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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 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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 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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" 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dir=\"auto\"\u003eMathUtilities\u003c/h1\u003e\u003ca id=\"user-content-mathutilities\" class=\"anchor\" aria-label=\"Permalink: MathUtilities\" href=\"#mathutilities\"\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 grab bag of some of the neat math and physics tricks that I've amassed over the last few years, implemented in Unity's C#. You're free to use the code here however you like.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Deform\"\u003eGeneralized Mesh Deformation\u003c/a\u003e\u003c/h2\u003e\u003ca id=\"user-content-generalized-mesh-deformation\" class=\"anchor\" aria-label=\"Permalink: Generalized Mesh Deformation\" href=\"#generalized-mesh-deformation\"\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/367e009ee8cf1465fa69441036197120803e4330df04b346f7dbb82f917e0342/68747470733a2f2f692e696d6775722e636f6d2f387162387335742e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/367e009ee8cf1465fa69441036197120803e4330df04b346f7dbb82f917e0342/68747470733a2f2f692e696d6775722e636f6d2f387162387335742e676966\" data-animated-image=\"\" data-canonical-src=\"https://i.imgur.com/8qb8s5t.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eAn extremely fast, general mesh-deformation algorithm with arbitrary point-placement and configurable rigidity. Has the desirable property that it acts like a rigid Kabsch when the \"weight\" (ductility) is set to 0, but smoothly blends in deformation as the weight is increased.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe implementation is similar to Linear Blend Skinning, but the skinning weights are automatically calculated with \u003ca href=\"https://en.wikipedia.org/wiki/Inverse_distance_weighting\" rel=\"nofollow\"\u003eInverse Distance Weighting\u003c/a\u003e (in cartesian \u003cem\u003eor\u003c/em\u003e surface space), and the \"bone rotations\" are calculated to optimally preserve the angular relationships between the control points.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Volume\"\u003eSigned Distance Field Texture Rendering\u003c/a\u003e\u003c/h2\u003e\u003ca id=\"user-content-signed-distance-field-texture-rendering\" class=\"anchor\" aria-label=\"Permalink: Signed Distance Field Texture Rendering\" href=\"#signed-distance-field-texture-rendering\"\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/45803628a93bc164882c50407356a88f89b1a1a95efa32a85ab1c969c02005bb/68747470733a2f2f692e696d6775722e636f6d2f6b374a3049376b2e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/45803628a93bc164882c50407356a88f89b1a1a95efa32a85ab1c969c02005bb/68747470733a2f2f692e696d6775722e636f6d2f6b374a3049376b2e676966\" data-animated-image=\"\" data-canonical-src=\"https://i.imgur.com/k7J0I7k.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eA simple example for raymarching and painting/blitting-to volumetric distance field textures. Distance Fields are regions of space that store the distance to the nearest surface at each point in space. This property allows them to represent and render solid shapes of arbitrary geometry and topology.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThis particular implementation also stores the normalized vector toward the nearest point (the normal or gradient of the field). This is useful for physics queries and lighting (without taking the numerical derivative).\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/blob/master/Assets/Kabsch/Kabsch.cs\"\u003eKabsch\u003c/a\u003e\u003c/h2\u003e\u003ca id=\"user-content-kabsch\" class=\"anchor\" aria-label=\"Permalink: Kabsch\" href=\"#kabsch\"\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/d5a59fbc066d390c40ad2c171963a0e846d05b7c89c005fe5d05f3bbdde8ee60/687474703a2f2f692e696d6775722e636f6d2f327168526d744e2e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/d5a59fbc066d390c40ad2c171963a0e846d05b7c89c005fe5d05f3bbdde8ee60/687474703a2f2f692e696d6775722e636f6d2f327168526d744e2e676966\" data-animated-image=\"\" data-canonical-src=\"http://i.imgur.com/2qhRmtN.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eAlso known as Procrustes Analysis, this algorithm can take in an arbitrary set of point-pairs and find the globally optimal rigid translation and rotation to minimize the distance between those point pairs. Incredibly useful, and very cheap. Uses Matthias Muller's polar decomposition solver in place of SVD, as outlined here: \u003ca href=\"https://animation.rwth-aachen.de/media/papers/2016-MIG-StableRotation.pdf\" rel=\"nofollow\"\u003ehttps://animation.rwth-aachen.de/media/papers/2016-MIG-StableRotation.pdf\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eUpdate: Added an example for averaging arbitrary numbers of quaternions; possibly more accurate than a normalized lerp (averaging the quaternion components in linear space and then normalizing).\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/blob/master/Assets/LeastSquares/LeastSquaresFitting.cs#L24\"\u003eLeast Squares Line/Plane Fitting\u003c/a\u003e \u003ca href=\"https://i.imgur.com/xTGKnx5\" rel=\"nofollow\"\u003e[Pic]\u003c/a\u003e\u003c/h2\u003e\u003ca id=\"user-content-least-squares-lineplane-fitting-pic\" class=\"anchor\" aria-label=\"Permalink: Least Squares Line/Plane Fitting[Pic]\" href=\"#least-squares-lineplane-fitting-pic\"\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/a6a540aac9d43d25bf762f9e6e1a1194c25ab27d661c7e3a328888752834e8ad/68747470733a2f2f692e696d6775722e636f6d2f4e4274596949712e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/a6a540aac9d43d25bf762f9e6e1a1194c25ab27d661c7e3a328888752834e8ad/68747470733a2f2f692e696d6775722e636f6d2f4e4274596949712e676966\" data-animated-image=\"\" data-canonical-src=\"https://i.imgur.com/NBtYiIq.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eA generic helper utility for fitting lines and planes to point-sets in 3D. Uses a novel* matrix-less formulation to solve for the orthogonal lines and planes of best fit; algorithm is without singularities and is extensible to arbitrary dimensionality.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e*(as far as I know; I have not seen anyone attempt orthogonal regression without using an SVD (which may be unnecessarily expensive for just the line/plane of best fit))\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/StereographicCamera\"\u003eStereographic/Fisheye Camera\u003c/a\u003e \u003ca href=\"https://en.wikipedia.org/wiki/Stereographic_projection\" rel=\"nofollow\"\u003e[Wiki]\u003c/a\u003e\u003c/h2\u003e\u003ca id=\"user-content-stereographicfisheye-camera-wiki\" class=\"anchor\" aria-label=\"Permalink: Stereographic/Fisheye Camera[Wiki]\" href=\"#stereographicfisheye-camera-wiki\"\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/97ce710369ab40336c0c6ac10243d3fd8e90e78faa8e5a72da1e7bbb8553a419/687474703a2f2f692e696d6775722e636f6d2f4d4f36524c5a712e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/97ce710369ab40336c0c6ac10243d3fd8e90e78faa8e5a72da1e7bbb8553a419/687474703a2f2f692e696d6775722e636f6d2f4d4f36524c5a712e676966\" data-animated-image=\"\" data-canonical-src=\"http://i.imgur.com/MO6RLZq.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eA prefab that concatenates and warps the images from four cameras into one 180 degree fisheye view, projected stereographically.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Verlet\"\u003eVerlet Softbody, Rigidbody, and Chain\u003c/a\u003e\u003c/h2\u003e\u003ca id=\"user-content-verlet-softbody-rigidbody-and-chain\" class=\"anchor\" aria-label=\"Permalink: Verlet Softbody, Rigidbody, and Chain\" href=\"#verlet-softbody-rigidbody-and-chain\"\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/e4da7350b7f43d4ebc01354418183b6ba027c9b77dc58d9068fd0a0e336e0fb3/687474703a2f2f692e696d6775722e636f6d2f79316a59417a772e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/e4da7350b7f43d4ebc01354418183b6ba027c9b77dc58d9068fd0a0e336e0fb3/687474703a2f2f692e696d6775722e636f6d2f79316a59417a772e676966\" data-animated-image=\"\" data-canonical-src=\"http://i.imgur.com/y1jYAzw.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e \u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://camo.githubusercontent.com/7a5107713f2d1cf2bf5d4d09add67947d18514608a42ec5a2f0034cbdfa16a5e/687474703a2f2f692e696d6775722e636f6d2f786c41686b4c342e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/7a5107713f2d1cf2bf5d4d09add67947d18514608a42ec5a2f0034cbdfa16a5e/687474703a2f2f692e696d6775722e636f6d2f786c41686b4c342e676966\" data-animated-image=\"\" data-canonical-src=\"http://i.imgur.com/xlAhkL4.