pb-2 pb-lg-4 px-lg-4"> <div class="HeaderMenu-column"> <div class="border-bottom pb-3 pb-lg-0 pb-lg-3 mb-3 mb-lg-0 mb-lg-3"> <ul class="list-style-none f5" > <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description" data-analytics-event="{"location":"navbar","action":"enterprise_platform","context":"enterprise","tag":"link","label":"enterprise_platform_link_enterprise_navbar"}" href="/enterprise"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-stack color-fg-subtle mr-3"> <path d="M11.063 1.456a1.749 1.749 0 0 1 1.874 0l8.383 5.316a1.751 1.751 0 0 1 0 2.956l-8.383 5.316a1.749 1.749 0 0 1-1.874 0L2.68 9.728a1.751 1.751 0 0 1 0-2.956Zm1.071 1.267a.25.25 0 0 0-.268 0L3.483 8.039a.25.25 0 0 0 0 .422l8.383 5.316a.25.25 0 0 0 .268 0l8.383-5.316a.25.25 0 0 0 0-.422Z"></path><path d="M1.867 12.324a.75.75 0 0 1 1.035-.232l8.964 5.685a.25.25 0 0 0 .268 0l8.964-5.685a.75.75 0 0 1 .804 1.267l-8.965 5.685a1.749 1.749 0 0 1-1.874 0l-8.965-5.685a.75.75 0 0 1-.231-1.035Z"></path><path d="M1.867 16.324a.75.75 0 0 1 1.035-.232l8.964 5.685a.25.25 0 0 0 .268 0l8.964-5.685a.75.75 0 0 1 .804 1.267l-8.965 5.685a1.749 1.749 0 0 1-1.874 0l-8.965-5.685a.75.75 0 0 1-.231-1.035Z"></path> </svg> <div> <div class="color-fg-default h4">Enterprise platform</div> AI-powered developer platform </div> </a></li> </ul> </div> <div class="border-bottom pb-3 pb-lg-0 border-bottom-0"> <span class="d-block h4 color-fg-default my-1" id="enterprise-available-add-ons-heading">Available add-ons</span> <ul class="list-style-none f5" aria-labelledby="enterprise-available-add-ons-heading"> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description pb-lg-3" data-analytics-event="{"location":"navbar","action":"github_advanced_security","context":"enterprise","tag":"link","label":"github_advanced_security_link_enterprise_navbar"}" href="https://github.com/security/advanced-security"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-shield-check color-fg-subtle mr-3"> <path d="M16.53 9.78a.75.75 0 0 0-1.06-1.06L11 13.19l-1.97-1.97a.75.75 0 0 0-1.06 1.06l2.5 2.5a.75.75 0 0 0 1.06 0l5-5Z"></path><path d="m12.54.637 8.25 2.675A1.75 1.75 0 0 1 22 4.976V10c0 6.19-3.771 10.704-9.401 12.83a1.704 1.704 0 0 1-1.198 0C5.77 20.705 2 16.19 2 10V4.976c0-.758.489-1.43 1.21-1.664L11.46.637a1.748 1.748 0 0 1 1.08 0Zm-.617 1.426-8.25 2.676a.249.249 0 0 0-.173.237V10c0 5.46 3.28 9.483 8.43 11.426a.199.199 0 0 0 .14 0C17.22 19.483 20.5 15.461 20.5 10V4.976a.25.25 0 0 0-.173-.237l-8.25-2.676a.253.253 0 0 0-.154 0Z"></path> </svg> <div> <div class="color-fg-default h4">GitHub Advanced Security</div> Enterprise-grade security features </div> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description pb-lg-3" data-analytics-event="{"location":"navbar","action":"copilot_for_business","context":"enterprise","tag":"link","label":"copilot_for_business_link_enterprise_navbar"}" href="/features/copilot/copilot-business"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-copilot color-fg-subtle mr-3"> <path d="M23.922 16.992c-.861 1.495-5.859 5.023-11.922 5.023-6.063 0-11.061-3.528-11.922-5.023A.641.641 0 0 1 0 16.736v-2.869a.841.841 0 0 1 .053-.22c.372-.935 1.347-2.292 2.605-2.656.167-.429.414-1.055.644-1.517a10.195 10.195 0 0 1-.052-1.086c0-1.331.282-2.499 1.132-3.368.397-.406.89-.717 1.474-.952 1.399-1.136 3.392-2.093 6.122-2.093 2.731 0 4.767.957 6.166 2.093.584.235 1.077.546 1.474.952.85.869 1.132 2.037 1.132 3.368 0 .368-.014.733-.052 1.086.23.462.477 1.088.644 1.517 1.258.364 2.233 1.721 2.605 2.656a.832.832 0 0 1 .053.22v2.869a.641.641 0 0 1-.078.256ZM12.172 11h-.344a4.323 4.323 0 0 1-.355.508C10.703 12.455 9.555 13 7.965 13c-1.725 0-2.989-.359-3.782-1.259a2.005 2.005 0 0 1-.085-.104L4 11.741v6.585c1.435.779 4.514 2.179 8 2.179 3.486 0 6.565-1.4 8-2.179v-6.585l-.098-.104s-.033.045-.085.104c-.793.9-2.057 1.259-3.782 1.259-1.59 0-2.738-.545-3.508-1.492a4.323 4.323 0 0 1-.355-.508h-.016.016Zm.641-2.935c.136 1.057.403 1.913.878 2.497.442.544 1.134.938 2.344.938 1.573 0 2.292-.337 2.657-.751.384-.435.558-1.15.558-2.361 0-1.14-.243-1.847-.705-2.319-.477-.488-1.319-.862-2.824-1.025-1.487-.161-2.192.138-2.533.529-.269.307-.437.808-.438 1.578v.021c0 .265.021.562.063.893Zm-1.626 0c.042-.331.063-.628.063-.894v-.02c-.001-.77-.169-1.271-.438-1.578-.341-.391-1.046-.69-2.533-.529-1.505.163-2.347.537-2.824 1.025-.462.472-.705 1.179-.705 2.319 0 1.211.175 1.926.558 2.361.365.414 1.084.751 2.657.751 1.21 0 1.902-.394 2.344-.938.475-.584.742-1.44.878-2.497Z"></path><path d="M14.5 14.25a1 1 0 0 1 1 1v2a1 1 0 0 1-2 0v-2a1 1 0 0 1 1-1Zm-5 0a1 1 0 0 1 1 1v2a1 1 0 0 1-2 0v-2a1 1 0 0 1 1-1Z"></path> </svg> <div> <div class="color-fg-default h4">Copilot for business</div> Enterprise-grade AI features </div> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description" data-analytics-event="{"location":"navbar","action":"premium_support","context":"enterprise","tag":"link","label":"premium_support_link_enterprise_navbar"}" href="/premium-support"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-comment-discussion color-fg-subtle mr-3"> <path d="M1.75 1h12.5c.966 0 1.75.784 1.75 1.75v9.5A1.75 1.75 0 0 1 14.25 14H8.061l-2.574 2.573A1.458 1.458 0 0 1 3 15.543V14H1.75A1.75 1.75 0 0 1 0 12.25v-9.5C0 1.784.784 1 1.75 1ZM1.5 2.75v9.5c0 .138.112.25.25.25h2a.75.75 0 0 1 .75.75v2.19l2.72-2.72a.749.749 0 0 1 .53-.22h6.5a.25.25 0 0 0 .25-.25v-9.5a.25.25 0 0 0-.25-.25H1.75a.25.25 0 0 0-.25.25Z"></path><path d="M22.5 8.75a.25.25 0 0 0-.25-.25h-3.5a.75.75 0 0 1 0-1.5h3.5c.966 0 1.75.784 1.75 1.75v9.5A1.75 1.75 0 0 1 22.25 20H21v1.543a1.457 1.457 0 0 1-2.487 1.03L15.939 20H10.75A1.75 1.75 0 0 1 9 18.25v-1.465a.75.75 0 0 1 1.5 0v1.465c0 .138.112.25.25.25h5.5a.75.75 0 0 1 .53.22l2.72 2.72v-2.19a.75.75 0 0 1 .75-.75h2a.25.25 0 0 0 .25-.25v-9.5Z"></path> </svg> <div> <div class="color-fg-default h4">Premium Support</div> Enterprise-grade 24/7 support </div> </a></li> </ul> </div> </div> </div> </li> <li class="HeaderMenu-item position-relative flex-wrap flex-justify-between flex-items-center d-block d-lg-flex flex-lg-nowrap flex-lg-items-center js-details-container js-header-menu-item"> <a class="HeaderMenu-link no-underline px-0 px-lg-2 py-3 py-lg-2 d-block d-lg-inline-block" data-analytics-event="{"location":"navbar","action":"pricing","context":"global","tag":"link","label":"pricing_link_global_navbar"}" href="https://github.com/pricing">Pricing</a> </li> </ul> </nav> <div class="d-flex flex-column flex-lg-row width-full flex-justify-end flex-lg-items-center text-center mt-3 mt-lg-0 text-lg-left ml-lg-3"> <qbsearch-input class="search-input" data-scope="repo:kroitor/gjk.c" data-custom-scopes-path="/search/custom_scopes" data-delete-custom-scopes-csrf="xYhpqnuHN77yiBJpdD265QJYllVxzE5hOA2Y00waUBRlc_Pp-HvmTwAaUhFl2omqX_kJ1icTaqSvfAAxtYK5uA" data-max-custom-scopes="10" data-header-redesign-enabled="false" data-initial-value="" data-blackbird-suggestions-path="/search/suggestions" data-jump-to-suggestions-path="/_graphql/GetSuggestedNavigationDestinations" data-current-repository="kroitor/gjk.c" data-current-org="" data-current-owner="kroitor" data-logged-in="false" data-copilot-chat-enabled="false" data-nl-search-enabled="false" data-retain-scroll-position="true"> <div class="search-input-container search-with-dialog position-relative d-flex flex-row flex-items-center mr-4 rounded" data-action="click:qbsearch-input#searchInputContainerClicked" > <button type="button" class="header-search-button placeholder input-button form-control d-flex flex-1 flex-self-stretch flex-items-center no-wrap width-full py-0 pl-2 pr-0 text-left border-0 box-shadow-none" data-target="qbsearch-input.inputButton" aria-label="Search or jump to…" aria-haspopup="dialog" placeholder="Search or jump to..." data-hotkey=s,/ autocapitalize="off" data-analytics-event="{"location":"navbar","action":"searchbar","context":"global","tag":"input","label":"searchbar_input_global_navbar"}" data-action="click:qbsearch-input#handleExpand" > <div class="mr-2 color-fg-muted"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-search"> <path d="M10.68 11.74a6 6 0 0 1-7.922-8.982 6 6 0 0 1 8.982 7.922l3.04 3.04a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215ZM11.5 7a4.499 4.499 0 1 0-8.997 0A4.499 4.499 0 0 0 11.5 7Z"></path> </svg> </div> <span class="flex-1" data-target="qbsearch-input.inputButtonText">Search or jump to...</span> <div class="d-flex" data-target="qbsearch-input.hotkeyIndicator"> <svg xmlns="http://www.w3.org/2000/svg" width="22" height="20" aria-hidden="true" class="mr-1"><path fill="none" stroke="#979A9C" opacity=".4" d="M3.5.5h12c1.7 0 3 1.3 3 3v13c0 1.7-1.3 3-3 3h-12c-1.7 0-3-1.3-3-3v-13c0-1.7 1.3-3 3-3z"></path><path fill="#979A9C" d="M11.8 6L8 15.1h-.9L10.8 6h1z"></path></svg> </div> </button> <input type="hidden" name="type" class="js-site-search-type-field"> <div class="Overlay--hidden " data-modal-dialog-overlay> <modal-dialog data-action="close:qbsearch-input#handleClose cancel:qbsearch-input#handleClose" data-target="qbsearch-input.searchSuggestionsDialog" role="dialog" id="search-suggestions-dialog" aria-modal="true" aria-labelledby="search-suggestions-dialog-header" data-view-component="true" class="Overlay Overlay--width-large Overlay--height-auto"> <h1 id="search-suggestions-dialog-header" class="sr-only">Search code, repositories, users, issues, pull requests...</h1> <div class="Overlay-body Overlay-body--paddingNone"> <div data-view-component="true"> <div class="search-suggestions position-fixed width-full color-shadow-large border color-fg-default color-bg-default overflow-hidden d-flex flex-column query-builder-container" style="border-radius: 12px;" data-target="qbsearch-input.queryBuilderContainer" hidden > <!-- '"` --><!-- </textarea></xmp> --></option></form><form id="query-builder-test-form" action="" accept-charset="UTF-8" method="get"> <query-builder data-target="qbsearch-input.queryBuilder" id="query-builder-query-builder-test" data-filter-key=":" data-view-component="true" class="QueryBuilder search-query-builder"> <div class="FormControl FormControl--fullWidth"> <label id="query-builder-test-label" for="query-builder-test" class="FormControl-label sr-only"> Search </label> <div class="QueryBuilder-StyledInput width-fit " data-target="query-builder.styledInput" > <span id="query-builder-test-leadingvisual-wrap" class="FormControl-input-leadingVisualWrap QueryBuilder-leadingVisualWrap"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-search FormControl-input-leadingVisual"> <path d="M10.68 11.74a6 6 0 0 1-7.922-8.982 6 6 0 0 1 8.982 7.922l3.04 3.04a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215ZM11.5 7a4.499 4.499 0 1 0-8.997 0A4.499 4.499 0 0 0 11.5 7Z"></path> </svg> </span> <div data-target="query-builder.styledInputContainer" class="QueryBuilder-StyledInputContainer"> <div aria-hidden="true" class="QueryBuilder-StyledInputContent" data-target="query-builder.styledInputContent" ></div> <div class="QueryBuilder-InputWrapper"> <div aria-hidden="true" class="QueryBuilder-Sizer" data-target="query-builder.sizer"></div> <input id="query-builder-test" name="query-builder-test" value="" autocomplete="off" type="text" role="combobox" spellcheck="false" aria-expanded="false" aria-describedby="validation-c5efe5f8-a18e-467a-8eaf-436be3d4651e" data-target="query-builder.input" data-action=" input:query-builder#inputChange blur:query-builder#inputBlur keydown:query-builder#inputKeydown focus:query-builder#inputFocus " data-view-component="true" class="FormControl-input QueryBuilder-Input FormControl-medium" /> </div> </div> <span class="sr-only" id="query-builder-test-clear">Clear</span> <button role="button" id="query-builder-test-clear-button" aria-labelledby="query-builder-test-clear query-builder-test-label" data-target="query-builder.clearButton" data-action=" click:query-builder#clear focus:query-builder#clearButtonFocus blur:query-builder#clearButtonBlur " variant="small" hidden="hidden" type="button" data-view-component="true" class="Button Button--iconOnly Button--invisible Button--medium mr-1 px-2 py-0 d-flex flex-items-center rounded-1 color-fg-muted"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-x-circle-fill Button-visual"> <path d="M2.343 13.657A8 8 0 1 1 13.658 2.343 8 8 0 0 1 2.343 13.657ZM6.03 4.97a.751.751 0 0 0-1.042.018.751.751 0 0 0-.018 1.042L6.94 8 4.97 9.97a.749.749 0 0 0 .326 1.275.749.749 0 0 0 .734-.215L8 9.06l1.97 1.97a.749.749 0 0 0 1.275-.326.749.749 0 0 0-.215-.734L9.06 8l1.97-1.97a.749.749 0 0 0-.326-1.275.749.749 0 0 0-.734.215L8 6.94Z"></path> </svg> </button> </div> <template id="search-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-search"> <path d="M10.68 11.74a6 6 0 0 1-7.922-8.982 6 6 0 0 1 8.982 7.922l3.04 3.04a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215ZM11.5 7a4.499 4.499 0 1 0-8.997 0A4.499 4.499 0 0 0 11.5 7Z"></path> </svg> </template> <template id="code-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-code"> <path d="m11.28 3.22 4.25 4.25a.75.75 0 0 1 0 1.06l-4.25 4.25a.749.749 0 0 1-1.275-.326.749.749 0 0 1 .215-.734L13.94 8l-3.72-3.72a.749.749 0 0 1 .326-1.275.749.749 0 0 1 .734.215Zm-6.56 0a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042L2.06 8l3.72 3.72a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215L.47 8.53a.75.75 0 0 1 0-1.06Z"></path> </svg> </template> <template id="file-code-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-file-code"> <path d="M4 1.75C4 .784 4.784 0 5.75 0h5.586c.464 0 .909.184 1.237.513l2.914 2.914c.329.328.513.773.513 1.237v8.586A1.75 1.75 0 0 1 14.25 15h-9a.75.75 0 0 1 0-1.5h9a.25.25 0 0 0 .25-.25V6h-2.75A1.75 1.75 0 0 1 10 4.25V1.5H5.75a.25.25 0 0 0-.25.25v2.5a.75.75 0 0 1-1.5 0Zm1.72 4.97a.75.75 0 0 1 1.06 0l2 2a.75.75 0 0 1 0 1.06l-2 2a.749.749 0 0 1-1.275-.326.749.749 0 0 1 .215-.734l1.47-1.47-1.47-1.47a.75.75 0 0 1 0-1.06ZM3.28 7.78 1.81 9.25l1.47 1.47a.751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018l-2-2a.75.75 0 0 1 0-1.06l2-2a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042Zm8.22-6.218V4.25c0 .138.112.25.25.25h2.688l-.011-.013-2.914-2.914-.013-.011Z"></path> </svg> </template> <template id="history-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-history"> <path d="m.427 1.927 1.215 1.215a8.002 8.002 0 1 1-1.6 5.685.75.75 0 1 1 1.493-.154 6.5 6.5 0 1 0 1.18-4.458l1.358 1.358A.25.25 0 0 1 3.896 6H.25A.25.25 0 0 1 0 5.75V2.104a.25.25 0 0 1 .427-.177ZM7.75 4a.75.75 0 0 1 .75.75v2.992l2.028.812a.75.75 0 0 1-.557 1.392l-2.5-1A.751.751 0 0 1 7 8.25v-3.5A.75.75 0 0 1 7.75 4Z"></path> </svg> </template> <template id="repo-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-repo"> <path d="M2 2.5A2.5 2.5 0 0 1 4.5 0h8.75a.75.75 0 0 1 .75.75v12.5a.75.75 0 0 1-.75.75h-2.5a.75.75 0 0 1 0-1.5h1.75v-2h-8a1 1 0 0 0-.714 1.7.75.75 0 1 1-1.072 1.05A2.495 2.495 0 0 1 2 11.5Zm10.5-1h-8a1 1 0 0 0-1 1v6.708A2.486 2.486 0 0 1 4.5 9h8ZM5 12.25a.25.25 0 0 1 .25-.25h3.5a.25.25 0 0 1 .25.25v3.25a.25.25 0 0 1-.4.2l-1.45-1.087a.249.249 0 0 0-.3 0L5.4 15.7a.25.25 0 0 1-.4-.2Z"></path> </svg> </template> <template id="bookmark-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-bookmark"> <path d="M3 2.75C3 1.784 3.784 1 4.75 1h6.5c.966 0 1.75.784 1.75 1.75v11.5a.75.75 0 0 1-1.227.579L8 11.722l-3.773 3.107A.751.751 0 0 1 3 14.25Zm1.75-.25a.25.25 0 0 0-.25.25v9.91l3.023-2.489a.75.75 0 0 1 .954 0l3.023 2.49V2.75a.25.25 0 0 0-.25-.25Z"></path> </svg> </template> <template id="plus-circle-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-plus-circle"> <path d="M8 0a8 8 0 1 1 0 16A8 8 0 0 1 8 0ZM1.5 8a6.5 6.5 0 1 0 13 0 6.5 6.5 0 0 0-13 0Zm7.25-3.25v2.5h2.5a.75.75 0 0 1 0 1.5h-2.5v2.5a.75.75 0 0 1-1.5 0v-2.5h-2.5a.75.75 0 0 1 0-1.5h2.5v-2.5a.75.75 0 0 1 1.5 0Z"></path> </svg> </template> <template id="circle-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-dot-fill"> <path d="M8 4a4 4 0 1 1 0 8 4 4 0 0 1 0-8Z"></path> </svg> </template> <template id="trash-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-trash"> <path d="M11 1.75V3h2.25a.75.75 0 0 1 0 1.5H2.75a.75.75 0 0 1 0-1.5H5V1.75C5 .784 5.784 0 6.75 0h2.5C10.216 0 11 .784 11 1.75ZM4.496 6.675l.66 6.6a.25.25 0 0 0 .249.225h5.19a.25.25 0 0 0 .249-.225l.66-6.6a.75.75 0 0 1 1.492.149l-.66 6.6A1.748 1.748 0 0 1 10.595 15h-5.19a1.75 1.75 0 0 1-1.741-1.575l-.66-6.6a.75.75 0 1 1 1.492-.15ZM6.5 1.75V3h3V1.75a.25.25 0 0 0-.25-.25h-2.5a.25.25 0 0 0-.25.25Z"></path> </svg> </template> <template id="team-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-people"> <path d="M2 5.5a3.5 3.5 0 1 1 5.898 2.549 5.508 5.508 0 0 1 3.034 4.084.75.75 0 1 1-1.482.235 4 4 0 0 0-7.9 0 .75.75 0 0 1-1.482-.236A5.507 5.507 0 0 1 3.102 8.05 3.493 3.493 0 0 1 2 5.5ZM11 4a3.001 3.001 0 0 1 2.22 5.018 5.01 5.01 0 0 1 2.56 3.012.749.749 0 0 1-.885.954.752.752 0 0 1-.549-.514 3.507 3.507 0 0 0-2.522-2.372.75.75 0 0 1-.574-.73v-.352a.75.75 0 0 1 .416-.672A1.5 1.5 0 0 0 11 5.5.75.75 0 0 1 11 4Zm-5.5-.5a2 2 0 1 0-.001 3.999A2 2 0 0 0 5.5 3.5Z"></path> </svg> </template> <template id="project-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-project"> <path d="M1.75 0h12.5C15.216 0 16 .784 16 1.75v12.5A1.75 1.75 0 0 1 14.25 16H1.75A1.75 1.75 0 0 1 0 14.25V1.75C0 .784.784 0 1.75 0ZM1.5 1.75v12.5c0 .138.112.25.25.25h12.5a.25.25 0 0 0 .25-.25V1.75a.25.25 0 0 0-.25-.25H1.75a.25.25 0 0 0-.25.25ZM11.75 3a.75.75 0 0 1 .75.75v7.5a.75.75 0 0 1-1.5 0v-7.5a.75.75 0 0 1 .75-.75Zm-8.25.75a.75.75 0 0 1 1.5 0v5.5a.75.75 0 0 1-1.5 0ZM8 3a.75.75 0 0 1 .75.75v3.5a.75.75 0 0 1-1.5 0v-3.5A.75.75 0 0 1 8 3Z"></path> </svg> </template> <template id="pencil-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-pencil"> <path d="M11.013 1.427a1.75 1.75 0 0 1 2.474 0l1.086 1.086a1.75 1.75 0 0 1 0 2.474l-8.61 8.61c-.21.21-.47.364-.756.445l-3.251.93a.75.75 0 0 1-.927-.928l.929-3.25c.081-.286.235-.547.445-.758l8.61-8.61Zm.176 4.823L9.75 4.81l-6.286 6.287a.253.253 0 0 0-.064.108l-.558 1.953 1.953-.558a.253.253 0 0 0 .108-.064Zm1.238-3.763a.25.25 0 0 0-.354 0L10.811 3.75l1.439 1.44 1.263-1.263a.25.25 0 0 0 0-.354Z"></path> </svg> </template> <template id="copilot-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-copilot"> <path d="M7.998 15.035c-4.562 0-7.873-2.914-7.998-3.749V9.338c.085-.628.677-1.686 1.588-2.065.013-.07.024-.143.036-.218.029-.183.06-.384.126-.612-.201-.508-.254-1.084-.254-1.656 0-.87.128-1.769.693-2.484.579-.733 1.494-1.124 2.724-1.261 1.206-.134 2.262.034 2.944.765.05.053.096.108.139.165.044-.057.094-.112.143-.165.682-.731 1.738-.899 2.944-.765 1.23.137 2.145.528 2.724 1.261.566.715.693 1.614.693 2.484 0 .572-.053 1.148-.254 1.656.066.228.098.429.126.612.012.076.024.148.037.218.924.385 1.522 1.471 1.591 2.095v1.872c0 .766-3.351 3.795-8.002 3.795Zm0-1.485c2.28 0 4.584-1.11 5.002-1.433V7.862l-.023-.116c-.49.21-1.075.291-1.727.291-1.146 0-2.059-.327-2.71-.991A3.222 3.222 0 0 1 8 6.303a3.24 3.24 0 0 1-.544.743c-.65.664-1.563.991-2.71.991-.652 0-1.236-.081-1.727-.291l-.023.116v4.255c.419.323 2.722 1.433 5.002 1.433ZM6.762 2.83c-.193-.206-.637-.413-1.682-.297-1.019.113-1.479.404-1.713.7-.247.312-.369.789-.369 1.554 0 .793.129 1.171.308 1.371.162.181.519.379 1.442.379.853 0 1.339-.235 1.638-.54.315-.322.527-.827.617-1.553.117-.935-.037-1.395-.241-1.614Zm4.155-.297c-1.044-.116-1.488.091-1.681.297-.204.219-.359.679-.242 1.614.091.726.303 1.231.618 1.553.299.305.784.54 1.638.54.922 0 1.28-.198 1.442-.379.179-.2.308-.578.308-1.371 0-.765-.123-1.242-.37-1.554-.233-.296-.693-.587-1.713-.7Z"></path><path d="M6.25 9.037a.75.75 0 0 1 .75.75v1.501a.75.75 0 0 1-1.5 0V9.787a.75.75 0 0 1 .75-.75Zm4.25.75v1.501a.75.75 0 0 1-1.5 0V9.787a.75.75 0 0 1 1.5 0Z"></path> </svg> </template> <template id="copilot-error-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-copilot-error"> <path d="M16 11.24c0 .112-.072.274-.21.467L13 9.688V7.862l-.023-.116c-.49.21-1.075.291-1.727.291-.198 0-.388-.009-.571-.029L6.833 5.226a4.01 4.01 0 0 0 .17-.782c.117-.935-.037-1.395-.241-1.614-.193-.206-.637-.413-1.682-.297-.683.076-1.115.231-1.395.415l-1.257-.91c.579-.564 1.413-.877 2.485-.996 1.206-.134 2.262.034 2.944.765.05.053.096.108.139.165.044-.057.094-.112.143-.165.682-.731 1.738-.899 2.944-.765 1.23.137 2.145.528 2.724 1.261.566.715.693 1.614.693 2.484 0 .572-.053 1.148-.254 1.656.066.228.098.429.126.612.012.076.024.148.037.218.924.385 1.522 1.471 1.591 2.095Zm-5.083-8.707c-1.044-.116-1.488.091-1.681.297-.204.219-.359.679-.242 1.614.091.726.303 1.231.618 1.553.299.305.784.54 1.638.54.922 0 1.28-.198 1.442-.379.179-.2.308-.578.308-1.371 0-.765-.123-1.242-.37-1.554-.233-.296-.693-.587-1.713-.7Zm2.511 11.074c-1.393.776-3.272 1.428-5.43 1.428-4.562 0-7.873-2.914-7.998-3.749V9.338c.085-.628.677-1.686 1.588-2.065.013-.07.024-.143.036-.218.029-.183.06-.384.126-.612-.18-.455-.241-.963-.252-1.475L.31 4.107A.747.747 0 0 1 0 3.509V3.49a.748.748 0 0 1 .625-.73c.156-.026.306.047.435.139l14.667 10.578a.592.592 0 0 1 .227.264.752.752 0 0 1 .046.249v.022a.75.75 0 0 1-1.19.596Zm-1.367-.991L5.635 7.964a5.128 5.128 0 0 1-.889.073c-.652 0-1.236-.081-1.727-.291l-.023.116v4.255c.419.323 2.722 1.433 5.002 1.433 1.539 0 3.089-.505 4.063-.934Z"></path> </svg> </template> <template id="workflow-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-workflow"> <path d="M0 1.75C0 .784.784 0 1.75 0h3.5C6.216 0 7 .784 7 1.75v3.5A1.75 1.75 0 0 1 5.25 7H4v4a1 1 0 0 0 1 1h4v-1.25C9 9.784 9.784 9 10.75 9h3.5c.966 0 1.75.784 1.75 1.75v3.5A1.75 1.75 0 0 1 14.25 16h-3.5A1.75 1.75 0 0 1 9 14.25v-.75H5A2.5 2.5 0 0 1 2.5 11V7h-.75A1.75 1.75 0 0 1 0 5.25Zm1.75-.25a.25.25 0 0 0-.25.25v3.5c0 .138.112.25.25.25h3.5a.25.25 0 0 0 .25-.25v-3.5a.25.25 0 0 0-.25-.25Zm9 9a.25.25 0 0 0-.25.25v3.5c0 .138.112.25.25.25h3.5a.25.25 0 0 0 .25-.25v-3.5a.25.25 0 0 0-.25-.25Z"></path> </svg> </template> <template id="book-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-book"> <path d="M0 1.75A.75.75 0 0 1 .75 1h4.253c1.227 0 2.317.59 3 1.501A3.743 3.743 0 0 1 11.006 1h4.245a.75.75 0 0 1 .75.75v10.5a.75.75 0 0 1-.75.75h-4.507a2.25 2.25 0 0 0-1.591.659l-.622.621a.75.75 0 0 1-1.06 0l-.622-.621A2.25 2.25 0 0 0 5.258 13H.75a.75.75 0 0 1-.75-.75Zm7.251 10.324.004-5.073-.002-2.253A2.25 2.25 0 0 0 5.003 2.5H1.5v9h3.757a3.75 3.75 0 0 1 1.994.574ZM8.755 4.75l-.004 7.322a3.752 3.752 0 0 1 1.992-.572H14.5v-9h-3.495a2.25 2.25 0 0 0-2.25 2.25Z"></path> </svg> </template> <template id="code-review-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-code-review"> <path d="M1.75 1h12.5c.966 0 1.75.784 1.75 1.75v8.5A1.75 1.75 0 0 1 14.25 13H8.061l-2.574 2.573A1.458 1.458 0 0 1 3 14.543V13H1.75A1.75 1.75 0 0 1 0 11.25v-8.5C0 1.784.784 1 1.75 1ZM1.5 2.75v8.5c0 .138.112.25.25.25h2a.75.75 0 0 1 .75.75v2.19l2.72-2.72a.749.749 0 0 1 .53-.22h6.5a.25.25 0 0 0 .25-.25v-8.5a.25.25 0 0 0-.25-.25H1.75a.25.25 0 0 0-.25.25Zm5.28 1.72a.75.75 0 0 1 0 1.06L5.31 7l1.47 1.47a.751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018l-2-2a.75.75 0 0 1 0-1.06l2-2a.75.75 0 0 1 1.06 0Zm2.44 0a.75.75 0 0 1 1.06 0l2 2a.75.75 0 0 1 0 1.06l-2 2a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042L10.69 7 9.22 5.53a.75.75 0 0 1 0-1.06Z"></path> </svg> </template> <template id="codespaces-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-codespaces"> <path d="M0 11.25c0-.966.784-1.75 1.75-1.75h12.5c.966 0 1.75.784 1.75 1.75v3A1.75 1.75 0 0 1 14.25 16H1.75A1.75 1.75 0 0 1 0 14.25Zm2-9.5C2 .784 2.784 0 3.75 0h8.5C13.216 0 14 .784 14 1.75v5a1.75 1.75 0 0 1-1.75 1.75h-8.5A1.75 1.75 0 0 1 2 6.75Zm1.75-.25a.25.25 0 0 0-.25.25v5c0 .138.112.25.25.25h8.5a.25.25 0 0 0 .25-.25v-5a.25.25 0 0 0-.25-.25Zm-2 9.5a.25.25 0 0 0-.25.25v3c0 .138.112.25.25.25h12.5a.25.25 0 0 0 .25-.25v-3a.25.25 0 0 0-.25-.25Z"></path><path d="M7 12.75a.75.75 0 0 1 .75-.75h4.5a.75.75 0 0 1 0 1.5h-4.5a.75.75 0 0 1-.75-.75Zm-4 0a.75.75 0 0 1 .75-.75h.5a.75.75 0 0 1 0 1.5h-.5a.75.75 0 0 1-.75-.75Z"></path> </svg> </template> <template id="comment-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-comment"> <path d="M1 2.75C1 1.784 1.784 1 2.75 1h10.5c.966 0 1.75.784 1.75 1.75v7.5A1.75 1.75 0 0 1 13.25 12H9.06l-2.573 2.573A1.458 1.458 0 0 1 4 13.543V12H2.75A1.75 1.75 0 0 1 1 10.25Zm1.75-.25a.25.25 0 0 0-.25.25v7.5c0 .138.112.25.25.25h2a.75.75 0 0 1 .75.75v2.19l2.72-2.72a.749.749 0 0 1 .53-.22h4.5a.25.25 0 0 0 .25-.25v-7.5a.25.25 0 0 0-.25-.25Z"></path> </svg> </template> <template id="comment-discussion-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-comment-discussion"> <path d="M1.75 1h8.5c.966 0 1.75.784 1.75 1.75v5.5A1.75 1.75 0 0 1 10.25 10H7.061l-2.574 2.573A1.458 1.458 0 0 1 2 11.543V10h-.25A1.75 1.75 0 0 1 0 8.25v-5.5C0 1.784.784 1 1.75 1ZM1.5 2.75v5.5c0 .138.112.25.25.25h1a.75.75 0 0 1 .75.75v2.19l2.72-2.72a.749.749 0 0 1 .53-.22h3.5a.25.25 0 0 0 .25-.25v-5.5a.25.25 0 0 0-.25-.25h-8.5a.25.25 0 0 0-.25.25Zm13 2a.25.25 0 0 0-.25-.25h-.5a.75.75 0 0 1 0-1.5h.5c.966 0 1.75.784 1.75 1.75v5.5A1.75 1.75 0 0 1 14.25 12H14v1.543a1.458 1.458 0 0 1-2.487 1.03L9.22 12.28a.749.749 0 0 1 .326-1.275.749.749 0 0 1 .734.215l2.22 2.22v-2.19a.75.75 0 0 1 .75-.75h1a.25.25 0 0 0 .25-.25Z"></path> </svg> </template> <template id="organization-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-organization"> <path d="M1.75 16A1.75 1.75 0 0 1 0 14.25V1.75C0 .784.784 0 1.75 0h8.5C11.216 0 12 .784 12 1.75v12.5c0 .085-.006.168-.018.25h2.268a.25.25 0 0 0 .25-.25V8.285a.25.25 0 0 0-.111-.208l-1.055-.703a.749.749 0 1 1 .832-1.248l1.055.703c.487.325.779.871.779 1.456v5.965A1.75 1.75 0 0 1 14.25 16h-3.5a.766.766 0 0 1-.197-.026c-.099.017-.2.026-.303.026h-3a.75.75 0 0 1-.75-.75V14h-1v1.25a.75.75 0 0 1-.75.75Zm-.25-1.75c0 .138.112.25.25.25H4v-1.25a.75.75 0 0 1 .75-.75h2.5a.75.75 0 0 1 .75.75v1.25h2.25a.25.25 0 0 0 .25-.25V1.75a.25.25 0 0 0-.25-.25h-8.5a.25.25 0 0 0-.25.25ZM3.75 6h.5a.75.75 0 0 1 0 1.5h-.5a.75.75 0 0 1 0-1.5ZM3 3.75A.75.75 0 0 1 3.75 3h.5a.75.75 0 0 1 0 1.5h-.5A.75.75 0 0 1 3 3.75Zm4 3A.75.75 0 0 1 7.75 6h.5a.75.75 0 0 1 0 1.5h-.5A.75.75 0 0 1 7 6.75ZM7.75 3h.5a.75.75 0 0 1 0 1.5h-.5a.75.75 0 0 1 0-1.5ZM3 9.75A.75.75 0 0 1 3.75 9h.5a.75.75 0 0 1 0 1.5h-.5A.75.75 0 0 1 3 9.75ZM7.75 9h.5a.75.75 0 0 1 0 1.5h-.5a.75.75 0 0 1 0-1.5Z"></path> </svg> </template> <template id="rocket-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-rocket"> <path d="M14.064 0h.186C15.216 0 16 .784 16 1.75v.186a8.752 8.752 0 0 1-2.564 6.186l-.458.459c-.314.314-.641.616-.979.904v3.207c0 .608-.315 1.172-.833 1.49l-2.774 1.707a.749.749 0 0 1-1.11-.418l-.954-3.102a1.214 1.214 0 0 1-.145-.125L3.754 9.816a1.218 1.218 0 0 1-.124-.145L.528 8.717a.749.749 0 0 1-.418-1.11l1.71-2.774A1.748 1.748 0 0 1 3.31 4h3.204c.288-.338.59-.665.904-.979l.459-.458A8.749 8.749 0 0 1 14.064 0ZM8.938 3.623h-.002l-.458.458c-.76.76-1.437 1.598-2.02 2.5l-1.5 2.317 2.143 2.143 2.317-1.5c.902-.583 1.74-1.26 2.499-2.02l.459-.458a7.25 7.25 0 0 0 2.123-5.127V1.75a.25.25 0 0 0-.25-.25h-.186a7.249 7.249 0 0 0-5.125 2.123ZM3.56 14.56c-.732.732-2.334 1.045-3.005 1.148a.234.234 0 0 1-.201-.064.234.234 0 0 1-.064-.201c.103-.671.416-2.273 1.15-3.003a1.502 1.502 0 1 1 2.12 2.12Zm6.94-3.935c-.088.06-.177.118-.266.175l-2.35 1.521.548 1.783 1.949-1.2a.25.25 0 0 0 .119-.213ZM3.678 8.116 5.2 5.766c.058-.09.117-.178.176-.266H3.309a.25.25 0 0 0-.213.119l-1.2 1.95ZM12 5a1 1 0 1 1-2 0 1 1 0 0 1 2 0Z"></path> </svg> </template> <template id="shield-check-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-shield-check"> <path d="m8.533.133 5.25 1.68A1.75 1.75 0 0 1 15 3.48V7c0 1.566-.32 3.182-1.303 4.682-.983 1.498-2.585 2.813-5.032 3.855a1.697 1.697 0 0 1-1.33 0c-2.447-1.042-4.049-2.357-5.032-3.855C1.32 10.182 1 8.566 1 7V3.48a1.75 1.75 0 0 1 1.217-1.667l5.25-1.68a1.748 1.748 0 0 1 1.066 0Zm-.61 1.429.001.001-5.25 1.68a.251.251 0 0 0-.174.237V7c0 1.36.275 2.666 1.057 3.859.784 1.194 2.121 2.342 4.366 3.298a.196.196 0 0 0 .154 0c2.245-.957 3.582-2.103 4.366-3.297C13.225 9.666 13.5 8.358 13.5 7V3.48a.25.25 0 0 0-.174-.238l-5.25-1.68a.25.25 0 0 0-.153 0ZM11.28 6.28l-3.5 3.5a.75.75 0 0 1-1.06 0l-1.5-1.5a.749.749 0 0 1 .326-1.275.749.749 0 0 1 .734.215l.97.97 2.97-2.97a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042Z"></path> </svg> </template> <template id="heart-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-heart"> <path d="m8 14.25.345.666a.75.75 0 0 1-.69 0l-.008-.004-.018-.01a7.152 7.152 0 0 1-.31-.17 22.055 22.055 0 0 1-3.434-2.414C2.045 10.731 0 8.35 0 5.5 0 2.836 2.086 1 4.25 1 5.797 1 7.153 1.802 8 3.02 8.847 1.802 10.203 1 11.75 1 13.914 1 16 2.836 16 5.5c0 2.85-2.045 5.231-3.885 6.818a22.066 22.066 0 0 1-3.744 2.584l-.018.01-.006.003h-.002ZM4.25 2.5c-1.336 0-2.75 1.164-2.75 3 0 2.15 1.58 4.144 3.365 5.682A20.58 20.58 0 0 0 8 13.393a20.58 20.58 0 0 0 3.135-2.211C12.92 9.644 14.5 7.65 14.5 5.5c0-1.836-1.414-3-2.75-3-1.373 0-2.609.986-3.029 2.456a.749.749 0 0 1-1.442 0C6.859 3.486 5.623 2.5 4.25 2.5Z"></path> </svg> </template> <template id="server-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-server"> <path d="M1.75 1h12.5c.966 0 1.75.784 1.75 1.75v4c0 .372-.116.717-.314 1 .198.283.314.628.314 1v4a1.75 1.75 0 0 1-1.75 1.75H1.75A1.75 1.75 0 0 1 0 12.75v-4c0-.358.109-.707.314-1a1.739 1.739 0 0 1-.314-1v-4C0 1.784.784 1 1.75 1ZM1.5 2.75v4c0 .138.112.25.25.25h12.5a.25.25 0 0 0 .25-.25v-4a.25.25 0 0 0-.25-.25H1.75a.25.25 0 0 0-.25.25Zm.25 5.75a.25.25 0 0 0-.25.25v4c0 .138.112.25.25.25h12.5a.25.25 0 0 0 .25-.25v-4a.25.25 0 0 0-.25-.25ZM7 4.75A.75.75 0 0 1 7.75 4h4.5a.75.75 0 0 1 0 1.5h-4.5A.75.75 0 0 1 7 4.75ZM7.75 10h4.5a.75.75 0 0 1 0 1.5h-4.5a.75.75 0 0 1 0-1.5ZM3 4.75A.75.75 0 0 1 3.75 4h.5a.75.75 0 0 1 0 1.5h-.5A.75.75 0 0 1 3 4.75ZM3.75 10h.5a.75.75 0 0 1 0 1.5h-.5a.75.75 0 0 1 0-1.5Z"></path> </svg> </template> <template id="globe-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-globe"> <path d="M8 0a8 8 0 1 1 0 16A8 8 0 0 1 8 0ZM5.78 8.75a9.64 9.64 0 0 0 1.363 4.177c.255.426.542.832.857 1.215.245-.296.551-.705.857-1.215A9.64 9.64 0 0 0 10.22 8.75Zm4.44-1.5a9.64 9.64 0 0 0-1.363-4.177c-.307-.51-.612-.919-.857-1.215a9.927 9.927 0 0 0-.857 1.215A9.64 9.64 0 0 0 5.78 7.25Zm-5.944 1.5H1.543a6.507 6.507 0 0 0 4.666 5.5c-.123-.181-.24-.365-.352-.552-.715-1.192-1.437-2.874-1.581-4.948Zm-2.733-1.5h2.733c.144-2.074.866-3.756 1.58-4.948.12-.197.237-.381.353-.552a6.507 6.507 0 0 0-4.666 5.5Zm10.181 1.5c-.144 2.074-.866 3.756-1.58 4.948-.12.197-.237.381-.353.552a6.507 6.507 0 0 0 4.666-5.5Zm2.733-1.5a6.507 6.507 0 0 0-4.666-5.5c.123.181.24.365.353.552.714 1.192 1.436 2.874 1.58 4.948Z"></path> </svg> </template> <template id="issue-opened-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-issue-opened"> <path d="M8 9.5a1.5 1.5 0 1 0 0-3 1.5 1.5 0 0 0 0 3Z"></path><path d="M8 0a8 8 0 1 1 0 16A8 8 0 0 1 8 0ZM1.5 8a6.5 6.5 0 1 0 13 0 6.5 6.5 0 0 0-13 0Z"></path> </svg> </template> <template id="device-mobile-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-device-mobile"> <path d="M3.75 0h8.5C13.216 0 14 .784 14 1.75v12.5A1.75 1.75 0 0 1 12.25 16h-8.5A1.75 1.75 0 0 1 2 14.25V1.75C2 .784 2.784 0 3.75 0ZM3.5 1.75v12.5c0 .138.112.25.25.25h8.5a.25.25 0 0 0 .25-.25V1.75a.25.25 0 0 0-.25-.25h-8.5a.25.25 0 0 0-.25.25ZM8 13a1 1 0 1 1 0-2 1 1 0 0 1 0 2Z"></path> </svg> </template> <template id="package-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-package"> <path d="m8.878.392 5.25 3.045c.54.314.872.89.872 1.514v6.098a1.75 1.75 0 0 1-.872 1.514l-5.25 3.045a1.75 1.75 0 0 1-1.756 0l-5.25-3.045A1.75 1.75 0 0 1 1 11.049V4.951c0-.624.332-1.201.872-1.514L7.122.392a1.75 1.75 0 0 1 1.756 0ZM7.875 1.69l-4.63 2.685L8 7.133l4.755-2.758-4.63-2.685a.248.248 0 0 0-.25 0ZM2.5 5.677v5.372c0 .09.047.171.125.216l4.625 2.683V8.432Zm6.25 8.271 4.625-2.683a.25.25 0 0 0 .125-.216V5.677L8.75 8.432Z"></path> </svg> </template> <template id="credit-card-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-credit-card"> <path d="M10.75 9a.75.75 0 0 0 0 1.5h1.5a.75.75 0 0 0 0-1.5h-1.5Z"></path><path d="M0 3.75C0 2.784.784 2 1.75 2h12.5c.966 0 1.75.784 1.75 1.75v8.5A1.75 1.75 0 0 1 14.25 14H1.75A1.75 1.75 0 0 1 0 12.25ZM14.5 6.5h-13v5.75c0 .138.112.25.25.25h12.5a.25.25 0 0 0 .25-.25Zm0-2.75a.25.25 0 0 0-.25-.25H1.75a.25.25 0 0 0-.25.25V5h13Z"></path> </svg> </template> <template id="play-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-play"> <path d="M8 0a8 8 0 1 1 0 16A8 8 0 0 1 8 0ZM1.5 8a6.5 6.5 0 1 0 13 0 6.5 6.5 0 0 0-13 0Zm4.879-2.773 4.264 2.559a.25.25 0 0 1 0 .428l-4.264 2.559A.25.25 0 0 1 6 10.559V5.442a.25.25 0 0 1 .379-.215Z"></path> </svg> </template> <template id="gift-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-gift"> <path d="M2 2.75A2.75 2.75 0 0 1 4.75 0c.983 0 1.873.42 2.57 1.232.268.318.497.668.68 1.042.183-.375.411-.725.68-1.044C9.376.42 10.266 0 11.25 0a2.75 2.75 0 0 1 2.45 4h.55c.966 0 1.75.784 1.75 1.75v2c0 .698-.409 1.301-1 1.582v4.918A1.75 1.75 0 0 1 13.25 16H2.75A1.75 1.75 0 0 1 1 14.25V9.332C.409 9.05 0 8.448 0 7.75v-2C0 4.784.784 4 1.75 4h.55c-.192-.375-.3-.8-.3-1.25ZM7.25 9.5H2.5v4.75c0 .138.112.25.25.25h4.5Zm1.5 0v5h4.5a.25.25 0 0 0 .25-.25V9.5Zm0-4V8h5.5a.25.25 0 0 0 .25-.25v-2a.25.25 0 0 0-.25-.25Zm-7 0a.25.25 0 0 0-.25.25v2c0 .138.112.25.25.25h5.5V5.5h-5.5Zm3-4a1.25 1.25 0 0 0 0 2.5h2.309c-.233-.818-.542-1.401-.878-1.793-.43-.502-.915-.707-1.431-.707ZM8.941 4h2.309a1.25 1.25 0 0 0 0-2.5c-.516 0-1 .205-1.43.707-.337.392-.646.975-.879 1.793Z"></path> </svg> </template> <template id="code-square-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-code-square"> <path d="M0 1.75C0 .784.784 0 1.75 0h12.5C15.216 0 16 .784 16 1.75v12.5A1.75 1.75 0 0 1 14.25 16H1.75A1.75 1.75 0 0 1 0 14.25Zm1.75-.25a.25.25 0 0 0-.25.25v12.5c0 .138.112.25.25.25h12.5a.25.25 0 0 0 .25-.25V1.75a.25.25 0 0 0-.25-.25Zm7.47 3.97a.75.75 0 0 1 1.06 0l2 2a.75.75 0 0 1 0 1.06l-2 2a.749.749 0 0 1-1.275-.326.749.749 0 0 1 .215-.734L10.69 8 9.22 6.53a.75.75 0 0 1 0-1.06ZM6.78 6.53 5.31 8l1.47 1.47a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215l-2-2a.75.75 0 0 1 0-1.06l2-2a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042Z"></path> </svg> </template> <template id="device-desktop-icon"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-device-desktop"> <path d="M14.25 1c.966 0 1.75.784 1.75 1.75v7.5A1.75 1.75 0 0 1 14.25 12h-3.727c.099 1.041.52 1.872 1.292 2.757A.752.752 0 0 1 11.25 16h-6.5a.75.75 0 0 1-.565-1.243c.772-.885 1.192-1.716 1.292-2.757H1.75A1.75 1.75 0 0 1 0 10.25v-7.5C0 1.784.784 1 1.75 1ZM1.75 2.5a.25.25 0 0 0-.25.25v7.5c0 .138.112.25.25.25h12.5a.25.25 0 0 0 .25-.25v-7.5a.25.25 0 0 0-.25-.25ZM9.018 12H6.982a5.72 5.72 0 0 1-.765 2.5h3.566a5.72 5.72 0 0 1-.765-2.5Z"></path> </svg> </template> <div class="position-relative"> <ul role="listbox" class="ActionListWrap QueryBuilder-ListWrap" aria-label="Suggestions" data-action=" combobox-commit:query-builder#comboboxCommit mousedown:query-builder#resultsMousedown " data-target="query-builder.resultsList" data-persist-list=false id="query-builder-test-results" ></ul> </div> <div class="FormControl-inlineValidation" id="validation-c5efe5f8-a18e-467a-8eaf-436be3d4651e" hidden="hidden"> <span class="FormControl-inlineValidation--visual"> <svg aria-hidden="true" height="12" viewBox="0 0 12 12" version="1.1" width="12" data-view-component="true" class="octicon octicon-alert-fill"> <path d="M4.855.708c.5-.896 1.79-.896 2.29 0l4.675 8.351a1.312 1.312 0 0 1-1.146 1.954H1.33A1.313 1.313 0 0 1 .183 9.058ZM7 7V3H5v4Zm-1 3a1 1 0 1 0 0-2 1 1 0 0 0 0 2Z"></path> </svg> </span> <span></span> </div> </div> <div data-target="query-builder.screenReaderFeedback" aria-live="polite" aria-atomic="true" class="sr-only"></div> </query-builder></form> <div class="d-flex flex-row color-fg-muted px-3 text-small color-bg-default search-feedback-prompt"> <a target="_blank" href="https://docs.github.com/search-github/github-code-search/understanding-github-code-search-syntax" data-view-component="true" class="Link color-fg-accent text-normal ml-2">Search syntax tips</a> <div class="d-flex flex-1"></div> </div> </div> </div> </div> </modal-dialog></div> </div> <div data-action="click:qbsearch-input#retract" class="dark-backdrop position-fixed" hidden data-target="qbsearch-input.darkBackdrop"></div> <div class="color-fg-default"> <dialog-helper> <dialog data-target="qbsearch-input.feedbackDialog" data-action="close:qbsearch-input#handleDialogClose cancel:qbsearch-input#handleDialogClose" id="feedback-dialog" aria-modal="true" aria-labelledby="feedback-dialog-title" aria-describedby="feedback-dialog-description" data-view-component="true" class="Overlay Overlay-whenNarrow Overlay--size-medium Overlay--motion-scaleFade Overlay--disableScroll"> <div data-view-component="true" class="Overlay-header"> <div class="Overlay-headerContentWrap"> <div class="Overlay-titleWrap"> <h1 class="Overlay-title " id="feedback-dialog-title"> Provide feedback </h1> </div> <div class="Overlay-actionWrap"> <button data-close-dialog-id="feedback-dialog" aria-label="Close" type="button" data-view-component="true" class="close-button Overlay-closeButton"><svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-x"> <path d="M3.72 3.72a.75.75 0 0 1 1.06 0L8 6.94l3.22-3.22a.749.749 0 0 1 1.275.326.749.749 0 0 1-.215.734L9.06 8l3.22 3.22a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215L8 9.06l-3.22 3.22a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042L6.94 8 3.72 4.78a.75.75 0 0 1 0-1.06Z"></path> </svg></button> </div> </div> </div> <scrollable-region data-labelled-by="feedback-dialog-title"> <div data-view-component="true" class="Overlay-body"> <!-- '"` --><!-- </textarea></xmp> --></option></form><form id="code-search-feedback-form" data-turbo="false" action="/search/feedback" accept-charset="UTF-8" method="post"><input type="hidden" data-csrf="true" name="authenticity_token" value="0HlOt4iJV9n1ToNsvvo7N2EnB75sLjsi0PQvSJdQE4SZpyztJfKOzqk157wXcfSfapSEHxj52dk2cRura6qQLA==" /> <p>We read every piece of feedback, and take your input very seriously.</p> <textarea name="feedback" class="form-control width-full mb-2" style="height: 120px" id="feedback"></textarea> <input name="include_email" id="include_email" aria-label="Include my email address so I can be contacted" class="form-control mr-2" type="checkbox"> <label for="include_email" style="font-weight: normal">Include my email address so I can be contacted</label> </form></div> </scrollable-region> <div data-view-component="true" class="Overlay-footer Overlay-footer--alignEnd"> <button data-close-dialog-id="feedback-dialog" type="button" data-view-component="true" class="btn"> Cancel </button> <button form="code-search-feedback-form" data-action="click:qbsearch-input#submitFeedback" type="submit" data-view-component="true" class="btn-primary btn"> Submit feedback </button> </div> </dialog></dialog-helper> <custom-scopes data-target="qbsearch-input.customScopesManager"> <dialog-helper> <dialog data-target="custom-scopes.customScopesModalDialog" data-action="close:qbsearch-input#handleDialogClose cancel:qbsearch-input#handleDialogClose" id="custom-scopes-dialog" aria-modal="true" aria-labelledby="custom-scopes-dialog-title" aria-describedby="custom-scopes-dialog-description" data-view-component="true" class="Overlay Overlay-whenNarrow Overlay--size-medium Overlay--motion-scaleFade Overlay--disableScroll"> <div data-view-component="true" class="Overlay-header Overlay-header--divided"> <div class="Overlay-headerContentWrap"> <div class="Overlay-titleWrap"> <h1 class="Overlay-title " id="custom-scopes-dialog-title"> Saved searches </h1> <h2 id="custom-scopes-dialog-description" class="Overlay-description">Use saved searches to filter your results more quickly</h2> </div> <div class="Overlay-actionWrap"> <button data-close-dialog-id="custom-scopes-dialog" aria-label="Close" type="button" data-view-component="true" class="close-button Overlay-closeButton"><svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-x"> <path d="M3.72 3.72a.75.75 0 0 1 1.06 0L8 6.94l3.22-3.22a.749.749 0 0 1 1.275.326.749.749 0 0 1-.215.734L9.06 8l3.22 3.22a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215L8 9.06l-3.22 3.22a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042L6.94 8 3.72 4.78a.75.75 0 0 1 0-1.06Z"></path> </svg></button> </div> </div> </div> <scrollable-region data-labelled-by="custom-scopes-dialog-title"> <div data-view-component="true" class="Overlay-body"> <div data-target="custom-scopes.customScopesModalDialogFlash"></div> <div hidden class="create-custom-scope-form" data-target="custom-scopes.createCustomScopeForm"> <!-- '"` --><!