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNumerous examples of using verlet integration (\u003ca href=\"http://matthias-mueller-fischer.ch/publications/posBasedDyn.pdf\" rel=\"nofollow\"\u003ea subset of Position Based Dynamics\u003c/a\u003e) to simulate soft/rigid bodies, cloth, and particle linkage chains.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/blob/master/Assets/Kalman/KalmanFilter.cs\"\u003eKalman Filter\u003c/a\u003e \u003ca href=\"https://en.wikipedia.org/wiki/Kalman_filter#Details\" rel=\"nofollow\"\u003e[Wiki]\u003c/a\u003e\u003c/h2\u003e\u003ca id=\"user-content-kalman-filter-wiki\" class=\"anchor\" aria-label=\"Permalink: Kalman Filter[Wiki]\" href=\"#kalman-filter-wiki\"\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/54937394d69482c2ec3e6ce8e79a9e998cd4d2a9e86f07f7124d73b7eb1f9256/687474703a2f2f692e696d6775722e636f6d2f534c354a4a4d762e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/54937394d69482c2ec3e6ce8e79a9e998cd4d2a9e86f07f7124d73b7eb1f9256/687474703a2f2f692e696d6775722e636f6d2f534c354a4a4d762e676966\" data-animated-image=\"\" data-canonical-src=\"http://i.imgur.com/SL5JJMv.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eA textbook implementation of a Kalman filter (transcribed from wikipedia) Kalman filters are a form of bayesian filtering; they are capable of taking in information from multiple sources and \"fusing\" them into a signal that is cleaner/more accurate than any of the constituent signals. A properly tuned Kalman filter is the mathematically \"optimal\" technique for turning noisy data into clean data in real time (as long as the noise follows a gaussian distribution and the data varies linearly). In practice one must deal with biases and non-linearly varying quantities.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Constraints\"\u003eConstraints\u003c/a\u003e/\u003ca href=\"https://github.com/zalo/MathUtilities/blob/master/Assets/IK/CCDIK/CCDIKJoint.cs\"\u003eInverse Kinematics\u003c/a\u003e\u003c/h2\u003e\u003ca id=\"user-content-constraintsinverse-kinematics\" class=\"anchor\" aria-label=\"Permalink: Constraints/Inverse Kinematics\" href=\"#constraintsinverse-kinematics\"\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/64a22a93ee13231e03ff2030a18adf2be1ebcc152e0802eb2d9b5e47bc019f06/687474703a2f2f692e696d6775722e636f6d2f75796d4a66314c2e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/64a22a93ee13231e03ff2030a18adf2be1ebcc152e0802eb2d9b5e47bc019f06/687474703a2f2f692e696d6775722e636f6d2f75796d4a66314c2e676966\" data-animated-image=\"\" data-canonical-src=\"http://i.imgur.com/uymJf1L.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e \u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://camo.githubusercontent.com/1e759eea2f115aed7f6ab4b7d6bef2f5c511c83cf55f4f338a735ecb32bc72a4/687474703a2f2f692e696d6775722e636f6d2f6f7635386851482e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/1e759eea2f115aed7f6ab4b7d6bef2f5c511c83cf55f4f338a735ecb32bc72a4/687474703a2f2f692e696d6775722e636f6d2f6f7635386851482e676966\" data-animated-image=\"\" data-canonical-src=\"http://i.imgur.com/ov58hQH.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eA set of constraint functions that can be used to build \u003ca href=\"https://makeshifted.itch.io/dexter-arm-ik\" rel=\"nofollow\"\u003ean iterative inverse kinematics solver.\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca href=\"http://www.elysium-labs.com/robotics-corner/learn-robotics/introduction-to-robotics/kinematic-jacobian/\" rel=\"nofollow\"\u003eNice introductory Tutorial to the concept behind CCDIK and the Kinematic Jacobian\u003c/a\u003e\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eInverse-Kinematics: \u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/IK/Tutorials/CCDIK\"\u003eCCDIK Illustration\u003c/a\u003e vs \u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/IK/Tutorials/FABRIK\"\u003eFABRIK Illustration\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-inverse-kinematics-ccdik-illustration-vs-fabrik-illustration\" class=\"anchor\" aria-label=\"Permalink: Inverse-Kinematics: CCDIK Illustration vs FABRIK Illustration\" href=\"#inverse-kinematics-ccdik-illustration-vs-fabrik-illustration\"\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/c68dee3bb66d68a31badbba3ae2493e5c052b8f55495927d794bb784cbe0ef69/68747470733a2f2f692e696d6775722e636f6d2f7832416b5458322e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/c68dee3bb66d68a31badbba3ae2493e5c052b8f55495927d794bb784cbe0ef69/68747470733a2f2f692e696d6775722e636f6d2f7832416b5458322e676966\" data-animated-image=\"\" data-canonical-src=\"https://i.imgur.com/x2AkTX2.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e \u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://camo.githubusercontent.com/49845b1c743512668f34b66976a8edfe87c05b61ac62e1d8488a7a8b0ffd1293/68747470733a2f2f692e696d6775722e636f6d2f6b5077534855302e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/49845b1c743512668f34b66976a8edfe87c05b61ac62e1d8488a7a8b0ffd1293/68747470733a2f2f692e696d6775722e636f6d2f6b5077534855302e676966\" data-animated-image=\"\" data-canonical-src=\"https://i.imgur.com/kPwSHU0.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/IK/CCDIK/Configuration\"\u003eRobotic Configuration Space Visualization\u003c/a\u003e \u003ca href=\"https://i.imgur.com/kHdEZto.png\" rel=\"nofollow\"\u003e[Pic]\u003c/a\u003e and \u003ca href=\"https://github.com/zalo/MathUtilities/blob/master/Assets/IK/CCDIK/Collision/CCCDIK.cs\"\u003eCollision-Aware IK\u003c/a\u003e \u003ca href=\"https://i.imgur.com/2BGw3vt.gif\" rel=\"nofollow\"\u003e[Gif]\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-robotic-configuration-space-visualization-pic-and-collision-aware-ik-gif\" class=\"anchor\" aria-label=\"Permalink: Robotic Configuration Space Visualization[Pic] and Collision-Aware IK[Gif]\" href=\"#robotic-configuration-space-visualization-pic-and-collision-aware-ik-gif\"\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 \"Configuration Space\" can be visualized by graphing the penetration of a robot with it's environment (and itself) as a distance field, where each axis is the angle/configuration of an individual joint. By path finding through valid regions in this space, one is actually planning the motion of the robot from one configuration to another. The gradient of the configuration space can also be used for light depenetration of the robot from invalid configurations.\nHowever, because precomputing the configuration space is slow (and must be redone for objects in the environment), I developed a variant of CCDIK (CCCDIK :) ) which iteratively depenetrates itself from the environment by temporarily treating the contact point as a new end-effector.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eOther experiments:\u003c/h2\u003e\u003ca id=\"user-content-other-experiments\" class=\"anchor\" aria-label=\"Permalink: Other experiments:\" href=\"#other-experiments\"\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\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/blob/master/Assets/Amoeba/NelderMead.cs\"\u003eNelder-Mead (Amoeba) Numerical Optimizer\u003c/a\u003e \u003ca href=\"https://en.wikipedia.org/wiki/Nelder-Mead_method\" rel=\"nofollow\"\u003e[Wiki]\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-nelder-mead-amoeba-numerical-optimizer-wiki\" class=\"anchor\" aria-label=\"Permalink: Nelder-Mead (Amoeba) Numerical Optimizer[Wiki]\" href=\"#nelder-mead-amoeba-numerical-optimizer-wiki\"\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/39f046c1c86a2d2f533a3eacb1c8be0a1c68ae495470a4df5843382869fdd997/68747470733a2f2f692e696d6775722e636f6d2f494273466f4d642e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/39f046c1c86a2d2f533a3eacb1c8be0a1c68ae495470a4df5843382869fdd997/68747470733a2f2f692e696d6775722e636f6d2f494273466f4d642e676966\" data-animated-image=\"\" data-canonical-src=\"https://i.imgur.com/IBsFoMd.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eA general, n-dimensional implementation of Nelder and Mead's gradient-less numerical optimization method for minimizing cost functions. This is a popular optimization technique for problems with high-dimensionality and no gradient information. Included is an example of optimizing a 5-DoF IK system (far less efficient than CCDIK, but more flexible overall). Also contains a numerical gradient descent optimizer for comparison.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Assignment\"\u003eLinear Assignment\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-linear-assignment\" class=\"anchor\" aria-label=\"Permalink: Linear Assignment\" href=\"#linear-assignment\"\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 port of Roy Jonker's famous solution to the \u003ca href=\"https://en.wikipedia.org/wiki/Assignment_problem\" rel=\"nofollow\"\u003eLinear Assignment Problem\u003c/a\u003e. Allows you to take two arbitrary lists of objects (with a cost to pair objects in each of them to each other), and to find the globally optimal pairing betweeing objects in these lists. Extremely handy.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eSee the source file for Commercial Licensing Details.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/blob/master/Assets/SpatialHashing/SpatialHashing.cs\"\u003eSpatial Hashing\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-spatial-hashing\" class=\"anchor\" aria-label=\"Permalink: Spatial Hashing\" href=\"#spatial-hashing\"\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 reimplementation of \u003ca href=\"https://www.youtube.com/watch?v=D2M8jTtKi44\" rel=\"nofollow\"\u003eMatthias Muller Fischer's O(1) Spatial Hashing system\u003c/a\u003e for Particle-Particle collision lookups. Generally results in a \u0026gt;20x speedup over naive O(n^2) collision checking.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/blob/master/Assets/Skinning/LinearBlendSkinning.cs\"\u003eLinear Blend Skinning\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-linear-blend-skinning\" class=\"anchor\" aria-label=\"Permalink: Linear Blend Skinning\" href=\"#linear-blend-skinning\"\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 reference implementation that demonstrates how to apply bone motions to a model using the data contained within a skinned mesh renderer. As they say, there is more than one way to skin a mesh.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Projection\"\u003ePer-Pixel Texture Reprojection\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-per-pixel-texture-reprojection\" class=\"anchor\" aria-label=\"Permalink: Per-Pixel Texture Reprojection\" href=\"#per-pixel-texture-reprojection\"\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\"\u003eDemonstrates how to set up a shader to project a texture onto scene geometry using a (disabled) camera as the projector frustum. Useful for \u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Projection/Illusion\"\u003eillusions\u003c/a\u003e.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/blob/master/Assets/Quasirandom/QuasirandomSequence.cs#L16\"\u003eQuasirandom Point Generation\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-quasirandom-point-generation\" class=\"anchor\" aria-label=\"Permalink: Quasirandom Point Generation\" href=\"#quasirandom-point-generation\"\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/1c2140d79813fcabe45b9926c24544b784f5f85be21d6c07009958345f1349f7/68747470733a2f2f692e696d6775722e636f6d2f6a6471345172622e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/1c2140d79813fcabe45b9926c24544b784f5f85be21d6c07009958345f1349f7/68747470733a2f2f692e696d6775722e636f6d2f6a6471345172622e676966\" data-animated-image=\"\" data-canonical-src=\"https://i.imgur.com/jdq4Qrb.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eContains a class for generating a near-uniform quasirandom distribution of points for arbitrary dimensionality and point counts.\nBased off of the article here: \u003ca href=\"http://extremelearning.com.au/unreasonable-effectiveness-of-quasirandom-sequences/\" rel=\"nofollow\"\u003ehttp://extremelearning.com.au/unreasonable-effectiveness-of-quasirandom-sequences/\u003c/a\u003e\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/blob/master/Assets/VectorField/ParticleDistanceField.cs#L10\"\u003eDistance Field Particle Displacement\u003c/a\u003e \u003ca href=\"https://imgur.com/0kWoSfJ\" rel=\"nofollow\"\u003e[Gif]\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-distance-field-particle-displacement-gif\" class=\"anchor\" aria-label=\"Permalink: Distance Field Particle Displacement[Gif]\" href=\"#distance-field-particle-displacement-gif\"\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\"\u003eShows an interesting technique for displacing particles to flow around distance fields (would work very well in UE4's Niagara).\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Minkowski\"\u003eMinkowski Difference Visualizer\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-minkowski-difference-visualizer\" class=\"anchor\" aria-label=\"Permalink: Minkowski Difference Visualizer\" href=\"#minkowski-difference-visualizer\"\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/6b701c1e1f8977ba5ce7066cb539bb93170f1801c62b823c9d3414a6d39f18fc/68747470733a2f2f692e696d6775722e636f6d2f5a6d415433536d2e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/6b701c1e1f8977ba5ce7066cb539bb93170f1801c62b823c9d3414a6d39f18fc/68747470733a2f2f692e696d6775722e636f6d2f5a6d415433536d2e676966\" data-animated-image=\"\" data-canonical-src=\"https://i.imgur.com/ZmAT3Sm.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eUses a compute shader to draw arbitrary 2D Minkowski \"Differences\" in real time. The Minkowski Sum (and its modification, the \"Minkowski Difference\") is a core operation in collision detection. This concept allows for a fully generalized way of determining whether any two objects are intersecting, and what the minimum translation is that separates them. The key is determining whether the origin (of the coordinate system used in the operation) is inside of the resulting Minkowski Difference shape. GJK is a collision detection technique that implements this check quickly for convex objects, with only a few samples of the implicit Minkowski Difference.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Raycast\"\u003eBidirectional Raycasting\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-bidirectional-raycasting\" class=\"anchor\" aria-label=\"Permalink: Bidirectional Raycasting\" href=\"#bidirectional-raycasting\"\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/61bfd783996090d78dc65320d948cebf132145c6423998b6b515faee6c6a5843/68747470733a2f2f692e696d6775722e636f6d2f755463595950462e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/61bfd783996090d78dc65320d948cebf132145c6423998b6b515faee6c6a5843/68747470733a2f2f692e696d6775722e636f6d2f755463595950462e676966\" data-animated-image=\"\" data-canonical-src=\"https://i.imgur.com/uTcYYPF.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eOperates like a standard raycast, but returns both entry and exit information. Useful for building wires that wrap around the environment and bullet entry/exit effects.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Control\"\u003eBang-Bang Kinetic Time-Optimal Movement Controller\u003c/a\u003e \u003ca href=\"https://i.imgur.com/ptwHgew.gif\" rel=\"nofollow\"\u003e[Gif]\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-bang-bang-kinetic-time-optimal-movement-controller-gif\" class=\"anchor\" aria-label=\"Permalink: Bang-Bang Kinetic Time-Optimal Movement Controller[Gif]\" href=\"#bang-bang-kinetic-time-optimal-movement-controller-gif\"\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 special heuristic formula to compute the time-optimal movement trajectory for a double-integrating mass with limited thrust.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eFormula taken from this excellent course: \u003ca href=\"http://underactuated.csail.mit.edu/underactuated.html?chapter=9\" rel=\"nofollow\"\u003ehttp://underactuated.csail.mit.edu/underactuated.html?chapter=9\u003c/a\u003e\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Bezier\"\u003eB�zier Trajectory Deformation\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-bzier-trajectory-deformation\" class=\"anchor\" aria-label=\"Permalink: B�zier Trajectory Deformation\" href=\"#bzier-trajectory-deformation\"\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/5b3c7cd822862b4629be408c0f71e78808141c0bc0b89aa73be4f1e61a57fe6d/68747470733a2f2f692e696d6775722e636f6d2f546d33613962792e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/5b3c7cd822862b4629be408c0f71e78808141c0bc0b89aa73be4f1e61a57fe6d/68747470733a2f2f692e696d6775722e636f6d2f546d33613962792e676966\" data-animated-image=\"\" data-canonical-src=\"https://i.imgur.com/Tm3a9by.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThis is a technique for augmenting the end-point of trajectories composed of discrete segments, using a rigid-as-possible/\"Blossoming\" (B�zier-like) interpolation scheme. Includes an implementation for both 3D and 6D trajectories.