-- </textarea></xmp> --></option></form><form id="custom-scopes-dialog-form" data-turbo="false" action="/search/custom_scopes" accept-charset="UTF-8" method="post"><input type="hidden" data-csrf="true" name="authenticity_token" value="WpQ1cOXU/1QAqpo3kPyNaMOtkubH4yjDQ1zB6x2rIeAzdCf/Xq0/V28zfQcnaqTOtU0FCUT2SC8x/iaueKJwIg==" /> <div data-target="custom-scopes.customScopesModalDialogFlash"></div> <input type="hidden" id="custom_scope_id" name="custom_scope_id" data-target="custom-scopes.customScopesIdField"> <div class="form-group"> <label for="custom_scope_name">Name</label> <auto-check src="/search/custom_scopes/check_name" required only-validate-on-blur="false"> <input type="text" name="custom_scope_name" id="custom_scope_name" data-target="custom-scopes.customScopesNameField" class="form-control" autocomplete="off" placeholder="github-ruby" required maxlength="50"> <input type="hidden" data-csrf="true" value="CZccj0c46G9ui6eaF0JlmTT9qds2uhogXWfWBqL6A5GtqqjLSplxSy12S589FcXGLZMk/IY3Mjpz3p0D8Zy1oQ==" /> </auto-check> </div> <div class="form-group"> <label for="custom_scope_query">Query</label> <input type="text" name="custom_scope_query" id="custom_scope_query" data-target="custom-scopes.customScopesQueryField" class="form-control" autocomplete="off" placeholder="(repo:mona/a OR repo:mona/b) AND lang:python" required maxlength="500"> </div> <p class="text-small color-fg-muted"> To see all available qualifiers, see our <a class="Link--inTextBlock" href="https://docs.github.com/search-github/github-code-search/understanding-github-code-search-syntax">documentation</a>. </p> </form> </div> <div data-target="custom-scopes.manageCustomScopesForm"> <div data-target="custom-scopes.list"></div> </div> </div> </scrollable-region> <div data-view-component="true" class="Overlay-footer Overlay-footer--alignEnd Overlay-footer--divided"> <button data-action="click:custom-scopes#customScopesCancel" type="button" data-view-component="true" class="btn"> Cancel </button> <button form="custom-scopes-dialog-form" data-action="click:custom-scopes#customScopesSubmit" data-target="custom-scopes.customScopesSubmitButton" type="submit" data-view-component="true" class="btn-primary btn"> Create saved search </button> </div> </dialog></dialog-helper> </custom-scopes> </div> </qbsearch-input> <div class="position-relative HeaderMenu-link-wrap d-lg-inline-block"> <a href="/login?return_to=https%3A%2F%2Fgithub.com%2Fkroitor%2Fgjk.c" class="HeaderMenu-link HeaderMenu-link--sign-in HeaderMenu-button flex-shrink-0 no-underline d-none d-lg-inline-flex border border-lg-0 rounded rounded-lg-0 px-2 py-1" style="margin-left: 12px;" data-hydro-click="{"event_type":"authentication.click","payload":{"location_in_page":"site header menu","repository_id":null,"auth_type":"SIGN_UP","originating_url":"https://github.com/kroitor/gjk.c","user_id":null}}" data-hydro-click-hmac="2ac8eb781f277273bc83a3a02d65530e3b132e22dc514c809d62270b9616962f" data-analytics-event="{"category":"Marketing nav","action":"click to go to homepage","label":"ref_page:Marketing;ref_cta:Sign in;ref_loc:Header"}" > Sign in </a> </div> <a href="/signup?ref_cta=Sign+up&ref_loc=header+logged+out&ref_page=%2F%3Cuser-name%3E%2F%3Crepo-name%3E&source=header-repo&source_repo=kroitor%2Fgjk.c" class="HeaderMenu-link HeaderMenu-link--sign-up HeaderMenu-button flex-shrink-0 d-flex d-lg-inline-flex no-underline border color-border-default rounded px-2 py-1" data-hydro-click="{"event_type":"authentication.click","payload":{"location_in_page":"site header menu","repository_id":null,"auth_type":"SIGN_UP","originating_url":"https://github.com/kroitor/gjk.c","user_id":null}}" data-hydro-click-hmac="2ac8eb781f277273bc83a3a02d65530e3b132e22dc514c809d62270b9616962f" data-analytics-event="{"category":"Sign up","action":"click to sign up for account","label":"ref_page:/<user-name>/<repo-name>;ref_cta:Sign up;ref_loc:header logged out"}" > Sign up </a> <button type="button" class="sr-only js-header-menu-focus-trap d-block d-lg-none">Reseting focus</button> </div> </div> </div> </div> </header> <div hidden="hidden" data-view-component="true" class="js-stale-session-flash stale-session-flash flash flash-warn flash-full"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-alert"> <path d="M6.457 1.047c.659-1.234 2.427-1.234 3.086 0l6.082 11.378A1.75 1.75 0 0 1 14.082 15H1.918a1.75 1.75 0 0 1-1.543-2.575Zm1.763.707a.25.25 0 0 0-.44 0L1.698 13.132a.25.25 0 0 0 .22.368h12.164a.25.25 0 0 0 .22-.368Zm.53 3.996v2.5a.75.75 0 0 1-1.5 0v-2.5a.75.75 0 0 1 1.5 0ZM9 11a1 1 0 1 1-2 0 1 1 0 0 1 2 0Z"></path> </svg> <span class="js-stale-session-flash-signed-in" hidden>You signed in with another tab or window. <a class="Link--inTextBlock" href="">Reload</a> to refresh your session.</span> <span class="js-stale-session-flash-signed-out" hidden>You signed out in another tab or window. <a class="Link--inTextBlock" href="">Reload</a> to refresh your session.</span> <span class="js-stale-session-flash-switched" hidden>You switched accounts on another tab or window. <a class="Link--inTextBlock" href="">Reload</a> to refresh your session.</span> <button id="icon-button-d8ebcf8b-5e70-4f80-a423-1994d08f8fdc" aria-labelledby="tooltip-558a3943-022c-4ae1-b890-cc6c5dabec38" type="button" data-view-component="true" class="Button Button--iconOnly Button--invisible Button--medium flash-close js-flash-close"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-x Button-visual"> <path d="M3.72 3.72a.75.75 0 0 1 1.06 0L8 6.94l3.22-3.22a.749.749 0 0 1 1.275.326.749.749 0 0 1-.215.734L9.06 8l3.22 3.22a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215L8 9.06l-3.22 3.22a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042L6.94 8 3.72 4.78a.75.75 0 0 1 0-1.06Z"></path> </svg> </button><tool-tip id="tooltip-558a3943-022c-4ae1-b890-cc6c5dabec38" for="icon-button-d8ebcf8b-5e70-4f80-a423-1994d08f8fdc" popover="manual" data-direction="s" data-type="label" data-view-component="true" class="sr-only position-absolute">Dismiss alert</tool-tip> </div> </div> <div id="start-of-content" class="show-on-focus"></div> <div id="js-flash-container" class="flash-container" data-turbo-replace> <template class="js-flash-template"> <div class="flash flash-full {{ className }}"> <div > <button autofocus class="flash-close js-flash-close" type="button" aria-label="Dismiss this message"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-x"> <path d="M3.72 3.72a.75.75 0 0 1 1.06 0L8 6.94l3.22-3.22a.749.749 0 0 1 1.275.326.749.749 0 0 1-.215.734L9.06 8l3.22 3.22a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215L8 9.06l-3.22 3.22a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042L6.94 8 3.72 4.78a.75.75 0 0 1 0-1.06Z"></path> </svg> </button> <div aria-atomic="true" role="alert" class="js-flash-alert"> <div>{{ message }}</div> </div> </div> </div> </template> </div> <div class="application-main " data-commit-hovercards-enabled data-discussion-hovercards-enabled data-issue-and-pr-hovercards-enabled data-project-hovercards-enabled > <div itemscope itemtype="http://schema.org/SoftwareSourceCode" class=""> <main id="js-repo-pjax-container" > <div id="repository-container-header" class="pt-3 hide-full-screen" style="background-color: var(--page-header-bgColor, var(--color-page-header-bg));" data-turbo-replace> <div class="d-flex flex-nowrap flex-justify-end mb-3 px-3 px-lg-5" style="gap: 1rem;"> <div class="flex-auto min-width-0 width-fit"> <div class=" d-flex flex-wrap flex-items-center wb-break-word f3 text-normal"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-repo color-fg-muted mr-2"> <path d="M2 2.5A2.5 2.5 0 0 1 4.5 0h8.75a.75.75 0 0 1 .75.75v12.5a.75.75 0 0 1-.75.75h-2.5a.75.75 0 0 1 0-1.5h1.75v-2h-8a1 1 0 0 0-.714 1.7.75.75 0 1 1-1.072 1.05A2.495 2.495 0 0 1 2 11.5Zm10.5-1h-8a1 1 0 0 0-1 1v6.708A2.486 2.486 0 0 1 4.5 9h8ZM5 12.25a.25.25 0 0 1 .25-.25h3.5a.25.25 0 0 1 .25.25v3.25a.25.25 0 0 1-.4.2l-1.45-1.087a.249.249 0 0 0-.3 0L5.4 15.7a.25.25 0 0 1-.4-.2Z"></path> </svg> <span class="author flex-self-stretch" itemprop="author"> <a class="url fn" rel="author" data-hovercard-type="user" data-hovercard-url="/users/kroitor/hovercard" data-octo-click="hovercard-link-click" data-octo-dimensions="link_type:self" href="/kroitor"> kroitor </a> </span> <span class="mx-1 flex-self-stretch color-fg-muted">/</span> <strong itemprop="name" class="mr-2 flex-self-stretch"> <a data-pjax="#repo-content-pjax-container" data-turbo-frame="repo-content-turbo-frame" href="/kroitor/gjk.c">gjk.c</a> </strong> <span></span><span class="Label Label--secondary v-align-middle mr-1">Public</span> </div> </div> <div id="repository-details-container" class="flex-shrink-0" data-turbo-replace style="max-width: 70%;"> <ul class="pagehead-actions flex-shrink-0 d-none d-md-inline" style="padding: 2px 0;"> <li> <a href="/login?return_to=%2Fkroitor%2Fgjk.c" rel="nofollow" id="repository-details-watch-button" data-hydro-click="{"event_type":"authentication.click","payload":{"location_in_page":"notification subscription menu watch","repository_id":null,"auth_type":"LOG_IN","originating_url":"https://github.com/kroitor/gjk.c","user_id":null}}" data-hydro-click-hmac="c55588df3b070771f0ea206a61c5bfeb41fd77b0ac6a419d2cd629ae9ff02a11" aria-label="You must be signed in to change notification settings" data-view-component="true" class="btn-sm btn"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-bell mr-2"> <path d="M8 16a2 2 0 0 0 1.985-1.75c.017-.137-.097-.25-.235-.25h-3.5c-.138 0-.252.113-.235.25A2 2 0 0 0 8 16ZM3 5a5 5 0 0 1 10 0v2.947c0 .05.015.098.042.139l1.703 2.555A1.519 1.519 0 0 1 13.482 13H2.518a1.516 1.516 0 0 1-1.263-2.36l1.703-2.554A.255.255 0 0 0 3 7.947Zm5-3.5A3.5 3.5 0 0 0 4.5 5v2.947c0 .346-.102.683-.294.97l-1.703 2.556a.017.017 0 0 0-.003.01l.001.006c0 .002.002.004.004.006l.006.004.007.001h10.964l.007-.001.006-.004.004-.006.001-.007a.017.017 0 0 0-.003-.01l-1.703-2.554a1.745 1.745 0 0 1-.294-.97V5A3.5 3.5 0 0 0 8 1.5Z"></path> </svg>Notifications </a> <tool-tip id="tooltip-deb95f2e-8299-4bac-adcf-103bac418238" for="repository-details-watch-button" popover="manual" data-direction="s" data-type="description" data-view-component="true" class="sr-only position-absolute">You must be signed in to change notification settings</tool-tip> </li> <li> <a icon="repo-forked" id="fork-button" href="/login?return_to=%2Fkroitor%2Fgjk.c" rel="nofollow" data-hydro-click="{"event_type":"authentication.click","payload":{"location_in_page":"repo details fork button","repository_id":48733330,"auth_type":"LOG_IN","originating_url":"https://github.com/kroitor/gjk.c","user_id":null}}" data-hydro-click-hmac="07142487d646e1d44b4dd6df38d7d6922c8c0e8e408b2d136f56b9603faec170" data-view-component="true" class="btn-sm btn"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-repo-forked mr-2"> <path d="M5 5.372v.878c0 .414.336.75.75.75h4.5a.75.75 0 0 0 .75-.75v-.878a2.25 2.25 0 1 1 1.5 0v.878a2.25 2.25 0 0 1-2.25 2.25h-1.5v2.128a2.251 2.251 0 1 1-1.5 0V8.5h-1.5A2.25 2.25 0 0 1 3.5 6.25v-.878a2.25 2.25 0 1 1 1.5 0ZM5 3.25a.75.75 0 1 0-1.5 0 .75.75 0 0 0 1.5 0Zm6.75.75a.75.75 0 1 0 0-1.5.75.75 0 0 0 0 1.5Zm-3 8.75a.75.75 0 1 0-1.5 0 .75.75 0 0 0 1.5 0Z"></path> </svg>Fork <span id="repo-network-counter" data-pjax-replace="true" data-turbo-replace="true" title="83" data-view-component="true" class="Counter">83</span> </a> </li> <li> <div data-view-component="true" class="BtnGroup d-flex"> <a href="/login?return_to=%2Fkroitor%2Fgjk.c" rel="nofollow" data-hydro-click="{"event_type":"authentication.click","payload":{"location_in_page":"star button","repository_id":48733330,"auth_type":"LOG_IN","originating_url":"https://github.com/kroitor/gjk.c","user_id":null}}" data-hydro-click-hmac="741b22b6ff31a3bbac0940943d6a87d719abca602a5e2a0abbfc10136e92b894" aria-label="You must be signed in to star a repository" data-view-component="true" class="tooltipped tooltipped-sw btn-sm btn"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-star v-align-text-bottom d-inline-block mr-2"> <path d="M8 .25a.75.75 0 0 1 .673.418l1.882 3.815 4.21.612a.75.75 0 0 1 .416 1.279l-3.046 2.97.719 4.192a.751.751 0 0 1-1.088.791L8 12.347l-3.766 1.98a.75.75 0 0 1-1.088-.79l.72-4.194L.818 6.374a.75.75 0 0 1 .416-1.28l4.21-.611L7.327.668A.75.75 0 0 1 8 .25Zm0 2.445L6.615 5.5a.75.75 0 0 1-.564.41l-3.097.45 2.24 2.184a.75.75 0 0 1 .216.664l-.528 3.084 2.769-1.456a.75.75 0 0 1 .698 0l2.77 1.456-.53-3.084a.75.75 0 0 1 .216-.664l2.24-2.183-3.096-.45a.75.75 0 0 1-.564-.41L8 2.694Z"></path> </svg><span data-view-component="true" class="d-inline"> Star </span> <span id="repo-stars-counter-star" aria-label="869 users starred this repository" data-singular-suffix="user starred this repository" data-plural-suffix="users starred this repository" data-turbo-replace="true" title="869" data-view-component="true" class="Counter js-social-count">869</span> </a></div> </li> </ul> </div> </div> <div id="responsive-meta-container" data-turbo-replace> <div class="d-block d-md-none mb-2 px-3 px-md-4 px-lg-5"> <p class="f4 mb-3 "> Gilbert-Johnson-Keerthi (GJK) collision detection algorithm in 200 lines of clean plain C </p> <h3 class="sr-only">License</h3> <div class="mb-2"> <a href="/kroitor/gjk.c/blob/master/LICENSE.txt" class="Link--muted" data-analytics-event="{"category":"Repository Overview","action":"click","label":"location:sidebar;file:license"}" > <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-law mr-2"> <path d="M8.75.75V2h.985c.304 0 .603.08.867.231l1.29.736c.038.022.08.033.124.033h2.234a.75.75 0 0 1 0 1.5h-.427l2.111 4.692a.75.75 0 0 1-.154.838l-.53-.53.529.531-.001.002-.002.002-.006.006-.006.005-.01.01-.045.04c-.21.176-.441.327-.686.45C14.556 10.78 13.88 11 13 11a4.498 4.498 0 0 1-2.023-.454 3.544 3.544 0 0 1-.686-.45l-.045-.04-.016-.015-.006-.006-.004-.004v-.001a.75.75 0 0 1-.154-.838L12.178 4.5h-.162c-.305 0-.604-.079-.868-.231l-1.29-.736a.245.245 0 0 0-.124-.033H8.75V13h2.5a.75.75 0 0 1 0 1.5h-6.5a.75.75 0 0 1 0-1.5h2.5V3.5h-.984a.245.245 0 0 0-.124.033l-1.289.737c-.265.15-.564.23-.869.23h-.162l2.112 4.692a.75.75 0 0 1-.154.838l-.53-.53.529.531-.001.002-.002.002-.006.006-.016.015-.045.04c-.21.176-.441.327-.686.45C4.556 10.78 3.88 11 3 11a4.498 4.498 0 0 1-2.023-.454 3.544 3.544 0 0 1-.686-.45l-.045-.04-.016-.015-.006-.006-.004-.004v-.001a.75.75 0 0 1-.154-.838L2.178 4.5H1.75a.75.75 0 0 1 0-1.5h2.234a.249.249 0 0 0 .125-.033l1.288-.737c.265-.15.564-.23.869-.23h.984V.75a.75.75 0 0 1 1.5 0Zm2.945 8.477c.285.135.718.273 1.305.273s1.02-.138 1.305-.273L13 6.327Zm-10 0c.285.135.718.273 1.305.273s1.02-.138 1.305-.273L3 6.327Z"></path> </svg> WTFPL license </a> </div> <div class="mb-3"> <a class="Link--secondary no-underline mr-3" href="/kroitor/gjk.c/stargazers"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-star mr-1"> <path d="M8 .25a.75.75 0 0 1 .673.418l1.882 3.815 4.21.612a.75.75 0 0 1 .416 1.279l-3.046 2.97.719 4.192a.751.751 0 0 1-1.088.791L8 12.347l-3.766 1.98a.75.75 0 0 1-1.088-.79l.72-4.194L.818 6.374a.75.75 0 0 1 .416-1.28l4.21-.611L7.327.668A.75.75 0 0 1 8 .25Zm0 2.445L6.615 5.5a.75.75 0 0 1-.564.41l-3.097.45 2.24 2.184a.75.75 0 0 1 .216.664l-.528 3.084 2.769-1.456a.75.75 0 0 1 .698 0l2.77 1.456-.53-3.084a.75.75 0 0 1 .216-.664l2.24-2.183-3.096-.45a.75.75 0 0 1-.564-.41L8 2.694Z"></path> </svg> <span class="text-bold">869</span> stars </a> <a class="Link--secondary no-underline mr-3" href="/kroitor/gjk.c/forks"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-repo-forked mr-1"> <path d="M5 5.372v.878c0 .414.336.75.75.75h4.5a.75.75 0 0 0 .75-.75v-.878a2.25 2.25 0 1 1 1.5 0v.878a2.25 2.25 0 0 1-2.25 2.25h-1.5v2.128a2.251 2.251 0 1 1-1.5 0V8.5h-1.5A2.25 2.25 0 0 1 3.5 6.25v-.878a2.25 2.25 0 1 1 1.5 0ZM5 3.25a.75.75 0 1 0-1.5 0 .75.75 0 0 0 1.5 0Zm6.75.75a.75.75 0 1 0 0-1.5.75.75 0 0 0 0 1.5Zm-3 8.75a.75.75 0 1 0-1.5 0 .75.75 0 0 0 1.5 0Z"></path> </svg> <span class="text-bold">83</span> forks </a> <a class="Link--secondary no-underline mr-3 d-inline-block" href="/kroitor/gjk.c/branches"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-git-branch mr-1"> <path d="M9.5 3.25a2.25 2.25 0 1 1 3 2.122V6A2.5 2.5 0 0 1 10 8.5H6a1 1 0 0 0-1 1v1.128a2.251 2.251 0 1 1-1.5 0V5.372a2.25 2.25 0 1 1 1.5 0v1.836A2.493 2.493 0 0 1 6 7h4a1 1 0 0 0 1-1v-.628A2.25 2.25 0 0 1 9.5 3.25Zm-6 0a.75.75 0 1 0 1.5 0 .75.75 0 0 0-1.5 0Zm8.25-.75a.75.75 0 1 0 0 1.5.75.75 0 0 0 0-1.5ZM4.25 12a.75.75 0 1 0 0 1.5.75.75 0 0 0 0-1.5Z"></path> </svg> <span>Branches</span> </a> <a class="Link--secondary no-underline d-inline-block" href="/kroitor/gjk.c/tags"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-tag mr-1"> <path d="M1 7.775V2.75C1 1.784 1.784 1 2.75 1h5.025c.464 0 .91.184 1.238.513l6.25 6.25a1.75 1.75 0 0 1 0 2.474l-5.026 5.026a1.75 1.75 0 0 1-2.474 0l-6.25-6.25A1.752 1.752 0 0 1 1 7.775Zm1.5 0c0 .066.026.13.073.177l6.25 6.25a.25.25 0 0 0 .354 0l5.025-5.025a.25.25 0 0 0 0-.354l-6.25-6.25a.25.25 0 0 0-.177-.073H2.75a.25.25 0 0 0-.25.25ZM6 5a1 1 0 1 1 0 2 1 1 0 0 1 0-2Z"></path> </svg> <span>Tags</span> </a> <a class="Link--secondary no-underline d-inline-block" href="/kroitor/gjk.c/activity"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-pulse mr-1"> <path d="M6 2c.306 0 .582.187.696.471L10 10.731l1.304-3.26A.751.751 0 0 1 12 7h3.25a.75.75 0 0 1 0 1.5h-2.742l-1.812 4.528a.751.751 0 0 1-1.392 0L6 4.77 4.696 8.03A.75.75 0 0 1 4 8.5H.75a.75.75 0 0 1 0-1.5h2.742l1.812-4.529A.751.751 0 0 1 6 2Z"></path> </svg> <span>Activity</span> </a> </div> <div class="d-flex flex-wrap gap-2"> <div class="flex-1"> <div data-view-component="true" class="BtnGroup d-flex"> <a href="/login?return_to=%2Fkroitor%2Fgjk.c" rel="nofollow" data-hydro-click="{"event_type":"authentication.click","payload":{"location_in_page":"star button","repository_id":48733330,"auth_type":"LOG_IN","originating_url":"https://github.com/kroitor/gjk.c","user_id":null}}" data-hydro-click-hmac="741b22b6ff31a3bbac0940943d6a87d719abca602a5e2a0abbfc10136e92b894" aria-label="You must be signed in to star a repository" data-view-component="true" class="tooltipped tooltipped-sw btn-sm btn btn-block"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-star v-align-text-bottom d-inline-block mr-2"> <path d="M8 .25a.75.75 0 0 1 .673.418l1.882 3.815 4.21.612a.75.75 0 0 1 .416 1.279l-3.046 2.97.719 4.192a.751.751 0 0 1-1.088.791L8 12.347l-3.766 1.98a.75.75 0 0 1-1.088-.79l.72-4.194L.818 6.374a.75.75 0 0 1 .416-1.28l4.21-.611L7.327.668A.75.75 0 0 1 8 .25Zm0 2.445L6.615 5.5a.75.75 0 0 1-.564.41l-3.097.45 2.24 2.184a.75.75 0 0 1 .216.664l-.528 3.084 2.769-1.456a.75.75 0 0 1 .698 0l2.77 1.456-.53-3.084a.75.75 0 0 1 .216-.664l2.24-2.183-3.096-.45a.75.75 0 0 1-.564-.41L8 2.694Z"></path> </svg><span data-view-component="true" class="d-inline"> Star </span> </a></div> </div> <div class="flex-1"> <a href="/login?return_to=%2Fkroitor%2Fgjk.c" rel="nofollow" id="files-overview-watch-button" data-hydro-click="{"event_type":"authentication.click","payload":{"location_in_page":"notification subscription menu watch","repository_id":null,"auth_type":"LOG_IN","originating_url":"https://github.com/kroitor/gjk.c","user_id":null}}" data-hydro-click-hmac="c55588df3b070771f0ea206a61c5bfeb41fd77b0ac6a419d2cd629ae9ff02a11" aria-label="You must be signed in to change notification settings" data-view-component="true" class="btn-sm btn btn-block"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-bell mr-2"> <path d="M8 16a2 2 0 0 0 1.985-1.75c.017-.137-.097-.25-.235-.25h-3.5c-.138 0-.252.113-.235.25A2 2 0 0 0 8 16ZM3 5a5 5 0 0 1 10 0v2.947c0 .05.015.098.042.139l1.703 2.555A1.519 1.519 0 0 1 13.482 13H2.518a1.516 1.516 0 0 1-1.263-2.36l1.703-2.554A.255.255 0 0 0 3 7.947Zm5-3.5A3.5 3.5 0 0 0 4.5 5v2.947c0 .346-.102.683-.294.97l-1.703 2.556a.017.017 0 0 0-.003.01l.001.006c0 .002.002.004.004.006l.006.004.007.001h10.964l.007-.001.006-.004.004-.006.001-.007a.017.017 0 0 0-.003-.01l-1.703-2.554a1.745 1.745 0 0 1-.294-.97V5A3.5 3.5 0 0 0 8 1.5Z"></path> </svg>Notifications </a> <tool-tip id="tooltip-7d24b455-57c9-4967-b2a4-0c735d785c77" for="files-overview-watch-button" popover="manual" data-direction="s" data-type="description" data-view-component="true" class="sr-only position-absolute">You must be signed in to change notification settings</tool-tip> </div> <span> </span> </div> </div> </div> <nav data-pjax="#js-repo-pjax-container" aria-label="Repository" data-view-component="true" class="js-repo-nav js-sidenav-container-pjax js-responsive-underlinenav overflow-hidden UnderlineNav px-3 px-md-4 px-lg-5"> <ul data-view-component="true" class="UnderlineNav-body list-style-none"> <li data-view-component="true" class="d-inline-flex"> <a id="code-tab" href="/kroitor/gjk.c" data-tab-item="i0code-tab" data-selected-links="repo_source repo_downloads repo_commits repo_releases repo_tags repo_branches repo_packages repo_deployments repo_attestations /kroitor/gjk.c" data-pjax="#repo-content-pjax-container" data-turbo-frame="repo-content-turbo-frame" data-hotkey="g c" data-analytics-event="{"category":"Underline navbar","action":"Click tab","label":"Code","target":"UNDERLINE_NAV.TAB"}" aria-current="page" data-view-component="true" class="UnderlineNav-item no-wrap js-responsive-underlinenav-item js-selected-navigation-item selected"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-code UnderlineNav-octicon d-none d-sm-inline"> <path d="m11.28 3.22 4.25 4.25a.75.75 0 0 1 0 1.06l-4.25 4.25a.749.749 0 0 1-1.275-.326.749.749 0 0 1 .215-.734L13.94 8l-3.72-3.72a.749.749 0 0 1 .326-1.275.749.749 0 0 1 .734.215Zm-6.56 0a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042L2.06 8l3.72 3.72a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215L.47 8.53a.75.75 0 0 1 0-1.06Z"></path> </svg> <span data-content="Code">Code</span> <span id="code-repo-tab-count" data-pjax-replace="" data-turbo-replace="" title="Not available" data-view-component="true" class="Counter"></span> </a></li> <li data-view-component="true" class="d-inline-flex"> <a id="issues-tab" href="/kroitor/gjk.c/issues" data-tab-item="i1issues-tab" data-selected-links="repo_issues repo_labels repo_milestones /kroitor/gjk.c/issues" data-pjax="#repo-content-pjax-container" data-turbo-frame="repo-content-turbo-frame" data-hotkey="g i" data-analytics-event="{"category":"Underline navbar","action":"Click tab","label":"Issues","target":"UNDERLINE_NAV.TAB"}" data-view-component="true" class="UnderlineNav-item no-wrap js-responsive-underlinenav-item js-selected-navigation-item"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-issue-opened UnderlineNav-octicon d-none d-sm-inline"> <path d="M8 9.5a1.5 1.5 0 1 0 0-3 1.5 1.5 0 0 0 0 3Z"></path><path d="M8 0a8 8 0 1 1 0 16A8 8 0 0 1 8 0ZM1.5 8a6.5 6.5 0 1 0 13 0 6.5 6.5 0 0 0-13 0Z"></path> </svg> <span data-content="Issues">Issues</span> <span id="issues-repo-tab-count" data-pjax-replace="" data-turbo-replace="" title="2" data-view-component="true" class="Counter">2</span> </a></li> <li data-view-component="true" class="d-inline-flex"> <a id="pull-requests-tab" href="/kroitor/gjk.c/pulls" data-tab-item="i2pull-requests-tab" data-selected-links="repo_pulls checks /kroitor/gjk.c/pulls" data-pjax="#repo-content-pjax-container" data-turbo-frame="repo-content-turbo-frame" data-hotkey="g p" data-analytics-event="{"category":"Underline navbar","action":"Click tab","label":"Pull requests","target":"UNDERLINE_NAV.TAB"}" data-view-component="true" class="UnderlineNav-item no-wrap js-responsive-underlinenav-item js-selected-navigation-item"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-git-pull-request UnderlineNav-octicon d-none d-sm-inline"> <path d="M1.5 3.25a2.25 2.25 0 1 1 3 2.122v5.256a2.251 2.251 0 1 1-1.5 0V5.372A2.25 2.25 0 0 1 1.5 3.25Zm5.677-.177L9.573.677A.25.25 0 0 1 10 .854V2.5h1A2.5 2.5 0 0 1 13.5 5v5.628a2.251 2.251 0 1 1-1.5 0V5a1 1 0 0 0-1-1h-1v1.646a.25.25 0 0 1-.427.177L7.177 3.427a.25.25 0 0 1 0-.354ZM3.75 2.5a.75.75 0 1 0 0 1.5.75.75 0 0 0 0-1.5Zm0 9.5a.75.75 0 1 0 0 1.5.75.75 0 0 0 0-1.5Zm8.25.75a.75.75 0 1 0 1.5 0 .75.75 0 0 0-1.5 0Z"></path> </svg> <span data-content="Pull requests">Pull requests</span> <span id="pull-requests-repo-tab-count" data-pjax-replace="" data-turbo-replace="" title="0" hidden="hidden" data-view-component="true" class="Counter">0</span> </a></li> <li data-view-component="true" class="d-inline-flex"> <a id="actions-tab" href="/kroitor/gjk.c/actions" data-tab-item="i3actions-tab" data-selected-links="repo_actions /kroitor/gjk.c/actions" data-pjax="#repo-content-pjax-container" data-turbo-frame="repo-content-turbo-frame" data-hotkey="g a" data-analytics-event="{"category":"Underline navbar","action":"Click tab","label":"Actions","target":"UNDERLINE_NAV.TAB"}" data-view-component="true" class="UnderlineNav-item no-wrap js-responsive-underlinenav-item js-selected-navigation-item"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-play UnderlineNav-octicon d-none d-sm-inline"> <path d="M8 0a8 8 0 1 1 0 16A8 8 0 0 1 8 0ZM1.5 8a6.5 6.5 0 1 0 13 0 6.5 6.5 0 0 0-13 0Zm4.879-2.773 4.264 2.559a.25.25 0 0 1 0 .428l-4.264 2.559A.25.25 0 0 1 6 10.559V5.442a.25.25 0 0 1 .379-.215Z"></path> </svg> <span data-content="Actions">Actions</span> <span id="actions-repo-tab-count" data-pjax-replace="" data-turbo-replace="" title="Not available" data-view-component="true" class="Counter"></span> </a></li> <li data-view-component="true" class="d-inline-flex"> <a id="projects-tab" href="/kroitor/gjk.c/projects" data-tab-item="i4projects-tab" data-selected-links="repo_projects new_repo_project repo_project /kroitor/gjk.c/projects" data-pjax="#repo-content-pjax-container" data-turbo-frame="repo-content-turbo-frame" data-hotkey="g b" data-analytics-event="{"category":"Underline navbar","action":"Click tab","label":"Projects","target":"UNDERLINE_NAV.TAB"}" data-view-component="true" class="UnderlineNav-item no-wrap js-responsive-underlinenav-item js-selected-navigation-item"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-table UnderlineNav-octicon d-none d-sm-inline"> <path d="M0 1.75C0 .784.784 0 1.75 0h12.5C15.216 0 16 .784 16 1.75v12.5A1.75 1.75 0 0 1 14.25 16H1.75A1.75 1.75 0 0 1 0 14.25ZM6.5 6.5v8h7.75a.25.25 0 0 0 .25-.25V6.5Zm8-1.5V1.75a.25.25 0 0 0-.25-.25H6.5V5Zm-13 1.5v7.75c0 .138.112.25.25.25H5v-8ZM5 5V1.5H1.75a.25.25 0 0 0-.25.25V5Z"></path> </svg> <span data-content="Projects">Projects</span> <span id="projects-repo-tab-count" data-pjax-replace="" data-turbo-replace="" title="0" hidden="hidden" data-view-component="true" class="Counter">0</span> </a></li> <li data-view-component="true" class="d-inline-flex"> <a id="security-tab" href="/kroitor/gjk.c/security" data-tab-item="i5security-tab" data-selected-links="security overview alerts policy token_scanning code_scanning /kroitor/gjk.c/security" data-pjax="#repo-content-pjax-container" data-turbo-frame="repo-content-turbo-frame" data-hotkey="g s" data-analytics-event="{"category":"Underline navbar","action":"Click tab","label":"Security","target":"UNDERLINE_NAV.TAB"}" data-view-component="true" class="UnderlineNav-item no-wrap js-responsive-underlinenav-item js-selected-navigation-item"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-shield UnderlineNav-octicon d-none d-sm-inline"> <path d="M7.467.133a1.748 1.748 0 0 1 1.066 0l5.25 1.68A1.75 1.75 0 0 1 15 3.48V7c0 1.566-.32 3.182-1.303 4.682-.983 1.498-2.585 2.813-5.032 3.855a1.697 1.697 0 0 1-1.33 0c-2.447-1.042-4.049-2.357-5.032-3.855C1.32 10.182 1 8.566 1 7V3.48a1.75 1.75 0 0 1 1.217-1.667Zm.61 1.429a.25.25 0 0 0-.153 0l-5.25 1.68a.25.25 0 0 0-.174.238V7c0 1.358.275 2.666 1.057 3.86.784 1.194 2.121 2.34 4.366 3.297a.196.196 0 0 0 .154 0c2.245-.956 3.582-2.104 4.366-3.298C13.225 9.666 13.5 8.36 13.5 7V3.48a.251.251 0 0 0-.174-.237l-5.25-1.68ZM8.75 4.75v3a.75.75 0 0 1-1.5 0v-3a.75.75 0 0 1 1.5 0ZM9 10.5a1 1 0 1 1-2 0 1 1 0 0 1 2 0Z"></path> </svg> <span data-content="Security">Security</span> <include-fragment src="/kroitor/gjk.c/security/overall-count" accept="text/fragment+html"></include-fragment> </a></li> <li data-view-component="true" class="d-inline-flex"> <a id="insights-tab" href="/kroitor/gjk.c/pulse" data-tab-item="i6insights-tab" data-selected-links="repo_graphs repo_contributors dependency_graph dependabot_updates pulse people community /kroitor/gjk.c/pulse" data-pjax="#repo-content-pjax-container" data-turbo-frame="repo-content-turbo-frame" data-analytics-event="{"category":"Underline navbar","action":"Click tab","label":"Insights","target":"UNDERLINE_NAV.TAB"}" data-view-component="true" class="UnderlineNav-item no-wrap js-responsive-underlinenav-item js-selected-navigation-item"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-graph UnderlineNav-octicon d-none d-sm-inline"> <path d="M1.5 1.75V13.5h13.75a.75.75 0 0 1 0 1.5H.75a.75.75 0 0 1-.75-.75V1.75a.75.75 0 0 1 1.5 0Zm14.28 2.53-5.25 5.25a.75.75 0 0 1-1.06 0L7 7.06 4.28 9.78a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042l3.25-3.25a.75.75 0 0 1 1.06 0L10 7.94l4.72-4.72a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042Z"></path> </svg> <span data-content="Insights">Insights</span> <span id="insights-repo-tab-count" data-pjax-replace="" data-turbo-replace="" title="Not available" data-view-component="true" class="Counter"></span> </a></li> </ul> <div style="visibility:hidden;" data-view-component="true" class="UnderlineNav-actions js-responsive-underlinenav-overflow position-absolute pr-3 pr-md-4 pr-lg-5 right-0"> <action-menu data-select-variant="none" data-view-component="true"> <focus-group direction="vertical" mnemonics retain> <button id="action-menu-f15cba40-ba97-4b5f-a072-a086b1bfb48c-button" popovertarget="action-menu-f15cba40-ba97-4b5f-a072-a086b1bfb48c-overlay" aria-controls="action-menu-f15cba40-ba97-4b5f-a072-a086b1bfb48c-list" aria-haspopup="true" aria-labelledby="tooltip-f6d979c5-7382-4e04-b263-78f77e2ae665" type="button" data-view-component="true" class="Button Button--iconOnly Button--secondary Button--medium UnderlineNav-item"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-kebab-horizontal Button-visual"> <path d="M8 9a1.5 1.5 0 1 0 0-3 1.5 1.5 0 0 0 0 3ZM1.5 9a1.5 1.5 0 1 0 0-3 1.5 1.5 0 0 0 0 3Zm13 0a1.5 1.5 0 1 0 0-3 1.5 1.5 0 0 0 0 3Z"></path> </svg> </button><tool-tip id="tooltip-f6d979c5-7382-4e04-b263-78f77e2ae665" for="action-menu-f15cba40-ba97-4b5f-a072-a086b1bfb48c-button" popover="manual" data-direction="s" data-type="label" data-view-component="true" class="sr-only position-absolute">Additional navigation options</tool-tip> <anchored-position data-target="action-menu.overlay" id="action-menu-f15cba40-ba97-4b5f-a072-a086b1bfb48c-overlay" anchor="action-menu-f15cba40-ba97-4b5f-a072-a086b1bfb48c-button" align="start" side="outside-bottom" anchor-offset="normal" popover="auto" data-view-component="true"> <div data-view-component="true" class="Overlay Overlay--size-auto"> <div data-view-component="true" class="Overlay-body Overlay-body--paddingNone"> <action-list> <div data-view-component="true"> <ul aria-labelledby="action-menu-f15cba40-ba97-4b5f-a072-a086b1bfb48c-button" id="action-menu-f15cba40-ba97-4b5f-a072-a086b1bfb48c-list" role="menu" data-view-component="true" class="ActionListWrap--inset ActionListWrap"> <li hidden="hidden" data-menu-item="i0code-tab" data-targets="action-list.items" role="none" data-view-component="true" class="ActionListItem"> <a tabindex="-1" id="item-84d0df56-6733-49e3-816a-08fdcda5bb4c" href="/kroitor/gjk.c" role="menuitem" data-view-component="true" class="ActionListContent ActionListContent--visual16"> <span class="ActionListItem-visual ActionListItem-visual--leading"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-code"> <path d="m11.28 3.22 4.25 4.25a.75.75 0 0 1 0 1.06l-4.25 4.25a.749.749 0 0 1-1.275-.326.749.749 0 0 1 .215-.734L13.94 8l-3.72-3.72a.749.749 0 0 1 .326-1.275.749.749 0 0 1 .734.215Zm-6.56 0a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042L2.06 8l3.72 3.72a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215L.47 8.53a.75.75 0 0 1 0-1.06Z"></path> </svg> </span> <span data-view-component="true" class="ActionListItem-label"> Code </span> </a> </li> <li hidden="hidden" data-menu-item="i1issues-tab" data-targets="action-list.items" role="none" data-view-component="true" class="ActionListItem"> <a tabindex="-1" id="item-2b895615-bac1-468c-898f-e813b0f6d4bb" href="/kroitor/gjk.c/issues" role="menuitem" data-view-component="true" class="ActionListContent ActionListContent--visual16"> <span class="ActionListItem-visual ActionListItem-visual--leading"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-issue-opened"> <path d="M8 9.5a1.5 1.5 0 1 0 0-3 1.5 1.5 0 0 0 0 3Z"></path><path d="M8 0a8 8 0 1 1 0 16A8 8 0 0 1 8 0ZM1.5 8a6.5 6.5 0 1 0 13 0 6.5 6.5 0 0 0-13 0Z"></path> </svg> </span> <span data-view-component="true" class="ActionListItem-label"> Issues </span> </a> </li> <li hidden="hidden" data-menu-item="i2pull-requests-tab" data-targets="action-list.items" role="none" data-view-component="true" class="ActionListItem"> <a tabindex="-1" id="item-39547051-eef5-4ce1-8580-ed81a0782f1c" href="/kroitor/gjk.c/pulls" role="menuitem" data-view-component="true" class="ActionListContent ActionListContent--visual16"> <span class="ActionListItem-visual ActionListItem-visual--leading"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-git-pull-request"> <path d="M1.5 3.25a2.25 2.25 0 1 1 3 2.122v5.256a2.251 2.251 0 1 1-1.5 0V5.372A2.25 2.25 0 0 1 1.5 3.25Zm5.677-.177L9.573.677A.25.25 0 0 1 10 .854V2.5h1A2.5 2.5 0 0 1 13.5 5v5.628a2.251 2.251 0 1 1-1.5 0V5a1 1 0 0 0-1-1h-1v1.646a.25.25 0 0 1-.427.177L7.177 3.427a.25.25 0 0 1 0-.354ZM3.75 2.5a.75.75 0 1 0 0 1.5.75.75 0 0 0 0-1.5Zm0 9.5a.75.75 0 1 0 0 1.5.75.75 0 0 0 0-1.5Zm8.25.75a.75.75 0 1 0 1.5 0 .75.75 0 0 0-1.5 0Z"></path> </svg> </span> <span data-view-component="true" class="ActionListItem-label"> Pull requests </span> </a> </li> <li hidden="hidden" data-menu-item="i3actions-tab" data-targets="action-list.items" role="none" data-view-component="true" class="ActionListItem"> <a tabindex="-1" id="item-8e7bc4ff-b63c-44cb-a851-86ca3e2447ab" href="/kroitor/gjk.c/actions" role="menuitem" data-view-component="true" class="ActionListContent ActionListContent--visual16"> <span class="ActionListItem-visual ActionListItem-visual--leading"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-play"> <path d="M8 0a8 8 0 1 1 0 16A8 8 0 0 1 8 0ZM1.5 8a6.5 6.5 0 1 0 13 0 6.5 6.5 0 0 0-13 0Zm4.879-2.773 4.264 2.559a.25.25 0 0 1 0 .428l-4.264 2.559A.25.25 0 0 1 6 10.559V5.442a.25.25 0 0 1 .379-.215Z"></path> </svg> </span> <span data-view-component="true" class="ActionListItem-label"> Actions </span> </a> </li> <li hidden="hidden" data-menu-item="i4projects-tab" data-targets="action-list.items" role="none" data-view-component="true" class="ActionListItem"> <a tabindex="-1" id="item-81dd3305-f454-41cf-ae09-bc983b0d3b91" href="/kroitor/gjk.c/projects" role="menuitem" data-view-component="true" class="ActionListContent ActionListContent--visual16"> <span class="ActionListItem-visual ActionListItem-visual--leading"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-table"> <path d="M0 1.75C0 .784.784 0 1.75 0h12.5C15.216 0 16 .784 16 1.75v12.5A1.75 1.75 0 0 1 14.25 16H1.75A1.75 1.75 0 0 1 0 14.25ZM6.5 6.5v8h7.75a.25.25 0 0 0 .25-.25V6.5Zm8-1.5V1.75a.25.25 0 0 0-.25-.25H6.5V5Zm-13 1.5v7.75c0 .138.112.25.25.25H5v-8ZM5 5V1.5H1.75a.25.25 0 0 0-.25.25V5Z"></path> </svg> </span> <span data-view-component="true" class="ActionListItem-label"> Projects </span> </a> </li> <li hidden="hidden" data-menu-item="i5security-tab" data-targets="action-list.items" role="none" data-view-component="true" class="ActionListItem"> <a tabindex="-1" id="item-e62c4bf2-fe20-42d2-8f82-62747dceed44" href="/kroitor/gjk.c/security" role="menuitem" data-view-component="true" class="ActionListContent ActionListContent--visual16"> <span class="ActionListItem-visual ActionListItem-visual--leading"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-shield"> <path d="M7.467.133a1.748 1.748 0 0 1 1.066 0l5.25 1.68A1.75 1.75 0 0 1 15 3.48V7c0 1.566-.32 3.182-1.303 4.682-.983 1.498-2.585 2.813-5.032 3.855a1.697 1.697 0 0 1-1.33 0c-2.447-1.042-4.049-2.357-5.032-3.855C1.32 10.182 1 8.566 1 7V3.48a1.75 1.75 0 0 1 1.217-1.667Zm.61 1.429a.25.25 0 0 0-.153 0l-5.25 1.68a.25.25 0 0 0-.174.238V7c0 1.358.275 2.666 1.057 3.86.784 1.194 2.121 2.34 4.366 3.297a.196.196 0 0 0 .154 0c2.245-.956 3.582-2.104 4.366-3.298C13.225 9.666 13.5 8.36 13.5 7V3.48a.251.251 0 0 0-.174-.237l-5.25-1.68ZM8.75 4.75v3a.75.75 0 0 1-1.5 0v-3a.75.75 0 0 1 1.5 0ZM9 10.5a1 1 0 1 1-2 0 1 1 0 0 1 2 0Z"></path> </svg> </span> <span data-view-component="true" class="ActionListItem-label"> Security </span> </a> </li> <li hidden="hidden" data-menu-item="i6insights-tab" data-targets="action-list.items" role="none" data-view-component="true" class="ActionListItem"> <a tabindex="-1" id="item-198442ab-7d5b-467f-9d6d-e569b9e599e0" href="/kroitor/gjk.c/pulse" role="menuitem" data-view-component="true" class="ActionListContent ActionListContent--visual16"> <span class="ActionListItem-visual ActionListItem-visual--leading"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-graph"> <path d="M1.5 1.75V13.5h13.75a.75.75 0 0 1 0 1.5H.75a.75.75 0 0 1-.75-.75V1.75a.75.75 0 0 1 1.5 0Zm14.28 2.53-5.25 5.25a.75.75 0 0 1-1.06 0L7 7.06 4.28 9.78a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042l3.25-3.25a.75.75 0 0 1 1.06 0L10 7.94l4.72-4.72a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042Z"></path> </svg> </span> <span data-view-component="true" class="ActionListItem-label"> Insights </span> </a> </li> </ul> </div></action-list> </div> </div></anchored-position> </focus-group> </action-menu></div> </nav> </div> <turbo-frame id="repo-content-turbo-frame" target="_top" data-turbo-action="advance" class=""> <div id="repo-content-pjax-container" class="repository-content " > <h1 class='sr-only'>kroitor/gjk.c</h1> <div class="clearfix container-xl px-md-4 px-lg-5 px-3"> <div> <div style="max-width: 100%" data-view-component="true" class="Layout Layout--flowRow-until-md react-repos-overview-margin Layout--sidebarPosition-end Layout--sidebarPosition-flowRow-end"> <div data-view-component="true" class="Layout-main"> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_dompurify_dist_purify_es_mjs-dd1d3ea6a436.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/vendors-node_modules_tanstack_query-core_build_modern_queryObserver_js-node_modules_tanstack_-defd52-843b41414e0e.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/ui_packages_aria-live_aria-live_ts-ui_packages_history_history_ts-ui_packages_promise-with-re-01dc80-b13b6c1d97b0.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/ui_packages_paths_index_ts-8a20a6d3af54.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/ui_packages_ref-selector_RefSelector_tsx-7496afc3784d.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/ui_packages_commit-attribution_index_ts-ui_packages_commit-checks-status_index_ts-ui_packages-762eaa-bac5b6fc3f70.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/ui_packages_code-view-shared_hooks_use-canonical-object_ts-ui_packages_code-view-shared_hooks-c2dbff-923a12b3d018.js"></script> <script crossorigin="anonymous" defer="defer" type="application/javascript" src="https://github.githubassets.com/assets/repos-overview-60297eccc5de.js"></script> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/primer-react.a490b7c9fa319e5cb069.module.css" /> <link crossorigin="anonymous" media="all" rel="stylesheet" href="https://github.githubassets.com/assets/repos-overview.0ee7cac3ab511a65d9f9.module.css" /> <react-partial partial-name="repos-overview" data-ssr="true" data-attempted-ssr="true" > <script type="application/json" data-target="react-partial.embeddedData">{"props":{"initialPayload":{"allShortcutsEnabled":false,"path":"/","repo":{"id":48733330,"defaultBranch":"master","name":"gjk.c","ownerLogin":"kroitor","currentUserCanPush":false,"isFork":false,"isEmpty":false,"createdAt":"2015-12-29T07:12:03.000Z","ownerAvatar":"https://avatars.githubusercontent.com/u/1294454?v=4","public":true,"private":false,"isOrgOwned":false},"currentUser":null,"refInfo":{"name":"master","listCacheKey":"v0:1451373542.0","canEdit":false,"refType":"branch","currentOid":"287d86ac9d6a422f0bce5ff58be6e82f8350a966"},"tree":{"items":[{"name":"python","path":"python","contentType":"directory"},{"name":".gitattributes","path":".gitattributes","contentType":"file"},{"name":"LICENSE.txt","path":"LICENSE.txt","contentType":"file"},{"name":"README.md","path":"README.md","contentType":"file"},{"name":"gjk.c","path":"gjk.c","contentType":"file"},{"name":"gjk1d.html","path":"gjk1d.html","contentType":"file"}],"templateDirectorySuggestionUrl":null,"readme":null,"totalCount":6,"showBranchInfobar":false},"fileTree":null,"fileTreeProcessingTime":null,"foldersToFetch":[],"treeExpanded":false,"symbolsExpanded":false,"isOverview":true,"overview":{"banners":{"shouldRecommendReadme":false,"isPersonalRepo":false,"showUseActionBanner":false,"actionSlug":null,"actionId":null,"showProtectBranchBanner":false,"publishBannersInfo":{"dismissActionNoticePath":"/settings/dismiss-notice/publish_action_from_repo","releasePath":"/kroitor/gjk.c/releases/new?marketplace=true","showPublishActionBanner":false},"interactionLimitBanner":null,"showInvitationBanner":false,"inviterName":null,"actionsMigrationBannerInfo":{"releaseTags":[],"showImmutableActionsMigrationBanner":false,"initialMigrationStatus":null}},"codeButton":{"contactPath":"/contact","isEnterprise":false,"local":{"protocolInfo":{"httpAvailable":true,"sshAvailable":null,"httpUrl":"https://github.com/kroitor/gjk.c.git","showCloneWarning":null,"sshUrl":null,"sshCertificatesRequired":null,"sshCertificatesAvailable":null,"ghCliUrl":"gh repo clone kroitor/gjk.c","defaultProtocol":"http","newSshKeyUrl":"/settings/ssh/new","setProtocolPath":"/users/set_protocol"},"platformInfo":{"cloneUrl":"https://desktop.github.com","showVisualStudioCloneButton":false,"visualStudioCloneUrl":"https://windows.github.com","showXcodeCloneButton":false,"xcodeCloneUrl":"xcode://clone?repo=https%3A%2F%2Fgithub.com%2Fkroitor%2Fgjk.c","zipballUrl":"/kroitor/gjk.c/archive/refs/heads/master.zip"}},"newCodespacePath":"/codespaces/new?hide_repo_select=true\u0026repo=48733330"},"popovers":{"rename":null,"renamedParentRepo":null},"commitCount":"514","overviewFiles":[{"displayName":"README.md","repoName":"gjk.c","refName":"master","path":"README.md","preferredFileType":"readme","tabName":"README","richText":"\u003carticle class=\"markdown-body entry-content container-lg\" itemprop=\"text\"\u003e\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch1 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003egjk.c – Gilbert-Johnson-Keerthi In Plain C\u003c/h1\u003e\u003ca id=\"user-content-gjkc--gilbert-johnson-keerthi-in-plain-c\" class=\"anchor\" aria-label=\"Permalink: gjk.c – Gilbert-Johnson-Keerthi In Plain C\" href=\"#gjkc--gilbert-johnson-keerthi-in-plain-c\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThis is a rough but fast implementation of GJK collision detection algorithm in plain C. Just one C file, less than 200 lines, no dependencies. It is in 2D for now, full 3D version is upcoming... This 2D-version uses Minkowski sums and builds a triangle-simplex in Minkowski space to tell if two arbitrary convex polygons are colliding. 