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eInspired by this paper: \u003ca href=\"https://april.eecs.umich.edu/media/pdfs/olson2006icra.pdf\" rel=\"nofollow\"\u003ehttps://april.eecs.umich.edu/media/pdfs/olson2006icra.pdf\u003c/a\u003e\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Pivot\"\u003eRigid Transform Pivot Point\u003c/a\u003e \u003ca href=\"https://imgur.com/5iCUe2A\" rel=\"nofollow\"\u003e[Gif]\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-rigid-transform-pivot-point-gif\" class=\"anchor\" aria-label=\"Permalink: Rigid Transform Pivot Point[Gif]\" href=\"#rigid-transform-pivot-point-gif\"\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 appears to be a unique geometric solution for calculating the pivot point of a rigid transformation in 2D and 3D (where applicable). Is computationally efficient, geometrically intuitive, and can be extended to arbitrary dimensions.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/WeightedAverage\"\u003eWeighted Average Smoothing Spline\u003c/a\u003e \u003ca href=\"https://i.imgur.com/NgNkZJ2.gif\" rel=\"nofollow\"\u003e[Gif]\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-weighted-average-smoothing-spline-gif\" class=\"anchor\" aria-label=\"Permalink: Weighted Average Smoothing Spline[Gif]\" href=\"#weighted-average-smoothing-spline-gif\"\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\"\u003eAn example demonstrating how using a sliding window weighted average can yield a continuous smoothing spline for noisy 6 DoF data.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/ThickPlane\"\u003eThick Tesellated Plane Generator\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-thick-tesellated-plane-generator\" class=\"anchor\" aria-label=\"Permalink: Thick Tesellated Plane Generator\" href=\"#thick-tesellated-plane-generator\"\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\"\u003eUseful for generating solid meshes that are essentially deformed planes (shapes that are common in free-form optics).\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Capsule\"\u003eCapsule Mesh Generator\u003c/a\u003e \u003ca href=\"https://www.shadertoy.com/view/4l2cRW\" rel=\"nofollow\"\u003e[Shadertoy]\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-capsule-mesh-generator-shadertoy\" class=\"anchor\" aria-label=\"Permalink: Capsule Mesh Generator[Shadertoy]\" href=\"#capsule-mesh-generator-shadertoy\"\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\"\u003eUseful for generating capsule meshes with an arbitrary length and radius.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Sphere\"\u003eRaytraced Sphere\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-raytraced-sphere\" class=\"anchor\" aria-label=\"Permalink: Raytraced Sphere\" href=\"#raytraced-sphere\"\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 little shader that raytraces a pixel-perfect sphere on your object/billboard, for when you want to be pixel-bound rather than vertex bound...\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eAlso contains a small demo for finding tangential points to circles and spheres.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/blob/master/Assets/Serialization/SerializedGameObject.cs\"\u003eRuntime .json GameObject Serialization\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-runtime-json-gameobject-serialization\" class=\"anchor\" aria-label=\"Permalink: Runtime .json GameObject Serialization\" href=\"#runtime-json-gameobject-serialization\"\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 serialization utility that smartly serializes and deserializes game object hierarchies, engine components, monobehaviours, references, and more at runtime using reflection. Useful for a save/load system or a runtime editor.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/blob/master/Assets/Kalman/Matrix.cs\"\u003eMatrix Class\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-matrix-class\" class=\"anchor\" aria-label=\"Permalink: Matrix Class\" href=\"#matrix-class\"\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 generic matrix class and a set of basic matrix operations (multiplication, addition, subtraction, inversion, cholesky decomposition, and more). Written to support the implementation of the Kalman Filter. Usage of any of these operations will allocate a new array (garbage), so be careful about using this on performance constrained systems.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Probe\"\u003eHaptic Probe\u003c/a\u003e \u003ca href=\"http://i.imgur.com/Ljy3U8y.gif\" rel=\"nofollow\"\u003e[Gif]\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-haptic-probe-gif\" class=\"anchor\" aria-label=\"Permalink: Haptic Probe[Gif]\" href=\"#haptic-probe-gif\"\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\"\u003eAn example implementing a haptic probe, where the force on the effector of the haptic controller is equal to the linear and angular displacement of the green cube to the gray one.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/BundleAdustment\"\u003eBundle Adjustment\u003c/a\u003e \u003ca href=\"https://i.imgur.com/rRBd6g9.gif\" rel=\"nofollow\"\u003e[Gif]\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-bundle-adjustment-gif\" class=\"anchor\" aria-label=\"Permalink: Bundle Adjustment[Gif]\" href=\"#bundle-adjustment-gif\"\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\"\u003eMy attempts at implementing \u003ca href=\"https://en.wikipedia.org/wiki/Bundle_adjustment\" rel=\"nofollow\"\u003eBundle Adjustment\u003c/a\u003e (an algorithm which attempts to solve for the relative motion between two camera images, given the motion of a set of feature-points between the images). The stereo-case now converges to a unique 6-DoF pose (in most situations). This implementation is highly parallelizable.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Torque\"\u003eTorque Extension for HingeJoints\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-torque-extension-for-hingejoints\" class=\"anchor\" aria-label=\"Permalink: Torque Extension for HingeJoints\" href=\"#torque-extension-for-hingejoints\"\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 adds a function to apply torque \"through\" HingeJoints and to Rigidbodies in general.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/blob/master/Assets/Torus/Torus.cs#L21\"\u003en-Torus Encoding\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-n-torus-encoding\" class=\"anchor\" aria-label=\"Permalink: n-Torus Encoding\" href=\"#n-torus-encoding\"\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\"\u003eAn untested function for encoding an n-dimensional vector to the surface of an n+1 dimensional torus. Originally intended to allow Neural Networks to learn continuous cyclic functions without ballooning the dimensionality to 2n (ie trivial x -\u0026gt; (sin(x), cos(x)) case).\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/AccelerationDamping\"\u003eAcceleration Dampened Rigidbodies\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-acceleration-dampened-rigidbodies\" class=\"anchor\" aria-label=\"Permalink: Acceleration Dampened Rigidbodies\" href=\"#acceleration-dampened-rigidbodies\"\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\"\u003eAttempts to simulate soft-spongy contact by damping the accelerations that are be applied to an object. Phenomena like friction cause it to exhibit strange artifacts...\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Platformer\"\u003e2D Platforming Character\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-2d-platforming-character\" class=\"anchor\" aria-label=\"Permalink: 2D Platforming Character\" href=\"#2d-platforming-character\"\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/9a0e27688adef012825533887e6301a766042d3cbd49cf4cbafe5c0c0f7c5f09/687474703a2f2f692e696d6775722e636f6d2f7749654b7178702e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/9a0e27688adef012825533887e6301a766042d3cbd49cf4cbafe5c0c0f7c5f09/687474703a2f2f692e696d6775722e636f6d2f7749654b7178702e676966\" data-animated-image=\"\" data-canonical-src=\"http://i.imgur.com/wIeKqxp.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eUses a neat trick where, if the anchor of a spring joint is moved, both connected rigidbodies are physically affected. This allows one to easily simulate \"muscles\".\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"https://github.com/zalo/MathUtilities/tree/master/Assets/Rolling\"\u003eRolling Cubes\u003c/a\u003e\u003c/h3\u003e\u003ca id=\"user-content-rolling-cubes\" class=\"anchor\" aria-label=\"Permalink: Rolling Cubes\" href=\"#rolling-cubes\"\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/2e80044c04fbcf2ea2c33f38886678dd64cebf7d29f79a4b45d48adbe8b55280/687474703a2f2f692e696d6775722e636f6d2f5433703845704b2e676966\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/2e80044c04fbcf2ea2c33f38886678dd64cebf7d29f79a4b45d48adbe8b55280/687474703a2f2f692e696d6775722e636f6d2f5433703845704b2e676966\" data-animated-image=\"\" data-canonical-src=\"http://i.imgur.com/T3p8EpK.gif\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eFun for the whole family!\u003c/p\u003e\n\u003c/article\u003e","loaded":true,"timedOut":false,"errorMessage":null,"headerInfo":{"toc":[{"level":1,"text":"MathUtilities","anchor":"mathutilities","htmlText":"MathUtilities"},{"level":2,"text":"Generalized Mesh Deformation","anchor":"generalized-mesh-deformation","htmlText":"Generalized Mesh Deformation"},{"level":2,"text":"Signed Distance Field Texture Rendering","anchor":"signed-distance-field-texture-rendering","htmlText":"Signed Distance Field Texture Rendering"},{"level":2,"text":"Kabsch","anchor":"kabsch","htmlText":"Kabsch"},{"level":2,"text":"Least Squares Line/Plane Fitting[Pic]","anchor":"least-squares-lineplane-fitting-pic","htmlText":"Least Squares Line/Plane Fitting[Pic]"},{"level":2,"text":"Stereographic/Fisheye Camera[Wiki]","anchor":"stereographicfisheye-camera-wiki","htmlText":"Stereographic/Fisheye Camera[Wiki]"},{"level":2,"text":"Verlet Softbody, Rigidbody, and Chain","anchor":"verlet-softbody-rigidbody-and-chain","htmlText":"Verlet 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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">MathUtilities</h1><a id="user-content-mathutilities" class="anchor" aria-label="Permalink: MathUtilities" href="#mathutilities"><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 grab bag of some of the neat math and physics tricks that I've amassed over the last few years, implemented in Unity's C#. You're free to use the code here however you like.</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Deform">Generalized Mesh Deformation</a></h2><a id="user-content-generalized-mesh-deformation" class="anchor" aria-label="Permalink: Generalized Mesh Deformation" href="#generalized-mesh-deformation"><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/367e009ee8cf1465fa69441036197120803e4330df04b346f7dbb82f917e0342/68747470733a2f2f692e696d6775722e636f6d2f387162387335742e676966"><img src="https://camo.githubusercontent.com/367e009ee8cf1465fa69441036197120803e4330df04b346f7dbb82f917e0342/68747470733a2f2f692e696d6775722e636f6d2f387162387335742e676966" data-animated-image="" data-canonical-src="https://i.imgur.com/8qb8s5t.gif" style="max-width: 100%;"></a></p> <p dir="auto">An extremely fast, general mesh-deformation algorithm with arbitrary point-placement and configurable rigidity. Has the desirable property that it acts like a rigid Kabsch when the "weight" (ductility) is set to 0, but smoothly blends in deformation as the weight is increased.</p> <p dir="auto">The implementation is similar to Linear Blend Skinning, but the skinning weights are automatically calculated with <a href="https://en.wikipedia.org/wiki/Inverse_distance_weighting" rel="nofollow">Inverse Distance Weighting</a> (in cartesian <em>or</em> surface space), and the "bone rotations" are calculated to optimally preserve the angular relationships between the control points.</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Volume">Signed Distance Field Texture Rendering</a></h2><a id="user-content-signed-distance-field-texture-rendering" class="anchor" aria-label="Permalink: Signed Distance Field Texture Rendering" href="#signed-distance-field-texture-rendering"><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/45803628a93bc164882c50407356a88f89b1a1a95efa32a85ab1c969c02005bb/68747470733a2f2f692e696d6775722e636f6d2f6b374a3049376b2e676966"><img src="https://camo.githubusercontent.com/45803628a93bc164882c50407356a88f89b1a1a95efa32a85ab1c969c02005bb/68747470733a2f2f692e696d6775722e636f6d2f6b374a3049376b2e676966" data-animated-image="" data-canonical-src="https://i.imgur.com/k7J0I7k.gif" style="max-width: 100%;"></a></p> <p dir="auto">A simple example for raymarching and painting/blitting-to volumetric distance field textures. Distance Fields are regions of space that store the distance to the nearest surface at each point in space. This property allows them to represent and render solid shapes of arbitrary geometry and topology.</p> <p dir="auto">This particular implementation also stores the normalized vector toward the nearest point (the normal or gradient of the field). This is useful for physics queries and lighting (without taking the numerical derivative).</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/blob/master/Assets/Kabsch/Kabsch.cs">Kabsch</a></h2><a id="user-content-kabsch" class="anchor" aria-label="Permalink: Kabsch" href="#kabsch"><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/d5a59fbc066d390c40ad2c171963a0e846d05b7c89c005fe5d05f3bbdde8ee60/687474703a2f2f692e696d6775722e636f6d2f327168526d744e2e676966"><img src="https://camo.githubusercontent.com/d5a59fbc066d390c40ad2c171963a0e846d05b7c89c005fe5d05f3bbdde8ee60/687474703a2f2f692e696d6775722e636f6d2f327168526d744e2e676966" data-animated-image="" data-canonical-src="http://i.imgur.com/2qhRmtN.gif" style="max-width: 100%;"></a></p> <p dir="auto">Also known as Procrustes Analysis, this algorithm can take in an arbitrary set of point-pairs and find the globally optimal rigid translation and rotation to minimize the distance between those point pairs. Incredibly useful, and very cheap. Uses Matthias Muller's polar decomposition solver in place of SVD, as outlined here: <a href="https://animation.rwth-aachen.de/media/papers/2016-MIG-StableRotation.pdf" rel="nofollow">https://animation.rwth-aachen.de/media/papers/2016-MIG-StableRotation.pdf</a></p> <p dir="auto">Update: Added an example for averaging arbitrary numbers of quaternions; possibly more accurate than a normalized lerp (averaging the quaternion components in linear space and then normalizing).</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/blob/master/Assets/LeastSquares/LeastSquaresFitting.cs#L24">Least Squares Line/Plane Fitting</a> <a href="https://i.imgur.com/xTGKnx5" rel="nofollow">[Pic]</a></h2><a id="user-content-least-squares-lineplane-fitting-pic" class="anchor" aria-label="Permalink: Least Squares Line/Plane Fitting[Pic]" href="#least-squares-lineplane-fitting-pic"><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/a6a540aac9d43d25bf762f9e6e1a1194c25ab27d661c7e3a328888752834e8ad/68747470733a2f2f692e696d6775722e636f6d2f4e4274596949712e676966"><img src="https://camo.githubusercontent.com/a6a540aac9d43d25bf762f9e6e1a1194c25ab27d661c7e3a328888752834e8ad/68747470733a2f2f692e696d6775722e636f6d2f4e4274596949712e676966" data-animated-image="" data-canonical-src="https://i.imgur.com/NBtYiIq.gif" style="max-width: 100%;"></a></p> <p dir="auto">A generic helper utility for fitting lines and planes to point-sets in 3D. Uses a novel* matrix-less formulation to solve for the orthogonal lines and planes of best fit; algorithm is without singularities and is extensible to arbitrary dimensionality.</p> <p dir="auto">*(as far as I know; I have not seen anyone attempt orthogonal regression without using an SVD (which may be unnecessarily expensive for just the line/plane of best fit))</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/StereographicCamera">Stereographic/Fisheye Camera</a> <a href="https://en.wikipedia.org/wiki/Stereographic_projection" rel="nofollow">[Wiki]</a></h2><a id="user-content-stereographicfisheye-camera-wiki" class="anchor" aria-label="Permalink: Stereographic/Fisheye Camera[Wiki]" href="#stereographicfisheye-camera-wiki"><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/97ce710369ab40336c0c6ac10243d3fd8e90e78faa8e5a72da1e7bbb8553a419/687474703a2f2f692e696d6775722e636f6d2f4d4f36524c5a712e676966"><img src="https://camo.githubusercontent.com/97ce710369ab40336c0c6ac10243d3fd8e90e78faa8e5a72da1e7bbb8553a419/687474703a2f2f692e696d6775722e636f6d2f4d4f36524c5a712e676966" data-animated-image="" data-canonical-src="http://i.imgur.com/MO6RLZq.gif" style="max-width: 100%;"></a></p> <p dir="auto">A prefab that concatenates and warps the images from four cameras into one 180 degree fisheye view, projected stereographically.