3D-version will be roughly the same, but will build a tetrahedron-simplex inside a 3-dimensional Minkowski space. It currently only tells if there is a collision or not. C-code for distance and contact points coming soon.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eDisclaimer\u003c/h2\u003e\u003ca id=\"user-content-disclaimer\" class=\"anchor\" aria-label=\"Permalink: Disclaimer\" href=\"#disclaimer\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eFuck licenses and copyright. I made it for learning purposes, this is public knowledge and it's absolutely free for any usage.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eUsage Example\u003c/h2\u003e\u003ca id=\"user-content-usage-example\" class=\"anchor\" aria-label=\"Permalink: Usage Example\" href=\"#usage-example\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThis is an illustration of the example case from \u003ca href=\"http://www.dyn4j.org/2010/04/gjk-gilbert-johnson-keerthi/\" rel=\"nofollow\"\u003edyn4j\u003c/a\u003e.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://user-images.githubusercontent.com/1294454/133015814-8e2fd47d-0f58-430d-8403-016cc5649cdb.png\"\u003e\u003cimg src=\"https://user-images.githubusercontent.com/1294454/133015814-8e2fd47d-0f58-430d-8403-016cc5649cdb.png\" alt=\"Example case from dyn4j\" title=\"Example case from dyn4j\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe two tested polygons are defined as arrays of plain C vector struct type. This implementation of GJK doesn't really care about the order of the vertices in the arrays, as it treats all sets of points as \u003cem\u003econvex polygons\u003c/em\u003e.\u003c/p\u003e\n\u003cdiv class=\"highlight highlight-source-c notranslate position-relative overflow-auto\" dir=\"auto\" data-snippet-clipboard-copy-content=\"struct _vec2 { float x; float y; };\ntypedef struct _vec2 vec2;\n\n...\n\nint main(int argc, const char * argv[]) {\n \n // test case from dyn4j\n\n vec2 vertices1[] = {\n { 4, 11 },\n { 4, 5 },\n { 9, 9 },\n };\n \n vec2 vertices2[] = {\n { 5, 7 },\n { 7, 3 },\n { 10, 2 },\n { 12, 7 },\n };\n\n size_t count1 = sizeof (vertices1) / sizeof (vec2); // == 3\n size_t count2 = sizeof (vertices2) / sizeof (vec2); // == 4\n \n int collisionDetected = gjk (vertices1, count1, vertices2, count2);\n \n printf (collisionDetected ? \u0026quot;Bodies collide!\\n\u0026quot; : \u0026quot;No collision\\n\u0026quot;);\n return 0;\n}\"\u003e\u003cpre\u003e\u003cspan class=\"pl-k\"\u003estruct\u003c/span\u003e \u003cspan class=\"pl-smi\"\u003e_vec2\u003c/span\u003e { \u003cspan class=\"pl-smi\"\u003efloat\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003ex\u003c/span\u003e; \u003cspan class=\"pl-smi\"\u003efloat\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003ey\u003c/span\u003e; };\n\u003cspan class=\"pl-k\"\u003etypedef\u003c/span\u003e \u003cspan class=\"pl-k\"\u003estruct\u003c/span\u003e \u003cspan class=\"pl-smi\"\u003e_vec2\u003c/span\u003e \u003cspan class=\"pl-smi\"\u003evec2\u003c/span\u003e;\n\n...\n\n\u003cspan class=\"pl-smi\"\u003eint\u003c/span\u003e \u003cspan class=\"pl-en\"\u003emain\u003c/span\u003e(\u003cspan class=\"pl-smi\"\u003eint\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003eargc\u003c/span\u003e, \u003cspan class=\"pl-k\"\u003econst\u003c/span\u003e \u003cspan class=\"pl-smi\"\u003echar\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e*\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003eargv\u003c/span\u003e[]) {\n \n \u003cspan class=\"pl-c\"\u003e// test case from dyn4j\u003c/span\u003e\n\n \u003cspan class=\"pl-smi\"\u003evec2\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003evertices1\u003c/span\u003e[] \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e {\n { \u003cspan class=\"pl-c1\"\u003e4\u003c/span\u003e, \u003cspan class=\"pl-c1\"\u003e11\u003c/span\u003e },\n { \u003cspan class=\"pl-c1\"\u003e4\u003c/span\u003e, \u003cspan class=\"pl-c1\"\u003e5\u003c/span\u003e },\n { \u003cspan class=\"pl-c1\"\u003e9\u003c/span\u003e, \u003cspan class=\"pl-c1\"\u003e9\u003c/span\u003e },\n };\n \n \u003cspan class=\"pl-smi\"\u003evec2\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003evertices2\u003c/span\u003e[] \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e {\n { \u003cspan class=\"pl-c1\"\u003e5\u003c/span\u003e, \u003cspan class=\"pl-c1\"\u003e7\u003c/span\u003e },\n { \u003cspan class=\"pl-c1\"\u003e7\u003c/span\u003e, \u003cspan class=\"pl-c1\"\u003e3\u003c/span\u003e },\n { \u003cspan class=\"pl-c1\"\u003e10\u003c/span\u003e, \u003cspan class=\"pl-c1\"\u003e2\u003c/span\u003e },\n { \u003cspan class=\"pl-c1\"\u003e12\u003c/span\u003e, \u003cspan class=\"pl-c1\"\u003e7\u003c/span\u003e },\n };\n\n \u003cspan class=\"pl-smi\"\u003esize_t\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ecount1\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-k\"\u003esizeof\u003c/span\u003e (\u003cspan class=\"pl-s1\"\u003evertices1\u003c/span\u003e) / \u003cspan class=\"pl-k\"\u003esizeof\u003c/span\u003e (\u003cspan class=\"pl-s1\"\u003evec2\u003c/span\u003e); \u003cspan class=\"pl-c\"\u003e// == 3\u003c/span\u003e\n \u003cspan class=\"pl-smi\"\u003esize_t\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ecount2\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-k\"\u003esizeof\u003c/span\u003e (\u003cspan class=\"pl-s1\"\u003evertices2\u003c/span\u003e) / \u003cspan class=\"pl-k\"\u003esizeof\u003c/span\u003e (\u003cspan class=\"pl-s1\"\u003evec2\u003c/span\u003e); \u003cspan class=\"pl-c\"\u003e// == 4\u003c/span\u003e\n \n \u003cspan class=\"pl-smi\"\u003eint\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ecollisionDetected\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-en\"\u003egjk\u003c/span\u003e (\u003cspan class=\"pl-s1\"\u003evertices1\u003c/span\u003e, \u003cspan class=\"pl-s1\"\u003ecount1\u003c/span\u003e, \u003cspan class=\"pl-s1\"\u003evertices2\u003c/span\u003e, \u003cspan class=\"pl-s1\"\u003ecount2\u003c/span\u003e);\n \n \u003cspan class=\"pl-en\"\u003eprintf\u003c/span\u003e (\u003cspan class=\"pl-s1\"\u003ecollisionDetected\u003c/span\u003e ? \u003cspan class=\"pl-s\"\u003e\"Bodies collide!\\n\"\u003c/span\u003e : \u003cspan class=\"pl-s\"\u003e\"No collision\\n\"\u003c/span\u003e);\n \u003cspan class=\"pl-k\"\u003ereturn\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e0\u003c/span\u003e;\n}\u003c/pre\u003e\u003c/div\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eHow It Works\u003c/h2\u003e\u003ca id=\"user-content-how-it-works\" class=\"anchor\" aria-label=\"Permalink: How It Works\" href=\"#how-it-works\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThe goal of this explanation is to help people visualize the logic of GJK. To explain it we have to oversimplify certain things. There's no complicated math except basic arithmetic and a little bit of vector math, nothing more, so that a 3rd-grader could understand it. It is actually not very difficult to have GJK algorithm explained in a proper understandable way from ground up.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eAt the very top level GJK tells if two arbitrary shapes are intersecting (colliding) or not. The algorithm is used to calculate the depth of intersection (collision distance). A collision occurs when two shapes try to occupy the same points in space at the same time. The space can be of any nature. It might be your in-game world simulation, or a calculation on a table of statistical data, or a built-in navigation system for a robot or any other application you can imagine. You can use it for calculating collisions of solid bodies and numeric intersections of any kind. You can have as many dimensions as you want, the amount of dimensions does not really matter, the logic is the same for 1D, 2D, 3D, etc... With GJK you can even calculate collisions in 4D if you're able to comprehend this.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe algorithm itself does not require any hard math whatsoever, it's actually very intuitive and easy. It takes an arithmetic difference of two shapes by subtracting all points of one shape from all points of another shape. If two shapes share a common point, subtracting that point from itself results in a difference of zero. So, if zero is found in the result then there's a collision. In fact, the algorithm does not have to calculate all differences for all possible pairs of points, only a small subset of significant points is examined, therefore it works very fast while being very accurate.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eIn order to understand GJK one has to build an imaginary visualization of what is going on under the hood. Once you see the picture in your head, you can implement it and even tweak it for your needs.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eA 1D Intro\u003c/h3\u003e\u003ca id=\"user-content-a-1d-intro\" class=\"anchor\" aria-label=\"Permalink: A 1D Intro\" href=\"#a-1d-intro\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eLet's start with a naive example of computing a difference of two shapes in one dimension. A segment of a number line is an example of 1D-shape. Imagine we have two segments on the number line: segment \u003ccode\u003e[1,3]\u003c/code\u003e and segment \u003ccode\u003e[2,4]\u003c/code\u003e:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/21998716/8f47b72e-dc47-11e6-836e-06d523a84105.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/21998716/8f47b72e-dc47-11e6-836e-06d523a84105.jpg\" alt=\"Segment [1,3] on the number line\" title=\"Segment [1,3] on the number line\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/21998717/8f487718-dc47-11e6-88f2-57bf590a0ee4.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/21998717/8f487718-dc47-11e6-88f2-57bf590a0ee4.jpg\" alt=\"Segment [2,4] on the number line\" title=\"Segment [2,4] on the number line\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eZero is our point of reference on the number line, so we call it \u003cem\u003ethe Origin\u003c/em\u003e. Easy enough.\nIt is obvious that our segments occupy some common region of our 1D-space, so they must be intersecting or colliding, you can tell that just by looking at the representation of both segments in same 1D-space (on the same line). Let's confirm arithmetically that these segments indeed intersect. We do that by subtracting all points of segment \u003ccode\u003e[2,4]\u003c/code\u003e from all points of segment \u003ccode\u003e[1,3]\u003c/code\u003e to see what we get on a number line.\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"1 - 2 = -1\n1 - 3 = -2\n1 - 4 = -3\n\n2 - 2 = 0\n2 - 3 = -1\n2 - 4 = -2\n\n3 - 2 = 1\n3 - 3 = 0\n3 - 4 = -1\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003e1 - 2 = -1\n1 - 3 = -2\n1 - 4 = -3\n\n2 - 2 = 0\n2 - 3 = -1\n2 - 4 = -2\n\n3 - 2 = 1\n3 - 3 = 0\n3 - 4 = -1\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eWe got a lot of numbers, many of them more than once. The resulting set of points (numbers) is larger than each of our initial sets of points of two shapes. Let's plot these resulting points in our 1D-space and look at the shape of resulting segment on the number line:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/21998715/8f465a96-dc47-11e6-9aa5-e30f59453685.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/21998715/8f465a96-dc47-11e6-9aa5-e30f59453685.jpg\" alt=\"Segment [-3,1] on the number line\" title=\"Segment [-3,1] on the number line\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eSo after all we have resulting segment \u003ccode\u003e[-3,1]\u003c/code\u003e that covers points \u003ccode\u003e-3\u003c/code\u003e, \u003ccode\u003e-2\u003c/code\u003e, \u003ccode\u003e-1\u003c/code\u003e, \u003ccode\u003e0\u003c/code\u003e and \u003ccode\u003e1\u003c/code\u003e. Because two initial shapes had some points in common the resulting segment contains a zero. This comes from a simple fact, that when you subtract a point (which is a number) from itself you inevitably end up with a zero. Note, that our initial segments had points 2 and 3 in common. When we subtracted 2 from 2 we got 0. When we subtracted 3 from 3 we also got a zero. This is quite obvious. So, if two shapes have at least one common point, because you subtract that point from itself, the resulting set must contain zero (the Origin) at least once. This is the key of GJK which says: if the Origin is contained inside the resulting set then initial shapes must have collided or kissed at least. Once you get it, you can then apply it to any number of dimensions.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNow let's take a look at a counter-example, say we have two segments \u003ccode\u003e[-2,-1]\u003c/code\u003e and \u003ccode\u003e[1,3]\u003c/code\u003e:\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/21998714/8f462616-dc47-11e6-9eec-b13454c7a67a.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/21998714/8f462616-dc47-11e6-9eec-b13454c7a67a.jpg\" alt=\"Segment [-2,-1] on the number line\" title=\"Segment [-2,-1] on the number line\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/21998716/8f47b72e-dc47-11e6-836e-06d523a84105.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/21998716/8f47b72e-dc47-11e6-836e-06d523a84105.jpg\" alt=\"Segment [1,3] on the number line\" title=\"Segment [1,3] on the number line\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWe can visually ensure that segment \u003ccode\u003e[-2,-1]\u003c/code\u003e occupies a different region of number line than that of segment \u003ccode\u003e[1,3]\u003c/code\u003e. There's a gap between our initial shapes, so these two shapes do not intersect in our 1D-space, therefore there's no collision. Let's prove that arithmetically by subtracting all points of any of the two segments from all points of the other.\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"1 - (-1) = 1 + 1 = 2\n1 - (-2) = 1 + 2 = 3\n\n2 - (-1) = 2 + 1 = 3\n2 - (-2) = 2 + 2 = 4\n\n3 - (-1) = 3 + 1 = 4\n3 - (-2) = 3 + 2 = 5 \"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003e1 - (-1) = 1 + 1 = 2\n1 - (-2) = 1 + 2 = 3\n\n2 - (-1) = 2 + 1 = 3\n2 - (-2) = 2 + 2 = 4\n\n3 - (-1) = 3 + 1 = 4\n3 - (-2) = 3 + 2 = 5 \n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eAnd we again draw the resulting segment on a number line in our imaginary 1D-space:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/21998713/8f45117c-dc47-11e6-8e2a-a759b36a559d.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/21998713/8f45117c-dc47-11e6-8e2a-a759b36a559d.jpg\" alt=\"Segment [2,5] on the number line\" title=\"Segment [2,5] on the number line\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWe got another bigger segment \u003ccode\u003e[2,5]\u003c/code\u003e that represents a difference of all the points of two initial segments but this time it does not contain the Origin. That is, the resulting set of points does not include zero, because initial segments did not have any points in common so they indeed occupy different regions of our number line and don't intersect.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNow, if our initial shapes were too big (long initial segments) we would have to calculate too many differences from too many pairs of points. But it's actually easy to see, that we only need to calculate the intersection between the endpoints of two segments, ignoring all the inside points of both segments.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eConsider two segments \u003ccode\u003e[10,20]\u003c/code\u003e and \u003ccode\u003e[5,40]\u003c/code\u003e:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/21999180/deb68374-dc49-11e6-9ef4-ea631b35178c.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/21999180/deb68374-dc49-11e6-9ef4-ea631b35178c.jpg\" alt=\"Segment [10,20] on the number line\" title=\"Segment [10,20] on the number line\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/21999179/de9cceac-dc49-11e6-8ddf-656ec1829c43.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/21999179/de9cceac-dc49-11e6-8ddf-656ec1829c43.jpg\" alt=\"Segment [5,40] on the number line\" title=\"Segment [5,40] on the number line\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNow subtract their four endpoints from each other:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"10 - 5 = 5\n10 - 40 = -30\n\n20 - 5 = 15\n20 - 40 = -20\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003e10 - 5 = 5\n10 - 40 = -30\n\n20 - 5 = 15\n20 - 40 = -20\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThe resulting segment \u003ccode\u003e[-30,15]\u003c/code\u003e would look like this:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/21999267/39333c70-dc4a-11e6-91ec-12a08ebb6f34.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/21999267/39333c70-dc4a-11e6-91ec-12a08ebb6f34.jpg\" alt=\"Segment [-30,15] on the number line\" title=\"Segment [-30,15] on the number line\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNotice that when we subtract the \u003cem\u003eopposite points\u003c/em\u003e of initial shapes from one another, the resulting difference number lands either to the left or to the right of the Origin (which is zero) on the number line. Say, we take the leftmost point of segment \u003ccode\u003e[5,40]\u003c/code\u003e (number \u003ccode\u003e5\u003c/code\u003e) and subtract it from the rightmost point of segment \u003ccode\u003e[10,20]\u003c/code\u003e (number \u003ccode\u003e20\u003c/code\u003e) the resulting number \u003ccode\u003e20 - 5 = 15\u003c/code\u003e is positive, so it lands to the right of zero on the number line. Then we take the rightmost point of segment \u003ccode\u003e[5,40]\u003c/code\u003e (number \u003ccode\u003e40\u003c/code\u003e this time) and subtract it from the leftmost point of segment \u003ccode\u003e[10,20]\u003c/code\u003e (number \u003ccode\u003e10\u003c/code\u003e) the resulting number \u003ccode\u003e10 - 40 = -30\u003c/code\u003e is negative, so it lands to the left of zero on the number line that is exactly opposite to the first positive difference point \u003ccode\u003e15\u003c/code\u003e. We say these points are \u003cem\u003eopposite\u003c/em\u003e directionwise. Therefore the Origin is between two resulting points \u003cem\u003eenclosing\u003c/em\u003e it.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWe ignored all insignificant internal points and only took the endpoints of initial segments into account thus reducing our calculation to four basic arithmetic operations (subtractions). We did that by switching to a simpler representation of a segment (only two endpoints instead of all points contained inside an initial segment). A simpler representation of the difference of two shapes is called a \u003cem\u003esimplex\u003c/em\u003e. It literally means 'the simplest possible'. The simplest possible \u003cem\u003ething\u003c/em\u003e on a one-dimensional number-line is a number (a single point in 1D-space is called a \u003cem\u003e0-simplex\u003c/em\u003e). A segment is the simplest possible shape sufficient to contain multiple points of a number line (a segment of two points in 1D space is called a \u003cem\u003e1-simplex\u003c/em\u003e). Even if one segment covers the other segment in its entirety (a segment fully contains another segment) – you can still detect an intersection of them in space. And it does not matter which one you're subtracting from, the resulting set will still contain the Origin at zero.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eGJK also works if two initial shapes don't intersect but just barely touch. Say, we have two shapes, segment \u003ccode\u003e[1,2]\u003c/code\u003e and segment \u003ccode\u003e[2,3]\u003c/code\u003e:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/21999597/fbec71fe-dc4b-11e6-8afa-371f87457339.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/21999597/fbec71fe-dc4b-11e6-8afa-371f87457339.jpg\" alt=\"Segment [1,2] on the number line\" title=\"Segment [1,2] on the number line\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/21999596/fbebff44-dc4b-11e6-811a-f832ab3eb9a4.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/21999596/fbebff44-dc4b-11e6-811a-f832ab3eb9a4.jpg\" alt=\"Segment [2,3] on the number line\" title=\"Segment [2,3] on the number line\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThese two segments only have one point in common. Subtracting their endpoints from each other gives:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"1 - 2 = -1\n1 - 3 = -2\n\n2 - 2 = 0\n2 - 3 = -1\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003e1 - 2 = -1\n1 - 3 = -2\n\n2 - 2 = 0\n2 - 3 = -1\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eAnd the resulting difference looks like:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/21999595/fbe9927c-dc4b-11e6-8521-816589df8552.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/21999595/fbe9927c-dc4b-11e6-8521-816589df8552.jpg\" alt=\"Segment [-2,0] on the number line\" title=\"Segment [-2,0] on the number line\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNotice, that in case of only one common point the Origin is not inside resulting segment, but is actually one of its endpoints (the rightmost in this example). When your resulting segment has the Origin at one of its endpoints that means that your initial shapes do not collide, but merely touch at a single point (two segments have only one common point of intersection).\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWhat GJK really says is: if you're able to build a simplex that contains (includes) the Origin then your shapes have at least one or more points of intersection (occupy same points in space).\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWORK IN PROGRESS, A live demo of GJK in a 1D-space and a video of GJK in action coming up soon )\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eMoving On To 2D\u003c/h3\u003e\u003ca id=\"user-content-moving-on-to-2d\" class=\"anchor\" aria-label=\"Permalink: Moving On To 2D\" href=\"#moving-on-to-2d\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eNow let's take a look at the picture in 2D. Our 2D-space is now an xy-plane (which is represented by two orthogonal number lines instead of a single number line). Every point in our 2D-space now has two xy-coordinates instead of one number, that is, each point is now a 2D-vector. Suppose we have two basic 2D-shapes – a rectangle \u003ccode\u003eABCD\u003c/code\u003e intersecting a triangle \u003ccode\u003eEFG\u003c/code\u003e on a plane.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22000794/450f715a-dc52-11e6-910f-4b548ab97c2d.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22000794/450f715a-dc52-11e6-910f-4b548ab97c2d.jpg\" alt=\"Rectangle ABCD and triangle EFG on 2D xy-plane\" title=\"Rectangle ABCD and triangle EFG on 2D xy-plane\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThese shapes are represented by the following sets of points (2D-vectors, which are pairs of xy-coordinates):\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"A (1, 3)\nB (5, 3)\nC (5, 1)\nD (1, 1)\n\nE (2, 4)\nF (4, 4)\nG (3, 2)\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003eA (1, 3)\nB (5, 3)\nC (5, 1)\nD (1, 1)\n\nE (2, 4)\nF (4, 4)\nG (3, 2)\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eNow, because we have much more numbers here, the arithmetic becomes a little more involved, but it's still very easy – we literally subtract all points of one shape from all points of another shape one by one.\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"A - E = (1 - 2, 3 - 4) = (-1, -1)\nA - F = (1 - 4, 3 - 4) = (-2, -1)\nA - G = (1 - 3, 3 - 2) = (-2, 1)\n\nB - E = (5 - 2, 3 - 4) = ( 3, -1)\nB - F = (5 - 4, 3 - 4) = ( 1, -1)\nB - G = (5 - 3, 3 - 2) = ( 2, 1)\n\nC - E = (5 - 2, 1 - 4) = ( 3, -3)\nC - F = (5 - 4, 1 - 4) = ( 1, -3)\nC - G = (5 - 3, 1 - 2) = ( 2, -1)\n\nD - E = (1 - 2, 1 - 4) = (-1, -3)\nD - F = (1 - 4, 1 - 4) = (-3, -3)\nD - G = (1 - 3, 1 - 2) = (-2, -1)\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003eA - E = (1 - 2, 3 - 4) = (-1, -1)\nA - F = (1 - 4, 3 - 4) = (-2, -1)\nA - G = (1 - 3, 3 - 2) = (-2, 1)\n\nB - E = (5 - 2, 3 - 4) = ( 3, -1)\nB - F = (5 - 4, 3 - 4) = ( 1, -1)\nB - G = (5 - 3, 3 - 2) = ( 2, 1)\n\nC - E = (5 - 2, 1 - 4) = ( 3, -3)\nC - F = (5 - 4, 1 - 4) = ( 1, -3)\nC - G = (5 - 3, 1 - 2) = ( 2, -1)\n\nD - E = (1 - 2, 1 - 4) = (-1, -3)\nD - F = (1 - 4, 1 - 4) = (-3, -3)\nD - G = (1 - 3, 1 - 2) = (-2, -1)\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eAfter plotting all of 12 resulting points in our 2D-space and connecting the \u003cem\u003eoutermost\u003c/em\u003e points with lines we get the following difference shape:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22000793/450e7e80-dc52-11e6-972c-4843338c3fad.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22000793/450e7e80-dc52-11e6-972c-4843338c3fad.jpg\" alt=\"Rectangle ABCD minus triangle EFG on 2D xy-plane\" title=\"Rectangle ABCD minus triangle EFG on 2D xy-plane\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eGJK says that if we're able to enclose the Origin within the resulting shape then two initial shapes must have collided.\nWe immediately see that this shape actually contains the Origin. Therefore we can visually confirm that our initial rectangle \u003ccode\u003eABCD\u003c/code\u003e indeed intersects our initial triangle \u003ccode\u003eEFG\u003c/code\u003e.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eAlso note, that we subtracted all points from all other points (calculated all combinations of pairs) which yields some points inside our resulting shape. But as with 1D, we will optimise the algorithm to skip all internal points entirely.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eBuilding The Simplex\u003c/h3\u003e\u003ca id=\"user-content-building-the-simplex\" class=\"anchor\" aria-label=\"Permalink: Building The Simplex\" href=\"#building-the-simplex\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eNow, remember, in 1D to determine whether the resulting segment contains the Origin we check if a primitive inequality \u003ccode\u003eleftEndpoint \u0026lt; Origin \u0026lt; rightEndpoint\u003c/code\u003e holds. We can think of it as if the segment \u003cem\u003esurrounds\u003c/em\u003e the Origin from all sides of 1D space (from both left and right, as there are only 2 relative sides in one dimension on our sketch). In 1D for an object to be able to surround any point (the Origin is the zero point on a number line), that object must itself consist of at least two of its own points (in other words, it must be a segment defined by its two points on a number line). Therefore the 1D-version is trying to build a simplex of two endpoints (a segment on a number line). And then it checks whether the Origin is contained within the resulting segment.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22043033/dd0fea8a-dd1e-11e6-9ec7-dd2c6103291f.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22043033/dd0fea8a-dd1e-11e6-9ec7-dd2c6103291f.jpg\" alt=\"Simplices in various spaces\" title=\"Simplices in various spaces\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eIn 2D the process is similar. We can immediately tell if the Origin is inside the resulting polygon shape just by a brief visual inspection. This is done by the natural wiring of human brains. But a machine cannot distinguish \u003cem\u003einside\u003c/em\u003e from \u003cem\u003eoutside\u003c/em\u003e unless you teach it how to do that. It's a lot easier for a machine to see if a point is contained inside a simple shape like a triangle or a circle, rather than a more complex shape like a general polygon with N vertices or some arbitrary irregular shape. Moreover, the algorithm has to do that in a fast but accurate way in order to be suitable for general-purpose collision detection.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eAnd this is where the true power of simplicity of GJK comes into play. Think this way: you need at least two points in 1D to surround the Origin. So a 1D segment of two points is a 1-simplex. But in 2D two points are not enough to surround anything on a plane. Two points define a single straight line on a plane, but a straight line cannot enclose anything, because it's a line and it is straight. Therefore in 2D you need at least three points connected by segments, that is a triangle, to be able to enclose at least some area of the plane. The simplest possible shape that can \u003cem\u003eenclose\u003c/em\u003e something in 2D is a triangle also known as \u003cem\u003e2-simplex\u003c/em\u003e.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNow our task of arithmetically enclosing the Origin within our resulting shape simplifies a little bit. We have to find such three points from our resulting set of points, that make a triangle that encloses the Origin. In other words, we need to build a 2-simplex that would satisfy our criteria. GJK can build a triangle and test if a certain point lies within that triangle, so we just made this task solvable by a machine.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eAfter we subtracted our initial shapes one from another we got a resulting set of all of the points of a new shape that represents the difference of initial shapes. The most straightforward way to build such an Origin-enclosing triangle from a given set of points is to start taking triples of points (combinations of three points) to see if they form a triangle with the Origin inside it. If a triple of points makes such a triangle, then we can conclude that the difference of two shapes contains the Origin, so initial shapes must have collided or intersected. If not, we try some other triple, and that is done in a loop until we run out of points. If none of the triples did enclose the Origin, then no such triangle was found and there was no collision at all.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eSo, if we randomly select any three of our points and connect them with line segments, we will probably end up with a triangle similar to one of the following:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22035222/38bca25c-dd00-11e6-9236-c93f8ba6d050.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22035222/38bca25c-dd00-11e6-9236-c93f8ba6d050.jpg\" alt=\"2-Simplex on a coordinate plane\" title=\"2-Simplex on a coordinate plane\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22035223/38d78054-dd00-11e6-99e3-e2e0c40dfe2c.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22035223/38d78054-dd00-11e6-99e3-e2e0c40dfe2c.jpg\" alt=\"2-Simplex on a coordinate plane\" title=\"2-Simplex on a coordinate plane\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22035224/38e5afda-dd00-11e6-903b-69b98087e5da.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22035224/38e5afda-dd00-11e6-903b-69b98087e5da.jpg\" alt=\"2-Simplex on a coordinate plane\" title=\"2-Simplex on a coordinate plane\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22035443/12889bb2-dd01-11e6-9d75-234dbf351722.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22035443/12889bb2-dd01-11e6-9d75-234dbf351722.jpg\" alt=\"2-Simplex on a coordinate plane\" title=\"2-Simplex on a coordinate plane\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eAll of these triangles satisfy our criteria (all of them contain the Origin), so if we randomly select one of these and find the Origin inside, then we can immediately tell that there was a collision. But if we selected a triple of points that does not contain the Origin we might end up with something like this:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22040185/12e3d646-dd13-11e6-9f41-61f671e254ae.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22040185/12e3d646-dd13-11e6-9f41-61f671e254ae.jpg\" alt=\"Bad 2-Simplex on a coordinate plane\" title=\"Bad 2-Simplex on a coordinate plane\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22040186/12e6f074-dd13-11e6-9dfc-6831d4f284d1.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22040186/12e6f074-dd13-11e6-9dfc-6831d4f284d1.jpg\" alt=\"Bad 2-Simplex on a coordinate plane\" title=\"Bad 2-Simplex on a coordinate plane\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eAs long as the choice of points for our 2-simplex is random we might have to try all possible triples in worst case. Random walk is not the best strategy for finding a triangle that satisfies our criteria, there's a better algorithm for that ;) The goal of the algorithm is to find a combination of the best three points that enclose the Origin out of all possible triples of points, if such a combination exists at all. If such a triple does not exist then there is some distance between our initial shapes. The search for a simplex is the core of GJK.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eIn order to build the simplex efficiently the algorithm introduces a special routine that calculates the difference of two points of initial shapes. It is called a \u003cem\u003esupport function\u003c/em\u003e and it is the workhorse of the search.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch4 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eThe Support Function\u003c/h4\u003e\u003ca id=\"user-content-the-support-function\" class=\"anchor\" aria-label=\"Permalink: The Support Function\" href=\"#the-support-function\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eRemember that in a 1D-space to obtain the resulting 1-simplex you subtract points from one another. The algorithm can skip 'internal' points and only compute the difference of outermost opposite points. The support function calculates the difference of opposite points in a more general way. As a bonus it also allows \u003cem\u003eflat vs curved\u003c/em\u003e collisions! With it you can detect intersections of ellipses, circles, curves and splines in 2D and rounded shapes and more complex objects in 3D (which is cool).\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eLet's think of opposite points in two dimensions. Take a look at these examples of opposite points of a 2D-shape:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22091779/cf553a08-de09-11e6-8161-c6abbbc9bd1b.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22091779/cf553a08-de09-11e6-8161-c6abbbc9bd1b.jpg\" alt=\"Opposite points of a shape in 2D\" title=\"Opposite points of a shape in 2D\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22092331/f897542e-de0d-11e6-9127-c5c4f178b56a.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22092331/f897542e-de0d-11e6-9127-c5c4f178b56a.jpg\" alt=\"Opposite points of a shape in 2D\" title=\"Opposite points of a shape in 2D\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThat is trivial. Now imagine you take not one but two arbitrary shapes and pick a random point of the first shape then pick a second point on the opposite side of the other shape. You might end up with something similar to this:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22092276/8fad1020-de0d-11e6-8287-3f43d05530ea.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22092276/8fad1020-de0d-11e6-8287-3f43d05530ea.jpg\" alt=\"Opposite points of two intersecting shapes in 2D\" title=\"Opposite points of two intersecting shapes in 2D\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22092386/564bbb28-de0e-11e6-835d-e31e65788e3a.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22092386/564bbb28-de0e-11e6-835d-e31e65788e3a.jpg\" alt=\"Opposite points of two non-intersecting shapes in 2D\" title=\"Opposite points of two non-intersecting shapes in 2D\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eA difference of two points yields another point of resulting shape. That point is literally the distance vector. So, if you take a point of a shape and then choose a point of the other shape and then subtract the two points from one another, you get exact distance and direction between your shapes (at specific points).\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNote, that when you subtract \u003cem\u003eopposite\u003c/em\u003e points of two shapes your resulting point-vector will always land somewhere on the contour (an outermost edge) of your resulting shape. Below is an illustration of how a difference of two opposite points (on the left) finally lands on the contour of resulting shape (on the right).\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22177875/724dd816-e038-11e6-8ed0-f28746cffc35.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22177875/724dd816-e038-11e6-8ed0-f28746cffc35.jpg\" alt=\"Difference of opposite points projected into 2D Minkowski Space\" title=\"Difference of opposite points projected into 2D Minkowski Space\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe left side of the picture shows our simulated world space. On the right is our 2D \u003cem\u003eMinkowski space\u003c/em\u003e. In GJK you can think of Minkowski space as if it was an imaginary world of shape differences. By subtracting two points in real world we therefore obtain a new point in another surreal world. It is also often called a \u003cem\u003emapping\u003c/em\u003e or a \u003cem\u003eprojection\u003c/em\u003e of a real-world intersection into Minkowski space (into the world of differences). The result in Minkowski space looks like a weird inside-out union as if one shape is \u003cem\u003e\"swept\"\u003c/em\u003e along the contour of the other. But this is how all intersections really look like in 2D.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eIt's easy to show arithmetically that the resulting point is obtained by taking the difference of the two opposing points:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"A(x1, y1) - B(x2, y2) = C(x1 - x2, y1 - y2)\n\n A(3, 1) - B(1, -1) = C(3 - 1, 1 - (-1)) = C(2, 2)\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003eA(x1, y1) - B(x2, y2) = C(x1 - x2, y1 - y2)\n\n A(3, 1) - B(1, -1) = C(3 - 1, 1 - (-1)) = C(2, 2)\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eResulting point \u003ccode\u003e(2, 2)\u003c/code\u003e ends up exactly on the contour of our difference shape. If initial points are opposite their difference will always be one of the outermost (\u003cem\u003e\"external\"\u003c/em\u003e) points of the resulting shape. Also notice, that resulting distance vector \u003ccode\u003e(2, 2)\u003c/code\u003e is in one-to-one correspondence to the distance between the two initial opposite points. They are literally the same.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe support function of GJK maps a difference of two real-world points into Minkowski space. It seeks for two opposite points which are furthest apart along a given direction and returns their difference. In seek of opposite points along a direction the support function does the following:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22177903/6e2dc3b2-e039-11e6-8492-7eb58f658560.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22177903/6e2dc3b2-e039-11e6-8492-7eb58f658560.jpg\" alt=\"The GJK support function in seek of opposite points along a given direction\" title=\"The GJK support function in seek of opposite points along a given direction\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col dir=\"auto\"\u003e\n\u003cli\u003eStart at the Origin and choose any direction you like (denoted as \u003ccode\u003eD\u003c/code\u003e on the image above). A direction is itself a vector, pointing somewhere away from the Origin. It can be random, you choose whatever you want for a start, later you'll see why initial direction doesn't really matter. The direction vector always starts at the Origin and is sometimes written as \u003ccode\u003eOD = D(x,y) - O(0,0) = D(x,y)\u003c/code\u003e (that is a direction from \u003ccode\u003eO\u003c/code\u003e towards \u003ccode\u003eD\u003c/code\u003e).\u003c/li\u003e\n\u003cli\u003eTake the first of two shapes. It does not matter which one of the two is first. From the Origin seek for the furthest point of first shape in direction \u003ccode\u003eD\u003c/code\u003e. To find the furthest point in direction \u003ccode\u003eD\u003c/code\u003e calculate the dot product of all the point-vectors of the first shape with direction \u003ccode\u003eD\u003c/code\u003e. This is the same as taking magnitudes (or distances) from the Origin to each point of the first shape in direction \u003ccode\u003eD\u003c/code\u003e. It is also often called \u003cem\u003eprojecting a point-vector onto a direction-vector\u003c/em\u003e. The distance to point \u003ccode\u003eA\u003c/code\u003e along direction \u003ccode\u003eD\u003c/code\u003e is the projection of vector \u003ccode\u003eA\u003c/code\u003e onto vector \u003ccode\u003eD\u003c/code\u003e. The most distant point \u003ccode\u003eA\u003c/code\u003e along \u003ccode\u003eD\u003c/code\u003e will have the greatest dot product of \u003ccode\u003eA\u003c/code\u003e and \u003ccode\u003eD\u003c/code\u003e. This way our first most distant point from the Origin along direction \u003ccode\u003eD\u003c/code\u003e is found. Note that a dot product can result in a negative projected vector.\u003c/li\u003e\n\u003cli\u003eNext an opposite point of the \u003cem\u003eother\u003c/em\u003e shape must be found. To do that the support function switches the direction \u003ccode\u003eD\u003c/code\u003e to the opposite, literally flipping it \u003ccode\u003eD = -D\u003c/code\u003e. The process from previous step is repeated, but this time with the other shape in a reverse direction \u003ccode\u003e-D\u003c/code\u003e. It looks for a point of the second shape that is most distant from the Origin along direction \u003ccode\u003e-D\u003c/code\u003e. To find the most distant point it calculates all dot products of all the point-vectors of the second shape and direction-vector \u003ccode\u003e-D\u003c/code\u003e. Then it takes the point which has the greatest dot product with \u003ccode\u003e-D\u003c/code\u003e. This way the second point \u003ccode\u003eB\u003c/code\u003e is found that is most distant from the Origin and is also opposite to the first point.\u003c/li\u003e\n\u003cli\u003eOnce two opposite points \u003ccode\u003eA\u003c/code\u003e and \u003ccode\u003eB\u003c/code\u003e have been found, subtract one of them from the other and done. It doesn't matter which one you are subtracting from. The resulting vector is a mapping of their difference into 2D Minkowski space.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp dir=\"auto\"\u003eIn other words, the support function takes an arbitrary line and two opposite points furthest from the Origin along that line, one point from each of the initial shapes. This is similar to subtracting segments in 1D. The support function finds the endpoints of two shapes along some direction-line and returns their difference that is another point in Minkowski space. On a one-dimensional number line the resulting point is a 1D-number. One a two-dimensional coordinate plane the resulting point is a 2D-vector (it has two coordinates).\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNow its easy to show that it does not matter which initial direction you choose to start with. The support function does not care about given initial direction at all. For example, if we choose a different arbitrary \u003ccode\u003eD\u003c/code\u003e, we might end up with something like this:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22178263/d12adcbc-e042-11e6-8a74-7c667c703f0d.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22178263/d12adcbc-e042-11e6-8a74-7c667c703f0d.jpg\" alt=\"The GJK support function in seek of opposite points along another direction\" title=\"The GJK support function in seek of opposite points along another direction\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eIt doesn't matter which direction \u003ccode\u003eD\u003c/code\u003e you choose to seek for opposite points \u003ccode\u003eA\u003c/code\u003e and \u003ccode\u003eB\u003c/code\u003e. By taking their difference you get a point \u003ccode\u003eC\u003c/code\u003e somewhere on a contour in Minkowski space. And it's easy to verify arithmetically that the intersection of opposite points \u003ccode\u003eA\u003c/code\u003e and \u003ccode\u003eB\u003c/code\u003e along any direction \u003ccode\u003eD\u003c/code\u003e still yields one of many points on the contour of our resulting shape:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"A(2, -2) - B(-1, 2) = C(2 - (-1), -2 - 2) = C(3, -4)\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003eA(2, -2) - B(-1, 2) = C(2 - (-1), -2 - 2) = C(3, -4)\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eSo, in general the support function can work in any given direction and in any given space. You give it two shapes and a direction, and then it finds two opposite points in your space and returns their intersection in Minkowski space. In mathematics the intersection is usually denoted with symbol \u003ccode\u003e∩\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"highlight highlight-source-c notranslate position-relative overflow-auto\" dir=\"auto\" data-snippet-clipboard-copy-content=\"C = support (shape1, shape2, D); // return a point of (shape1 ∩ shape2) along arbitrary direction D\"\u003e\u003cpre\u003e\u003cspan class=\"pl-c1\"\u003eC\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-en\"\u003esupport\u003c/span\u003e (\u003cspan class=\"pl-s1\"\u003eshape1\u003c/span\u003e, \u003cspan class=\"pl-s1\"\u003eshape2\u003c/span\u003e, \u003cspan class=\"pl-c1\"\u003eD\u003c/span\u003e); \u003cspan class=\"pl-c\"\u003e// return a point of (shape1 ∩ shape2) along arbitrary direction D\u003c/span\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThe intersection of two points always yields a third point. In 1D a point is a number. Thus, an intersection of two numbers or two points in 1D gives a third number or another 1D-point. In 2D an intersection of two vectors or two points gives a third vector or another 2D-point.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch5 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eA Word On Math\u003c/h5\u003e\u003ca id=\"user-content-a-word-on-math\" class=\"anchor\" aria-label=\"Permalink: A Word On Math\" href=\"#a-word-on-math\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eTo calculate the difference of two numbers we subtract one of them from the other like this: \u003ccode\u003eA - B = C\u003c/code\u003e. In 1D which number of the two is A and which one is B does not matter, the algorithm works either way. This is because in 1D you have only one possible direction that is the actual number line itself. But in general when dealing with 2D or 3D coordinate vectors (which is usually the case in many applications) the order of subtraction actually does matter. A more accurate way of representing the difference of two vectors in Minkowski space is to take one vector and sum it with a negated version of the other vector, so that \u003ccode\u003eA + (-B) = C\u003c/code\u003e. Because of this fact the Minkowski support function is often defined as a sum of the first point-vector with the negated version of the second point-vector. After negating the second vector you simply add it to the first one to get their arithmetic total result. This is why the support function is called \u003cem\u003eMinkowski addition\u003c/em\u003e or \u003cem\u003eMinkowski sum\u003c/em\u003e and you will probably never hear of \u003cem\u003eMinkowski subtraction\u003c/em\u003e nor \u003cem\u003eMinkowski difference\u003c/em\u003e.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eHere's how an implementation of the Minkowski sum support function for any space of arbitrary amount of dimensions might look like in pseudocode:\u003c/p\u003e\n\u003cdiv class=\"highlight highlight-source-c notranslate position-relative overflow-auto\" dir=\"auto\" data-snippet-clipboard-copy-content=\"//-----------------------------------------------------------------------------\n// Return an arithmetic sum of two point-vectors\n// A 1D-vector has only one component (one coordinate on a number line)\n// In general a vector has one or more components\n\nvec sum (vec a, vec b) {\n return a + b; // [ a.x + b.x, a.y + b.y, a.z + b.z, ... ]\n}\n\n//-----------------------------------------------------------------------------\n// Dot product is the sum of all corresponding components of both vectors multiplied \n\nfloat dotProduct (vec a, vec b) {\n return a * b; // a.x * b.x + a.y * b.y + a.z * b.z + ...\n}\n\n//-----------------------------------------------------------------------------\n// Get furthest vertex along a certain direction d\n// This is the same as finding max dot product with d\n// In 1D direction is always 1 or -1 (to the left or to the right of the Origin)\n\nsize_t indexOfFurthestPoint (const vec * vertices, size_t count, vec d) {\n size_t index = 0;\n float maxProduct = dotProduct (d, vertices[index]);\n for (size_t i = 1; i \u0026lt; count; i++) {\n float product = dotProduct (d, vertices[i]); // may be negative\n if (product \u0026gt; maxProduct) {\n maxProduct = product;\n index = i;\n }\n }\n return index;\n}\n\n//-----------------------------------------------------------------------------\n// Minkowski sum support function for GJK\n\nvec support (const vec * vertices1, size_t count1, // first shape\n const vec * vertices2, size_t count2, // second shape\n vec d) { // direction\n\n // get furthest point of first body along an arbitrary direction\n size_t i = indexOfFurthestPoint (vertices1, count1, d);\n \n // get furthest point of second body along the opposite direction\n // note that this time direction is negated\n size_t j = indexOfFurthestPoint (vertices2, count2, -d);\n\n // return the Minkowski sum of two points to see if bodies 'overlap'\n // note that the second point-vector is negated, a + (-b) = c\n return sum (vertices1[i], -vertices2[j]); \n}\"\u003e\u003cpre\u003e\u003cspan class=\"pl-c\"\u003e//-----------------------------------------------------------------------------\u003c/span\u003e\n\u003cspan class=\"pl-c\"\u003e// Return an arithmetic sum of two point-vectors\u003c/span\u003e\n\u003cspan class=\"pl-c\"\u003e// A 1D-vector has only one component (one coordinate on a number line)\u003c/span\u003e\n\u003cspan class=\"pl-c\"\u003e// In general a vector has one or more components\u003c/span\u003e\n\n\u003cspan class=\"pl-smi\"\u003evec\u003c/span\u003e \u003cspan class=\"pl-en\"\u003esum\u003c/span\u003e (\u003cspan class=\"pl-smi\"\u003evec\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ea\u003c/span\u003e, \u003cspan class=\"pl-smi\"\u003evec\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003eb\u003c/span\u003e) {\n \u003cspan class=\"pl-k\"\u003ereturn\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ea\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e+\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003eb\u003c/span\u003e; \u003cspan class=\"pl-c\"\u003e// [ a.x + b.x, a.y + b.y, a.z + b.z, ... ]\u003c/span\u003e\n}\n\n\u003cspan class=\"pl-c\"\u003e//-----------------------------------------------------------------------------\u003c/span\u003e\n\u003cspan class=\"pl-c\"\u003e// Dot product is the sum of all corresponding components of both vectors multiplied \u003c/span\u003e\n\n\u003cspan class=\"pl-smi\"\u003efloat\u003c/span\u003e \u003cspan class=\"pl-en\"\u003edotProduct\u003c/span\u003e (\u003cspan class=\"pl-smi\"\u003evec\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ea\u003c/span\u003e, \u003cspan class=\"pl-smi\"\u003evec\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003eb\u003c/span\u003e) {\n \u003cspan class=\"pl-k\"\u003ereturn\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ea\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e*\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003eb\u003c/span\u003e; \u003cspan class=\"pl-c\"\u003e// a.x * b.x + a.y * b.y + a.z * b.z + ...\u003c/span\u003e\n}\n\n\u003cspan class=\"pl-c\"\u003e//-----------------------------------------------------------------------------\u003c/span\u003e\n\u003cspan class=\"pl-c\"\u003e// Get furthest vertex along a certain direction d\u003c/span\u003e\n\u003cspan class=\"pl-c\"\u003e// This is the same as finding max dot product with d\u003c/span\u003e\n\u003cspan class=\"pl-c\"\u003e// In 1D direction is always 1 or -1 (to the left or to the right of the Origin)\u003c/span\u003e\n\n\u003cspan class=\"pl-smi\"\u003esize_t\u003c/span\u003e \u003cspan class=\"pl-en\"\u003eindexOfFurthestPoint\u003c/span\u003e (\u003cspan class=\"pl-k\"\u003econst\u003c/span\u003e \u003cspan class=\"pl-smi\"\u003evec\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e*\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003evertices\u003c/span\u003e, \u003cspan class=\"pl-smi\"\u003esize_t\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ecount\u003c/span\u003e, \u003cspan class=\"pl-smi\"\u003evec\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ed\u003c/span\u003e) {\n \u003cspan class=\"pl-smi\"\u003esize_t\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003eindex\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e0\u003c/span\u003e;\n \u003cspan class=\"pl-smi\"\u003efloat\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003emaxProduct\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-en\"\u003edotProduct\u003c/span\u003e (\u003cspan class=\"pl-s1\"\u003ed\u003c/span\u003e, \u003cspan class=\"pl-s1\"\u003evertices\u003c/span\u003e[\u003cspan class=\"pl-s1\"\u003eindex\u003c/span\u003e]);\n \u003cspan class=\"pl-k\"\u003efor\u003c/span\u003e (\u003cspan class=\"pl-smi\"\u003esize_t\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ei\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e1\u003c/span\u003e; \u003cspan class=\"pl-s1\"\u003ei\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e\u0026lt;\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ecount\u003c/span\u003e; \u003cspan class=\"pl-s1\"\u003ei\u003c/span\u003e\u003cspan class=\"pl-c1\"\u003e++\u003c/span\u003e) {\n \u003cspan class=\"pl-smi\"\u003efloat\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003eproduct\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-en\"\u003edotProduct\u003c/span\u003e (\u003cspan class=\"pl-s1\"\u003ed\u003c/span\u003e, \u003cspan class=\"pl-s1\"\u003evertices\u003c/span\u003e[\u003cspan class=\"pl-s1\"\u003ei\u003c/span\u003e]); \u003cspan class=\"pl-c\"\u003e// may be negative\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003eif\u003c/span\u003e (\u003cspan class=\"pl-s1\"\u003eproduct\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e\u0026gt;\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003emaxProduct\u003c/span\u003e) {\n \u003cspan class=\"pl-s1\"\u003emaxProduct\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003eproduct\u003c/span\u003e;\n \u003cspan class=\"pl-s1\"\u003eindex\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ei\u003c/span\u003e;\n }\n }\n \u003cspan class=\"pl-k\"\u003ereturn\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003eindex\u003c/span\u003e;\n}\n\n\u003cspan class=\"pl-c\"\u003e//-----------------------------------------------------------------------------\u003c/span\u003e\n\u003cspan class=\"pl-c\"\u003e// Minkowski sum support function for GJK\u003c/span\u003e\n\n\u003cspan class=\"pl-smi\"\u003evec\u003c/span\u003e \u003cspan class=\"pl-en\"\u003esupport\u003c/span\u003e (\u003cspan class=\"pl-k\"\u003econst\u003c/span\u003e \u003cspan class=\"pl-smi\"\u003evec\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e*\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003evertices1\u003c/span\u003e, \u003cspan class=\"pl-smi\"\u003esize_t\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ecount1\u003c/span\u003e, \u003cspan class=\"pl-c\"\u003e// first shape\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003econst\u003c/span\u003e \u003cspan class=\"pl-smi\"\u003evec\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e*\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003evertices2\u003c/span\u003e, \u003cspan class=\"pl-smi\"\u003esize_t\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ecount2\u003c/span\u003e, \u003cspan class=\"pl-c\"\u003e// second shape\u003c/span\u003e\n \u003cspan class=\"pl-smi\"\u003evec\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ed\u003c/span\u003e) { \u003cspan class=\"pl-c\"\u003e// direction\u003c/span\u003e\n\n \u003cspan class=\"pl-c\"\u003e// get furthest point of first body along an arbitrary direction\u003c/span\u003e\n \u003cspan class=\"pl-smi\"\u003esize_t\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ei\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-en\"\u003eindexOfFurthestPoint\u003c/span\u003e (\u003cspan class=\"pl-s1\"\u003evertices1\u003c/span\u003e, \u003cspan class=\"pl-s1\"\u003ecount1\u003c/span\u003e, \u003cspan class=\"pl-s1\"\u003ed\u003c/span\u003e);\n \n \u003cspan class=\"pl-c\"\u003e// get furthest point of second body along the opposite direction\u003c/span\u003e\n \u003cspan class=\"pl-c\"\u003e// note that this time direction is negated\u003c/span\u003e\n \u003cspan class=\"pl-smi\"\u003esize_t\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ej\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-en\"\u003eindexOfFurthestPoint\u003c/span\u003e (\u003cspan class=\"pl-s1\"\u003evertices2\u003c/span\u003e, \u003cspan class=\"pl-s1\"\u003ecount2\u003c/span\u003e, \u003cspan class=\"pl-c1\"\u003e-\u003c/span\u003e\u003cspan class=\"pl-s1\"\u003ed\u003c/span\u003e);\n\n \u003cspan class=\"pl-c\"\u003e// return the Minkowski sum of two points to see if bodies 'overlap'\u003c/span\u003e\n \u003cspan class=\"pl-c\"\u003e// note that the second point-vector is negated, a + (-b) = c\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003ereturn\u003c/span\u003e \u003cspan class=\"pl-en\"\u003esum\u003c/span\u003e (\u003cspan class=\"pl-s1\"\u003evertices1\u003c/span\u003e[\u003cspan class=\"pl-s1\"\u003ei\u003c/span\u003e], \u003cspan class=\"pl-c1\"\u003e-\u003c/span\u003e\u003cspan class=\"pl-s1\"\u003evertices2\u003c/span\u003e[\u003cspan class=\"pl-s1\"\u003ej\u003c/span\u003e]); \n}\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eIt does not really care how many dimensions are there in space. So given a direction and two shapes the support function always returns another point regardless of how many dimensions you have. It works the same for 1D, 2D, 3D, etc... The point returned by the support function is always on the contour of intersection. This is because the initial points involved in the intersection are opposite.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eRemember, the whole purpose of having a support function was to help us quickly build the simplex for GJK. The support function is used in the search for a 2-simplex that encloses the Origin in 2D. Now that we have such a function, we can move on and build the rest of the algorithm on top of it.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch4 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eThe Evolution\u003c/h4\u003e\u003ca id=\"user-content-the-evolution\" class=\"anchor\" aria-label=\"Permalink: The Evolution\" href=\"#the-evolution\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThis is where the actual logic of GJK kicks in. The general plan of GJK is:\u003c/p\u003e\n\u003col dir=\"auto\"\u003e\n\u003cli\u003eFind best points for a simplex with the help of the support function.\u003c/li\u003e\n\u003cli\u003eCheck if a simplex of those points encloses the Origin from all sides.\u003c/li\u003e\n\u003cli\u003eIf it does, there is a collision – hooray and thanks for the support, support function )\u003c/li\u003e\n\u003cli\u003eIf it doesn't, well... try other points, why not?\u003c/li\u003e\n\u003cli\u003eIf a simplex cannot be built at all no matter how many times you try – it might be a \u003cem\u003edegenerate case\u003c/em\u003e (explained later).\u003c/li\u003e\n\u003cli\u003eIf it's not a degenerate case, then there's no collision.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp dir=\"auto\"\u003eAll of that is done in a loop. The algorithm tries different points and checks if they satisfy the condition. The whole process is called \u003cem\u003eevolution\u003c/em\u003e. If it succeeds to enclose the Origin in another branch of evolution (upon another loop iteration), then the intersection is detected. If not, it tries again until finally it either succeeds or fails.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eSome people might be reasonably worried of the possibility that the evolution of GJK runs out of control and continues on and on without ever stopping. It might even become intelligent some day and who knows what could happen... So they add a limit of iterations into their implementations which is a countdown that cuts power off when the game is over, forcing the algorithm to stop. It's like an emergency halt or a safety button in case something goes wrong with float number precision or whatever. But that is actually not necessary in general.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eIn search for a 2-simplex the algorithm has to obtain 3 points one by one to make a triangle. The resulting set of points is initially empty. The process of finding three points is done step-by-step. First, the algorithm finds the first point and adds it to the resulting set. Then it finds and adds the second point, and then it finds and adds the third one. So, while being built, the 2-simplex kinda \u003cem\u003eevolves\u003c/em\u003e actually passing through all stages of evolution from a 0-simplex (one point, the first one in the resulting set), through a 1-simplex (segment of two points, with the second point added to the set), to a 2-simplex (three points, with the third point added to the resulting set).\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22043033/dd0fea8a-dd1e-11e6-9ec7-dd2c6103291f.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22043033/dd0fea8a-dd1e-11e6-9ec7-dd2c6103291f.jpg\" alt=\"Simplices in various spaces\" title=\"Simplices in various spaces\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eA 2-simplex (a triangle) contains or consists of 1-simplices (segments). A 1-simplex (a segment) consists of even simpler 0-simplices (points). It's clear that each new dimension adds one more point to the simplex. So, in general:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"0-simplex = nothing + 1 point = 1 point \n1-simplex = 0-simplex + 1 point = 2 points \n2-simplex = 1-simplex + 1 point = 3 points\n3-simplex = 2-simplex + 1 point = 4 points\n ... = ... + 1 point = ...\nN-simplex = (N-1)-simplex + 1 point = N+1 points\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003e0-simplex = nothing + 1 point = 1 point \n1-simplex = 0-simplex + 1 point = 2 points \n2-simplex = 1-simplex + 1 point = 3 points\n3-simplex = 2-simplex + 1 point = 4 points\n ... = ... + 1 point = ...\nN-simplex = (N-1)-simplex + 1 point = N+1 points\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eGJK evolves the simplex from the very beginning each time, restarting the evolution when no further progress is possible in current branch. So a simplex passes through all stages of its evolution upon each iteration of the main loop.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eLet us do a single iteration of the main evolution loop by an example, step-by-step. Below is an illustration of a pair of slightly different intersecting shapes (for example), and their intersection Minkowski sum on the surreal side to the right. The Minkowski sum is unknown to our algorithm at the moment, as we haven't calculated all the differences of points yet and, in fact, we won't need the full resulting shape, we only need best three points to surround the Origin and that's it. So the Minkowski sum is currently unknown and we're drawing it in full on the right just for visual explicacy:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/25145390/bd3d06f0-2479-11e7-91d4-23194591181c.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/25145390/bd3d06f0-2479-11e7-91d4-23194591181c.jpg\" alt=\"GJK evolution iteration example\" title=\"GJK evolution iteration example\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eTo build a 2-simplex in 2D we need three points from our resulting Minkowski set that would enclose the Origin within a triangle and we have our nice support function for finding best points for that. Before we begin the search for three points we reserve a place for each one of them and label or tag each place with capital letters \u003ccode\u003eC\u003c/code\u003e, \u003ccode\u003eB\u003c/code\u003e and \u003ccode\u003eA\u003c/code\u003e in that order. Their naming might be a little confusing, as we've used \u003ccode\u003eA\u003c/code\u003e and \u003ccode\u003eB\u003c/code\u003e to denote opposite points previously in this text, but it's an ancient tradition, sorry, can't do much about that. These will be placeholders for the points we might find in the search process. Initially our resulting set of points is empty (all placeholders are empty).\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe first point \u003ccode\u003eC\u003c/code\u003e is the easiest to obtain. You just choose a random direction \u003ccode\u003eD\u003c/code\u003e and call the support function which calculates that point for you as we did earlier. Initial direction \u003ccode\u003eD\u003c/code\u003e is chosen randomly and passed along with the shapes to the support function. The support function then finds opposite points of two shapes along the given direction (labelled below as \u003ccode\u003ea\u003c/code\u003e and \u003ccode\u003eb\u003c/code\u003e in lowercase for less confusion) and returns their Minkowski sum which is point \u003ccode\u003eC\u003c/code\u003e on the right, the first point of our simplex (which is a 0-simplex for now).\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eOnce you get the first resulting point, you place it in placeholder \u003ccode\u003eC\u003c/code\u003e. From now on we will be referring to that point by that label or tag \u003ccode\u003eC\u003c/code\u003e. Don't forget, that \u003ccode\u003eC\u003c/code\u003e is a point on the difference contour line in Minkowski space. Now we have a 0-simplex (a point) in our resulting set and it is the first stage of current iteration of evolution. Note, that the support function looks in both \u003ccode\u003eD\u003c/code\u003e and its opposite \u003ccode\u003e-D\u003c/code\u003e in search for points \u003ccode\u003ea\u003c/code\u003e and \u003ccode\u003eb\u003c/code\u003e. This is what we get after executing the first step and obtaining point \u003ccode\u003eC\u003c/code\u003e in Minkowski space:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/25108220/124100ae-23dd-11e7-9f63-9eb125f8acd7.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/25108220/124100ae-23dd-11e7-9f63-9eb125f8acd7.jpg\" alt=\"Obtaining the first point C (GJK iterative evolution)\" title=\"Obtaining the first point C (GJK iterative evolution)\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eOnce we got the first difference of most distant opposite points along a certain direction-line we need to choose a different direction to find the next point of our simplex in Minkowski resulting set. Remember, that the magic of GJK says that we have to surround the Origin from all sides to detect a collision. As with 1D number line where a segment can surround the Origin with two of its endpoints, in 2D we can surround the Origin from all sides with three \u003cem\u003eendpoints\u003c/em\u003e or vertices of a triangle. So our goal is to find such a 2-simplex that would enclose the Origin within itself.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe logic behind this can be described in the following way: imagine we \u003cem\u003estand at\u003c/em\u003e point \u003ccode\u003eC\u003c/code\u003e and look towards the Origin from that perspective. We want to find next point that would be \u003cem\u003ebeyond\u003c/em\u003e the Origin as seen by us. If there's no such point then we are not reaching far enough and we will not be able to surround the Origin with a triangle using current point \u003ccode\u003eC\u003c/code\u003e.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eIf we cannot reach for a point beyond the Origin from current standpoint \u003ccode\u003eC\u003c/code\u003e, then there's probably some distance between initial shapes (at least those two shapes don't intersect along direction \u003ccode\u003eD\u003c/code\u003e). So, in that case no collision is detected. The halting criteria of GJK algorithm says that current iteration stops if our simplex fails to include (or surround) the Origin, handling these cases will be explained in more detail later in this text.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eStanding at point \u003ccode\u003eC\u003c/code\u003e we need to check if there's a point that is further away from us than the Origin is, in the direction from us towards the Origin. The direction from point \u003ccode\u003eC\u003c/code\u003e towards the Origin \u003ccode\u003eO\u003c/code\u003e is the direction \u003ccode\u003eCO\u003c/code\u003e which is the reversed opposite of direction from Origin to point \u003ccode\u003eC\u003c/code\u003e, so that \u003ccode\u003eCO == -OC\u003c/code\u003e. Therefore we should be looking in direction \u003ccode\u003eCO\u003c/code\u003e next as we hope to find some point beyond the Origin there.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/25163108/87781846-24d0-11e7-93f0-0c82a89c6bb9.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/25163108/87781846-24d0-11e7-93f0-0c82a89c6bb9.jpg\" alt=\"Choosing a direction for 2nd stage of GJK evolution\" title=\"Choosing a direction for 2nd stage of GJK evolution\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe idea of \u003cem\u003elooking beyond Origin from your current standpoint\u003c/em\u003e is the key principle of the search for simplex in GJK. It helps to quickly find the biggest simplex by taking the most distant opposite points of Minkowski sum. The bigger the simplex the higher the chances that it will surround the Origin upon some early iteration.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe second point \u003ccode\u003eB\u003c/code\u003e is quite easy to get. From now on we set new direction \u003ccode\u003eD = -OC\u003c/code\u003e which is equal to \u003ccode\u003eCO\u003c/code\u003e, and we will refer to it as our new direction \u003ccode\u003eD\u003c/code\u003e without the negative sign. The support function finds the other two most distant opposite points \u003ccode\u003ea\u003c/code\u003e and \u003ccode\u003eb\u003c/code\u003e and it looks in both directions \u003ccode\u003eD\u003c/code\u003e and \u003ccode\u003e-D\u003c/code\u003e (yes, again) in the process. Note, that new direction vector \u003ccode\u003eD\u003c/code\u003e is different from the first initial direction vector \u003ccode\u003eD\u003c/code\u003e as it now points in the opposite way.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eYou call the support function with your new opposite direction \u003ccode\u003eD\u003c/code\u003e and get the second point which is labelled or tagged as capital \u003ccode\u003eB\u003c/code\u003e. And now there's a 1-simplex (a segment of two points, \u003ccode\u003eC\u003c/code\u003e and \u003ccode\u003eB\u003c/code\u003e) in the resulting set, and it is the second stage of that same iteration of evolution. Here's what we get after the second step:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/25157552/c0fc12a0-24aa-11e7-88d5-adfbd48d1ee4.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/25157552/c0fc12a0-24aa-11e7-88d5-adfbd48d1ee4.jpg\" alt=\"Obtaining the second point B (GJK iterative evolution)\" title=\"Obtaining the second point B (GJK iterative evolution)\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eBy calling the support function the second time with the opposite direction you get a second point \u003ccode\u003eB\u003c/code\u003e which is another point on the same contour in Minkowski space. That second point \u003ccode\u003eB\u003c/code\u003e is exactly opposite to the first point \u003ccode\u003eC\u003c/code\u003e, because initial directions passed into the support function were opposite, right?\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eHaving point \u003ccode\u003eB\u003c/code\u003e we need to verify that it is indeed beyond Origin as seen from point \u003ccode\u003eC\u003c/code\u003e. In other words, we have to check if point \u003ccode\u003eB\u003c/code\u003e is further away from point \u003ccode\u003eC\u003c/code\u003e than the Origin is (in direction from \u003ccode\u003eC\u003c/code\u003e towards the Origin). If a dot product of \u003ccode\u003eCO ⋅ OB\u003c/code\u003e is positive, then point \u003ccode\u003eB\u003c/code\u003e is really beyond the Origin as seen from point \u003ccode\u003eC\u003c/code\u003e and the evolution continues, otherwise there's no collision and the evolution stops or restarts in a different direction. In geometry the sign of dot product of two vectors basically tells if their orientation is roughly the same.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ccode\u003eif ((CO ⋅ OB) \u0026gt; 0) // test if point B is beyond Origin as seen from point C\u003c/code\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThis test for \u003cem\u003ea point beyond Origin\u003c/em\u003e can also be explained as follows: we know that point \u003ccode\u003eC\u003c/code\u003e is located on the contour of Minkowski sum (this is a feature of our support function) and point \u003ccode\u003eB\u003c/code\u003e is on the opposite side of that same contour, so the Origin must be in between \u003ccode\u003eC\u003c/code\u003e and \u003ccode\u003eB\u003c/code\u003e, otherwise we're not reaching far enough to surround it, and if we cannot surround it, then there's probably some distance between the two initial shapes, therefore no intersection.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eRemember that the algorithm doesn't know anything about the whole set of Minkowski points, it currently only knows about points \u003ccode\u003eC\u003c/code\u003e and \u003ccode\u003eB\u003c/code\u003e that were found during first two stages of the evolution. Now let's take a look at the Minkowski space as seen by GJK algorithm at this point. The left part of the image below shows the 1-simplex of two points (segment \u003ccode\u003eCB\u003c/code\u003e) in Minkowski space, and on the right we see the same segment with the full set of Minkowski sum points added for visual reference and for explanation purposes (the algorithm only sees the left side of this picture, not the right side):\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/25153389/06b3b44e-2495-11e7-9a2f-f5c87c32c82b.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/25153389/06b3b44e-2495-11e7-9a2f-f5c87c32c82b.jpg\" alt=\"The GJK simplex after completing the 2nd stage of evolution\" title=\"The GJK simplex after completing the 2nd stage of evolution\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe algorithm's current view into surreal Minkowski world is on the left side of the image above. There's 1-simplex segment \u003ccode\u003eCB\u003c/code\u003e of two points. The simplex has evolved from being a single point \u003ccode\u003eC\u003c/code\u003e 0-simplex into a 1-simplex or a segment of two points \u003ccode\u003eCB\u003c/code\u003e, in other words, our simplex has become a little more complex and obtained an extra dimension )\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eBefore we proceed to the third point \u003ccode\u003eA\u003c/code\u003e we might need to gain even more intuition about what is going on here. Seeing that segment \u003ccode\u003eCB\u003c/code\u003e on the image above and seeing the whole set of Minkowski sum points on the right side, humans can quickly visually locate the third point to enclose the Origin. Easy. But for a machine to know where to look for the third point, we have to make a critical decision in which direction it should be looking next.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe logic of GJK goes this way: imagine we \u003cem\u003estand anywhere on the segment\u003c/em\u003e \u003ccode\u003eCB\u003c/code\u003e and look towards the Origin from that perspective. We want to find such a third point that would be at least \u003cem\u003ebeyond\u003c/em\u003e the Origin (as seen by us relatively to the segment \u003ccode\u003eCB\u003c/code\u003e that we're \u003cem\u003estanding on\u003c/em\u003e). So, again, if we cannot find a third point beyond the Origin then we will not be able to surround it using current segment \u003ccode\u003eCB\u003c/code\u003e and there's no collision at least on this iteration.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/25166261/26cc765a-24e3-11e7-9f6e-87d64be6ca12.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/25166261/26cc765a-24e3-11e7-9f6e-87d64be6ca12.jpg\" alt=\"Choosing a direction for 3rd stage of GJK evolution\" title=\"Choosing a direction for 3rd stage of GJK evolution\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe image above shows the direction to look for the last point of triangle simplex. To find the exact direction vector, you just take one of perpendiculars to the segment \u003ccode\u003eCB\u003c/code\u003e. If you tilt your head to the right a little while looking at the segment \u003ccode\u003eCB\u003c/code\u003e, you will immediately notice the following simple fact: any segment kinda cuts the coordinate plane into two halves, to the left and to the right of the segment. And the Origin always ends up either on one side of the segment \u003ccode\u003eCB\u003c/code\u003e or on the other side. All we have to do to find the next direction vector is just take a perpendicular (often called \u003cem\u003ea normal\u003c/em\u003e) to \u003ccode\u003eCB\u003c/code\u003e that points towards the Origin.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThere are two possible directions perpendicular to segment \u003ccode\u003eCB\u003c/code\u003e, the one pointing towards the Origin and the one pointing the opposite way (away from the Origin). In order to get the one pointing towards the Origin, we can do a simple vector math trick, called \u003cem\u003e\u003ca href=\"https://en.wikipedia.org/wiki/Triple_product#Vector_triple_product\" rel=\"nofollow\"\u003evector triple product\u003c/a\u003e\u003c/em\u003e. It works like this: take a cross product of the segment \u003ccode\u003eCB\u003c/code\u003e with a segment \u003ccode\u003eCO\u003c/code\u003e (from endpoint to the Origin) and then take a cross product of the result with the segment \u003ccode\u003eCB\u003c/code\u003e again, basically two cross products done sequentially, involving the same segment \u003ccode\u003eCB\u003c/code\u003e twice:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ccode\u003eCB ⨯ CO ⨯ CB\u003c/code\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThere's also a very fast formula for this (an expanded equation) aka \u003cem\u003etriple product expansion\u003c/em\u003e with only a few multiplications and subtractions for calculating the resulting perpendicular:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ccode\u003ea ⨯ (b ⨯ c) = b(a ⋅ c) - c(a ⋅ b)\u003c/code\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eYou have to be precise with the orientation of vectors when using the expansion formula, but if done carefully and correctly, this always gives a perpendicular to segment \u003ccode\u003eCB\u003c/code\u003e pointing towards the Origin.