</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Verlet">Verlet Softbody, Rigidbody, and Chain</a></h2><a id="user-content-verlet-softbody-rigidbody-and-chain" class="anchor" aria-label="Permalink: Verlet Softbody, Rigidbody, and Chain" href="#verlet-softbody-rigidbody-and-chain"><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/e4da7350b7f43d4ebc01354418183b6ba027c9b77dc58d9068fd0a0e336e0fb3/687474703a2f2f692e696d6775722e636f6d2f79316a59417a772e676966"><img src="https://camo.githubusercontent.com/e4da7350b7f43d4ebc01354418183b6ba027c9b77dc58d9068fd0a0e336e0fb3/687474703a2f2f692e696d6775722e636f6d2f79316a59417a772e676966" data-animated-image="" data-canonical-src="http://i.imgur.com/y1jYAzw.gif" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://camo.githubusercontent.com/7a5107713f2d1cf2bf5d4d09add67947d18514608a42ec5a2f0034cbdfa16a5e/687474703a2f2f692e696d6775722e636f6d2f786c41686b4c342e676966"><img src="https://camo.githubusercontent.com/7a5107713f2d1cf2bf5d4d09add67947d18514608a42ec5a2f0034cbdfa16a5e/687474703a2f2f692e696d6775722e636f6d2f786c41686b4c342e676966" data-animated-image="" data-canonical-src="http://i.imgur.com/xlAhkL4.gif" style="max-width: 100%;"></a></p> <p dir="auto">Numerous examples of using verlet integration (<a href="http://matthias-mueller-fischer.ch/publications/posBasedDyn.pdf" rel="nofollow">a subset of Position Based Dynamics</a>) to simulate soft/rigid bodies, cloth, and particle linkage chains.</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/blob/master/Assets/Kalman/KalmanFilter.cs">Kalman Filter</a> <a href="https://en.wikipedia.org/wiki/Kalman_filter#Details" rel="nofollow">[Wiki]</a></h2><a id="user-content-kalman-filter-wiki" class="anchor" aria-label="Permalink: Kalman Filter[Wiki]" href="#kalman-filter-wiki"><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/54937394d69482c2ec3e6ce8e79a9e998cd4d2a9e86f07f7124d73b7eb1f9256/687474703a2f2f692e696d6775722e636f6d2f534c354a4a4d762e676966"><img src="https://camo.githubusercontent.com/54937394d69482c2ec3e6ce8e79a9e998cd4d2a9e86f07f7124d73b7eb1f9256/687474703a2f2f692e696d6775722e636f6d2f534c354a4a4d762e676966" data-animated-image="" data-canonical-src="http://i.imgur.com/SL5JJMv.gif" style="max-width: 100%;"></a></p> <p dir="auto">A textbook implementation of a Kalman filter (transcribed from wikipedia) Kalman filters are a form of bayesian filtering; they are capable of taking in information from multiple sources and "fusing" them into a signal that is cleaner/more accurate than any of the constituent signals. A properly tuned Kalman filter is the mathematically "optimal" technique for turning noisy data into clean data in real time (as long as the noise follows a gaussian distribution and the data varies linearly). In practice one must deal with biases and non-linearly varying quantities.</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Constraints">Constraints</a>/<a href="https://github.com/zalo/MathUtilities/blob/master/Assets/IK/CCDIK/CCDIKJoint.cs">Inverse Kinematics</a></h2><a id="user-content-constraintsinverse-kinematics" class="anchor" aria-label="Permalink: Constraints/Inverse Kinematics" href="#constraintsinverse-kinematics"><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/64a22a93ee13231e03ff2030a18adf2be1ebcc152e0802eb2d9b5e47bc019f06/687474703a2f2f692e696d6775722e636f6d2f75796d4a66314c2e676966"><img src="https://camo.githubusercontent.com/64a22a93ee13231e03ff2030a18adf2be1ebcc152e0802eb2d9b5e47bc019f06/687474703a2f2f692e696d6775722e636f6d2f75796d4a66314c2e676966" data-animated-image="" data-canonical-src="http://i.imgur.com/uymJf1L.gif" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://camo.githubusercontent.com/1e759eea2f115aed7f6ab4b7d6bef2f5c511c83cf55f4f338a735ecb32bc72a4/687474703a2f2f692e696d6775722e636f6d2f6f7635386851482e676966"><img src="https://camo.githubusercontent.com/1e759eea2f115aed7f6ab4b7d6bef2f5c511c83cf55f4f338a735ecb32bc72a4/687474703a2f2f692e696d6775722e636f6d2f6f7635386851482e676966" data-animated-image="" data-canonical-src="http://i.imgur.com/ov58hQH.gif" style="max-width: 100%;"></a></p> <p dir="auto">A set of constraint functions that can be used to build <a href="https://makeshifted.itch.io/dexter-arm-ik" rel="nofollow">an iterative inverse kinematics solver.</a></p> <p dir="auto"><a href="http://www.elysium-labs.com/robotics-corner/learn-robotics/introduction-to-robotics/kinematic-jacobian/" rel="nofollow">Nice introductory Tutorial to the concept behind CCDIK and the Kinematic Jacobian</a></p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">Inverse-Kinematics: <a href="https://github.com/zalo/MathUtilities/tree/master/Assets/IK/Tutorials/CCDIK">CCDIK Illustration</a> vs <a href="https://github.com/zalo/MathUtilities/tree/master/Assets/IK/Tutorials/FABRIK">FABRIK Illustration</a></h3><a id="user-content-inverse-kinematics-ccdik-illustration-vs-fabrik-illustration" class="anchor" aria-label="Permalink: Inverse-Kinematics: CCDIK Illustration vs FABRIK Illustration" href="#inverse-kinematics-ccdik-illustration-vs-fabrik-illustration"><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/c68dee3bb66d68a31badbba3ae2493e5c052b8f55495927d794bb784cbe0ef69/68747470733a2f2f692e696d6775722e636f6d2f7832416b5458322e676966"><img src="https://camo.githubusercontent.com/c68dee3bb66d68a31badbba3ae2493e5c052b8f55495927d794bb784cbe0ef69/68747470733a2f2f692e696d6775722e636f6d2f7832416b5458322e676966" data-animated-image="" data-canonical-src="https://i.imgur.com/x2AkTX2.gif" style="max-width: 100%;"></a> <a target="_blank" rel="noopener noreferrer nofollow" href="https://camo.githubusercontent.com/49845b1c743512668f34b66976a8edfe87c05b61ac62e1d8488a7a8b0ffd1293/68747470733a2f2f692e696d6775722e636f6d2f6b5077534855302e676966"><img src="https://camo.githubusercontent.com/49845b1c743512668f34b66976a8edfe87c05b61ac62e1d8488a7a8b0ffd1293/68747470733a2f2f692e696d6775722e636f6d2f6b5077534855302e676966" data-animated-image="" data-canonical-src="https://i.imgur.com/kPwSHU0.gif" style="max-width: 100%;"></a></p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/IK/CCDIK/Configuration">Robotic Configuration Space Visualization</a> <a href="https://i.imgur.com/kHdEZto.png" rel="nofollow">[Pic]</a> and <a href="https://github.com/zalo/MathUtilities/blob/master/Assets/IK/CCDIK/Collision/CCCDIK.cs">Collision-Aware IK</a> <a href="https://i.imgur.com/2BGw3vt.gif" rel="nofollow">[Gif]</a></h3><a id="user-content-robotic-configuration-space-visualization-pic-and-collision-aware-ik-gif" class="anchor" aria-label="Permalink: Robotic Configuration Space Visualization[Pic] and Collision-Aware IK[Gif]" href="#robotic-configuration-space-visualization-pic-and-collision-aware-ik-gif"><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 "Configuration Space" can be visualized by graphing the penetration of a robot with it's environment (and itself) as a distance field, where each axis is the angle/configuration of an individual joint. By path finding through valid regions in this space, one is actually planning the motion of the robot from one configuration to another. The gradient of the configuration space can also be used for light depenetration of the robot from invalid configurations. However, because precomputing the configuration space is slow (and must be redone for objects in the environment), I developed a variant of CCDIK (CCCDIK :) ) which iteratively depenetrates itself from the environment by temporarily treating the contact point as a new end-effector.</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Other experiments:</h2><a id="user-content-other-experiments" class="anchor" aria-label="Permalink: Other experiments:" href="#other-experiments"><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> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/blob/master/Assets/Amoeba/NelderMead.cs">Nelder-Mead (Amoeba) Numerical Optimizer</a> <a href="https://en.wikipedia.org/wiki/Nelder-Mead_method" rel="nofollow">[Wiki]</a></h3><a id="user-content-nelder-mead-amoeba-numerical-optimizer-wiki" class="anchor" aria-label="Permalink: Nelder-Mead (Amoeba) Numerical Optimizer[Wiki]" href="#nelder-mead-amoeba-numerical-optimizer-wiki"><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/39f046c1c86a2d2f533a3eacb1c8be0a1c68ae495470a4df5843382869fdd997/68747470733a2f2f692e696d6775722e636f6d2f494273466f4d642e676966"><img src="https://camo.githubusercontent.com/39f046c1c86a2d2f533a3eacb1c8be0a1c68ae495470a4df5843382869fdd997/68747470733a2f2f692e696d6775722e636f6d2f494273466f4d642e676966" data-animated-image="" data-canonical-src="https://i.imgur.com/IBsFoMd.gif" style="max-width: 100%;"></a></p> <p dir="auto">A general, n-dimensional implementation of Nelder and Mead's gradient-less numerical optimization method for minimizing cost functions. This is a popular optimization technique for problems with high-dimensionality and no gradient information. Included is an example of optimizing a 5-DoF IK system (far less efficient than CCDIK, but more flexible overall). Also contains a numerical gradient descent optimizer for comparison.