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWe set our final direction \u003ccode\u003eD\u003c/code\u003e to that perpendicular to \u003ccode\u003eCB\u003c/code\u003e pointing towards Origin. That will be a direction to search for point A of our triangle simplex. This is also the last stage of the evolution of the 2-simplex. We call our support function and pass new direction \u003ccode\u003eD\u003c/code\u003e along with the two shapes. The image below shows the result after calling the support function for the 3rd time:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/25314806/523d6236-2853-11e7-8c50-5e35404ff9a6.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/25314806/523d6236-2853-11e7-8c50-5e35404ff9a6.jpg\" alt=\"The GJK simplex after completing the 3rd stage of evolution\" title=\"The GJK simplex after completing the 3rd stage of evolution\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWe obtained point A and now we have a complete triangle of three points in total. Our simplex has evolved from a 0-simplex point through a 1-simplex segment into a 2-simplex triangle. Also notice, that we did not calculate all points of Minkowski sum, we only calculated three of all possible points. This is one of the reasons why this algorithm actually works very fast in narrow-phase collision detection. It just doesn't have to calculate all points, so it does not know anything about the full Minkowski set. Here's how it sees the triangle with point A added to the simplex in memory (remember, it only knows about the left side of the image below, but not about the right side which is there for explanation purposes):\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/25315541/3a5b5ad6-285f-11e7-8783-3acb05b25606.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/25315541/3a5b5ad6-285f-11e7-8783-3acb05b25606.jpg\" alt=\"The GJK simplex after completing the 3rd stage of evolution\" title=\"The GJK simplex after completing the 3rd stage of evolution\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNow that we have all three points for a complete triangle (a full 2-simplex), the only task that's left is to test if the Origin is really contained inside that triangle. In other words, if our resulting simplex surrounds the Origin from all sides, then the whole Minkowski sum also does surround the Origin as well, and it follows that there was a collision between two initial shapes. The most straightforward way to test if a point is inside a triangle (the Origin is a point at zero) is to do a bunch of dot product operations.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eFor each edge (each side) of a triangle you need two things: a normal vector (a perpendicular) to that side towards the Origin and a vector from the Origin to opposite vertex of the triangle. Then you check if dot product of the two vectors is positive (greater than zero)...\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWORK IN PROGRESS, to be continued soon. A live demo of GJK in a 2D-space and a video of GJK in action coming up )\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e...\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch5 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eVoronoi Para Nos\u003c/h5\u003e\u003ca id=\"user-content-voronoi-para-nos\" class=\"anchor\" aria-label=\"Permalink: Voronoi Para Nos\" href=\"#voronoi-para-nos\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eWORK IN PROGRESS, to be continued soon... )\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e...\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch5 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eA Touch Of Degenerate Case\u003c/h5\u003e\u003ca id=\"user-content-a-touch-of-degenerate-case\" class=\"anchor\" aria-label=\"Permalink: A Touch Of Degenerate Case\" href=\"#a-touch-of-degenerate-case\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eA collision is when two bodies occupy the same points in space. In two dimensions a collision is either an intersection of two shapes (when shapes kinda \"overlap\") or they might not intersect but instead one shape could just touch the other, and that is also considered to be a collision. There's even more details to the nature of collisions, because there are different types of touch...\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eA full-on collision when a shape overlaps or penetrates another shape usually looks like this:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22180045/d361a334-e075-11e6-8436-b756a6a5cfcb.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22180045/d361a334-e075-11e6-8436-b756a6a5cfcb.jpg\" alt=\"A usual case of overlapping or penetrating collision\" title=\"A usual case of overlapping or penetrating collision\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWhen 2D-shapes overlap there is usually at least one way to build a 2-simplex that encloses the Origin. But there can be other types of non-penetrating collisions without overlapping, when shapes touch edge-to-edge or meet at one single point. In GJK these collisions are usually called \u003cem\u003edegenerate case\u003c/em\u003e. They are not collisions but contacts in common sense. By design GJK handles all degenerate cases absolutely fine.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eHere are some examples of what a degenerate case collision (a touch) in GJK is:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22180044/d36159f6-e075-11e6-869a-06b14c96cedf.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22180044/d36159f6-e075-11e6-869a-06b14c96cedf.jpg\" alt=\"A GJK degenerate case of non-penetrating collision (a touch) in 2D\" title=\"A GJK degenerate case of non-penetrating collision (a touch) in 2D\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22180046/d3625888-e075-11e6-9432-7ff7566eb509.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22180046/d3625888-e075-11e6-9432-7ff7566eb509.jpg\" alt=\"A GJK degenerate case of non-penetrating collision (a touch) in 2D\" title=\"A GJK degenerate case of non-penetrating collision (a touch) in 2D\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22180043/d352cbd4-e075-11e6-95f3-956725d0cb27.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22180043/d352cbd4-e075-11e6-95f3-956725d0cb27.jpg\" alt=\"A GJK degenerate case of non-penetrating collision (a touch) in 2D\" title=\"A GJK degenerate case of non-penetrating collision (a touch) in 2D\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\n\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22180042/d33b626e-e075-11e6-97fa-360e24fd0739.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22180042/d33b626e-e075-11e6-97fa-360e24fd0739.jpg\" alt=\"A GJK degenerate case of non-penetrating collision (a touch) in 2D\" title=\"A GJK degenerate case of non-penetrating collision (a touch) in 2D\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eIt may seem like a lot of special cases to handle, but in fact, GJK already does that intrinsically.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eA non-overlapping collision will yield a Minkowski sum that has the Origin on its \u003cem\u003eedge\u003c/em\u003e or \u003cem\u003econtour\u003c/em\u003e. Remember, in 1D, when two segments have only one common point, the Origin lands on the endpoint of resulting segment. In 2D, when there is only one common point, the Origin in Minkowski space will be located on the contour of your resulting 2D-intersection shape. If two shapes share a common face, then the Origin will be on one of the sides of triange simplex.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eSo, the Origin can either be inside the Minkowski sum, or it can be on the edge of the sum. And the trick is to check if the Origin is actually one of the points of your simplex. The Origin can also end up exactly on one of the sides of your simplex, between two points of the simplex.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eBasically, to each stage of evolution you add a check, to see if the Origin is already included in your not-fully-evolved simplex or not. That is enough to detect all degenerate cases. If the Origin is already included in your half-built simplex upon some early stage of evolution, then you don't have to evolve it further. Then you can stop and signal a collision (or no collision, at your discretion). You can differentiate between a penetrating collision or not, by counting degenerate collisions or not counting them as being collisions.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWORK IN PROGRESS, to be continued soon... )\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eRoundness And Curvature\u003c/h3\u003e\u003ca id=\"user-content-roundness-and-curvature\" class=\"anchor\" aria-label=\"Permalink: Roundness And Curvature\" href=\"#roundness-and-curvature\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eWhat's very remarkable of this algorithm, is that the support function can be also defined for a huge variety of shapes. If every point of a shape can be derived from a simple formula (aka \u003cem\u003eparametric equation\u003c/em\u003e) then your support function can use that equation to calculate differences of points. When geometry is defined in a parametric way there is no need to keep all points in memory. And you can also scale everything up and down as you like, do boolean combinations of shapes and much more... That's what vector graphics is about, but it's a whole other topic in itself, we won't dive deep into it here.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe key point is: a parametric equation can be an equation of a circle \u003ccode\u003e((x - h)/r)² + ((y - k)/r)² = 1\u003c/code\u003e or an equation of an ellipse \u003ccode\u003e(x - h)²/a² + (y - k)²/b² = 1\u003c/code\u003e, or some \u003cem\u003espline\u003c/em\u003e (!), or even a combination of polygons and conic sections. So, you can detect collisions and intersections of round shapes very accurately up to an exact point of contact. GJK is not restricted to polygons, you can do perfect circle collisions, and as we'll see later in the 3D section, spherical collisions are also possible as well.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe only restriction that still holds is that your shapes should be convex, not concave, so, any line crossing your shape should not intersect its contour more than twice. As long as your shapes are fat and bulgy, you'll be fine. But there's a hint: you can always make a concave shape from multiple convex shapes.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWORK IN PROGRESS, to be continued soon... )\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eAdding 3D\u003c/h3\u003e\u003ca id=\"user-content-adding-3d\" class=\"anchor\" aria-label=\"Permalink: Adding 3D\" href=\"#adding-3d\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eA careful reader might have already noticed a pattern in the algorithm...\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://cloud.githubusercontent.com/assets/1294454/22039058/c30ea898-dd0e-11e6-8e15-62a5b612036d.jpg\"\u003e\u003cimg src=\"https://cloud.githubusercontent.com/assets/1294454/22039058/c30ea898-dd0e-11e6-8e15-62a5b612036d.jpg\" alt=\"Simplices in various spaces\" title=\"Simplices in various spaces\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eFor a 1D number line we need a 1-simplex of two points to enclose the Origin. If we can find such two points then our shapes do intersect indeed. If we cannot find such two points then our initial shapes must have some distance (non-zero difference) between them. For 2D coordinate plane a 2-simplex of three points (a triangle) is necessary to enclose the Origin. In 3D space we need a 3-simplex of four points (a tetrahedron). These are all examples of simplest possible shape primitives in given dimensions.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWORK IN PROGRESS, A live demo of GJK in a 3D-space and a video of GJK in action coming up soon )\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eCollision Details\u003c/h3\u003e\u003ca id=\"user-content-collision-details\" class=\"anchor\" aria-label=\"Permalink: Collision Details\" href=\"#collision-details\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThe version of GJK that gives a boolean answer to a yes/no collision test is somewhat simplified. The algorithm is able to not only detect the fact of intersection, but is also capable of giving back the exact depth of penetration and information about points of contact, so that the collision could be handled properly.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe simplified yes/no test is often called a \u003cem\u003ebastardized\u003c/em\u003e version of GJK algorithm in comparison to its original purpose of calculating detailed collisions. But the simplified version described in the text above is easier for understanding. Having understood the simple yes/no GJK test it is much easier to grasp the Gilbert-Johnson-Keerthi algorithm in its entirety along with a few useful concepts. We will proceed to cover the rest of GJK functionality below.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch4 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eDistance or Depth Of Penetration\u003c/h4\u003e\u003ca id=\"user-content-distance-or-depth-of-penetration\" class=\"anchor\" aria-label=\"Permalink: Distance or Depth Of Penetration\" href=\"#distance-or-depth-of-penetration\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eWORK IN PROGRESS, to be continued soon... )\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch4 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eContact Points\u003c/h4\u003e\u003ca id=\"user-content-contact-points\" class=\"anchor\" aria-label=\"Permalink: Contact Points\" href=\"#contact-points\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eWORK IN PROGRESS, to be continued soon... )\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eReferences (must see)\u003c/h2\u003e\u003ca id=\"user-content-references-must-see\" class=\"anchor\" aria-label=\"Permalink: References (must see)\" href=\"#references-must-see\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eMost of the info (along with reference implementation) was taken from dyn4j. It has a very thorough explanation of GJK and it is definitely a must visit for those who need to understand the intricacies of the algorithm.\u003c/p\u003e\n\u003col dir=\"auto\"\u003e\n\u003cli\u003e\u003ca href=\"http://www.dyn4j.org/2010/04/gjk-gilbert-johnson-keerthi/\" rel=\"nofollow\"\u003ehttp://www.dyn4j.org/2010/04/gjk-gilbert-johnson-keerthi/\u003c/a\u003e GJK description (+ a lot of other useful articles)\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"http://mollyrocket.com/849\" rel=\"nofollow\"\u003ehttp://mollyrocket.com/849\u003c/a\u003e Awesome old-school GJK / Minkowski space video by Casey Muratori\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://github.com/wnbittle/dyn4j\"\u003ehttps://github.com/wnbittle/dyn4j\u003c/a\u003e Quality source code for reference\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.youtube.com/watch?v=-GZOGJ26yxE\" rel=\"nofollow\"\u003ehttps://www.youtube.com/watch?v=-GZOGJ26yxE\u003c/a\u003e – A video of a seminar by \u003ca href=\"https://aero.engin.umich.edu/people/elmer-gilbert/\" rel=\"nofollow\"\u003eElmer Gilbert Professor Emeritus Aerospace Engineering, University of Michigan\u003c/a\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eP.S. 3D-version coming soon...\u003c/h2\u003e\u003ca id=\"user-content-ps-3d-version-coming-soon\" class=\"anchor\" aria-label=\"Permalink: P.S. 3D-version coming soon...\" href=\"#ps-3d-version-coming-soon\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://camo.githubusercontent.com/87ee43169ef2f48415d9721bab7160259b2f9d6ae311983bdc413cfd973acc85/687474703a2f2f7332312e706f7374696d672e6f72672f6461397478633375762f53637265656e5f53686f745f323031365f30315f31335f61745f30395f31335f31322e6a7067\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/87ee43169ef2f48415d9721bab7160259b2f9d6ae311983bdc413cfd973acc85/687474703a2f2f7332312e706f7374696d672e6f72672f6461397478633375762f53637265656e5f53686f745f323031365f30315f31335f61745f30395f31335f31322e6a7067\" alt=\"3D-version of GJK in plain C coming soon...\" title=\"3D-version of GJK in plain C coming soon...\" data-canonical-src=\"http://s21.postimg.org/da9txc3uv/Screen_Shot_2016_01_13_at_09_13_12.jpg\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003c/article\u003e","loaded":true,"timedOut":false,"errorMessage":null,"headerInfo":{"toc":[{"level":1,"text":"gjk.c – Gilbert-Johnson-Keerthi In Plain C","anchor":"gjkc--gilbert-johnson-keerthi-in-plain-c","htmlText":"gjk.c – Gilbert-Johnson-Keerthi In Plain C"},{"level":2,"text":"Disclaimer","anchor":"disclaimer","htmlText":"Disclaimer"},{"level":2,"text":"Usage Example","anchor":"usage-example","htmlText":"Usage Example"},{"level":2,"text":"How It Works","anchor":"how-it-works","htmlText":"How It Works"},{"level":3,"text":"A 1D Intro","anchor":"a-1d-intro","htmlText":"A 1D Intro"},{"level":3,"text":"Moving On To 2D","anchor":"moving-on-to-2d","htmlText":"Moving On To 2D"},{"level":3,"text":"Building The Simplex","anchor":"building-the-simplex","htmlText":"Building The Simplex"},{"level":4,"text":"The Support Function","anchor":"the-support-function","htmlText":"The Support Function"},{"level":5,"text":"A Word On Math","anchor":"a-word-on-math","htmlText":"A Word On Math"},{"level":4,"text":"The Evolution","anchor":"the-evolution","htmlText":"The Evolution"},{"level":5,"text":"Voronoi Para Nos","anchor":"voronoi-para-nos","htmlText":"Voronoi Para Nos"},{"level":5,"text":"A Touch Of Degenerate Case","anchor":"a-touch-of-degenerate-case","htmlText":"A Touch Of Degenerate Case"},{"level":3,"text":"Roundness And Curvature","anchor":"roundness-and-curvature","htmlText":"Roundness And Curvature"},{"level":3,"text":"Adding 3D","anchor":"adding-3d","htmlText":"Adding 3D"},{"level":3,"text":"Collision Details","anchor":"collision-details","htmlText":"Collision Details"},{"level":4,"text":"Distance or Depth Of Penetration","anchor":"distance-or-depth-of-penetration","htmlText":"Distance or Depth Of Penetration"},{"level":4,"text":"Contact Points","anchor":"contact-points","htmlText":"Contact Points"},{"level":2,"text":"References (must see)","anchor":"references-must-see","htmlText":"References (must see)"},{"level":2,"text":"P.S. 3D-version coming soon...","anchor":"ps-3d-version-coming-soon","htmlText":"P.S. 3D-version coming soon..."}],"siteNavLoginPath":"/login?return_to=https%3A%2F%2Fgithub.com%2Fkroitor%2Fgjk.c"}},{"displayName":"LICENSE.txt","repoName":"gjk.c","refName":"master","path":"LICENSE.txt","preferredFileType":"license","tabName":"WTFPL","richText":null,"loaded":false,"timedOut":false,"errorMessage":null,"headerInfo":{"toc":null,"siteNavLoginPath":"/login?return_to=https%3A%2F%2Fgithub.com%2Fkroitor%2Fgjk.c"}}],"overviewFilesProcessingTime":0}},"appPayload":{"helpUrl":"https://docs.github.com","findFileWorkerPath":"/assets-cdn/worker/find-file-worker-7d7eb7c71814.js","findInFileWorkerPath":"/assets-cdn/worker/find-in-file-worker-708ec8ade250.js","githubDevUrl":null,"enabled_features":{"copilot_workspace":null,"code_nav_ui_events":false,"overview_shared_code_dropdown_button":true,"react_blob_overlay":false,"accessible_code_button":true,"github_models_repo_integration":false}}}}</script> <div data-target="react-partial.reactRoot"><style data-styled="true" data-styled-version="5.3.11">.iVEunk{margin-top:16px;margin-bottom:16px;}/*!sc*/ .jzuOtQ{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-direction:column;-ms-flex-direction:column;flex-direction:column;-webkit-box-pack:justify;-webkit-justify-content:space-between;-ms-flex-pack:justify;justify-content:space-between;}/*!sc*/ .bGojzy{margin-bottom:0;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-direction:column;-ms-flex-direction:column;flex-direction:column;row-gap:16px;}/*!sc*/ .iNSVHo{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-box-pack:justify;-webkit-justify-content:space-between;-ms-flex-pack:justify;justify-content:space-between;-webkit-box-flex:1;-webkit-flex-grow:1;-ms-flex-positive:1;flex-grow:1;padding-bottom:16px;padding-top:8px;}/*!sc*/ .bVgnfw{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-direction:row;-ms-flex-direction:row;flex-direction:row;gap:8px;}/*!sc*/ @media screen and (max-width:320px){.bVgnfw{-webkit-box-flex:1;-webkit-flex-grow:1;-ms-flex-positive:1;flex-grow:1;}}/*!sc*/ .CEgMp{position:relative;}/*!sc*/ @media screen and (max-width:380px){.CEgMp .ref-selector-button-text-container{max-width:80px;}}/*!sc*/ @media screen and (max-width:320px){.CEgMp{-webkit-box-flex:1;-webkit-flex-grow:1;-ms-flex-positive:1;flex-grow:1;}.CEgMp .overview-ref-selector{width:100%;}.CEgMp .overview-ref-selector > span{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-box-pack:start;-webkit-justify-content:flex-start;-ms-flex-pack:start;justify-content:flex-start;}.CEgMp .overview-ref-selector > span > span[data-component="text"]{-webkit-box-flex:1;-webkit-flex-grow:1;-ms-flex-positive:1;flex-grow:1;}}/*!sc*/ .gMOVLe[data-size="medium"]{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;min-width:0;}/*!sc*/ .gMOVLe[data-size="medium"] svg{color:var(--fgColor-muted,var(--color-fg-muted,#656d76));}/*!sc*/ .gMOVLe[data-size="medium"] > span{width:inherit;}/*!sc*/ .gUkoLg{-webkit-box-pack:center;-webkit-justify-content:center;-ms-flex-pack:center;justify-content:center;}/*!sc*/ .bZBlpz{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;width:100%;}/*!sc*/ .lhTYNA{margin-right:4px;color:var(--fgColor-muted,var(--color-fg-muted,#656d76));}/*!sc*/ .ffLUq{font-size:14px;min-width:0;overflow:hidden;text-overflow:ellipsis;white-space:nowrap;}/*!sc*/ .bmcJak{min-width:0;}/*!sc*/ .fLXEGX{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;}/*!sc*/ @media screen and (max-width:1079px){.fLXEGX{display:none;}}/*!sc*/ .lmSMZJ[data-size="medium"]{color:var(--fgColor-muted,var(--color-fg-muted,#656d76));padding-left:4px;padding-right:4px;}/*!sc*/ .lmSMZJ[data-size="medium"] span[data-component="leadingVisual"]{margin-right:4px !important;}/*!sc*/ .dqfxud{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;}/*!sc*/ @media screen and (min-width:1080px){.dqfxud{display:none;}}/*!sc*/ @media screen and (max-width:543px){.dqfxud{display:none;}}/*!sc*/ .fGwBZA[data-size="medium"][data-no-visuals]{color:var(--fgColor-muted,var(--color-fg-muted,#656d76));}/*!sc*/ .jxTzTd{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;padding-left:8px;gap:8px;}/*!sc*/ .gqqBXN{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;gap:8px;}/*!sc*/ @media screen and (max-width:543px){.gqqBXN{display:none;}}/*!sc*/ .dzXgxt{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;}/*!sc*/ @media screen and (max-width:1011px){.dzXgxt{display:none;}}/*!sc*/ .iWFGlI{margin-left:8px;margin-right:8px;margin:0;}/*!sc*/ .vcvyP{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;min-width:160px;}/*!sc*/ .YUPas{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;}/*!sc*/ @media screen and (min-width:1012px){.YUPas{display:none;}}/*!sc*/ .izFOf{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;}/*!sc*/ @media screen and (min-width:544px){.izFOf{display:none;}}/*!sc*/ .vIPPs{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-direction:column;-ms-flex-direction:column;flex-direction:column;gap:16px;}/*!sc*/ .fdROMU{width:100%;border-collapse:separate;border-spacing:0;border:1px solid;border-color:var(--borderColor-default,var(--color-border-default,#d0d7de));border-radius:6px;table-layout:fixed;overflow:unset;}/*!sc*/ .jGKpsv{height:0px;line-height:0px;}/*!sc*/ .jGKpsv tr{height:0px;font-size:0px;}/*!sc*/ .jdgHnn{padding:16px;color:var(--fgColor-muted,var(--color-fg-muted,#656d76));font-size:12px;text-align:left;height:40px;}/*!sc*/ .jdgHnn th{padding-left:16px;background-color:var(--bgColor-muted,var(--color-canvas-subtle,#f6f8fa));}/*!sc*/ .bQivRW{width:100%;border-top-left-radius:6px;}/*!sc*/ @media screen and (min-width:544px){.bQivRW{display:none;}}/*!sc*/ .ldkMIO{width:40%;border-top-left-radius:6px;}/*!sc*/ @media screen and (max-width:543px){.ldkMIO{display:none;}}/*!sc*/ .jMbWeI{text-align:right;padding-right:16px;width:136px;border-top-right-radius:6px;}/*!sc*/ .gpqjiB{color:var(--fgColor-muted,var(--color-fg-muted,#656d76));font-size:12px;height:40px;}/*!sc*/ .dzCJzi{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-direction:row;-ms-flex-direction:row;flex-direction:row;-webkit-flex-wrap:wrap;-ms-flex-wrap:wrap;flex-wrap:wrap;-webkit-box-pack:justify;-webkit-justify-content:space-between;-ms-flex-pack:justify;justify-content:space-between;-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;gap:8px;min-width:273px;padding:8px;}/*!sc*/ @media screen and (min-width:544px){.dzCJzi{-webkit-flex-wrap:nowrap;-ms-flex-wrap:nowrap;flex-wrap:nowrap;}}/*!sc*/ .eNCcrz{text-align:center;vertical-align:center;height:40px;border-top:1px solid;border-color:var(--borderColor-default,var(--color-border-default,#d0d7de));}/*!sc*/ .bHTcCe{border-top:1px solid var(--borderColor-default,var(--color-border-default));cursor:pointer;}/*!sc*/ .csrIcr{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-box-flex:1;-webkit-flex-grow:1;-ms-flex-positive:1;flex-grow:1;gap:16px;}/*!sc*/ .bUQNHB{border:1px solid;border-color:var(--borderColor-default,var(--color-border-default,#d0d7de));border-radius:6px;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-direction:column;-ms-flex-direction:column;flex-direction:column;-webkit-box-flex:1;-webkit-flex-grow:1;-ms-flex-positive:1;flex-grow:1;}/*!sc*/ @media screen and (max-width:543px){.bUQNHB{margin-left:-16px;margin-right:-16px;max-width:calc(100% + 32px);}}/*!sc*/ @media screen and (min-width:544px){.bUQNHB{max-width:100%;}}/*!sc*/ .jPdcfu{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;border-bottom:1px solid;border-bottom-color:var(--borderColor-default,var(--color-border-default,#d0d7de));-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;padding-right:8px;position:-webkit-sticky;position:sticky;top:0;background-color:var(--bgColor-default,var(--color-canvas-default,#ffffff));z-index:1;border-top-left-radius:6px;border-top-right-radius:6px;}/*!sc*/ .iphEWz{-webkit-box-flex:1;-webkit-flex-grow:1;-ms-flex-positive:1;flex-grow:1;border-bottom:none;max-width:100%;padding-left:8px;padding-right:8px;}/*!sc*/ .hUCRAk{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-direction:column;-ms-flex-direction:column;flex-direction:column;-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;}/*!sc*/ .cwoBXV[data-size="medium"]{color:var(--fgColor-muted,var(--color-fg-subtle,#6e7781));padding-left:8px;padding-right:8px;}/*!sc*/ .QkQOb{padding:32px;overflow:auto;}/*!sc*/ data-styled.g1[id="Box-sc-g0xbh4-0"]{content:"iVEunk,jzuOtQ,bGojzy,iNSVHo,bVgnfw,CEgMp,gMOVLe,gUkoLg,bZBlpz,lhTYNA,ffLUq,bmcJak,fLXEGX,lmSMZJ,dqfxud,fGwBZA,jxTzTd,gqqBXN,dzXgxt,iWFGlI,vcvyP,YUPas,izFOf,vIPPs,fdROMU,jGKpsv,jdgHnn,bQivRW,ldkMIO,jMbWeI,gpqjiB,dzCJzi,eNCcrz,bHTcCe,csrIcr,bUQNHB,jPdcfu,iphEWz,hUCRAk,cwoBXV,QkQOb,"}/*!sc*/ .brGdpi{position:absolute;width:1px;height:1px;padding:0;margin:-1px;overflow:hidden;-webkit-clip:rect(0,0,0,0);clip:rect(0,0,0,0);white-space:nowrap;border-width:0;}/*!sc*/ data-styled.g5[id="_VisuallyHidden__VisuallyHidden-sc-11jhm7a-0"]{content:"brGdpi,"}/*!sc*/ .hWlpPn{position:relative;display:inline-block;}/*!sc*/ .hWlpPn::after{position:absolute;z-index:1000000;display:none;padding:0.5em 0.75em;font:normal normal 11px/1.5 -apple-system,BlinkMacSystemFont,"Segoe UI","Noto Sans",Helvetica,Arial,sans-serif,"Apple Color Emoji","Segoe UI Emoji";-webkit-font-smoothing:subpixel-antialiased;color:var(--tooltip-fgColor,var(--fgColor-onEmphasis,var(--color-fg-on-emphasis,#ffffff)));text-align:center;-webkit-text-decoration:none;text-decoration:none;text-shadow:none;text-transform:none;-webkit-letter-spacing:normal;-moz-letter-spacing:normal;-ms-letter-spacing:normal;letter-spacing:normal;word-wrap:break-word;white-space:pre;pointer-events:none;content:attr(aria-label);background:var(--tooltip-bgColor,var(--bgColor-emphasis,var(--color-neutral-emphasis-plus,#24292f)));border-radius:6px;opacity:0;}/*!sc*/ @-webkit-keyframes tooltip-appear{from{opacity:0;}to{opacity:1;}}/*!sc*/ @keyframes tooltip-appear{from{opacity:0;}to{opacity:1;}}/*!sc*/ .hWlpPn:hover::after,.hWlpPn:active::after,.hWlpPn:focus::after,.hWlpPn:focus-within::after{display:inline-block;-webkit-text-decoration:none;text-decoration:none;-webkit-animation-name:tooltip-appear;animation-name:tooltip-appear;-webkit-animation-duration:0.1s;animation-duration:0.1s;-webkit-animation-fill-mode:forwards;animation-fill-mode:forwards;-webkit-animation-timing-function:ease-in;animation-timing-function:ease-in;-webkit-animation-delay:0s;animation-delay:0s;}/*!sc*/ .hWlpPn.tooltipped-no-delay:hover::after,.hWlpPn.tooltipped-no-delay:active::after,.hWlpPn.tooltipped-no-delay:focus::after,.hWlpPn.tooltipped-no-delay:focus-within::after{-webkit-animation-delay:0s;animation-delay:0s;}/*!sc*/ .hWlpPn.tooltipped-multiline:hover::after,.hWlpPn.tooltipped-multiline:active::after,.hWlpPn.tooltipped-multiline:focus::after,.hWlpPn.tooltipped-multiline:focus-within::after{display:table-cell;}/*!sc*/ .hWlpPn.tooltipped-s::after,.hWlpPn.tooltipped-se::after,.hWlpPn.tooltipped-sw::after{top:100%;right:50%;margin-top:6px;}/*!sc*/ .hWlpPn.tooltipped-se::after{right:auto;left:50%;margin-left:-16px;}/*!sc*/ .hWlpPn.tooltipped-sw::after{margin-right:-16px;}/*!sc*/ .hWlpPn.tooltipped-n::after,.hWlpPn.tooltipped-ne::after,.hWlpPn.tooltipped-nw::after{right:50%;bottom:100%;margin-bottom:6px;}/*!sc*/ .hWlpPn.tooltipped-ne::after{right:auto;left:50%;margin-left:-16px;}/*!sc*/ .hWlpPn.tooltipped-nw::after{margin-right:-16px;}/*!sc*/ .hWlpPn.tooltipped-s::after,.hWlpPn.tooltipped-n::after{-webkit-transform:translateX(50%);-ms-transform:translateX(50%);transform:translateX(50%);}/*!sc*/ .hWlpPn.tooltipped-w::after{right:100%;bottom:50%;margin-right:6px;-webkit-transform:translateY(50%);-ms-transform:translateY(50%);transform:translateY(50%);}/*!sc*/ .hWlpPn.tooltipped-e::after{bottom:50%;left:100%;margin-left:6px;-webkit-transform:translateY(50%);-ms-transform:translateY(50%);transform:translateY(50%);}/*!sc*/ .hWlpPn.tooltipped-multiline::after{width:-webkit-max-content;width:-moz-max-content;width:max-content;max-width:250px;word-wrap:break-word;white-space:pre-line;border-collapse:separate;}/*!sc*/ .hWlpPn.tooltipped-multiline.tooltipped-s::after,.hWlpPn.tooltipped-multiline.tooltipped-n::after{right:auto;left:50%;-webkit-transform:translateX(-50%);-ms-transform:translateX(-50%);transform:translateX(-50%);}/*!sc*/ .hWlpPn.tooltipped-multiline.tooltipped-w::after,.hWlpPn.tooltipped-multiline.tooltipped-e::after{right:100%;}/*!sc*/ .hWlpPn.tooltipped-align-right-2::after{right:0;margin-right:0;}/*!sc*/ .hWlpPn.tooltipped-align-left-2::after{left:0;margin-left:0;}/*!sc*/ data-styled.g16[id="Tooltip__TooltipBase-sc-17tf59c-0"]{content:"hWlpPn,"}/*!sc*/ .liVpTx{display:inline-block;overflow:hidden;text-overflow:ellipsis;vertical-align:top;white-space:nowrap;max-width:125px;}/*!sc*/ data-styled.g18[id="Truncate__StyledTruncate-sc-23o1d2-0"]{content:"liVpTx,"}/*!sc*/ </style> <!-- --> <!-- --> <div class="Box-sc-g0xbh4-0 iVEunk"><div class="Box-sc-g0xbh4-0 jzuOtQ"><div class="Box-sc-g0xbh4-0 bGojzy"></div></div><div class="Box-sc-g0xbh4-0 iNSVHo"><div class="Box-sc-g0xbh4-0 bVgnfw"><div class="Box-sc-g0xbh4-0 CEgMp"><button type="button" aria-haspopup="true" aria-expanded="false" tabindex="0" aria-label="master branch" data-testid="anchor-button" class="Box-sc-g0xbh4-0 gMOVLe prc-Button-ButtonBase-c50BI overview-ref-selector width-full" data-loading="false" data-size="medium" data-variant="default" aria-describedby="branch-picker-repos-header-ref-selector-loading-announcement" id="branch-picker-repos-header-ref-selector"><span data-component="buttonContent" class="Box-sc-g0xbh4-0 gUkoLg prc-Button-ButtonContent-HKbr-"><span data-component="text" class="prc-Button-Label-pTQ3x"><div class="Box-sc-g0xbh4-0 bZBlpz"><div class="Box-sc-g0xbh4-0 lhTYNA"><svg aria-hidden="true" focusable="false" class="octicon octicon-git-branch" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M9.5 3.25a2.25 2.25 0 1 1 3 2.122V6A2.5 2.5 0 0 1 10 8.5H6a1 1 0 0 0-1 1v1.128a2.251 2.251 0 1 1-1.5 0V5.372a2.25 2.25 0 1 1 1.5 0v1.836A2.493 2.493 0 0 1 6 7h4a1 1 0 0 0 1-1v-.628A2.25 2.25 0 0 1 9.5 3.25Zm-6 0a.75.75 0 1 0 1.5 0 .75.75 0 0 0-1.5 0Zm8.25-.75a.75.75 0 1 0 0 1.5.75.75 0 0 0 0-1.5ZM4.25 12a.75.75 0 1 0 0 1.5.75.75 0 0 0 0-1.5Z"></path></svg></div><div class="Box-sc-g0xbh4-0 ffLUq ref-selector-button-text-container"><span class="Box-sc-g0xbh4-0 bmcJak prc-Text-Text-0ima0"> <!-- -->master</span></div></div></span><span data-component="trailingVisual" class="prc-Button-Visual-2epfX prc-Button-VisualWrap-Db-eB"><svg aria-hidden="true" focusable="false" class="octicon octicon-triangle-down" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="m4.427 7.427 3.396 3.396a.25.25 0 0 0 .354 0l3.396-3.396A.25.25 0 0 0 11.396 7H4.604a.25.25 0 0 0-.177.427Z"></path></svg></span></span></button><button hidden="" data-hotkey-scope="read-only-cursor-text-area"></button></div><div class="Box-sc-g0xbh4-0 fLXEGX"><a style="--button-color:fg.muted" type="button" href="/kroitor/gjk.c/branches" class="Box-sc-g0xbh4-0 lmSMZJ prc-Button-ButtonBase-c50BI" data-loading="false" data-size="medium" data-variant="invisible" aria-describedby=":Rclab:-loading-announcement"><span data-component="buttonContent" class="Box-sc-g0xbh4-0 gUkoLg prc-Button-ButtonContent-HKbr-"><span data-component="leadingVisual" class="prc-Button-Visual-2epfX prc-Button-VisualWrap-Db-eB"><svg aria-hidden="true" focusable="false" class="octicon octicon-git-branch" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M9.5 3.25a2.25 2.25 0 1 1 3 2.122V6A2.5 2.5 0 0 1 10 8.5H6a1 1 0 0 0-1 1v1.128a2.251 2.251 0 1 1-1.5 0V5.372a2.25 2.25 0 1 1 1.5 0v1.836A2.493 2.493 0 0 1 6 7h4a1 1 0 0 0 1-1v-.628A2.25 2.25 0 0 1 9.5 3.25Zm-6 0a.75.75 0 1 0 1.5 0 .75.75 0 0 0-1.5 0Zm8.25-.75a.75.75 0 1 0 0 1.5.75.75 0 0 0 0-1.5ZM4.25 12a.75.75 0 1 0 0 1.5.75.75 0 0 0 0-1.5Z"></path></svg></span><span data-component="text" class="prc-Button-Label-pTQ3x">Branches</span></span></a><a style="--button-color:fg.muted" type="button" href="/kroitor/gjk.c/tags" class="Box-sc-g0xbh4-0 lmSMZJ prc-Button-ButtonBase-c50BI" data-loading="false" data-size="medium" data-variant="invisible" aria-describedby=":Rklab:-loading-announcement"><span data-component="buttonContent" class="Box-sc-g0xbh4-0 gUkoLg prc-Button-ButtonContent-HKbr-"><span data-component="leadingVisual" class="prc-Button-Visual-2epfX prc-Button-VisualWrap-Db-eB"><svg aria-hidden="true" focusable="false" class="octicon octicon-tag" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M1 7.775V2.75C1 1.784 1.784 1 2.75 1h5.025c.464 0 .91.184 1.238.513l6.25 6.25a1.75 1.75 0 0 1 0 2.474l-5.026 5.026a1.75 1.75 0 0 1-2.474 0l-6.25-6.25A1.752 1.752 0 0 1 1 7.775Zm1.5 0c0 .066.026.13.073.177l6.25 6.25a.25.25 0 0 0 .354 0l5.025-5.025a.25.25 0 0 0 0-.354l-6.25-6.25a.25.25 0 0 0-.177-.073H2.75a.25.25 0 0 0-.25.25ZM6 5a1 1 0 1 1 0 2 1 1 0 0 1 0-2Z"></path></svg></span><span data-component="text" class="prc-Button-Label-pTQ3x">Tags</span></span></a></div><div class="Box-sc-g0xbh4-0 dqfxud"><a style="--button-color:fg.muted" type="button" aria-label="Go to Branches page" href="/kroitor/gjk.c/branches" class="Box-sc-g0xbh4-0 fGwBZA prc-Button-ButtonBase-c50BI" data-loading="false" data-no-visuals="true" data-size="medium" data-variant="invisible" aria-describedby=":Relab:-loading-announcement"><svg aria-hidden="true" focusable="false" class="octicon octicon-git-branch" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M9.5 3.25a2.25 2.25 0 1 1 3 2.122V6A2.5 2.5 0 0 1 10 8.5H6a1 1 0 0 0-1 1v1.128a2.251 2.251 0 1 1-1.5 0V5.372a2.25 2.25 0 1 1 1.5 0v1.836A2.493 2.493 0 0 1 6 7h4a1 1 0 0 0 1-1v-.628A2.25 2.25 0 0 1 9.5 3.25Zm-6 0a.75.75 0 1 0 1.5 0 .75.75 0 0 0-1.5 0Zm8.25-.75a.75.75 0 1 0 0 1.5.75.75 0 0 0 0-1.5ZM4.25 12a.75.75 0 1 0 0 1.5.75.75 0 0 0 0-1.5Z"></path></svg></a><a style="--button-color:fg.muted" type="button" aria-label="Go to Tags page" href="/kroitor/gjk.c/tags" class="Box-sc-g0xbh4-0 fGwBZA prc-Button-ButtonBase-c50BI" data-loading="false" data-no-visuals="true" data-size="medium" data-variant="invisible" aria-describedby=":Rmlab:-loading-announcement"><svg aria-hidden="true" focusable="false" class="octicon octicon-tag" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M1 7.775V2.75C1 1.784 1.784 1 2.75 1h5.025c.464 0 .91.184 1.238.513l6.25 6.25a1.75 1.75 0 0 1 0 2.474l-5.026 5.026a1.75 1.75 0 0 1-2.474 0l-6.25-6.25A1.752 1.752 0 0 1 1 7.775Zm1.5 0c0 .066.026.13.073.177l6.25 6.25a.25.25 0 0 0 .354 0l5.025-5.025a.25.25 0 0 0 0-.354l-6.25-6.25a.25.25 0 0 0-.177-.073H2.75a.25.25 0 0 0-.25.25ZM6 5a1 1 0 1 1 0 2 1 1 0 0 1 0-2Z"></path></svg></a></div></div><div class="Box-sc-g0xbh4-0 jxTzTd"><div class="Box-sc-g0xbh4-0 gqqBXN"><div class="Box-sc-g0xbh4-0 dzXgxt"><!--$--><div class="Box-sc-g0xbh4-0 iWFGlI"><span class="Box-sc-g0xbh4-0 vcvyP TextInput-wrapper prc-components-TextInputWrapper-i1ofR prc-components-TextInputBaseWrapper-ueK9q" data-leading-visual="true" data-trailing-visual="true" aria-busy="false"><span class="TextInput-icon" id=":R2j5ab:" aria-hidden="true"><svg aria-hidden="true" focusable="false" class="octicon octicon-search" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M10.68 11.74a6 6 0 0 1-7.922-8.982 6 6 0 0 1 8.982 7.922l3.04 3.04a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215ZM11.5 7a4.499 4.499 0 1 0-8.997 0A4.499 4.499 0 0 0 11.5 7Z"></path></svg></span><input type="text" aria-label="Go to file" role="combobox" aria-controls="file-results-list" aria-expanded="false" aria-haspopup="dialog" autoCorrect="off" spellcheck="false" placeholder="Go to file" aria-describedby=":R2j5ab: :R2j5abH1:" data-component="input" class="prc-components-Input-Ic-y8" value=""/><span class="TextInput-icon" id=":R2j5abH1:" aria-hidden="true"></span></span></div><!--/$--></div><div class="Box-sc-g0xbh4-0 YUPas"><button type="button" class="prc-Button-ButtonBase-c50BI" data-loading="false" data-no-visuals="true" data-size="medium" data-variant="default" aria-describedby=":Rr5ab:-loading-announcement"><span data-component="buttonContent" data-align="center" class="prc-Button-ButtonContent-HKbr-"><span data-component="text" class="prc-Button-Label-pTQ3x">Go to file</span></span></button></div><div class="react-directory-add-file-icon"></div><div class="react-directory-remove-file-icon"></div></div><button type="button" aria-haspopup="true" aria-expanded="false" tabindex="0" class="prc-Button-ButtonBase-c50BI" data-loading="false" data-size="medium" data-variant="primary" aria-describedby=":R55ab:-loading-announcement" id=":R55ab:"><span data-component="buttonContent" data-align="center" class="prc-Button-ButtonContent-HKbr-"><span data-component="leadingVisual" class="prc-Button-Visual-2epfX prc-Button-VisualWrap-Db-eB"><svg aria-hidden="true" focusable="false" class="octicon octicon-code hide-sm" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="m11.28 3.22 4.25 4.25a.75.75 0 0 1 0 1.06l-4.25 4.25a.749.749 0 0 1-1.275-.326.749.749 0 0 1 .215-.734L13.94 8l-3.72-3.72a.749.749 0 0 1 .326-1.275.749.749 0 0 1 .734.215Zm-6.56 0a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042L2.06 8l3.72 3.72a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215L.47 8.53a.75.75 0 0 1 0-1.06Z"></path></svg></span><span data-component="text" class="prc-Button-Label-pTQ3x">Code</span><span data-component="trailingVisual" class="prc-Button-Visual-2epfX prc-Button-VisualWrap-Db-eB"><svg aria-hidden="true" focusable="false" class="octicon octicon-triangle-down" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="m4.427 7.427 3.396 3.396a.25.25 0 0 0 .354 0l3.396-3.396A.25.25 0 0 0 11.396 7H4.604a.25.25 0 0 0-.177.427Z"></path></svg></span></span></button><div class="Box-sc-g0xbh4-0 izFOf"><button data-component="IconButton" type="button" aria-label="Open more actions menu" aria-haspopup="true" aria-expanded="false" tabindex="0" class="prc-Button-ButtonBase-c50BI prc-Button-IconButton-szpyj" data-loading="false" data-no-visuals="true" data-size="medium" data-variant="default" aria-describedby=":R75ab:-loading-announcement" id=":R75ab:"><svg aria-hidden="true" focusable="false" class="octicon octicon-kebab-horizontal" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M8 9a1.5 1.5 0 1 0 0-3 1.5 1.5 0 0 0 0 3ZM1.5 9a1.5 1.5 0 1 0 0-3 1.5 1.5 0 0 0 0 3Zm13 0a1.5 1.5 0 1 0 0-3 1.5 1.5 0 0 0 0 3Z"></path></svg></button></div></div></div><div class="Box-sc-g0xbh4-0 vIPPs"><div data-hpc="true"><button hidden="" data-testid="focus-next-element-button" data-hotkey="j"></button><button hidden="" data-testid="focus-previous-element-button" data-hotkey="k"></button><h2 class="sr-only ScreenReaderHeading-module__userSelectNone--vW4Cq prc-Heading-Heading-6CmGO" data-testid="screen-reader-heading" id="folders-and-files">Folders and files</h2><table aria-labelledby="folders-and-files" class="Box-sc-g0xbh4-0 fdROMU"><thead class="Box-sc-g0xbh4-0 jGKpsv"><tr class="Box-sc-g0xbh4-0 jdgHnn"><th colSpan="2" class="Box-sc-g0xbh4-0 bQivRW"><span class="text-bold">Name</span></th><th colSpan="1" class="Box-sc-g0xbh4-0 ldkMIO"><span class="text-bold">Name</span></th><th class="hide-sm"><div title="Last commit message" class="Truncate__StyledTruncate-sc-23o1d2-0 liVpTx width-fit"><span class="text-bold">Last commit message</span></div></th><th colSpan="1" class="Box-sc-g0xbh4-0 jMbWeI"><div title="Last commit date" class="Truncate__StyledTruncate-sc-23o1d2-0 liVpTx width-fit"><span class="text-bold">Last commit date</span></div></th></tr></thead><tbody><tr class="Box-sc-g0xbh4-0 gpqjiB"><td colSpan="3" class="bgColor-muted p-1 rounded-top-2"><div class="Box-sc-g0xbh4-0 dzCJzi"><h2 class="sr-only ScreenReaderHeading-module__userSelectNone--vW4Cq prc-Heading-Heading-6CmGO" data-testid="screen-reader-heading">Latest commit</h2><div style="width:120px" class="Skeleton Skeleton--text" data-testid="loading"> </div><div class="d-flex flex-shrink-0 gap-2"><div data-testid="latest-commit-details" class="d-none d-sm-flex flex-items-center"></div><div class="d-flex gap-2"><h2 class="sr-only ScreenReaderHeading-module__userSelectNone--vW4Cq prc-Heading-Heading-6CmGO" data-testid="screen-reader-heading">History</h2><a href="/kroitor/gjk.c/commits/master/" class="prc-Button-ButtonBase-c50BI d-none d-lg-flex LinkButton-module__code-view-link-button--xvCGA flex-items-center fgColor-default" data-loading="false" data-size="small" data-variant="invisible" aria-describedby=":Raqj8pab:-loading-announcement"><span data-component="buttonContent" data-align="center" class="prc-Button-ButtonContent-HKbr-"><span data-component="leadingVisual" class="prc-Button-Visual-2epfX prc-Button-VisualWrap-Db-eB"><svg aria-hidden="true" focusable="false" class="octicon octicon-history" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="m.427 1.927 1.215 1.215a8.002 8.002 0 1 1-1.6 5.685.75.75 0 1 1 1.493-.154 6.5 6.5 0 1 0 1.18-4.458l1.358 1.358A.25.25 0 0 1 3.896 6H.25A.25.25 0 0 1 0 5.75V2.104a.25.25 0 0 1 .427-.177ZM7.75 4a.75.75 0 0 1 .75.75v2.992l2.028.812a.75.75 0 0 1-.557 1.392l-2.5-1A.751.751 0 0 1 7 8.25v-3.5A.75.75 0 0 1 7.75 4Z"></path></svg></span><span data-component="text" class="prc-Button-Label-pTQ3x"><span class="fgColor-default">514 Commits</span></span></span></a><div class="d-sm-none"></div><div class="d-flex d-lg-none"><span role="tooltip" aria-label="514 Commits" id="history-icon-button-tooltip" class="Tooltip__TooltipBase-sc-17tf59c-0 hWlpPn tooltipped-n"><a href="/kroitor/gjk.c/commits/master/" class="prc-Button-ButtonBase-c50BI LinkButton-module__code-view-link-button--xvCGA flex-items-center fgColor-default" data-loading="false" data-size="small" data-variant="invisible" aria-describedby=":R1iqj8pab:-loading-announcement history-icon-button-tooltip"><span data-component="buttonContent" data-align="center" class="prc-Button-ButtonContent-HKbr-"><span data-component="leadingVisual" class="prc-Button-Visual-2epfX prc-Button-VisualWrap-Db-eB"><svg aria-hidden="true" focusable="false" class="octicon octicon-history" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="m.427 1.927 1.215 1.215a8.002 8.002 0 1 1-1.6 5.685.75.75 0 1 1 1.493-.154 6.5 6.5 0 1 0 1.18-4.458l1.358 1.358A.25.25 0 0 1 3.896 6H.25A.25.25 0 0 1 0 5.75V2.104a.25.25 0 0 1 .427-.177ZM7.75 4a.75.75 0 0 1 .75.75v2.992l2.028.812a.75.75 0 0 1-.557 1.392l-2.5-1A.751.751 0 0 1 7 8.25v-3.5A.75.75 0 0 1 7.75 4Z"></path></svg></span></span></a></span></div></div></div></div></td></tr><tr class="react-directory-row undefined" id="folder-row-0"><td class="react-directory-row-name-cell-small-screen" colSpan="2"><div class="react-directory-filename-column"><svg aria-hidden="true" focusable="false" class="octicon octicon-file-directory-fill icon-directory" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M1.75 1A1.75 1.75 0 0 0 0 2.75v10.5C0 14.216.784 15 1.75 15h12.5A1.75 1.75 0 0 0 16 13.25v-8.5A1.75 1.75 0 0 0 14.25 3H7.5a.25.25 0 0 1-.2-.1l-.9-1.2C6.07 1.26 5.55 1 5 1H1.75Z"></path></svg><div class="overflow-hidden"><div class="react-directory-filename-cell"><div class="react-directory-truncate"><a title="python" aria-label="python, (Directory)" class="Link--primary" href="/kroitor/gjk.c/tree/master/python">python</a></div></div></div></div></td><td class="react-directory-row-name-cell-large-screen" colSpan="1"><div class="react-directory-filename-column"><svg aria-hidden="true" focusable="false" class="octicon octicon-file-directory-fill icon-directory" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M1.75 1A1.75 1.75 0 0 0 0 2.75v10.5C0 14.216.784 15 1.75 15h12.5A1.75 1.75 0 0 0 16 13.25v-8.5A1.75 1.75 0 0 0 14.25 3H7.5a.25.25 0 0 1-.2-.1l-.9-1.2C6.07 1.26 5.55 1 5 1H1.75Z"></path></svg><div class="overflow-hidden"><div class="react-directory-filename-cell"><div class="react-directory-truncate"><a title="python" aria-label="python, (Directory)" class="Link--primary" href="/kroitor/gjk.c/tree/master/python">python</a></div></div></div></div></td><td class="react-directory-row-commit-cell"><div class="Skeleton Skeleton--text"> </div></td><td><div class="Skeleton Skeleton--text"> </div></td></tr><tr class="react-directory-row undefined" id="folder-row-1"><td class="react-directory-row-name-cell-small-screen" colSpan="2"><div class="react-directory-filename-column"><svg aria-hidden="true" focusable="false" class="octicon octicon-file color-fg-muted" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M2 1.75C2 .784 2.784 0 3.75 0h6.586c.464 0 .909.184 1.237.513l2.914 2.914c.329.328.513.773.513 1.237v9.586A1.75 1.75 0 0 1 13.25 16h-9.5A1.75 1.75 0 0 1 2 14.25Zm1.75-.25a.25.25 0 0 0-.25.25v12.5c0 .138.112.25.25.25h9.5a.25.25 0 0 0 .25-.25V6h-2.75A1.75 1.75 0 0 1 9 4.25V1.5Zm6.75.062V4.25c0 .138.112.25.25.25h2.688l-.011-.013-2.914-2.914-.013-.011Z"></path></svg><div class="overflow-hidden"><div class="react-directory-filename-cell"><div class="react-directory-truncate"><a title=".gitattributes" aria-label=".gitattributes, (File)" class="Link--primary" href="/kroitor/gjk.c/blob/master/.gitattributes">.gitattributes</a></div></div></div></div></td><td class="react-directory-row-name-cell-large-screen" colSpan="1"><div class="react-directory-filename-column"><svg aria-hidden="true" focusable="false" class="octicon octicon-file color-fg-muted" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M2 1.75C2 .784 2.784 0 3.75 0h6.586c.464 0 .909.184 1.237.513l2.914 2.914c.329.328.513.773.513 1.237v9.586A1.75 1.75 0 0 1 13.25 16h-9.5A1.75 1.75 0 0 1 2 14.25Zm1.75-.25a.25.25 0 0 0-.25.25v12.5c0 .138.112.25.25.25h9.5a.25.25 0 0 0 .25-.25V6h-2.75A1.75 1.75 0 0 1 9 4.25V1.5Zm6.75.062V4.25c0 .138.112.25.25.25h2.688l-.011-.013-2.914-2.914-.013-.011Z"></path></svg><div class="overflow-hidden"><div class="react-directory-filename-cell"><div class="react-directory-truncate"><a title=".gitattributes" aria-label=".gitattributes, (File)" class="Link--primary" href="/kroitor/gjk.c/blob/master/.gitattributes">.gitattributes</a></div></div></div></div></td><td class="react-directory-row-commit-cell"><div class="Skeleton Skeleton--text"> </div></td><td><div class="Skeleton Skeleton--text"> </div></td></tr><tr class="react-directory-row undefined" id="folder-row-2"><td class="react-directory-row-name-cell-small-screen" colSpan="2"><div class="react-directory-filename-column"><svg aria-hidden="true" focusable="false" class="octicon octicon-file color-fg-muted" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M2 1.75C2 .784 2.784 0 3.75 0h6.586c.464 0 .909.184 1.237.513l2.914 2.914c.329.328.513.773.513 1.237v9.586A1.75 1.75 0 0 1 13.25 16h-9.5A1.75 1.75 0 0 1 2 14.25Zm1.75-.25a.25.25 0 0 0-.25.25v12.5c0 .138.112.25.25.25h9.5a.25.25 0 0 0 .25-.25V6h-2.75A1.75 1.75 0 0 1 9 4.25V1.5Zm6.75.062V4.25c0 .138.112.25.25.25h2.688l-.011-.013-2.914-2.914-.013-.011Z"></path></svg><div class="overflow-hidden"><div class="react-directory-filename-cell"><div class="react-directory-truncate"><a title="LICENSE.txt" aria-label="LICENSE.txt, (File)" class="Link--primary" href="/kroitor/gjk.c/blob/master/LICENSE.txt">LICENSE.txt</a></div></div></div></div></td><td class="react-directory-row-name-cell-large-screen" colSpan="1"><div class="react-directory-filename-column"><svg aria-hidden="true" focusable="false" class="octicon octicon-file color-fg-muted" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M2 1.75C2 .784 2.784 0 3.75 0h6.586c.464 0 .909.184 1.237.513l2.914 2.914c.329.328.513.773.513 1.237v9.586A1.75 1.75 0 0 1 13.25 16h-9.5A1.75 1.75 0 0 1 2 14.25Zm1.75-.25a.25.25 0 0 0-.25.25v12.5c0 .138.112.25.25.25h9.5a.25.25 0 0 0 .25-.25V6h-2.75A1.75 1.75 0 0 1 9 4.25V1.5Zm6.75.062V4.25c0 .138.112.25.25.25h2.688l-.011-.013-2.914-2.914-.013-.011Z"></path></svg><div class="overflow-hidden"><div class="react-directory-filename-cell"><div class="react-directory-truncate"><a title="LICENSE.txt" aria-label="LICENSE.txt, (File)" class="Link--primary" href="/kroitor/gjk.c/blob/master/LICENSE.txt">LICENSE.txt</a></div></div></div></div></td><td class="react-directory-row-commit-cell"><div class="Skeleton Skeleton--text"> </div></td><td><div class="Skeleton Skeleton--text"> </div></td></tr><tr class="react-directory-row undefined" id="folder-row-3"><td class="react-directory-row-name-cell-small-screen" colSpan="2"><div class="react-directory-filename-column"><svg aria-hidden="true" focusable="false" class="octicon octicon-file color-fg-muted" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M2 1.75C2 .784 2.784 0 3.75 0h6.586c.464 0 .909.184 1.237.513l2.914 2.914c.329.328.513.773.513 1.237v9.586A1.75 1.75 0 0 1 13.25 16h-9.5A1.75 1.75 0 0 1 2 14.25Zm1.75-.25a.25.25 0 0 0-.25.25v12.5c0 .138.112.25.25.25h9.5a.25.25 0 0 0 .25-.25V6h-2.75A1.75 1.75 0 0 1 9 4.25V1.5Zm6.75.062V4.25c0 .138.112.25.25.25h2.688l-.011-.013-2.914-2.914-.013-.011Z"></path></svg><div class="overflow-hidden"><div class="react-directory-filename-cell"><div class="react-directory-truncate"><a title="README.md" aria-label="README.md, (File)" class="Link--primary" href="/kroitor/gjk.c/blob/master/README.md">README.md</a></div></div></div></div></td><td class="react-directory-row-name-cell-large-screen" colSpan="1"><div class="react-directory-filename-column"><svg aria-hidden="true" focusable="false" class="octicon octicon-file color-fg-muted" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M2 1.75C2 .784 2.784 0 3.75 0h6.586c.464 0 .909.184 1.237.513l2.914 2.914c.329.328.513.773.513 1.237v9.586A1.75 1.75 0 0 1 13.25 16h-9.5A1.75 1.75 0 0 1 2 14.25Zm1.75-.25a.25.25 0 0 0-.25.25v12.5c0 .138.112.25.25.25h9.5a.25.25 0 0 0 .25-.25V6h-2.75A1.75 1.75 0 0 1 9 4.25V1.5Zm6.75.062V4.25c0 .138.112.25.25.25h2.688l-.011-.013-2.914-2.914-.013-.011Z"></path></svg><div class="overflow-hidden"><div class="react-directory-filename-cell"><div class="react-directory-truncate"><a title="README.md" aria-label="README.md, (File)" class="Link--primary" href="/kroitor/gjk.c/blob/master/README.md">README.md</a></div></div></div></div></td><td class="react-directory-row-commit-cell"><div class="Skeleton Skeleton--text"> </div></td><td><div class="Skeleton Skeleton--text"> </div></td></tr><tr class="react-directory-row undefined" id="folder-row-4"><td class="react-directory-row-name-cell-small-screen" colSpan="2"><div class="react-directory-filename-column"><svg aria-hidden="true" focusable="false" class="octicon octicon-file color-fg-muted" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M2 1.75C2 .784 2.784 0 3.75 0h6.586c.464 0 .909.184 1.237.513l2.914 2.914c.329.328.513.773.513 1.237v9.586A1.75 1.75 0 0 1 13.25 16h-9.5A1.75 1.75 0 0 1 2 14.25Zm1.75-.25a.25.25 0 0 0-.25.25v12.5c0 .138.112.25.25.25h9.5a.25.25 0 0 0 .25-.25V6h-2.75A1.75 1.75 0 0 1 9 4.25V1.5Zm6.75.062V4.25c0 .138.112.25.25.25h2.688l-.011-.013-2.914-2.914-.013-.011Z"></path></svg><div class="overflow-hidden"><div class="react-directory-filename-cell"><div class="react-directory-truncate"><a title="gjk.c" aria-label="gjk.c, (File)" class="Link--primary" href="/kroitor/gjk.c/blob/master/gjk.c">gjk.c</a></div></div></div></div></td><td class="react-directory-row-name-cell-large-screen" colSpan="1"><div class="react-directory-filename-column"><svg aria-hidden="true" focusable="false" class="octicon octicon-file color-fg-muted" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M2 1.75C2 .784 2.784 0 3.75 0h6.586c.464 0 .909.184 1.237.513l2.914 2.914c.329.328.513.773.513 1.237v9.586A1.75 1.75 0 0 1 13.25 16h-9.5A1.75 1.75 0 0 1 2 14.25Zm1.75-.25a.25.25 0 0 0-.25.25v12.5c0 .138.112.25.25.25h9.5a.25.25 0 0 0 .25-.25V6h-2.75A1.75 1.75 0 0 1 9 4.25V1.5Zm6.75.062V4.25c0 .138.112.25.25.25h2.688l-.011-.013-2.914-2.914-.013-.011Z"></path></svg><div class="overflow-hidden"><div class="react-directory-filename-cell"><div class="react-directory-truncate"><a title="gjk.c" aria-label="gjk.c, (File)" class="Link--primary" href="/kroitor/gjk.c/blob/master/gjk.c">gjk.c</a></div></div></div></div></td><td class="react-directory-row-commit-cell"><div class="Skeleton Skeleton--text"> </div></td><td><div class="Skeleton Skeleton--text"> </div></td></tr><tr class="react-directory-row undefined" id="folder-row-5"><td class="react-directory-row-name-cell-small-screen" colSpan="2"><div class="react-directory-filename-column"><svg aria-hidden="true" focusable="false" class="octicon octicon-file color-fg-muted" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M2 1.75C2 .784 2.784 0 3.75 0h6.586c.464 0 .909.184 1.237.513l2.914 2.914c.329.328.513.773.513 1.237v9.586A1.75 1.75 0 0 1 13.25 16h-9.5A1.75 1.75 0 0 1 2 14.25Zm1.75-.25a.25.25 0 0 0-.25.25v12.5c0 .138.112.25.25.25h9.5a.25.25 0 0 0 .25-.25V6h-2.75A1.75 1.75 0 0 1 9 4.25V1.5Zm6.75.062V4.25c0 .138.112.25.25.25h2.688l-.011-.013-2.914-2.914-.013-.011Z"></path></svg><div class="overflow-hidden"><div class="react-directory-filename-cell"><div class="react-directory-truncate"><a title="gjk1d.html" aria-label="gjk1d.html, (File)" class="Link--primary" href="/kroitor/gjk.c/blob/master/gjk1d.html">gjk1d.html</a></div></div></div></div></td><td class="react-directory-row-name-cell-large-screen" colSpan="1"><div class="react-directory-filename-column"><svg aria-hidden="true" focusable="false" class="octicon octicon-file color-fg-muted" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M2 1.75C2 .784 2.784 0 3.75 0h6.586c.464 0 .909.184 1.237.513l2.914 2.914c.329.328.513.773.513 1.237v9.586A1.75 1.75 0 0 1 13.25 16h-9.5A1.75 1.75 0 0 1 2 14.25Zm1.75-.25a.25.25 0 0 0-.25.25v12.5c0 .138.112.25.25.25h9.5a.25.25 0 0 0 .25-.25V6h-2.75A1.75 1.75 0 0 1 9 4.25V1.5Zm6.75.062V4.25c0 .138.112.25.25.25h2.688l-.011-.013-2.914-2.914-.013-.011Z"></path></svg><div class="overflow-hidden"><div class="react-directory-filename-cell"><div class="react-directory-truncate"><a title="gjk1d.html" aria-label="gjk1d.html, (File)" class="Link--primary" href="/kroitor/gjk.c/blob/master/gjk1d.html">gjk1d.html</a></div></div></div></div></td><td class="react-directory-row-commit-cell"><div class="Skeleton Skeleton--text"> </div></td><td><div class="Skeleton Skeleton--text"> </div></td></tr><tr class="Box-sc-g0xbh4-0 eNCcrz d-none" data-testid="view-all-files-row"><td colSpan="3" class="Box-sc-g0xbh4-0 bHTcCe"><div><button class="prc-Link-Link-85e08">View all files</button></div></td></tr></tbody></table></div><div class="Box-sc-g0xbh4-0 csrIcr"><div class="Box-sc-g0xbh4-0 bUQNHB"><div itemscope="" itemType="https://schema.org/abstract" class="Box-sc-g0xbh4-0 jPdcfu"><h2 class="_VisuallyHidden__VisuallyHidden-sc-11jhm7a-0 brGdpi">Repository files navigation</h2><nav class="Box-sc-g0xbh4-0 iphEWz prc-components-UnderlineWrapper-oOh5J" aria-label="Repository files"><ul class="prc-components-UnderlineItemList-b23Hf" role="list"><li class="Box-sc-g0xbh4-0 hUCRAk"><a class="prc-components-UnderlineItem-lJsg-" href="#" aria-current="page"><span data-component="icon"><svg aria-hidden="true" focusable="false" class="octicon octicon-book" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M0 1.75A.75.75 0 0 1 .75 1h4.253c1.227 0 2.317.59 3 1.501A3.743 3.743 0 0 1 11.006 1h4.245a.75.75 0 0 1 .75.75v10.5a.75.75 0 0 1-.75.75h-4.507a2.25 2.25 0 0 0-1.591.659l-.622.621a.75.75 0 0 1-1.06 0l-.622-.621A2.25 2.25 0 0 0 5.258 13H.75a.75.75 0 0 1-.75-.75Zm7.251 10.324.004-5.073-.002-2.253A2.25 2.25 0 0 0 5.003 2.5H1.5v9h3.757a3.75 3.75 0 0 1 1.994.574ZM8.755 4.75l-.004 7.322a3.752 3.752 0 0 1 1.992-.572H14.5v-9h-3.495a2.25 2.25 0 0 0-2.25 2.25Z"></path></svg></span><span data-component="text" data-content="README">README</span></a></li><li class="Box-sc-g0xbh4-0 hUCRAk"><a class="prc-components-UnderlineItem-lJsg-" href="#"><span data-component="icon"><svg aria-hidden="true" focusable="false" class="octicon octicon-law" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M8.75.75V2h.985c.304 0 .603.08.867.231l1.29.736c.038.022.08.033.124.033h2.234a.75.75 0 0 1 0 1.5h-.427l2.111 4.692a.75.75 0 0 1-.154.838l-.53-.53.529.531-.001.002-.002.002-.006.006-.006.005-.01.01-.045.04c-.21.176-.441.327-.686.45C14.556 10.78 13.88 11 13 11a4.498 4.498 0 0 1-2.023-.454 3.544 3.544 0 0 1-.686-.45l-.045-.04-.016-.015-.006-.006-.004-.004v-.001a.75.75 0 0 1-.154-.838L12.178 4.5h-.162c-.305 0-.604-.079-.868-.231l-1.29-.736a.245.245 0 0 0-.124-.033H8.75V13h2.5a.75.75 0 0 1 0 1.5h-6.5a.75.75 0 0 1 0-1.5h2.5V3.5h-.984a.245.245 0 0 0-.124.033l-1.289.737c-.265.15-.564.23-.869.23h-.162l2.112 4.692a.75.75 0 0 1-.154.838l-.53-.53.529.531-.001.002-.002.002-.006.006-.016.015-.045.04c-.21.176-.441.327-.686.45C4.556 10.78 3.88 11 3 11a4.498 4.498 0 0 1-2.023-.454 3.544 3.544 0 0 1-.686-.45l-.045-.04-.016-.015-.006-.006-.004-.004v-.001a.75.75 0 0 1-.154-.838L2.178 4.5H1.75a.75.75 0 0 1 0-1.5h2.234a.249.249 0 0 0 .125-.033l1.288-.737c.265-.15.564-.23.869-.23h.984V.75a.75.75 0 0 1 1.5 0Zm2.945 8.477c.285.135.718.273 1.305.273s1.02-.138 1.305-.273L13 6.327Zm-10 0c.285.135.718.273 1.305.273s1.02-.138 1.305-.273L3 6.327Z"></path></svg></span><span data-component="text" data-content="WTFPL license">WTFPL license</span></a></li></ul></nav><button style="--button-color:fg.subtle" type="button" aria-label="Outline" aria-haspopup="true" aria-expanded="false" tabindex="0" class="Box-sc-g0xbh4-0 cwoBXV prc-Button-ButtonBase-c50BI" data-loading="false" data-size="medium" data-variant="invisible" aria-describedby=":Rr9ab:-loading-announcement" id=":Rr9ab:"><svg aria-hidden="true" focusable="false" class="octicon octicon-list-unordered" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M5.75 2.5h8.5a.75.75 0 0 1 0 1.5h-8.5a.75.75 0 0 1 0-1.5Zm0 5h8.5a.75.75 0 0 1 0 1.5h-8.5a.75.75 0 0 1 0-1.5Zm0 5h8.5a.75.75 0 0 1 0 1.5h-8.5a.75.75 0 0 1 0-1.5ZM2 14a1 1 0 1 1 0-2 1 1 0 0 1 0 2Zm1-6a1 1 0 1 1-2 0 1 1 0 0 1 2 0ZM2 4a1 1 0 1 1 0-2 1 1 0 0 1 0 2Z"></path></svg></button></div><div class="Box-sc-g0xbh4-0 QkQOb js-snippet-clipboard-copy-unpositioned undefined" data-hpc="true"><article class="markdown-body entry-content container-lg" itemprop="text"><div class="markdown-heading" dir="auto"><h1 tabindex="-1" class="heading-element" dir="auto">gjk.c – Gilbert-Johnson-Keerthi In Plain C</h1><a id="user-content-gjkc--gilbert-johnson-keerthi-in-plain-c" class="anchor" aria-label="Permalink: gjk.c – Gilbert-Johnson-Keerthi In Plain C" href="#gjkc--gilbert-johnson-keerthi-in-plain-c"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">This is a rough but fast implementation of GJK collision detection algorithm in plain C. Just one C file, less than 200 lines, no dependencies. It is in 2D for now, full 3D version is upcoming... This 2D-version uses Minkowski sums and builds a triangle-simplex in Minkowski space to tell if two arbitrary convex polygons are colliding. 3D-version will be roughly the same, but will build a tetrahedron-simplex inside a 3-dimensional Minkowski space. It currently only tells if there is a collision or not. C-code for distance and contact points coming soon.</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Disclaimer</h2><a id="user-content-disclaimer" class="anchor" aria-label="Permalink: Disclaimer" href="#disclaimer"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">Fuck licenses and copyright. I made it for learning purposes, this is public knowledge and it's absolutely free for any usage.</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Usage Example</h2><a id="user-content-usage-example" class="anchor" aria-label="Permalink: Usage Example" href="#usage-example"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">This is an illustration of the example case from <a href="http://www.dyn4j.org/2010/04/gjk-gilbert-johnson-keerthi/" rel="nofollow">dyn4j</a>.</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://user-images.githubusercontent.com/1294454/133015814-8e2fd47d-0f58-430d-8403-016cc5649cdb.png"><img src="https://user-images.githubusercontent.com/1294454/133015814-8e2fd47d-0f58-430d-8403-016cc5649cdb.png" alt="Example case from dyn4j" title="Example case from dyn4j" style="max-width: 100%;"></a></p> <p dir="auto">The two tested polygons are defined as arrays of plain C vector struct type. This implementation of GJK doesn't really care about the order of the vertices in the arrays, as it treats all sets of points as <em>convex polygons</em>.</p> <div class="highlight highlight-source-c notranslate position-relative overflow-auto" dir="auto" data-snippet-clipboard-copy-content="struct _vec2 { float x; float y; }; typedef struct _vec2 vec2; ... int main(int argc, const char * argv[]) { // test case from dyn4j vec2 vertices1[] = { { 4, 11 }, { 4, 5 }, { 9, 9 }, }; vec2 vertices2[] = { { 5, 7 }, { 7, 3 }, { 10, 2 }, { 12, 7 }, }; size_t count1 = sizeof (vertices1) / sizeof (vec2); // == 3 size_t count2 = sizeof (vertices2) / sizeof (vec2); // == 4 int collisionDetected = gjk (vertices1, count1, vertices2, count2); printf (collisionDetected ? "Bodies collide!\n" : "No collision\n"); return 0; }"><pre><span class="pl-k">struct</span> <span class="pl-smi">_vec2</span> { <span class="pl-smi">float</span> <span class="pl-c1">x</span>; <span class="pl-smi">float</span> <span class="pl-c1">y</span>; }; <span class="pl-k">typedef</span> <span class="pl-k">struct</span> <span class="pl-smi">_vec2</span> <span class="pl-smi">vec2</span>; ... <span class="pl-smi">int</span> <span class="pl-en">main</span>(<span class="pl-smi">int</span> <span class="pl-s1">argc</span>, <span class="pl-k">const</span> <span class="pl-smi">char</span> <span class="pl-c1">*</span> <span class="pl-s1">argv</span>[]) { <span class="pl-c">// test case from dyn4j</span> <span class="pl-smi">vec2</span> <span class="pl-s1">vertices1</span>[] <span class="pl-c1">=</span> { { <span class="pl-c1">4</span>, <span class="pl-c1">11</span> }, { <span class="pl-c1">4</span>, <span class="pl-c1">5</span> }, { <span class="pl-c1">9</span>, <span class="pl-c1">9</span> }, }; <span class="pl-smi">vec2</span> <span class="pl-s1">vertices2</span>[] <span class="pl-c1">=</span> { { <span class="pl-c1">5</span>, <span class="pl-c1">7</span> }, { <span class="pl-c1">7</span>, <span class="pl-c1">3</span> }, { <span class="pl-c1">10</span>, <span class="pl-c1">2</span> }, { <span class="pl-c1">12</span>, <span class="pl-c1">7</span> }, }; <span class="pl-smi">size_t</span> <span class="pl-s1">count1</span> <span class="pl-c1">=</span> <span class="pl-k">sizeof</span> (<span class="pl-s1">vertices1</span>) / <span class="pl-k">sizeof</span> (<span class="pl-s1">vec2</span>); <span class="pl-c">// == 3</span> <span class="pl-smi">size_t</span> <span class="pl-s1">count2</span> <span class="pl-c1">=</span> <span class="pl-k">sizeof</span> (<span class="pl-s1">vertices2</span>) / <span class="pl-k">sizeof</span> (<span class="pl-s1">vec2</span>); <span class="pl-c">// == 4</span> <span class="pl-smi">int</span> <span class="pl-s1">collisionDetected</span> <span class="pl-c1">=</span> <span class="pl-en">gjk</span> (<span class="pl-s1">vertices1</span>, <span class="pl-s1">count1</span>, <span class="pl-s1">vertices2</span>, <span class="pl-s1">count2</span>); <span class="pl-en">printf</span> (<span class="pl-s1">collisionDetected</span> ? <span class="pl-s">"Bodies collide!\n"</span> : <span class="pl-s">"No collision\n"</span>); <span class="pl-k">return</span> <span class="pl-c1">0</span>; }</pre></div> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">How It Works</h2><a id="user-content-how-it-works" class="anchor" aria-label="Permalink: How It Works" href="#how-it-works"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">The goal of this explanation is to help people visualize the logic of GJK. To explain it we have to oversimplify certain things. There's no complicated math except basic arithmetic and a little bit of vector math, nothing more, so that a 3rd-grader could understand it. It is actually not very difficult to have GJK algorithm explained in a proper understandable way from ground up.</p> <p dir="auto">At the very top level GJK tells if two arbitrary shapes are intersecting (colliding) or not. The algorithm is used to calculate the depth of intersection (collision distance). A collision occurs when two shapes try to occupy the same points in space at the same time. The space can be of any nature. It might be your in-game world simulation, or a calculation on a table of statistical data, or a built-in navigation system for a robot or any other application you can imagine. You can use it for calculating collisions of solid bodies and numeric intersections of any kind. You can have as many dimensions as you want, the amount of dimensions does not really matter, the logic is the same for 1D, 2D, 3D, etc... With GJK you can even calculate collisions in 4D if you're able to comprehend this.</p> <p dir="auto">The algorithm itself does not require any hard math whatsoever, it's actually very intuitive and easy. It takes an arithmetic difference of two shapes by subtracting all points of one shape from all points of another shape. If two shapes share a common point, subtracting that point from itself results in a difference of zero. So, if zero is found in the result then there's a collision. In fact, the algorithm does not have to calculate all differences for all possible pairs of points, only a small subset of significant points is examined, therefore it works very fast while being very accurate.</p> <p dir="auto">In order to understand GJK one has to build an imaginary visualization of what is going on under the hood. Once you see the picture in your head, you can implement it and even tweak it for your needs.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">A 1D Intro</h3><a id="user-content-a-1d-intro" class="anchor" aria-label="Permalink: A 1D Intro" href="#a-1d-intro"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">Let's start with a naive example of computing a difference of two shapes in one dimension. A segment of a number line is an example of 1D-shape. Imagine we have two segments on the number line: segment <code>[1,3]</code> and segment <code>[2,4]</code>:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/21998716/8f47b72e-dc47-11e6-836e-06d523a84105.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/21998716/8f47b72e-dc47-11e6-836e-06d523a84105.jpg" alt="Segment [1,3] on the number line" title="Segment [1,3] on the number line" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/21998717/8f487718-dc47-11e6-88f2-57bf590a0ee4.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/21998717/8f487718-dc47-11e6-88f2-57bf590a0ee4.jpg" alt="Segment [2,4] on the number line" title="Segment [2,4] on the number line" style="max-width: 100%;"></a></p> <p dir="auto">Zero is our point of reference on the number line, so we call it <em>the Origin</em>. Easy enough. It is obvious that our segments occupy some common region of our 1D-space, so they must be intersecting or colliding, you can tell that just by looking at the representation of both segments in same 1D-space (on the same line). Let's confirm arithmetically that these segments indeed intersect. We do that by subtracting all points of segment <code>[2,4]</code> from all points of segment <code>[1,3]</code> to see what we get on a number line.</p> <div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="1 - 2 = -1 1 - 3 = -2 1 - 4 = -3 2 - 2 = 0 2 - 3 = -1 2 - 4 = -2 3 - 2 = 1 3 - 3 = 0 3 - 4 = -1"><pre class="notranslate"><code>1 - 2 = -1 1 - 3 = -2 1 - 4 = -3 2 - 2 = 0 2 - 3 = -1 2 - 4 = -2 3 - 2 = 1 3 - 3 = 0 3 - 4 = -1 </code></pre></div> <p dir="auto">We got a lot of numbers, many of them more than once. The resulting set of points (numbers) is larger than each of our initial sets of points of two shapes. Let's plot these resulting points in our 1D-space and look at the shape of resulting segment on the number line:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/21998715/8f465a96-dc47-11e6-9aa5-e30f59453685.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/21998715/8f465a96-dc47-11e6-9aa5-e30f59453685.jpg" alt="Segment [-3,1] on the number line" title="Segment [-3,1] on the number line" style="max-width: 100%;"></a></p> <p dir="auto">So after all we have resulting segment <code>[-3,1]</code> that covers points <code>-3</code>, <code>-2</code>, <code>-1</code>, <code>0</code> and <code>1</code>. Because two initial shapes had some points in common the resulting segment contains a zero. This comes from a simple fact, that when you subtract a point (which is a number) from itself you inevitably end up with a zero. Note, that our initial segments had points 2 and 3 in common. When we subtracted 2 from 2 we got 0. When we subtracted 3 from 3 we also got a zero. This is quite obvious. So, if two shapes have at least one common point, because you subtract that point from itself, the resulting set must contain zero (the Origin) at least once. This is the key of GJK which says: if the Origin is contained inside the resulting set then initial shapes must have collided or kissed at least. Once you get it, you can then apply it to any number of dimensions.</p> <p dir="auto">Now let's take a look at a counter-example, say we have two segments <code>[-2,-1]</code> and <code>[1,3]</code>: <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/21998714/8f462616-dc47-11e6-9eec-b13454c7a67a.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/21998714/8f462616-dc47-11e6-9eec-b13454c7a67a.jpg" alt="Segment [-2,-1] on the number line" title="Segment [-2,-1] on the number line" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/21998716/8f47b72e-dc47-11e6-836e-06d523a84105.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/21998716/8f47b72e-dc47-11e6-836e-06d523a84105.jpg" alt="Segment [1,3] on the number line" title="Segment [1,3] on the number line" style="max-width: 100%;"></a></p> <p dir="auto">We can visually ensure that segment <code>[-2,-1]</code> occupies a different region of number line than that of segment <code>[1,3]</code>. There's a gap between our initial shapes, so these two shapes do not intersect in our 1D-space, therefore there's no collision. Let's prove that arithmetically by subtracting all points of any of the two segments from all points of the other.</p> <div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="1 - (-1) = 1 + 1 = 2 1 - (-2) = 1 + 2 = 3 2 - (-1) = 2 + 1 = 3 2 - (-2) = 2 + 2 = 4 3 - (-1) = 3 + 1 = 4 3 - (-2) = 3 + 2 = 5 "><pre class="notranslate"><code>1 - (-1) = 1 + 1 = 2 1 - (-2) = 1 + 2 = 3 2 - (-1) = 2 + 1 = 3 2 - (-2) = 2 + 2 = 4 3 - (-1) = 3 + 1 = 4 3 - (-2) = 3 + 2 = 5 </code></pre></div> <p dir="auto">And we again draw the resulting segment on a number line in our imaginary 1D-space:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/21998713/8f45117c-dc47-11e6-8e2a-a759b36a559d.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/21998713/8f45117c-dc47-11e6-8e2a-a759b36a559d.jpg" alt="Segment [2,5] on the number line" title="Segment [2,5] on the number line" style="max-width: 100%;"></a></p> <p dir="auto">We got another bigger segment <code>[2,5]</code> that represents a difference of all the points of two initial segments but this time it does not contain the Origin. That is, the resulting set of points does not include zero, because initial segments did not have any points in common so they indeed occupy different regions of our number line and don't intersect.</p> <p dir="auto">Now, if our initial shapes were too big (long initial segments) we would have to calculate too many differences from too many pairs of points. But it's actually easy to see, that we only need to calculate the intersection between the endpoints of two segments, ignoring all the inside points of both segments.</p> <p dir="auto">Consider two segments <code>[10,20]</code> and <code>[5,40]</code>:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/21999180/deb68374-dc49-11e6-9ef4-ea631b35178c.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/21999180/deb68374-dc49-11e6-9ef4-ea631b35178c.jpg" alt="Segment [10,20] on the number line" title="Segment [10,20] on the number line" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/21999179/de9cceac-dc49-11e6-8ddf-656ec1829c43.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/21999179/de9cceac-dc49-11e6-8ddf-656ec1829c43.jpg" alt="Segment [5,40] on the number line" title="Segment [5,40] on the number line" style="max-width: 100%;"></a></p> <p dir="auto">Now subtract their four endpoints from each other:</p> <div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="10 - 5 = 5 10 - 40 = -30 20 - 5 = 15 20 - 40 = -20"><pre class="notranslate"><code>10 - 5 = 5 10 - 40 = -30 20 - 5 = 15 20 - 40 = -20 </code></pre></div> <p dir="auto">The resulting segment <code>[-30,15]</code> would look like this:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/21999267/39333c70-dc4a-11e6-91ec-12a08ebb6f34.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/21999267/39333c70-dc4a-11e6-91ec-12a08ebb6f34.jpg" alt="Segment [-30,15] on the number line" title="Segment [-30,15] on the number line" style="max-width: 100%;"></a></p> <p dir="auto">Notice that when we subtract the <em>opposite points</em> of initial shapes from one another, the resulting difference number lands either to the left or to the right of the Origin (which is zero) on the number line. Say, we take the leftmost point of segment <code>[5,40]</code> (number <code>5</code>) and subtract it from the rightmost point of segment <code>[10,20]</code> (number <code>20</code>) the resulting number <code>20 - 5 = 15</code> is positive, so it lands to the right of zero on the number line. Then we take the rightmost point of segment <code>[5,40]</code> (number <code>40</code> this time) and subtract it from the leftmost point of segment <code>[10,20]</code> (number <code>10</code>) the resulting number <code>10 - 40 = -30</code> is negative, so it lands to the left of zero on the number line that is exactly opposite to the first positive difference point <code>15</code>. We say these points are <em>opposite</em> directionwise. Therefore the Origin is between two resulting points <em>enclosing</em> it.</p> <p dir="auto">We ignored all insignificant internal points and only took the endpoints of initial segments into account thus reducing our calculation to four basic arithmetic operations (subtractions). We did that by switching to a simpler representation of a segment (only two endpoints instead of all points contained inside an initial segment). A simpler representation of the difference of two shapes is called a <em>simplex</em>. It literally means 'the simplest possible'. The simplest possible <em>thing</em> on a one-dimensional number-line is a number (a single point in 1D-space is called a <em>0-simplex</em>). A segment is the simplest possible shape sufficient to contain multiple points of a number line (a segment of two points in 1D space is called a <em>1-simplex</em>). Even if one segment covers the other segment in its entirety (a segment fully contains another segment) – you can still detect an intersection of them in space. And it does not matter which one you're subtracting from, the resulting set will still contain the Origin at zero.</p> <p dir="auto">GJK also works if two initial shapes don't intersect but just barely touch. Say, we have two shapes, segment <code>[1,2]</code> and segment <code>[2,3]</code>:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/21999597/fbec71fe-dc4b-11e6-8afa-371f87457339.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/21999597/fbec71fe-dc4b-11e6-8afa-371f87457339.jpg" alt="Segment [1,2] on the number line" title="Segment [1,2] on the number line" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/21999596/fbebff44-dc4b-11e6-811a-f832ab3eb9a4.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/21999596/fbebff44-dc4b-11e6-811a-f832ab3eb9a4.jpg" alt="Segment [2,3] on the number line" title="Segment [2,3] on the number line" style="max-width: 100%;"></a></p> <p dir="auto">These two segments only have one point in common. Subtracting their endpoints from each other gives:</p> <div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="1 - 2 = -1 1 - 3 = -2 2 - 2 = 0 2 - 3 = -1"><pre class="notranslate"><code>1 - 2 = -1 1 - 3 = -2 2 - 2 = 0 2 - 3 = -1 </code></pre></div> <p dir="auto">And the resulting difference looks like:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/21999595/fbe9927c-dc4b-11e6-8521-816589df8552.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/21999595/fbe9927c-dc4b-11e6-8521-816589df8552.jpg" alt="Segment [-2,0] on the number line" title="Segment [-2,0] on the number line" style="max-width: 100%;"></a></p> <p dir="auto">Notice, that in case of only one common point the Origin is not inside resulting segment, but is actually one of its endpoints (the rightmost in this example). When your resulting segment has the Origin at one of its endpoints that means that your initial shapes do not collide, but merely touch at a single point (two segments have only one common point of intersection).</p> <p dir="auto">What GJK really says is: if you're able to build a simplex that contains (includes) the Origin then your shapes have at least one or more points of intersection (occupy same points in space).</p> <p dir="auto">WORK IN PROGRESS, A live demo of GJK in a 1D-space and a video of GJK in action coming up soon )</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">Moving On To 2D</h3><a id="user-content-moving-on-to-2d" class="anchor" aria-label="Permalink: Moving On To 2D" href="#moving-on-to-2d"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">Now let's take a look at the picture in 2D. Our 2D-space is now an xy-plane (which is represented by two orthogonal number lines instead of a single number line). Every point in our 2D-space now has two xy-coordinates instead of one number, that is, each point is now a 2D-vector. Suppose we have two basic 2D-shapes – a rectangle <code>ABCD</code> intersecting a triangle <code>EFG</code> on a plane.</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22000794/450f715a-dc52-11e6-910f-4b548ab97c2d.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22000794/450f715a-dc52-11e6-910f-4b548ab97c2d.jpg" alt="Rectangle ABCD and triangle EFG on 2D xy-plane" title="Rectangle ABCD and triangle EFG on 2D xy-plane" style="max-width: 100%;"></a></p> <p dir="auto">These shapes are represented by the following sets of points (2D-vectors, which are pairs of xy-coordinates):</p> <div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="A (1, 3) B (5, 3) C (5, 1) D (1, 1) E (2, 4) F (4, 4) G (3, 2)"><pre class="notranslate"><code>A (1, 3) B (5, 3) C (5, 1) D (1, 1) E (2, 4) F (4, 4) G (3, 2) </code></pre></div> <p dir="auto">Now, because we have much more numbers here, the arithmetic becomes a little more involved, but it's still very easy – we literally subtract all points of one shape from all points of another shape one by one.</p> <div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="A - E = (1 - 2, 3 - 4) = (-1, -1) A - F = (1 - 4, 3 - 4) = (-2, -1) A - G = (1 - 3, 3 - 2) = (-2, 1) B - E = (5 - 2, 3 - 4) = ( 3, -1) B - F = (5 - 4, 3 - 4) = ( 1, -1) B - G = (5 - 3, 3 - 2) = ( 2, 1) C - E = (5 - 2, 1 - 4) = ( 3, -3) C - F = (5 - 4, 1 - 4) = ( 1, -3) C - G = (5 - 3, 1 - 2) = ( 2, -1) D - E = (1 - 2, 1 - 4) = (-1, -3) D - F = (1 - 4, 1 - 4) = (-3, -3) D - G = (1 - 3, 1 - 2) = (-2, -1)"><pre class="notranslate"><code>A - E = (1 - 2, 3 - 4) = (-1, -1) A - F = (1 - 4, 3 - 4) = (-2, -1) A - G = (1 - 3, 3 - 2) = (-2, 1) B - E = (5 - 2, 3 - 4) = ( 3, -1) B - F = (5 - 4, 3 - 4) = ( 1, -1) B - G = (5 - 3, 3 - 2) = ( 2, 1) C - E = (5 - 2, 1 - 4) = ( 3, -3) C - F = (5 - 4, 1 - 4) = ( 1, -3) C - G = (5 - 3, 1 - 2) = ( 2, -1) D - E = (1 - 2, 1 - 4) = (-1, -3) D - F = (1 - 4, 1 - 4) = (-3, -3) D - G = (1 - 3, 1 - 2) = (-2, -1) </code></pre></div> <p dir="auto">After plotting all of 12 resulting points in our 2D-space and connecting the <em>outermost</em> points with lines we get the following difference shape:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22000793/450e7e80-dc52-11e6-972c-4843338c3fad.