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Assignment">Linear Assignment</a></h3><a id="user-content-linear-assignment" class="anchor" aria-label="Permalink: Linear Assignment" href="#linear-assignment"><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 port of Roy Jonker's famous solution to the <a href="https://en.wikipedia.org/wiki/Assignment_problem" rel="nofollow">Linear Assignment Problem</a>. Allows you to take two arbitrary lists of objects (with a cost to pair objects in each of them to each other), and to find the globally optimal pairing betweeing objects in these lists. Extremely handy.</p> <p dir="auto">See the source file for Commercial Licensing Details.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/blob/master/Assets/SpatialHashing/SpatialHashing.cs">Spatial Hashing</a></h3><a id="user-content-spatial-hashing" class="anchor" aria-label="Permalink: Spatial Hashing" href="#spatial-hashing"><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 reimplementation of <a href="https://www.youtube.com/watch?v=D2M8jTtKi44" rel="nofollow">Matthias Muller Fischer's O(1) Spatial Hashing system</a> for Particle-Particle collision lookups. Generally results in a >20x speedup over naive O(n^2) collision checking.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/blob/master/Assets/Skinning/LinearBlendSkinning.cs">Linear Blend Skinning</a></h3><a id="user-content-linear-blend-skinning" class="anchor" aria-label="Permalink: Linear Blend Skinning" href="#linear-blend-skinning"><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 reference implementation that demonstrates how to apply bone motions to a model using the data contained within a skinned mesh renderer. As they say, there is more than one way to skin a mesh.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Projection">Per-Pixel Texture Reprojection</a></h3><a id="user-content-per-pixel-texture-reprojection" class="anchor" aria-label="Permalink: Per-Pixel Texture Reprojection" href="#per-pixel-texture-reprojection"><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">Demonstrates how to set up a shader to project a texture onto scene geometry using a (disabled) camera as the projector frustum. Useful for <a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Projection/Illusion">illusions</a>.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/blob/master/Assets/Quasirandom/QuasirandomSequence.cs#L16">Quasirandom Point Generation</a></h3><a id="user-content-quasirandom-point-generation" class="anchor" aria-label="Permalink: Quasirandom Point Generation" href="#quasirandom-point-generation"><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/1c2140d79813fcabe45b9926c24544b784f5f85be21d6c07009958345f1349f7/68747470733a2f2f692e696d6775722e636f6d2f6a6471345172622e676966"><img src="https://camo.githubusercontent.com/1c2140d79813fcabe45b9926c24544b784f5f85be21d6c07009958345f1349f7/68747470733a2f2f692e696d6775722e636f6d2f6a6471345172622e676966" data-animated-image="" data-canonical-src="https://i.imgur.com/jdq4Qrb.gif" style="max-width: 100%;"></a></p> <p dir="auto">Contains a class for generating a near-uniform quasirandom distribution of points for arbitrary dimensionality and point counts. Based off of the article here: <a href="http://extremelearning.com.au/unreasonable-effectiveness-of-quasirandom-sequences/" rel="nofollow">http://extremelearning.com.au/unreasonable-effectiveness-of-quasirandom-sequences/</a></p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/blob/master/Assets/VectorField/ParticleDistanceField.cs#L10">Distance Field Particle Displacement</a> <a href="https://imgur.com/0kWoSfJ" rel="nofollow">[Gif]</a></h3><a id="user-content-distance-field-particle-displacement-gif" class="anchor" aria-label="Permalink: Distance Field Particle Displacement[Gif]" href="#distance-field-particle-displacement-gif"><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">Shows an interesting technique for displacing particles to flow around distance fields (would work very well in UE4's Niagara).</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Minkowski">Minkowski Difference Visualizer</a></h3><a id="user-content-minkowski-difference-visualizer" class="anchor" aria-label="Permalink: Minkowski Difference Visualizer" href="#minkowski-difference-visualizer"><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/6b701c1e1f8977ba5ce7066cb539bb93170f1801c62b823c9d3414a6d39f18fc/68747470733a2f2f692e696d6775722e636f6d2f5a6d415433536d2e676966"><img src="https://camo.githubusercontent.com/6b701c1e1f8977ba5ce7066cb539bb93170f1801c62b823c9d3414a6d39f18fc/68747470733a2f2f692e696d6775722e636f6d2f5a6d415433536d2e676966" data-animated-image="" data-canonical-src="https://i.imgur.com/ZmAT3Sm.gif" style="max-width: 100%;"></a></p> <p dir="auto">Uses a compute shader to draw arbitrary 2D Minkowski "Differences" in real time. The Minkowski Sum (and its modification, the "Minkowski Difference") is a core operation in collision detection. This concept allows for a fully generalized way of determining whether any two objects are intersecting, and what the minimum translation is that separates them. The key is determining whether the origin (of the coordinate system used in the operation) is inside of the resulting Minkowski Difference shape. GJK is a collision detection technique that implements this check quickly for convex objects, with only a few samples of the implicit Minkowski Difference.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Raycast">Bidirectional Raycasting</a></h3><a id="user-content-bidirectional-raycasting" class="anchor" aria-label="Permalink: Bidirectional Raycasting" href="#bidirectional-raycasting"><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/61bfd783996090d78dc65320d948cebf132145c6423998b6b515faee6c6a5843/68747470733a2f2f692e696d6775722e636f6d2f755463595950462e676966"><img src="https://camo.githubusercontent.com/61bfd783996090d78dc65320d948cebf132145c6423998b6b515faee6c6a5843/68747470733a2f2f692e696d6775722e636f6d2f755463595950462e676966" data-animated-image="" data-canonical-src="https://i.imgur.com/uTcYYPF.gif" style="max-width: 100%;"></a></p> <p dir="auto">Operates like a standard raycast, but returns both entry and exit information. Useful for building wires that wrap around the environment and bullet entry/exit effects.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Control">Bang-Bang Kinetic Time-Optimal Movement Controller</a> <a href="https://i.imgur.com/ptwHgew.gif" rel="nofollow">[Gif]</a></h3><a id="user-content-bang-bang-kinetic-time-optimal-movement-controller-gif" class="anchor" aria-label="Permalink: Bang-Bang Kinetic Time-Optimal Movement Controller[Gif]" href="#bang-bang-kinetic-time-optimal-movement-controller-gif"><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 special heuristic formula to compute the time-optimal movement trajectory for a double-integrating mass with limited thrust.</p> <p dir="auto">Formula taken from this excellent course: <a href="http://underactuated.csail.mit.edu/underactuated.html?chapter=9" rel="nofollow">http://underactuated.csail.mit.edu/underactuated.html?chapter=9</a></p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Bezier">B�zier Trajectory Deformation</a></h3><a id="user-content-bzier-trajectory-deformation" class="anchor" aria-label="Permalink: B�zier Trajectory Deformation" href="#bzier-trajectory-deformation"><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/5b3c7cd822862b4629be408c0f71e78808141c0bc0b89aa73be4f1e61a57fe6d/68747470733a2f2f692e696d6775722e636f6d2f546d33613962792e676966"><img src="https://camo.githubusercontent.com/5b3c7cd822862b4629be408c0f71e78808141c0bc0b89aa73be4f1e61a57fe6d/68747470733a2f2f692e696d6775722e636f6d2f546d33613962792e676966" data-animated-image="" data-canonical-src="https://i.imgur.com/Tm3a9by.gif" style="max-width: 100%;"></a></p> <p dir="auto">This is a technique for augmenting the end-point of trajectories composed of discrete segments, using a rigid-as-possible/"Blossoming" (B�zier-like) interpolation scheme. Includes an implementation for both 3D and 6D trajectories.</p> <p dir="auto">Inspired by this paper: <a href="https://april.eecs.umich.edu/media/pdfs/olson2006icra.pdf" rel="nofollow">https://april.eecs.umich.edu/media/pdfs/olson2006icra.pdf</a></p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Pivot">Rigid Transform Pivot Point</a> <a href="https://imgur.com/5iCUe2A" rel="nofollow">[Gif]</a></h3><a id="user-content-rigid-transform-pivot-point-gif" class="anchor" aria-label="Permalink: Rigid Transform Pivot Point[Gif]" href="#rigid-transform-pivot-point-gif"><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 appears to be a unique geometric solution for calculating the pivot point of a rigid transformation in 2D and 3D (where applicable). Is computationally efficient, geometrically intuitive, and can be extended to arbitrary dimensions.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/WeightedAverage">Weighted Average Smoothing Spline</a> <a href="https://i.imgur.com/NgNkZJ2.gif" rel="nofollow">[Gif]</a></h3><a id="user-content-weighted-average-smoothing-spline-gif" class="anchor" aria-label="Permalink: Weighted Average Smoothing Spline[Gif]" href="#weighted-average-smoothing-spline-gif"><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">An example demonstrating how using a sliding window weighted average can yield a continuous smoothing spline for noisy 6 DoF data.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/ThickPlane">Thick Tesellated Plane Generator</a></h3><a id="user-content-thick-tesellated-plane-generator" class="anchor" aria-label="Permalink: Thick Tesellated Plane Generator" href="#thick-tesellated-plane-generator"><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">Useful for generating solid meshes that are essentially deformed planes (shapes that are common in free-form optics).</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Capsule">Capsule Mesh Generator</a> <a href="https://www.shadertoy.com/view/4l2cRW" rel="nofollow">[Shadertoy]</a></h3><a id="user-content-capsule-mesh-generator-shadertoy" class="anchor" aria-label="Permalink: Capsule Mesh Generator[Shadertoy]" href="#capsule-mesh-generator-shadertoy"><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">Useful for generating capsule meshes with an arbitrary length and radius.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Sphere">Raytraced Sphere</a></h3><a id="user-content-raytraced-sphere" class="anchor" aria-label="Permalink: Raytraced Sphere" href="#raytraced-sphere"><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 little shader that raytraces a pixel-perfect sphere on your object/billboard, for when you want to be pixel-bound rather than vertex bound...</p> <p dir="auto">Also contains a small demo for finding tangential points to circles and spheres.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/blob/master/Assets/Serialization/SerializedGameObject.cs">Runtime .json GameObject Serialization</a></h3><a id="user-content-runtime-json-gameobject-serialization" class="anchor" aria-label="Permalink: Runtime .json GameObject Serialization" href="#runtime-json-gameobject-serialization"><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 serialization utility that smartly serializes and deserializes game object hierarchies, engine components, monobehaviours, references, and more at runtime using reflection. Useful for a save/load system or a runtime editor.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/blob/master/Assets/Kalman/Matrix.cs">Matrix Class</a></h3><a id="user-content-matrix-class" class="anchor" aria-label="Permalink: Matrix Class" href="#matrix-class"><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 generic matrix class and a set of basic matrix operations (multiplication, addition, subtraction, inversion, cholesky decomposition, and more). Written to support the implementation of the Kalman Filter. Usage of any of these operations will allocate a new array (garbage), so be careful about using this on performance constrained systems.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Probe">Haptic Probe</a> <a href="http://i.imgur.com/Ljy3U8y.gif" rel="nofollow">[Gif]</a></h3><a id="user-content-haptic-probe-gif" class="anchor" aria-label="Permalink: Haptic Probe[Gif]" href="#haptic-probe-gif"><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">An example implementing a haptic probe, where the force on the effector of the haptic controller is equal to the linear and angular displacement of the green cube to the gray one.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/BundleAdustment">Bundle Adjustment</a> <a href="https://i.imgur.com/rRBd6g9.gif" rel="nofollow">[Gif]</a></h3><a id="user-content-bundle-adjustment-gif" class="anchor" aria-label="Permalink: Bundle Adjustment[Gif]" href="#bundle-adjustment-gif"><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">My attempts at implementing <a href="https://en.wikipedia.org/wiki/Bundle_adjustment" rel="nofollow">Bundle Adjustment</a> (an algorithm which attempts to solve for the relative motion between two camera images, given the motion of a set of feature-points between the images). The stereo-case now converges to a unique 6-DoF pose (in most situations). This implementation is highly parallelizable.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Torque">Torque Extension for HingeJoints</a></h3><a id="user-content-torque-extension-for-hingejoints" class="anchor" aria-label="Permalink: Torque Extension for HingeJoints" href="#torque-extension-for-hingejoints"><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 adds a function to apply torque "through" HingeJoints and to Rigidbodies in general.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/blob/master/Assets/Torus/Torus.cs#L21">n-Torus Encoding</a></h3><a id="user-content-n-torus-encoding" class="anchor" aria-label="Permalink: n-Torus Encoding" href="#n-torus-encoding"><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">An untested function for encoding an n-dimensional vector to the surface of an n+1 dimensional torus. Originally intended to allow Neural Networks to learn continuous cyclic functions without ballooning the dimensionality to 2n (ie trivial x -> (sin(x), cos(x)) case).</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/AccelerationDamping">Acceleration Dampened Rigidbodies</a></h3><a id="user-content-acceleration-dampened-rigidbodies" class="anchor" aria-label="Permalink: Acceleration Dampened Rigidbodies" href="#acceleration-dampened-rigidbodies"><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">Attempts to simulate soft-spongy contact by damping the accelerations that are be applied to an object. Phenomena like friction cause it to exhibit strange artifacts...</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Platformer">2D Platforming Character</a></h3><a id="user-content-2d-platforming-character" class="anchor" aria-label="Permalink: 2D Platforming Character" href="#2d-platforming-character"><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/9a0e27688adef012825533887e6301a766042d3cbd49cf4cbafe5c0c0f7c5f09/687474703a2f2f692e696d6775722e636f6d2f7749654b7178702e676966"><img src="https://camo.githubusercontent.com/9a0e27688adef012825533887e6301a766042d3cbd49cf4cbafe5c0c0f7c5f09/687474703a2f2f692e696d6775722e636f6d2f7749654b7178702e676966" data-animated-image="" data-canonical-src="http://i.imgur.com/wIeKqxp.gif" style="max-width: 100%;"></a></p> <p dir="auto">Uses a neat trick where, if the anchor of a spring joint is moved, both connected rigidbodies are physically affected. This allows one to easily simulate "muscles".</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto"><a href="https://github.com/zalo/MathUtilities/tree/master/Assets/Rolling">Rolling Cubes</a></h3><a id="user-content-rolling-cubes" class="anchor" aria-label="Permalink: Rolling Cubes" href="#rolling-cubes"><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/2e80044c04fbcf2ea2c33f38886678dd64cebf7d29f79a4b45d48adbe8b55280/687474703a2f2f692e696d6775722e636f6d2f5433703845704b2e676966"><img src="https://camo.githubusercontent.com/2e80044c04fbcf2ea2c33f38886678dd64cebf7d29f79a4b45d48adbe8b55280/687474703a2f2f692e696d6775722e636f6d2f5433703845704b2e676966" data-animated-image="" data-canonical-src="http://i.imgur.com/T3p8EpK.gif" style="max-width: 100%;"></a></p> <p dir="auto">Fun for the whole family!</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="xJnVoJUzZBgTsdeifvnkYBrOGs/i0FDVT4ljdtmFREXSLbQMeoQktuAa7AWyBAL+reft8MISg7+L5BuUm1QzZw==" /> </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"> A collection of some of the neat math and physics tricks that I've collected over the last few years. </p> <h3 class="sr-only">Topics</h3> <div class="my-3"> <div class="f6"> <a href="/topics/camera" title="Topic: camera" data-view-component="true" class="topic-tag topic-tag-link"> camera </a> <a href="/topics/utilities" title="Topic: utilities" data-view-component="true" class="topic-tag topic-tag-link"> utilities </a> <a href="/topics/math" title="Topic: math" data-view-component="true" class="topic-tag topic-tag-link"> math </a> <a href="/topics/unity" title="Topic: unity" data-view-component="true" class="topic-tag topic-tag-link"> unity </a> <a href="/topics/physics" title="Topic: physics" data-view-component="true" class="topic-tag topic-tag-link"> physics </a> <a href="/topics/neat" title="Topic: neat" data-view-component="true" 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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>