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22000793/450e7e80-dc52-11e6-972c-4843338c3fad.jpg" alt="Rectangle ABCD minus triangle EFG on 2D xy-plane" title="Rectangle ABCD minus triangle EFG on 2D xy-plane" style="max-width: 100%;"></a></p> <p dir="auto">GJK says that if we're able to enclose the Origin within the resulting shape then two initial shapes must have collided. We immediately see that this shape actually contains the Origin. Therefore we can visually confirm that our initial rectangle <code>ABCD</code> indeed intersects our initial triangle <code>EFG</code>.</p> <p dir="auto">Also note, that we subtracted all points from all other points (calculated all combinations of pairs) which yields some points inside our resulting shape. But as with 1D, we will optimise the algorithm to skip all internal points entirely.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">Building The Simplex</h3><a id="user-content-building-the-simplex" class="anchor" aria-label="Permalink: Building The Simplex" href="#building-the-simplex"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">Now, remember, in 1D to determine whether the resulting segment contains the Origin we check if a primitive inequality <code>leftEndpoint < Origin < rightEndpoint</code> holds. We can think of it as if the segment <em>surrounds</em> the Origin from all sides of 1D space (from both left and right, as there are only 2 relative sides in one dimension on our sketch). In 1D for an object to be able to surround any point (the Origin is the zero point on a number line), that object must itself consist of at least two of its own points (in other words, it must be a segment defined by its two points on a number line). Therefore the 1D-version is trying to build a simplex of two endpoints (a segment on a number line). And then it checks whether the Origin is contained within the resulting segment.</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22043033/dd0fea8a-dd1e-11e6-9ec7-dd2c6103291f.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22043033/dd0fea8a-dd1e-11e6-9ec7-dd2c6103291f.jpg" alt="Simplices in various spaces" title="Simplices in various spaces" style="max-width: 100%;"></a></p> <p dir="auto">In 2D the process is similar. We can immediately tell if the Origin is inside the resulting polygon shape just by a brief visual inspection. This is done by the natural wiring of human brains. But a machine cannot distinguish <em>inside</em> from <em>outside</em> unless you teach it how to do that. It's a lot easier for a machine to see if a point is contained inside a simple shape like a triangle or a circle, rather than a more complex shape like a general polygon with N vertices or some arbitrary irregular shape. Moreover, the algorithm has to do that in a fast but accurate way in order to be suitable for general-purpose collision detection.</p> <p dir="auto">And this is where the true power of simplicity of GJK comes into play. Think this way: you need at least two points in 1D to surround the Origin. So a 1D segment of two points is a 1-simplex. But in 2D two points are not enough to surround anything on a plane. Two points define a single straight line on a plane, but a straight line cannot enclose anything, because it's a line and it is straight. Therefore in 2D you need at least three points connected by segments, that is a triangle, to be able to enclose at least some area of the plane. The simplest possible shape that can <em>enclose</em> something in 2D is a triangle also known as <em>2-simplex</em>.</p> <p dir="auto">Now our task of arithmetically enclosing the Origin within our resulting shape simplifies a little bit. We have to find such three points from our resulting set of points, that make a triangle that encloses the Origin. In other words, we need to build a 2-simplex that would satisfy our criteria. GJK can build a triangle and test if a certain point lies within that triangle, so we just made this task solvable by a machine.</p> <p dir="auto">After we subtracted our initial shapes one from another we got a resulting set of all of the points of a new shape that represents the difference of initial shapes. The most straightforward way to build such an Origin-enclosing triangle from a given set of points is to start taking triples of points (combinations of three points) to see if they form a triangle with the Origin inside it. If a triple of points makes such a triangle, then we can conclude that the difference of two shapes contains the Origin, so initial shapes must have collided or intersected. If not, we try some other triple, and that is done in a loop until we run out of points. If none of the triples did enclose the Origin, then no such triangle was found and there was no collision at all.</p> <p dir="auto">So, if we randomly select any three of our points and connect them with line segments, we will probably end up with a triangle similar to one of the following:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22035222/38bca25c-dd00-11e6-9236-c93f8ba6d050.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22035222/38bca25c-dd00-11e6-9236-c93f8ba6d050.jpg" alt="2-Simplex on a coordinate plane" title="2-Simplex on a coordinate plane" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22035223/38d78054-dd00-11e6-99e3-e2e0c40dfe2c.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22035223/38d78054-dd00-11e6-99e3-e2e0c40dfe2c.jpg" alt="2-Simplex on a coordinate plane" title="2-Simplex on a coordinate plane" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22035224/38e5afda-dd00-11e6-903b-69b98087e5da.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22035224/38e5afda-dd00-11e6-903b-69b98087e5da.jpg" alt="2-Simplex on a coordinate plane" title="2-Simplex on a coordinate plane" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22035443/12889bb2-dd01-11e6-9d75-234dbf351722.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22035443/12889bb2-dd01-11e6-9d75-234dbf351722.jpg" alt="2-Simplex on a coordinate plane" title="2-Simplex on a coordinate plane" style="max-width: 100%;"></a></p> <p dir="auto">All of these triangles satisfy our criteria (all of them contain the Origin), so if we randomly select one of these and find the Origin inside, then we can immediately tell that there was a collision. But if we selected a triple of points that does not contain the Origin we might end up with something like this:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22040185/12e3d646-dd13-11e6-9f41-61f671e254ae.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22040185/12e3d646-dd13-11e6-9f41-61f671e254ae.jpg" alt="Bad 2-Simplex on a coordinate plane" title="Bad 2-Simplex on a coordinate plane" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22040186/12e6f074-dd13-11e6-9dfc-6831d4f284d1.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22040186/12e6f074-dd13-11e6-9dfc-6831d4f284d1.jpg" alt="Bad 2-Simplex on a coordinate plane" title="Bad 2-Simplex on a coordinate plane" style="max-width: 100%;"></a></p> <p dir="auto">As long as the choice of points for our 2-simplex is random we might have to try all possible triples in worst case. Random walk is not the best strategy for finding a triangle that satisfies our criteria, there's a better algorithm for that ;) The goal of the algorithm is to find a combination of the best three points that enclose the Origin out of all possible triples of points, if such a combination exists at all. If such a triple does not exist then there is some distance between our initial shapes. The search for a simplex is the core of GJK.</p> <p dir="auto">In order to build the simplex efficiently the algorithm introduces a special routine that calculates the difference of two points of initial shapes. It is called a <em>support function</em> and it is the workhorse of the search.</p> <div class="markdown-heading" dir="auto"><h4 tabindex="-1" class="heading-element" dir="auto">The Support Function</h4><a id="user-content-the-support-function" class="anchor" aria-label="Permalink: The Support Function" href="#the-support-function"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">Remember that in a 1D-space to obtain the resulting 1-simplex you subtract points from one another. The algorithm can skip 'internal' points and only compute the difference of outermost opposite points. The support function calculates the difference of opposite points in a more general way. As a bonus it also allows <em>flat vs curved</em> collisions! With it you can detect intersections of ellipses, circles, curves and splines in 2D and rounded shapes and more complex objects in 3D (which is cool).</p> <p dir="auto">Let's think of opposite points in two dimensions. Take a look at these examples of opposite points of a 2D-shape:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22091779/cf553a08-de09-11e6-8161-c6abbbc9bd1b.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22091779/cf553a08-de09-11e6-8161-c6abbbc9bd1b.jpg" alt="Opposite points of a shape in 2D" title="Opposite points of a shape in 2D" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22092331/f897542e-de0d-11e6-9127-c5c4f178b56a.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22092331/f897542e-de0d-11e6-9127-c5c4f178b56a.jpg" alt="Opposite points of a shape in 2D" title="Opposite points of a shape in 2D" style="max-width: 100%;"></a></p> <p dir="auto">That is trivial. Now imagine you take not one but two arbitrary shapes and pick a random point of the first shape then pick a second point on the opposite side of the other shape. You might end up with something similar to this:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22092276/8fad1020-de0d-11e6-8287-3f43d05530ea.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22092276/8fad1020-de0d-11e6-8287-3f43d05530ea.jpg" alt="Opposite points of two intersecting shapes in 2D" title="Opposite points of two intersecting shapes in 2D" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22092386/564bbb28-de0e-11e6-835d-e31e65788e3a.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22092386/564bbb28-de0e-11e6-835d-e31e65788e3a.jpg" alt="Opposite points of two non-intersecting shapes in 2D" title="Opposite points of two non-intersecting shapes in 2D" style="max-width: 100%;"></a></p> <p dir="auto">A difference of two points yields another point of resulting shape. That point is literally the distance vector. So, if you take a point of a shape and then choose a point of the other shape and then subtract the two points from one another, you get exact distance and direction between your shapes (at specific points).</p> <p dir="auto">Note, that when you subtract <em>opposite</em> points of two shapes your resulting point-vector will always land somewhere on the contour (an outermost edge) of your resulting shape. Below is an illustration of how a difference of two opposite points (on the left) finally lands on the contour of resulting shape (on the right).</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22177875/724dd816-e038-11e6-8ed0-f28746cffc35.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22177875/724dd816-e038-11e6-8ed0-f28746cffc35.jpg" alt="Difference of opposite points projected into 2D Minkowski Space" title="Difference of opposite points projected into 2D Minkowski Space" style="max-width: 100%;"></a></p> <p dir="auto">The left side of the picture shows our simulated world space. On the right is our 2D <em>Minkowski space</em>. In GJK you can think of Minkowski space as if it was an imaginary world of shape differences. By subtracting two points in real world we therefore obtain a new point in another surreal world. It is also often called a <em>mapping</em> or a <em>projection</em> of a real-world intersection into Minkowski space (into the world of differences). The result in Minkowski space looks like a weird inside-out union as if one shape is <em>"swept"</em> along the contour of the other. But this is how all intersections really look like in 2D.</p> <p dir="auto">It's easy to show arithmetically that the resulting point is obtained by taking the difference of the two opposing points:</p> <div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="A(x1, y1) - B(x2, y2) = C(x1 - x2, y1 - y2) A(3, 1) - B(1, -1) = C(3 - 1, 1 - (-1)) = C(2, 2)"><pre class="notranslate"><code>A(x1, y1) - B(x2, y2) = C(x1 - x2, y1 - y2) A(3, 1) - B(1, -1) = C(3 - 1, 1 - (-1)) = C(2, 2) </code></pre></div> <p dir="auto">Resulting point <code>(2, 2)</code> ends up exactly on the contour of our difference shape. If initial points are opposite their difference will always be one of the outermost (<em>"external"</em>) points of the resulting shape. Also notice, that resulting distance vector <code>(2, 2)</code> is in one-to-one correspondence to the distance between the two initial opposite points. They are literally the same.</p> <p dir="auto">The support function of GJK maps a difference of two real-world points into Minkowski space. It seeks for two opposite points which are furthest apart along a given direction and returns their difference. In seek of opposite points along a direction the support function does the following:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22177903/6e2dc3b2-e039-11e6-8492-7eb58f658560.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22177903/6e2dc3b2-e039-11e6-8492-7eb58f658560.jpg" alt="The GJK support function in seek of opposite points along a given direction" title="The GJK support function in seek of opposite points along a given direction" style="max-width: 100%;"></a></p> <ol dir="auto"> <li>Start at the Origin and choose any direction you like (denoted as <code>D</code> on the image above). A direction is itself a vector, pointing somewhere away from the Origin. It can be random, you choose whatever you want for a start, later you'll see why initial direction doesn't really matter. The direction vector always starts at the Origin and is sometimes written as <code>OD = D(x,y) - O(0,0) = D(x,y)</code> (that is a direction from <code>O</code> towards <code>D</code>).</li> <li>Take the first of two shapes. It does not matter which one of the two is first. From the Origin seek for the furthest point of first shape in direction <code>D</code>. To find the furthest point in direction <code>D</code> calculate the dot product of all the point-vectors of the first shape with direction <code>D</code>. This is the same as taking magnitudes (or distances) from the Origin to each point of the first shape in direction <code>D</code>. It is also often called <em>projecting a point-vector onto a direction-vector</em>. The distance to point <code>A</code> along direction <code>D</code> is the projection of vector <code>A</code> onto vector <code>D</code>. The most distant point <code>A</code> along <code>D</code> will have the greatest dot product of <code>A</code> and <code>D</code>. This way our first most distant point from the Origin along direction <code>D</code> is found. Note that a dot product can result in a negative projected vector.</li> <li>Next an opposite point of the <em>other</em> shape must be found. To do that the support function switches the direction <code>D</code> to the opposite, literally flipping it <code>D = -D</code>. The process from previous step is repeated, but this time with the other shape in a reverse direction <code>-D</code>. It looks for a point of the second shape that is most distant from the Origin along direction <code>-D</code>. To find the most distant point it calculates all dot products of all the point-vectors of the second shape and direction-vector <code>-D</code>. Then it takes the point which has the greatest dot product with <code>-D</code>. This way the second point <code>B</code> is found that is most distant from the Origin and is also opposite to the first point.</li> <li>Once two opposite points <code>A</code> and <code>B</code> have been found, subtract one of them from the other and done. It doesn't matter which one you are subtracting from. The resulting vector is a mapping of their difference into 2D Minkowski space.</li> </ol> <p dir="auto">In other words, the support function takes an arbitrary line and two opposite points furthest from the Origin along that line, one point from each of the initial shapes. This is similar to subtracting segments in 1D. The support function finds the endpoints of two shapes along some direction-line and returns their difference that is another point in Minkowski space. On a one-dimensional number line the resulting point is a 1D-number. One a two-dimensional coordinate plane the resulting point is a 2D-vector (it has two coordinates).</p> <p dir="auto">Now its easy to show that it does not matter which initial direction you choose to start with. The support function does not care about given initial direction at all. For example, if we choose a different arbitrary <code>D</code>, we might end up with something like this:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22178263/d12adcbc-e042-11e6-8a74-7c667c703f0d.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22178263/d12adcbc-e042-11e6-8a74-7c667c703f0d.jpg" alt="The GJK support function in seek of opposite points along another direction" title="The GJK support function in seek of opposite points along another direction" style="max-width: 100%;"></a></p> <p dir="auto">It doesn't matter which direction <code>D</code> you choose to seek for opposite points <code>A</code> and <code>B</code>. By taking their difference you get a point <code>C</code> somewhere on a contour in Minkowski space. And it's easy to verify arithmetically that the intersection of opposite points <code>A</code> and <code>B</code> along any direction <code>D</code> still yields one of many points on the contour of our resulting shape:</p> <div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="A(2, -2) - B(-1, 2) = C(2 - (-1), -2 - 2) = C(3, -4)"><pre class="notranslate"><code>A(2, -2) - B(-1, 2) = C(2 - (-1), -2 - 2) = C(3, -4) </code></pre></div> <p dir="auto">So, in general the support function can work in any given direction and in any given space. You give it two shapes and a direction, and then it finds two opposite points in your space and returns their intersection in Minkowski space. In mathematics the intersection is usually denoted with symbol <code>∩</code>.</p> <div class="highlight highlight-source-c notranslate position-relative overflow-auto" dir="auto" data-snippet-clipboard-copy-content="C = support (shape1, shape2, D); // return a point of (shape1 ∩ shape2) along arbitrary direction D"><pre><span class="pl-c1">C</span> <span class="pl-c1">=</span> <span class="pl-en">support</span> (<span class="pl-s1">shape1</span>, <span class="pl-s1">shape2</span>, <span class="pl-c1">D</span>); <span class="pl-c">// return a point of (shape1 ∩ shape2) along arbitrary direction D</span></pre></div> <p dir="auto">The intersection of two points always yields a third point. In 1D a point is a number. Thus, an intersection of two numbers or two points in 1D gives a third number or another 1D-point. In 2D an intersection of two vectors or two points gives a third vector or another 2D-point.</p> <div class="markdown-heading" dir="auto"><h5 tabindex="-1" class="heading-element" dir="auto">A Word On Math</h5><a id="user-content-a-word-on-math" class="anchor" aria-label="Permalink: A Word On Math" href="#a-word-on-math"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">To calculate the difference of two numbers we subtract one of them from the other like this: <code>A - B = C</code>. In 1D which number of the two is A and which one is B does not matter, the algorithm works either way. This is because in 1D you have only one possible direction that is the actual number line itself. But in general when dealing with 2D or 3D coordinate vectors (which is usually the case in many applications) the order of subtraction actually does matter. A more accurate way of representing the difference of two vectors in Minkowski space is to take one vector and sum it with a negated version of the other vector, so that <code>A + (-B) = C</code>. Because of this fact the Minkowski support function is often defined as a sum of the first point-vector with the negated version of the second point-vector. After negating the second vector you simply add it to the first one to get their arithmetic total result. This is why the support function is called <em>Minkowski addition</em> or <em>Minkowski sum</em> and you will probably never hear of <em>Minkowski subtraction</em> nor <em>Minkowski difference</em>.</p> <p dir="auto">Here's how an implementation of the Minkowski sum support function for any space of arbitrary amount of dimensions might look like in pseudocode:</p> <div class="highlight highlight-source-c notranslate position-relative overflow-auto" dir="auto" data-snippet-clipboard-copy-content="//----------------------------------------------------------------------------- // Return an arithmetic sum of two point-vectors // A 1D-vector has only one component (one coordinate on a number line) // In general a vector has one or more components vec sum (vec a, vec b) { return a + b; // [ a.x + b.x, a.y + b.y, a.z + b.z, ... ] } //----------------------------------------------------------------------------- // Dot product is the sum of all corresponding components of both vectors multiplied float dotProduct (vec a, vec b) { return a * b; // a.x * b.x + a.y * b.y + a.z * b.z + ... } //----------------------------------------------------------------------------- // Get furthest vertex along a certain direction d // This is the same as finding max dot product with d // In 1D direction is always 1 or -1 (to the left or to the right of the Origin) size_t indexOfFurthestPoint (const vec * vertices, size_t count, vec d) { size_t index = 0; float maxProduct = dotProduct (d, vertices[index]); for (size_t i = 1; i < count; i++) { float product = dotProduct (d, vertices[i]); // may be negative if (product > maxProduct) { maxProduct = product; index = i; } } return index; } //----------------------------------------------------------------------------- // Minkowski sum support function for GJK vec support (const vec * vertices1, size_t count1, // first shape const vec * vertices2, size_t count2, // second shape vec d) { // direction // get furthest point of first body along an arbitrary direction size_t i = indexOfFurthestPoint (vertices1, count1, d); // get furthest point of second body along the opposite direction // note that this time direction is negated size_t j = indexOfFurthestPoint (vertices2, count2, -d); // return the Minkowski sum of two points to see if bodies 'overlap' // note that the second point-vector is negated, a + (-b) = c return sum (vertices1[i], -vertices2[j]); }"><pre><span class="pl-c">//-----------------------------------------------------------------------------</span> <span class="pl-c">// Return an arithmetic sum of two point-vectors</span> <span class="pl-c">// A 1D-vector has only one component (one coordinate on a number line)</span> <span class="pl-c">// In general a vector has one or more components</span> <span class="pl-smi">vec</span> <span class="pl-en">sum</span> (<span class="pl-smi">vec</span> <span class="pl-s1">a</span>, <span class="pl-smi">vec</span> <span class="pl-s1">b</span>) { <span class="pl-k">return</span> <span class="pl-s1">a</span> <span class="pl-c1">+</span> <span class="pl-s1">b</span>; <span class="pl-c">// [ a.x + b.x, a.y + b.y, a.z + b.z, ... ]</span> } <span class="pl-c">//-----------------------------------------------------------------------------</span> <span class="pl-c">// Dot product is the sum of all corresponding components of both vectors multiplied </span> <span class="pl-smi">float</span> <span class="pl-en">dotProduct</span> (<span class="pl-smi">vec</span> <span class="pl-s1">a</span>, <span class="pl-smi">vec</span> <span class="pl-s1">b</span>) { <span class="pl-k">return</span> <span class="pl-s1">a</span> <span class="pl-c1">*</span> <span class="pl-s1">b</span>; <span class="pl-c">// a.x * b.x + a.y * b.y + a.z * b.z + ...</span> } <span class="pl-c">//-----------------------------------------------------------------------------</span> <span class="pl-c">// Get furthest vertex along a certain direction d</span> <span class="pl-c">// This is the same as finding max dot product with d</span> <span class="pl-c">// In 1D direction is always 1 or -1 (to the left or to the right of the Origin)</span> <span class="pl-smi">size_t</span> <span class="pl-en">indexOfFurthestPoint</span> (<span class="pl-k">const</span> <span class="pl-smi">vec</span> <span class="pl-c1">*</span> <span class="pl-s1">vertices</span>, <span class="pl-smi">size_t</span> <span class="pl-s1">count</span>, <span class="pl-smi">vec</span> <span class="pl-s1">d</span>) { <span class="pl-smi">size_t</span> <span class="pl-s1">index</span> <span class="pl-c1">=</span> <span class="pl-c1">0</span>; <span class="pl-smi">float</span> <span class="pl-s1">maxProduct</span> <span class="pl-c1">=</span> <span class="pl-en">dotProduct</span> (<span class="pl-s1">d</span>, <span class="pl-s1">vertices</span>[<span class="pl-s1">index</span>]); <span class="pl-k">for</span> (<span class="pl-smi">size_t</span> <span class="pl-s1">i</span> <span class="pl-c1">=</span> <span class="pl-c1">1</span>; <span class="pl-s1">i</span> <span class="pl-c1"><</span> <span class="pl-s1">count</span>; <span class="pl-s1">i</span><span class="pl-c1">++</span>) { <span class="pl-smi">float</span> <span class="pl-s1">product</span> <span class="pl-c1">=</span> <span class="pl-en">dotProduct</span> (<span class="pl-s1">d</span>, <span class="pl-s1">vertices</span>[<span class="pl-s1">i</span>]); <span class="pl-c">// may be negative</span> <span class="pl-k">if</span> (<span class="pl-s1">product</span> <span class="pl-c1">></span> <span class="pl-s1">maxProduct</span>) { <span class="pl-s1">maxProduct</span> <span class="pl-c1">=</span> <span class="pl-s1">product</span>; <span class="pl-s1">index</span> <span class="pl-c1">=</span> <span class="pl-s1">i</span>; } } <span class="pl-k">return</span> <span class="pl-s1">index</span>; } <span class="pl-c">//-----------------------------------------------------------------------------</span> <span class="pl-c">// Minkowski sum support function for GJK</span> <span class="pl-smi">vec</span> <span class="pl-en">support</span> (<span class="pl-k">const</span> <span class="pl-smi">vec</span> <span class="pl-c1">*</span> <span class="pl-s1">vertices1</span>, <span class="pl-smi">size_t</span> <span class="pl-s1">count1</span>, <span class="pl-c">// first shape</span> <span class="pl-k">const</span> <span class="pl-smi">vec</span> <span class="pl-c1">*</span> <span class="pl-s1">vertices2</span>, <span class="pl-smi">size_t</span> <span class="pl-s1">count2</span>, <span class="pl-c">// second shape</span> <span class="pl-smi">vec</span> <span class="pl-s1">d</span>) { <span class="pl-c">// direction</span> <span class="pl-c">// get furthest point of first body along an arbitrary direction</span> <span class="pl-smi">size_t</span> <span class="pl-s1">i</span> <span class="pl-c1">=</span> <span class="pl-en">indexOfFurthestPoint</span> (<span class="pl-s1">vertices1</span>, <span class="pl-s1">count1</span>, <span class="pl-s1">d</span>); <span class="pl-c">// get furthest point of second body along the opposite direction</span> <span class="pl-c">// note that this time direction is negated</span> <span class="pl-smi">size_t</span> <span class="pl-s1">j</span> <span class="pl-c1">=</span> <span class="pl-en">indexOfFurthestPoint</span> (<span class="pl-s1">vertices2</span>, <span class="pl-s1">count2</span>, <span class="pl-c1">-</span><span class="pl-s1">d</span>); <span class="pl-c">// return the Minkowski sum of two points to see if bodies 'overlap'</span> <span class="pl-c">// note that the second point-vector is negated, a + (-b) = c</span> <span class="pl-k">return</span> <span class="pl-en">sum</span> (<span class="pl-s1">vertices1</span>[<span class="pl-s1">i</span>], <span class="pl-c1">-</span><span class="pl-s1">vertices2</span>[<span class="pl-s1">j</span>]); }</pre></div> <p dir="auto">It does not really care how many dimensions are there in space. So given a direction and two shapes the support function always returns another point regardless of how many dimensions you have. It works the same for 1D, 2D, 3D, etc... The point returned by the support function is always on the contour of intersection. This is because the initial points involved in the intersection are opposite.</p> <p dir="auto">Remember, the whole purpose of having a support function was to help us quickly build the simplex for GJK. The support function is used in the search for a 2-simplex that encloses the Origin in 2D. Now that we have such a function, we can move on and build the rest of the algorithm on top of it.</p> <div class="markdown-heading" dir="auto"><h4 tabindex="-1" class="heading-element" dir="auto">The Evolution</h4><a id="user-content-the-evolution" class="anchor" aria-label="Permalink: The Evolution" href="#the-evolution"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">This is where the actual logic of GJK kicks in. The general plan of GJK is:</p> <ol dir="auto"> <li>Find best points for a simplex with the help of the support function.</li> <li>Check if a simplex of those points encloses the Origin from all sides.</li> <li>If it does, there is a collision – hooray and thanks for the support, support function )</li> <li>If it doesn't, well... try other points, why not?</li> <li>If a simplex cannot be built at all no matter how many times you try – it might be a <em>degenerate case</em> (explained later).</li> <li>If it's not a degenerate case, then there's no collision.</li> </ol> <p dir="auto">All of that is done in a loop. The algorithm tries different points and checks if they satisfy the condition. The whole process is called <em>evolution</em>. If it succeeds to enclose the Origin in another branch of evolution (upon another loop iteration), then the intersection is detected. If not, it tries again until finally it either succeeds or fails.</p> <p dir="auto">Some people might be reasonably worried of the possibility that the evolution of GJK runs out of control and continues on and on without ever stopping. It might even become intelligent some day and who knows what could happen... So they add a limit of iterations into their implementations which is a countdown that cuts power off when the game is over, forcing the algorithm to stop. It's like an emergency halt or a safety button in case something goes wrong with float number precision or whatever. But that is actually not necessary in general.</p> <p dir="auto">In search for a 2-simplex the algorithm has to obtain 3 points one by one to make a triangle. The resulting set of points is initially empty. The process of finding three points is done step-by-step. First, the algorithm finds the first point and adds it to the resulting set. Then it finds and adds the second point, and then it finds and adds the third one. So, while being built, the 2-simplex kinda <em>evolves</em> actually passing through all stages of evolution from a 0-simplex (one point, the first one in the resulting set), through a 1-simplex (segment of two points, with the second point added to the set), to a 2-simplex (three points, with the third point added to the resulting set).</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22043033/dd0fea8a-dd1e-11e6-9ec7-dd2c6103291f.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22043033/dd0fea8a-dd1e-11e6-9ec7-dd2c6103291f.jpg" alt="Simplices in various spaces" title="Simplices in various spaces" style="max-width: 100%;"></a></p> <p dir="auto">A 2-simplex (a triangle) contains or consists of 1-simplices (segments). A 1-simplex (a segment) consists of even simpler 0-simplices (points). It's clear that each new dimension adds one more point to the simplex. So, in general:</p> <div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="0-simplex = nothing + 1 point = 1 point 1-simplex = 0-simplex + 1 point = 2 points 2-simplex = 1-simplex + 1 point = 3 points 3-simplex = 2-simplex + 1 point = 4 points ... = ... + 1 point = ... N-simplex = (N-1)-simplex + 1 point = N+1 points"><pre class="notranslate"><code>0-simplex = nothing + 1 point = 1 point 1-simplex = 0-simplex + 1 point = 2 points 2-simplex = 1-simplex + 1 point = 3 points 3-simplex = 2-simplex + 1 point = 4 points ... = ... + 1 point = ... N-simplex = (N-1)-simplex + 1 point = N+1 points </code></pre></div> <p dir="auto">GJK evolves the simplex from the very beginning each time, restarting the evolution when no further progress is possible in current branch. So a simplex passes through all stages of its evolution upon each iteration of the main loop.</p> <p dir="auto">Let us do a single iteration of the main evolution loop by an example, step-by-step. Below is an illustration of a pair of slightly different intersecting shapes (for example), and their intersection Minkowski sum on the surreal side to the right. The Minkowski sum is unknown to our algorithm at the moment, as we haven't calculated all the differences of points yet and, in fact, we won't need the full resulting shape, we only need best three points to surround the Origin and that's it. So the Minkowski sum is currently unknown and we're drawing it in full on the right just for visual explicacy:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/25145390/bd3d06f0-2479-11e7-91d4-23194591181c.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/25145390/bd3d06f0-2479-11e7-91d4-23194591181c.jpg" alt="GJK evolution iteration example" title="GJK evolution iteration example" style="max-width: 100%;"></a></p> <p dir="auto">To build a 2-simplex in 2D we need three points from our resulting Minkowski set that would enclose the Origin within a triangle and we have our nice support function for finding best points for that. Before we begin the search for three points we reserve a place for each one of them and label or tag each place with capital letters <code>C</code>, <code>B</code> and <code>A</code> in that order. Their naming might be a little confusing, as we've used <code>A</code> and <code>B</code> to denote opposite points previously in this text, but it's an ancient tradition, sorry, can't do much about that. These will be placeholders for the points we might find in the search process. Initially our resulting set of points is empty (all placeholders are empty).</p> <p dir="auto">The first point <code>C</code> is the easiest to obtain. You just choose a random direction <code>D</code> and call the support function which calculates that point for you as we did earlier. Initial direction <code>D</code> is chosen randomly and passed along with the shapes to the support function. The support function then finds opposite points of two shapes along the given direction (labelled below as <code>a</code> and <code>b</code> in lowercase for less confusion) and returns their Minkowski sum which is point <code>C</code> on the right, the first point of our simplex (which is a 0-simplex for now).</p> <p dir="auto">Once you get the first resulting point, you place it in placeholder <code>C</code>. From now on we will be referring to that point by that label or tag <code>C</code>. Don't forget, that <code>C</code> is a point on the difference contour line in Minkowski space. Now we have a 0-simplex (a point) in our resulting set and it is the first stage of current iteration of evolution. Note, that the support function looks in both <code>D</code> and its opposite <code>-D</code> in search for points <code>a</code> and <code>b</code>. This is what we get after executing the first step and obtaining point <code>C</code> in Minkowski space:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/25108220/124100ae-23dd-11e7-9f63-9eb125f8acd7.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/25108220/124100ae-23dd-11e7-9f63-9eb125f8acd7.jpg" alt="Obtaining the first point C (GJK iterative evolution)" title="Obtaining the first point C (GJK iterative evolution)" style="max-width: 100%;"></a></p> <p dir="auto">Once we got the first difference of most distant opposite points along a certain direction-line we need to choose a different direction to find the next point of our simplex in Minkowski resulting set. Remember, that the magic of GJK says that we have to surround the Origin from all sides to detect a collision. As with 1D number line where a segment can surround the Origin with two of its endpoints, in 2D we can surround the Origin from all sides with three <em>endpoints</em> or vertices of a triangle. So our goal is to find such a 2-simplex that would enclose the Origin within itself.</p> <p dir="auto">The logic behind this can be described in the following way: imagine we <em>stand at</em> point <code>C</code> and look towards the Origin from that perspective. We want to find next point that would be <em>beyond</em> the Origin as seen by us. If there's no such point then we are not reaching far enough and we will not be able to surround the Origin with a triangle using current point <code>C</code>.</p> <p dir="auto">If we cannot reach for a point beyond the Origin from current standpoint <code>C</code>, then there's probably some distance between initial shapes (at least those two shapes don't intersect along direction <code>D</code>). So, in that case no collision is detected. The halting criteria of GJK algorithm says that current iteration stops if our simplex fails to include (or surround) the Origin, handling these cases will be explained in more detail later in this text.</p> <p dir="auto">Standing at point <code>C</code> we need to check if there's a point that is further away from us than the Origin is, in the direction from us towards the Origin. The direction from point <code>C</code> towards the Origin <code>O</code> is the direction <code>CO</code> which is the reversed opposite of direction from Origin to point <code>C</code>, so that <code>CO == -OC</code>. Therefore we should be looking in direction <code>CO</code> next as we hope to find some point beyond the Origin there.</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/25163108/87781846-24d0-11e7-93f0-0c82a89c6bb9.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/25163108/87781846-24d0-11e7-93f0-0c82a89c6bb9.jpg" alt="Choosing a direction for 2nd stage of GJK evolution" title="Choosing a direction for 2nd stage of GJK evolution" style="max-width: 100%;"></a></p> <p dir="auto">The idea of <em>looking beyond Origin from your current standpoint</em> is the key principle of the search for simplex in GJK. It helps to quickly find the biggest simplex by taking the most distant opposite points of Minkowski sum. The bigger the simplex the higher the chances that it will surround the Origin upon some early iteration.</p> <p dir="auto">The second point <code>B</code> is quite easy to get. From now on we set new direction <code>D = -OC</code> which is equal to <code>CO</code>, and we will refer to it as our new direction <code>D</code> without the negative sign. The support function finds the other two most distant opposite points <code>a</code> and <code>b</code> and it looks in both directions <code>D</code> and <code>-D</code> (yes, again) in the process. Note, that new direction vector <code>D</code> is different from the first initial direction vector <code>D</code> as it now points in the opposite way.</p> <p dir="auto">You call the support function with your new opposite direction <code>D</code> and get the second point which is labelled or tagged as capital <code>B</code>. And now there's a 1-simplex (a segment of two points, <code>C</code> and <code>B</code>) in the resulting set, and it is the second stage of that same iteration of evolution. Here's what we get after the second step:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/25157552/c0fc12a0-24aa-11e7-88d5-adfbd48d1ee4.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/25157552/c0fc12a0-24aa-11e7-88d5-adfbd48d1ee4.jpg" alt="Obtaining the second point B (GJK iterative evolution)" title="Obtaining the second point B (GJK iterative evolution)" style="max-width: 100%;"></a></p> <p dir="auto">By calling the support function the second time with the opposite direction you get a second point <code>B</code> which is another point on the same contour in Minkowski space. That second point <code>B</code> is exactly opposite to the first point <code>C</code>, because initial directions passed into the support function were opposite, right?</p> <p dir="auto">Having point <code>B</code> we need to verify that it is indeed beyond Origin as seen from point <code>C</code>. In other words, we have to check if point <code>B</code> is further away from point <code>C</code> than the Origin is (in direction from <code>C</code> towards the Origin). If a dot product of <code>CO ⋅ OB</code> is positive, then point <code>B</code> is really beyond the Origin as seen from point <code>C</code> and the evolution continues, otherwise there's no collision and the evolution stops or restarts in a different direction. In geometry the sign of dot product of two vectors basically tells if their orientation is roughly the same.</p> <p dir="auto"><code>if ((CO ⋅ OB) > 0) // test if point B is beyond Origin as seen from point C</code></p> <p dir="auto">This test for <em>a point beyond Origin</em> can also be explained as follows: we know that point <code>C</code> is located on the contour of Minkowski sum (this is a feature of our support function) and point <code>B</code> is on the opposite side of that same contour, so the Origin must be in between <code>C</code> and <code>B</code>, otherwise we're not reaching far enough to surround it, and if we cannot surround it, then there's probably some distance between the two initial shapes, therefore no intersection.</p> <p dir="auto">Remember that the algorithm doesn't know anything about the whole set of Minkowski points, it currently only knows about points <code>C</code> and <code>B</code> that were found during first two stages of the evolution. Now let's take a look at the Minkowski space as seen by GJK algorithm at this point. The left part of the image below shows the 1-simplex of two points (segment <code>CB</code>) in Minkowski space, and on the right we see the same segment with the full set of Minkowski sum points added for visual reference and for explanation purposes (the algorithm only sees the left side of this picture, not the right side):</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/25153389/06b3b44e-2495-11e7-9a2f-f5c87c32c82b.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/25153389/06b3b44e-2495-11e7-9a2f-f5c87c32c82b.jpg" alt="The GJK simplex after completing the 2nd stage of evolution" title="The GJK simplex after completing the 2nd stage of evolution" style="max-width: 100%;"></a></p> <p dir="auto">The algorithm's current view into surreal Minkowski world is on the left side of the image above. There's 1-simplex segment <code>CB</code> of two points. The simplex has evolved from being a single point <code>C</code> 0-simplex into a 1-simplex or a segment of two points <code>CB</code>, in other words, our simplex has become a little more complex and obtained an extra dimension )</p> <p dir="auto">Before we proceed to the third point <code>A</code> we might need to gain even more intuition about what is going on here. Seeing that segment <code>CB</code> on the image above and seeing the whole set of Minkowski sum points on the right side, humans can quickly visually locate the third point to enclose the Origin. Easy. But for a machine to know where to look for the third point, we have to make a critical decision in which direction it should be looking next.</p> <p dir="auto">The logic of GJK goes this way: imagine we <em>stand anywhere on the segment</em> <code>CB</code> and look towards the Origin from that perspective. We want to find such a third point that would be at least <em>beyond</em> the Origin (as seen by us relatively to the segment <code>CB</code> that we're <em>standing on</em>). So, again, if we cannot find a third point beyond the Origin then we will not be able to surround it using current segment <code>CB</code> and there's no collision at least on this iteration.</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/25166261/26cc765a-24e3-11e7-9f6e-87d64be6ca12.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/25166261/26cc765a-24e3-11e7-9f6e-87d64be6ca12.jpg" alt="Choosing a direction for 3rd stage of GJK evolution" title="Choosing a direction for 3rd stage of GJK evolution" style="max-width: 100%;"></a></p> <p dir="auto">The image above shows the direction to look for the last point of triangle simplex. To find the exact direction vector, you just take one of perpendiculars to the segment <code>CB</code>. If you tilt your head to the right a little while looking at the segment <code>CB</code>, you will immediately notice the following simple fact: any segment kinda cuts the coordinate plane into two halves, to the left and to the right of the segment. And the Origin always ends up either on one side of the segment <code>CB</code> or on the other side. All we have to do to find the next direction vector is just take a perpendicular (often called <em>a normal</em>) to <code>CB</code> that points towards the Origin.</p> <p dir="auto">There are two possible directions perpendicular to segment <code>CB</code>, the one pointing towards the Origin and the one pointing the opposite way (away from the Origin). In order to get the one pointing towards the Origin, we can do a simple vector math trick, called <em><a href="https://en.wikipedia.org/wiki/Triple_product#Vector_triple_product" rel="nofollow">vector triple product</a></em>. It works like this: take a cross product of the segment <code>CB</code> with a segment <code>CO</code> (from endpoint to the Origin) and then take a cross product of the result with the segment <code>CB</code> again, basically two cross products done sequentially, involving the same segment <code>CB</code> twice:</p> <p dir="auto"><code>CB ⨯ CO ⨯ CB</code></p> <p dir="auto">There's also a very fast formula for this (an expanded equation) aka <em>triple product expansion</em> with only a few multiplications and subtractions for calculating the resulting perpendicular:</p> <p dir="auto"><code>a ⨯ (b ⨯ c) = b(a ⋅ c) - c(a ⋅ b)</code></p> <p dir="auto">You have to be precise with the orientation of vectors when using the expansion formula, but if done carefully and correctly, this always gives a perpendicular to segment <code>CB</code> pointing towards the Origin.</p> <p dir="auto">We set our final direction <code>D</code> to that perpendicular to <code>CB</code> pointing towards Origin. That will be a direction to search for point A of our triangle simplex. This is also the last stage of the evolution of the 2-simplex. We call our support function and pass new direction <code>D</code> along with the two shapes. The image below shows the result after calling the support function for the 3rd time:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/25314806/523d6236-2853-11e7-8c50-5e35404ff9a6.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/25314806/523d6236-2853-11e7-8c50-5e35404ff9a6.jpg" alt="The GJK simplex after completing the 3rd stage of evolution" title="The GJK simplex after completing the 3rd stage of evolution" style="max-width: 100%;"></a></p> <p dir="auto">We obtained point A and now we have a complete triangle of three points in total. Our simplex has evolved from a 0-simplex point through a 1-simplex segment into a 2-simplex triangle. Also notice, that we did not calculate all points of Minkowski sum, we only calculated three of all possible points. This is one of the reasons why this algorithm actually works very fast in narrow-phase collision detection. It just doesn't have to calculate all points, so it does not know anything about the full Minkowski set. Here's how it sees the triangle with point A added to the simplex in memory (remember, it only knows about the left side of the image below, but not about the right side which is there for explanation purposes):</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/25315541/3a5b5ad6-285f-11e7-8783-3acb05b25606.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/25315541/3a5b5ad6-285f-11e7-8783-3acb05b25606.jpg" alt="The GJK simplex after completing the 3rd stage of evolution" title="The GJK simplex after completing the 3rd stage of evolution" style="max-width: 100%;"></a></p> <p dir="auto">Now that we have all three points for a complete triangle (a full 2-simplex), the only task that's left is to test if the Origin is really contained inside that triangle. In other words, if our resulting simplex surrounds the Origin from all sides, then the whole Minkowski sum also does surround the Origin as well, and it follows that there was a collision between two initial shapes. The most straightforward way to test if a point is inside a triangle (the Origin is a point at zero) is to do a bunch of dot product operations.</p> <p dir="auto">For each edge (each side) of a triangle you need two things: a normal vector (a perpendicular) to that side towards the Origin and a vector from the Origin to opposite vertex of the triangle. Then you check if dot product of the two vectors is positive (greater than zero)...</p> <p dir="auto">WORK IN PROGRESS, to be continued soon. A live demo of GJK in a 2D-space and a video of GJK in action coming up )</p> <p dir="auto">...</p> <div class="markdown-heading" dir="auto"><h5 tabindex="-1" class="heading-element" dir="auto">Voronoi Para Nos</h5><a id="user-content-voronoi-para-nos" class="anchor" aria-label="Permalink: Voronoi Para Nos" href="#voronoi-para-nos"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">WORK IN PROGRESS, to be continued soon... )</p> <p dir="auto">...</p> <div class="markdown-heading" dir="auto"><h5 tabindex="-1" class="heading-element" dir="auto">A Touch Of Degenerate Case</h5><a id="user-content-a-touch-of-degenerate-case" class="anchor" aria-label="Permalink: A Touch Of Degenerate Case" href="#a-touch-of-degenerate-case"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">A collision is when two bodies occupy the same points in space. In two dimensions a collision is either an intersection of two shapes (when shapes kinda "overlap") or they might not intersect but instead one shape could just touch the other, and that is also considered to be a collision. There's even more details to the nature of collisions, because there are different types of touch...</p> <p dir="auto">A full-on collision when a shape overlaps or penetrates another shape usually looks like this:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22180045/d361a334-e075-11e6-8436-b756a6a5cfcb.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22180045/d361a334-e075-11e6-8436-b756a6a5cfcb.jpg" alt="A usual case of overlapping or penetrating collision" title="A usual case of overlapping or penetrating collision" style="max-width: 100%;"></a></p> <p dir="auto">When 2D-shapes overlap there is usually at least one way to build a 2-simplex that encloses the Origin. But there can be other types of non-penetrating collisions without overlapping, when shapes touch edge-to-edge or meet at one single point. In GJK these collisions are usually called <em>degenerate case</em>. They are not collisions but contacts in common sense. By design GJK handles all degenerate cases absolutely fine.</p> <p dir="auto">Here are some examples of what a degenerate case collision (a touch) in GJK is:</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22180044/d36159f6-e075-11e6-869a-06b14c96cedf.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22180044/d36159f6-e075-11e6-869a-06b14c96cedf.jpg" alt="A GJK degenerate case of non-penetrating collision (a touch) in 2D" title="A GJK degenerate case of non-penetrating collision (a touch) in 2D" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22180046/d3625888-e075-11e6-9432-7ff7566eb509.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22180046/d3625888-e075-11e6-9432-7ff7566eb509.jpg" alt="A GJK degenerate case of non-penetrating collision (a touch) in 2D" title="A GJK degenerate case of non-penetrating collision (a touch) in 2D" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22180043/d352cbd4-e075-11e6-95f3-956725d0cb27.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22180043/d352cbd4-e075-11e6-95f3-956725d0cb27.jpg" alt="A GJK degenerate case of non-penetrating collision (a touch) in 2D" title="A GJK degenerate case of non-penetrating collision (a touch) in 2D" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22180042/d33b626e-e075-11e6-97fa-360e24fd0739.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22180042/d33b626e-e075-11e6-97fa-360e24fd0739.jpg" alt="A GJK degenerate case of non-penetrating collision (a touch) in 2D" title="A GJK degenerate case of non-penetrating collision (a touch) in 2D" style="max-width: 100%;"></a></p> <p dir="auto">It may seem like a lot of special cases to handle, but in fact, GJK already does that intrinsically.</p> <p dir="auto">A non-overlapping collision will yield a Minkowski sum that has the Origin on its <em>edge</em> or <em>contour</em>. Remember, in 1D, when two segments have only one common point, the Origin lands on the endpoint of resulting segment. In 2D, when there is only one common point, the Origin in Minkowski space will be located on the contour of your resulting 2D-intersection shape. If two shapes share a common face, then the Origin will be on one of the sides of triange simplex.</p> <p dir="auto">So, the Origin can either be inside the Minkowski sum, or it can be on the edge of the sum. And the trick is to check if the Origin is actually one of the points of your simplex. The Origin can also end up exactly on one of the sides of your simplex, between two points of the simplex.</p> <p dir="auto">Basically, to each stage of evolution you add a check, to see if the Origin is already included in your not-fully-evolved simplex or not. That is enough to detect all degenerate cases. If the Origin is already included in your half-built simplex upon some early stage of evolution, then you don't have to evolve it further. Then you can stop and signal a collision (or no collision, at your discretion). You can differentiate between a penetrating collision or not, by counting degenerate collisions or not counting them as being collisions.</p> <p dir="auto">WORK IN PROGRESS, to be continued soon... )</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">Roundness And Curvature</h3><a id="user-content-roundness-and-curvature" class="anchor" aria-label="Permalink: Roundness And Curvature" href="#roundness-and-curvature"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">What's very remarkable of this algorithm, is that the support function can be also defined for a huge variety of shapes. If every point of a shape can be derived from a simple formula (aka <em>parametric equation</em>) then your support function can use that equation to calculate differences of points. When geometry is defined in a parametric way there is no need to keep all points in memory. And you can also scale everything up and down as you like, do boolean combinations of shapes and much more... That's what vector graphics is about, but it's a whole other topic in itself, we won't dive deep into it here.</p> <p dir="auto">The key point is: a parametric equation can be an equation of a circle <code>((x - h)/r)² + ((y - k)/r)² = 1</code> or an equation of an ellipse <code>(x - h)²/a² + (y - k)²/b² = 1</code>, or some <em>spline</em> (!), or even a combination of polygons and conic sections. So, you can detect collisions and intersections of round shapes very accurately up to an exact point of contact. GJK is not restricted to polygons, you can do perfect circle collisions, and as we'll see later in the 3D section, spherical collisions are also possible as well.</p> <p dir="auto">The only restriction that still holds is that your shapes should be convex, not concave, so, any line crossing your shape should not intersect its contour more than twice. As long as your shapes are fat and bulgy, you'll be fine. But there's a hint: you can always make a concave shape from multiple convex shapes.</p> <p dir="auto">WORK IN PROGRESS, to be continued soon... )</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">Adding 3D</h3><a id="user-content-adding-3d" class="anchor" aria-label="Permalink: Adding 3D" href="#adding-3d"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">A careful reader might have already noticed a pattern in the algorithm...</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://cloud.githubusercontent.com/assets/1294454/22039058/c30ea898-dd0e-11e6-8e15-62a5b612036d.jpg"><img src="https://cloud.githubusercontent.com/assets/1294454/22039058/c30ea898-dd0e-11e6-8e15-62a5b612036d.jpg" alt="Simplices in various spaces" title="Simplices in various spaces" style="max-width: 100%;"></a></p> <p dir="auto">For a 1D number line we need a 1-simplex of two points to enclose the Origin. If we can find such two points then our shapes do intersect indeed. If we cannot find such two points then our initial shapes must have some distance (non-zero difference) between them. For 2D coordinate plane a 2-simplex of three points (a triangle) is necessary to enclose the Origin. In 3D space we need a 3-simplex of four points (a tetrahedron). These are all examples of simplest possible shape primitives in given dimensions.</p> <p dir="auto">WORK IN PROGRESS, A live demo of GJK in a 3D-space and a video of GJK in action coming up soon )</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">Collision Details</h3><a id="user-content-collision-details" class="anchor" aria-label="Permalink: Collision Details" href="#collision-details"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">The version of GJK that gives a boolean answer to a yes/no collision test is somewhat simplified. The algorithm is able to not only detect the fact of intersection, but is also capable of giving back the exact depth of penetration and information about points of contact, so that the collision could be handled properly.</p> <p dir="auto">The simplified yes/no test is often called a <em>bastardized</em> version of GJK algorithm in comparison to its original purpose of calculating detailed collisions. But the simplified version described in the text above is easier for understanding. Having understood the simple yes/no GJK test it is much easier to grasp the Gilbert-Johnson-Keerthi algorithm in its entirety along with a few useful concepts. We will proceed to cover the rest of GJK functionality below.</p> <div class="markdown-heading" dir="auto"><h4 tabindex="-1" class="heading-element" dir="auto">Distance or Depth Of Penetration</h4><a id="user-content-distance-or-depth-of-penetration" class="anchor" aria-label="Permalink: Distance or Depth Of Penetration" href="#distance-or-depth-of-penetration"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">WORK IN PROGRESS, to be continued soon... )</p> <div class="markdown-heading" dir="auto"><h4 tabindex="-1" class="heading-element" dir="auto">Contact Points</h4><a id="user-content-contact-points" class="anchor" aria-label="Permalink: Contact Points" href="#contact-points"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">WORK IN PROGRESS, to be continued soon... )</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">References (must see)</h2><a id="user-content-references-must-see" class="anchor" aria-label="Permalink: References (must see)" href="#references-must-see"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">Most of the info (along with reference implementation) was taken from dyn4j. It has a very thorough explanation of GJK and it is definitely a must visit for those who need to understand the intricacies of the algorithm.</p> <ol dir="auto"> <li><a href="http://www.dyn4j.org/2010/04/gjk-gilbert-johnson-keerthi/" rel="nofollow">http://www.dyn4j.org/2010/04/gjk-gilbert-johnson-keerthi/</a> GJK description (+ a lot of other useful articles)</li> <li><a href="http://mollyrocket.com/849" rel="nofollow">http://mollyrocket.com/849</a> Awesome old-school GJK / Minkowski space video by Casey Muratori</li> <li><a href="https://github.com/wnbittle/dyn4j">https://github.com/wnbittle/dyn4j</a> Quality source code for reference</li> <li><a href="https://www.youtube.com/watch?v=-GZOGJ26yxE" rel="nofollow">https://www.youtube.com/watch?v=-GZOGJ26yxE</a> – A video of a seminar by <a href="https://aero.engin.umich.edu/people/elmer-gilbert/" rel="nofollow">Elmer Gilbert Professor Emeritus Aerospace Engineering, University of Michigan</a></li> </ol> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">P.S. 3D-version coming soon...</h2><a id="user-content-ps-3d-version-coming-soon" class="anchor" aria-label="Permalink: P.S. 3D-version coming soon..." href="#ps-3d-version-coming-soon"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://camo.githubusercontent.com/87ee43169ef2f48415d9721bab7160259b2f9d6ae311983bdc413cfd973acc85/687474703a2f2f7332312e706f7374696d672e6f72672f6461397478633375762f53637265656e5f53686f745f323031365f30315f31335f61745f30395f31335f31322e6a7067"><img src="https://camo.githubusercontent.com/87ee43169ef2f48415d9721bab7160259b2f9d6ae311983bdc413cfd973acc85/687474703a2f2f7332312e706f7374696d672e6f72672f6461397478633375762f53637265656e5f53686f745f323031365f30315f31335f61745f30395f31335f31322e6a7067" alt="3D-version of GJK in plain C coming soon..." title="3D-version of GJK in plain C coming soon..." data-canonical-src="http://s21.postimg.org/da9txc3uv/Screen_Shot_2016_01_13_at_09_13_12.jpg" style="max-width: 100%;"></a></p> </article></div></div></div></div></div> <!-- --> <!-- --> <script type="application/json" id="__PRIMER_DATA_:R0:__">{"resolvedServerColorMode":"day"}</script></div> </react-partial> <input type="hidden" data-csrf="true" value="Ni+rHFxsL50pjmBsU2qzFUAHZM31eXF+GperxnQYnp6OzCv57Wf47KUcyjw2P9SPJKcrOxWGTcXlfad7aywoQA==" /> </div> <div data-view-component="true" class="Layout-sidebar"> <div class="BorderGrid about-margin" data-pjax> <div class="BorderGrid-row"> <div class="BorderGrid-cell"> <div class="hide-sm hide-md"> <h2 class="mb-3 h4">About</h2> <p class="f4 my-3"> Gilbert-Johnson-Keerthi (GJK) collision detection algorithm in 200 lines of clean plain C </p> <h3 class="sr-only">Topics</h3> <div class="my-3"> <div class="f6"> <a href="/topics/physics" title="Topic: physics" data-view-component="true" class="topic-tag topic-tag-link"> physics </a> <a href="/topics/simplex" title="Topic: simplex" data-view-component="true" class="topic-tag topic-tag-link"> simplex </a> <a href="/topics/collision-detection" title="Topic: collision-detection" data-view-component="true" class="topic-tag topic-tag-link"> collision-detection </a> <a href="/topics/gjk" title="Topic: gjk" data-view-component="true" class="topic-tag topic-tag-link"> gjk </a> <a href="/topics/minkowski-space" title="Topic: minkowski-space" data-view-component="true" class="topic-tag topic-tag-link"> minkowski-space </a> </div> </div> <h3 class="sr-only">Resources</h3> <div class="mt-2"> <a class="Link--muted" data-analytics-event="{"category":"Repository Overview","action":"click","label":"location:sidebar;file:readme"}" href="#readme-ov-file"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-book mr-2"> <path d="M0 1.75A.75.75 0 0 1 .75 1h4.253c1.227 0 2.317.59 3 1.501A3.743 3.743 0 0 1 11.006 1h4.245a.75.75 0 0 1 .75.75v10.5a.75.75 0 0 1-.75.75h-4.507a2.25 2.25 0 0 0-1.591.659l-.622.621a.75.75 0 0 1-1.06 0l-.622-.621A2.25 2.25 0 0 0 5.258 13H.75a.75.75 0 0 1-.75-.75Zm7.251 10.324.004-5.073-.002-2.253A2.25 2.25 0 0 0 5.003 2.5H1.5v9h3.757a3.75 3.75 0 0 1 1.994.574ZM8.755 4.75l-.004 7.322a3.752 3.752 0 0 1 1.992-.572H14.5v-9h-3.495a2.25 2.25 0 0 0-2.25 2.25Z"></path> </svg> Readme </a> </div> <h3 class="sr-only">License</h3> <div class="mt-2"> <a href="#WTFPL-1-ov-file" class="Link--muted" data-analytics-event="{"category":"Repository Overview","action":"click","label":"location:sidebar;file:license"}" > <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-law mr-2"> <path d="M8.75.75V2h.985c.304 0 .603.08.867.231l1.29.736c.038.022.08.033.124.033h2.234a.75.75 0 0 1 0 1.5h-.427l2.111 4.692a.75.75 0 0 1-.154.838l-.53-.53.529.531-.001.002-.002.002-.006.006-.006.005-.01.01-.045.04c-.21.176-.441.327-.686.45C14.556 10.78 13.88 11 13 11a4.498 4.498 0 0 1-2.023-.454 3.544 3.544 0 0 1-.686-.45l-.045-.04-.016-.015-.006-.006-.004-.004v-.001a.75.75 0 0 1-.154-.838L12.178 4.5h-.162c-.305 0-.604-.079-.868-.231l-1.29-.736a.245.245 0 0 0-.124-.033H8.75V13h2.5a.75.75 0 0 1 0 1.5h-6.5a.75.75 0 0 1 0-1.5h2.5V3.5h-.984a.245.245 0 0 0-.124.033l-1.289.737c-.265.15-.564.23-.869.23h-.162l2.112 4.692a.75.75 0 0 1-.154.838l-.53-.53.529.531-.001.002-.002.002-.006.006-.016.015-.045.04c-.21.176-.441.327-.686.45C4.556 10.78 3.88 11 3 11a4.498 4.498 0 0 1-2.023-.454 3.544 3.544 0 0 1-.686-.45l-.045-.04-.016-.015-.006-.006-.004-.004v-.001a.75.75 0 0 1-.154-.838L2.178 4.5H1.75a.75.75 0 0 1 0-1.5h2.234a.249.249 0 0 0 .125-.033l1.288-.737c.265-.15.564-.23.869-.23h.984V.75a.75.75 0 0 1 1.5 0Zm2.945 8.477c.285.135.718.273 1.305.273s1.02-.138 1.305-.273L13 6.327Zm-10 0c.285.135.718.273 1.305.273s1.02-.138 1.305-.273L3 6.327Z"></path> </svg> WTFPL license </a> </div> <include-fragment src="/kroitor/gjk.c/hovercards/citation/sidebar_partial?tree_name=master"> </include-fragment> <div class="mt-2"> <a href="/kroitor/gjk.c/activity" data-view-component="true" class="Link Link--muted"><svg text="gray" aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-pulse mr-2"> <path d="M6 2c.306 0 .582.187.696.471L10 10.731l1.304-3.26A.751.751 0 0 1 12 7h3.25a.75.75 0 0 1 0 1.5h-2.742l-1.812 4.528a.751.751 0 0 1-1.392 0L6 4.77 4.696 8.03A.75.75 0 0 1 4 8.5H.75a.75.75 0 0 1 0-1.5h2.742l1.812-4.529A.751.751 0 0 1 6 2Z"></path> </svg> <span class="color-fg-muted">Activity</span></a> </div> <h3 class="sr-only">Stars</h3> <div class="mt-2"> <a href="/kroitor/gjk.c/stargazers" data-view-component="true" class="Link Link--muted"><svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-star mr-2"> <path d="M8 .25a.75.75 0 0 1 .673.418l1.882 3.815 4.21.612a.75.75 0 0 1 .416 1.279l-3.046 2.97.719 4.192a.751.751 0 0 1-1.088.791L8 12.347l-3.766 1.98a.75.75 0 0 1-1.088-.79l.72-4.194L.818 6.374a.75.75 0 0 1 .416-1.28l4.21-.611L7.327.668A.75.75 0 0 1 8 .25Zm0 2.445L6.615 5.5a.75.75 0 0 1-.564.41l-3.097.45 2.24 2.184a.75.75 0 0 1 .216.664l-.528 3.084 2.769-1.456a.75.75 0 0 1 .698 0l2.77 1.456-.53-3.084a.75.75 0 0 1 .216-.664l2.24-2.183-3.096-.45a.75.75 0 0 1-.564-.41L8 2.694Z"></path> </svg> <strong>869</strong> stars</a> </div> <h3 class="sr-only">Watchers</h3> <div class="mt-2"> <a href="/kroitor/gjk.c/watchers" data-view-component="true" class="Link Link--muted"><svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-eye mr-2"> <path d="M8 2c1.981 0 3.671.992 4.933 2.078 1.27 1.091 2.187 2.345 2.637 3.023a1.62 1.62 0 0 1 0 1.798c-.45.678-1.367 1.932-2.637 3.023C11.67 13.008 9.981 14 8 14c-1.981 0-3.671-.992-4.933-2.078C1.797 10.83.88 9.576.43 8.898a1.62 1.62 0 0 1 0-1.798c.45-.677 1.367-1.931 2.637-3.022C4.33 2.992 6.019 2 8 2ZM1.679 7.932a.12.12 0 0 0 0 .136c.411.622 1.241 1.75 2.366 2.717C5.176 11.758 6.527 12.5 8 12.5c1.473 0 2.825-.742 3.955-1.715 1.124-.967 1.954-2.096 2.366-2.717a.12.12 0 0 0 0-.136c-.412-.621-1.242-1.75-2.366-2.717C10.824 4.242 9.473 3.5 8 3.5c-1.473 0-2.825.742-3.955 1.715-1.124.967-1.954 2.096-2.366 2.717ZM8 10a2 2 0 1 1-.001-3.999A2 2 0 0 1 8 10Z"></path> </svg> <strong>33</strong> watching</a> </div> <h3 class="sr-only">Forks</h3> <div class="mt-2"> <a href="/kroitor/gjk.c/forks" data-view-component="true" class="Link Link--muted"><svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-repo-forked mr-2"> <path d="M5 5.372v.878c0 .414.336.75.75.75h4.5a.75.75 0 0 0 .75-.75v-.878a2.25 2.25 0 1 1 1.5 0v.878a2.25 2.25 0 0 1-2.25 2.25h-1.5v2.128a2.251 2.251 0 1 1-1.5 0V8.5h-1.5A2.25 2.25 0 0 1 3.5 6.25v-.878a2.25 2.25 0 1 1 1.5 0ZM5 3.25a.75.75 0 1 0-1.5 0 .75.75 0 0 0 1.5 0Zm6.75.75a.75.75 0 1 0 0-1.5.75.75 0 0 0 0 1.5Zm-3 8.75a.75.75 0 1 0-1.5 0 .75.75 0 0 0 1.5 0Z"></path> </svg> <strong>83</strong> forks</a> </div> <div class="mt-2"> <a class="Link--muted" href="/contact/report-content?content_url=https%3A%2F%2Fgithub.com%2Fkroitor%2Fgjk.c&report=kroitor+%28user%29"> Report repository </a> </div> </div> </div> </div> <div class="BorderGrid-row"> <div class="BorderGrid-cell"> <h2 class="h4 mb-3" data-pjax="#repo-content-pjax-container" data-turbo-frame="repo-content-turbo-frame"> <a href="/kroitor/gjk.c/releases" data-view-component="true" class="Link--primary no-underline Link">Releases</a></h2> <div class="text-small color-fg-muted">No releases published</div> </div> </div> <div class="BorderGrid-row"> <div class="BorderGrid-cell"> <h2 class="h4 mb-3"> <a href="/users/kroitor/packages?repo_name=gjk.c" data-view-component="true" class="Link--primary no-underline Link d-flex flex-items-center">Packages <span title="0" hidden="hidden" data-view-component="true" class="Counter ml-1">0</span></a></h2> <div class="text-small color-fg-muted" > No packages published <br> </div> </div> </div> <div class="BorderGrid-row" hidden> <div class="BorderGrid-cell"> <include-fragment src="/kroitor/gjk.c/used_by_list" accept="text/fragment+html"> </include-fragment> </div> </div> <div class="BorderGrid-row"> <div class="BorderGrid-cell"> <h2 class="h4 mb-3"> <a href="/kroitor/gjk.c/graphs/contributors" data-view-component="true" class="Link--primary no-underline Link d-flex flex-items-center">Contributors <span title="6" data-view-component="true" class="Counter ml-1">6</span></a></h2> <include-fragment src="/kroitor/gjk.c/contributors_list?count=6&current_repository=gjk.c&items_to_show=6" aria-busy="true" aria-label="Loading contributors"> <ul class="list-style-none d-flex flex-wrap mb-n2"> <li class="mb-2 "> <div class="Skeleton avatar avatar-user mr-2" style="width:32px;height:32px;"></div> </li> <li class="mb-2 "> <div class="Skeleton avatar avatar-user mr-2" style="width:32px;height:32px;"></div> </li> <li class="mb-2 "> <div class="Skeleton avatar avatar-user mr-2" style="width:32px;height:32px;"></div> </li> <li class="mb-2 "> <div class="Skeleton avatar avatar-user mr-2" style="width:32px;height:32px;"></div> </li> <li class="mb-2 "> <div class="Skeleton avatar avatar-user mr-2" style="width:32px;height:32px;"></div> </li> <li class="mb-2 "> <div class="Skeleton avatar avatar-user mr-2" style="width:32px;height:32px;"></div> </li> </ul> </include-fragment> </div> </div> <div class="BorderGrid-row"> <div class="BorderGrid-cell"> <h2 class="h4 mb-3">Languages</h2> <div class="mb-2"> <span data-view-component="true" class="Progress"> <span style="background-color:#555555 !important;;width: 86.1%;" itemprop="keywords" data-view-component="true" class="Progress-item color-bg-success-emphasis"></span> <span style="background-color:#3572A5 !important;;width: 13.9%;" itemprop="keywords" data-view-component="true" class="Progress-item color-bg-success-emphasis"></span> </span></div> <ul class="list-style-none"> <li class="d-inline"> <a class="d-inline-flex flex-items-center flex-nowrap Link--secondary no-underline text-small mr-3" href="/kroitor/gjk.c/search?l=c" data-ga-click="Repository, language stats search click, location:repo overview"> <svg style="color:#555555;" aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-dot-fill mr-2"> <path d="M8 4a4 4 0 1 1 0 8 4 4 0 0 1 0-8Z"></path> </svg> <span class="color-fg-default text-bold mr-1">C</span> <span>86.1%</span> </a> </li> <li class="d-inline"> <a class="d-inline-flex flex-items-center flex-nowrap Link--secondary no-underline text-small mr-3" href="/kroitor/gjk.c/search?l=python" data-ga-click="Repository, language stats search click, location:repo overview"> <svg style="color:#3572A5;" aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-dot-fill mr-2"> <path d="M8 4a4 4 0 1 1 0 8 4 4 0 0 1 0-8Z"></path> </svg> <span class="color-fg-default text-bold mr-1">Python</span> <span>13.9%</span> </a> </li> </ul> </div> </div> </div> </div> </div></div> </div> </div> </turbo-frame> </main> </div> </div> <footer class="footer pt-8 pb-6 f6 color-fg-muted p-responsive" role="contentinfo" > <h2 class='sr-only'>Footer</h2> <div class="d-flex flex-justify-center flex-items-center flex-column-reverse flex-lg-row flex-wrap flex-lg-nowrap"> <div class="d-flex flex-items-center flex-shrink-0 mx-2"> <a aria-label="Homepage" title="GitHub" class="footer-octicon mr-2" href="https://github.com"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-mark-github"> <path d="M12 1C5.9225 1 1 5.9225 1 12C1 16.8675 4.14875 20.9787 8.52125 22.4362C9.07125 22.5325 9.2775 22.2025 9.2775 21.9137C9.2775 21.6525 9.26375 20.7862 9.26375 19.865C6.5 20.3737 5.785 19.1912 5.565 18.5725C5.44125 18.2562 4.905 17.28 4.4375 17.0187C4.0525 16.8125 3.5025 16.3037 4.42375 16.29C5.29 16.2762 5.90875 17.0875 6.115 17.4175C7.105 19.0812 8.68625 18.6137 9.31875 18.325C9.415 17.61 9.70375 17.1287 10.02 16.8537C7.5725 16.5787 5.015 15.63 5.015 11.4225C5.015 10.2262 5.44125 9.23625 6.1425 8.46625C6.0325 8.19125 5.6475 7.06375 6.2525 5.55125C6.2525 5.55125 7.17375 5.2625 9.2775 6.67875C10.1575 6.43125 11.0925 6.3075 12.0275 6.3075C12.9625 6.3075 13.8975 6.43125 14.7775 6.67875C16.8813 5.24875 17.8025 5.55125 17.8025 5.55125C18.4075 7.06375 18.0225 8.19125 17.9125 8.46625C18.6138 9.23625 19.04 10.2125 19.04 11.4225C19.04 15.6437 16.4688 16.5787 14.0213 16.8537C14.42 17.1975 14.7638 17.8575 14.7638 18.8887C14.7638 20.36 14.75 21.5425 14.75 21.9137C14.75 22.2025 14.9563 22.5462 15.5063 22.4362C19.8513 20.9787 23 16.8537 23 12C23 5.9225 18.0775 1 12 1Z"></path> </svg> </a> <span> © 2025 GitHub, Inc. </span> </div> <nav aria-label="Footer"> <h3 class="sr-only" id="sr-footer-heading">Footer navigation</h3> <ul class="list-style-none d-flex flex-justify-center flex-wrap mb-2 mb-lg-0" aria-labelledby="sr-footer-heading"> <li class="mx-2"> <a data-analytics-event="{"category":"Footer","action":"go to Terms","label":"text:terms"}" href="https://docs.github.com/site-policy/github-terms/github-terms-of-service" data-view-component="true" class="Link--secondary Link">Terms</a> </li> <li class="mx-2"> <a data-analytics-event="{"category":"Footer","action":"go to privacy","label":"text:privacy"}" href="https://docs.github.com/site-policy/privacy-policies/github-privacy-statement" data-view-component="true" class="Link--secondary Link">Privacy</a> </li> <li class="mx-2"> <a data-analytics-event="{"category":"Footer","action":"go to security","label":"text:security"}" href="https://github.com/security" data-view-component="true" class="Link--secondary Link">Security</a> </li> <li class="mx-2"> <a data-analytics-event="{"category":"Footer","action":"go to status","label":"text:status"}" href="https://www.githubstatus.com/" data-view-component="true" class="Link--secondary Link">Status</a> </li> <li class="mx-2"> <a data-analytics-event="{"category":"Footer","action":"go to docs","label":"text:docs"}" href="https://docs.github.com/" data-view-component="true" class="Link--secondary Link">Docs</a> </li> <li class="mx-2"> <a data-analytics-event="{"category":"Footer","action":"go to contact","label":"text:contact"}" href="https://support.github.com?tags=dotcom-footer" data-view-component="true" class="Link--secondary Link">Contact</a> </li> <li class="mx-2" > <cookie-consent-link> <button type="button" class="Link--secondary underline-on-hover border-0 p-0 color-bg-transparent" data-action="click:cookie-consent-link#showConsentManagement" data-analytics-event="{"location":"footer","action":"cookies","context":"subfooter","tag":"link","label":"cookies_link_subfooter_footer"}" > Manage cookies </button> </cookie-consent-link> </li> <li class="mx-2"> <cookie-consent-link> <button type="button" class="Link--secondary underline-on-hover border-0 p-0 color-bg-transparent" data-action="click:cookie-consent-link#showConsentManagement" data-analytics-event="{"location":"footer","action":"dont_share_info","context":"subfooter","tag":"link","label":"dont_share_info_link_subfooter_footer"}" > Do not share my personal information </button> </cookie-consent-link> </li> </ul> </nav> </div> </footer> <ghcc-consent id="ghcc" class="position-fixed bottom-0 left-0" style="z-index: 999999" data-initial-cookie-consent-allowed="" data-cookie-consent-required="false"></ghcc-consent> <div id="ajax-error-message" class="ajax-error-message flash flash-error" hidden> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-alert"> <path d="M6.457 1.047c.659-1.234 2.427-1.234 3.086 0l6.082 11.378A1.75 1.75 0 0 1 14.082 15H1.918a1.75 1.75 0 0 1-1.543-2.575Zm1.763.707a.25.25 0 0 0-.44 0L1.698 13.132a.25.25 0 0 0 .22.368h12.164a.25.25 0 0 0 .22-.368Zm.53 3.996v2.5a.75.75 0 0 1-1.5 0v-2.5a.75.75 0 0 1 1.5 0ZM9 11a1 1 0 1 1-2 0 1 1 0 0 1 2 0Z"></path> </svg> <button type="button" class="flash-close js-ajax-error-dismiss" aria-label="Dismiss error"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-x"> <path d="M3.72 3.72a.75.75 0 0 1 1.06 0L8 6.94l3.22-3.22a.749.749 0 0 1 1.275.326.749.749 0 0 1-.215.734L9.06 8l3.22 3.22a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215L8 9.06l-3.22 3.22a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042L6.94 8 3.72 4.78a.75.75 0 0 1 0-1.06Z"></path> </svg> </button> You can’t perform that action at this time. </div> <template id="site-details-dialog"> <details class="details-reset details-overlay details-overlay-dark lh-default color-fg-default hx_rsm" open> <summary role="button" aria-label="Close dialog"></summary> <details-dialog class="Box Box--overlay d-flex flex-column anim-fade-in fast hx_rsm-dialog hx_rsm-modal"> <button class="Box-btn-octicon m-0 btn-octicon position-absolute right-0 top-0" type="button" aria-label="Close dialog" data-close-dialog> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-x"> <path d="M3.72 3.72a.75.75 0 0 1 1.06 0L8 6.94l3.22-3.22a.749.749 0 0 1 1.275.326.749.749 0 0 1-.215.734L9.06 8l3.22 3.22a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215L8 9.06l-3.22 3.22a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042L6.94 8 3.72 4.78a.75.75 0 0 1 0-1.06Z"></path> </svg> </button> <div class="octocat-spinner my-6 js-details-dialog-spinner"></div> </details-dialog> </details> </template> <div class="Popover js-hovercard-content position-absolute" style="display: none; outline: none;"> <div class="Popover-message Popover-message--bottom-left Popover-message--large Box color-shadow-large" style="width:360px;"> </div> </div> <template id="snippet-clipboard-copy-button"> <div class="zeroclipboard-container position-absolute right-0 top-0"> <clipboard-copy aria-label="Copy" class="ClipboardButton btn js-clipboard-copy m-2 p-0" data-copy-feedback="Copied!" data-tooltip-direction="w"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-copy js-clipboard-copy-icon m-2"> <path d="M0 6.75C0 5.784.784 5 1.75 5h1.5a.75.75 0 0 1 0 1.5h-1.5a.25.25 0 0 0-.25.25v7.5c0 .138.112.25.25.25h7.5a.25.25 0 0 0 .25-.25v-1.5a.75.75 0 0 1 1.5 0v1.5A1.75 1.75 0 0 1 9.25 16h-7.5A1.75 1.75 0 0 1 0 14.25Z"></path><path d="M5 1.75C5 .784 5.784 0 6.75 0h7.5C15.216 0 16 .784 16 1.75v7.5A1.75 1.75 0 0 1 14.25 11h-7.5A1.75 1.75 0 0 1 5 9.25Zm1.75-.25a.25.25 0 0 0-.25.25v7.5c0 .138.112.25.25.25h7.5a.25.25 0 0 0 .25-.25v-7.5a.25.25 0 0 0-.25-.25Z"></path> </svg> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-check js-clipboard-check-icon color-fg-success d-none m-2"> <path d="M13.78 4.22a.75.75 0 0 1 0 1.06l-7.25 7.25a.75.75 0 0 1-1.06 0L2.22 9.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018L6 10.94l6.72-6.72a.75.75 0 0 1 1.06 0Z"></path> </svg> </clipboard-copy> </div> </template> <template id="snippet-clipboard-copy-button-unpositioned"> <div class="zeroclipboard-container"> <clipboard-copy aria-label="Copy" class="ClipboardButton btn btn-invisible js-clipboard-copy m-2 p-0 d-flex flex-justify-center flex-items-center" data-copy-feedback="Copied!" data-tooltip-direction="w"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-copy js-clipboard-copy-icon"> <path d="M0 6.75C0 5.784.784 5 1.75 5h1.5a.75.75 0 0 1 0 1.5h-1.5a.25.25 0 0 0-.25.25v7.5c0 .138.112.25.25.25h7.5a.25.25 0 0 0 .25-.25v-1.5a.75.75 0 0 1 1.5 0v1.5A1.75 1.75 0 0 1 9.25 16h-7.5A1.75 1.75 0 0 1 0 14.25Z"></path><path d="M5 1.75C5 .784 5.784 0 6.75 0h7.5C15.216 0 16 .784 16 1.75v7.5A1.75 1.75 0 0 1 14.25 11h-7.5A1.75 1.75 0 0 1 5 9.25Zm1.75-.25a.25.25 0 0 0-.25.25v7.5c0 .138.112.25.25.25h7.5a.25.25 0 0 0 .25-.25v-7.5a.25.25 0 0 0-.25-.25Z"></path> </svg> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-check js-clipboard-check-icon color-fg-success d-none"> <path d="M13.78 4.22a.75.75 0 0 1 0 1.06l-7.25 7.25a.75.75 0 0 1-1.06 0L2.22 9.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018L6 10.94l6.72-6.72a.75.75 0 0 1 1.06 0Z"></path> </svg> </clipboard-copy> </div> </template> </div> <div id="js-global-screen-reader-notice" class="sr-only mt-n1" aria-live="polite" aria-atomic="true" ></div> <div id="js-global-screen-reader-notice-assertive" class="sr-only mt-n1" aria-live="assertive" aria-atomic="true"></div> </body> </html>