2D computer graphics

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</ul> </li> <li id="toc-Non-standard_orientation_of_the_coordinate_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-standard_orientation_of_the_coordinate_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Non-standard orientation of the coordinate system</span> </div> </a> <ul id="toc-Non-standard_orientation_of_the_coordinate_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Common_rotations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Common_rotations"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Common rotations</span> </div> </a> <ul id="toc-Common_rotations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scaling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scaling"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Scaling</span> </div> </a> <ul id="toc-Scaling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_homogeneous_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_homogeneous_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.7</span> <span>Using homogeneous coordinates</span> </div> </a> <ul id="toc-Using_homogeneous_coordinates-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Techniques" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Techniques"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Techniques</span> </div> </a> <button aria-controls="toc-Techniques-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Techniques subsection</span> </button> <ul id="toc-Techniques-sublist" class="vector-toc-list"> <li id="toc-Direct_painting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Direct_painting"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Direct painting</span> </div> </a> <ul id="toc-Direct_painting-sublist" class="vector-toc-list"> <li id="toc-Extended_color_models" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Extended_color_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Extended color models</span> </div> </a> <ul id="toc-Extended_color_models-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Layers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Layers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Layers</span> </div> </a> <ul id="toc-Layers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Hardware" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hardware"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Hardware</span> </div> </a> <ul id="toc-Hardware-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Software" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Software"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Software</span> </div> </a> <ul id="toc-Software-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Developmental_animation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Developmental_animation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Developmental animation</span> </div> </a> <ul id="toc-Developmental_animation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 27 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-27" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">27 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D8%B3%D9%88%D9%85%D8%A7%D8%AA_%D8%A7%D9%84%D8%AD%D8%A7%D8%B3%D9%88%D8%A8_%D8%AB%D9%86%D8%A7%D8%A6%D9%8A%D8%A9_%D8%A7%D9%84%D8%A3%D8%A8%D8%B9%D8%A7%D8%AF" title="رسومات الحاسوب ثنائية الأبعاد – Arabic" lang="ar" hreflang="ar" data-title="رسومات الحاسوب ثنائية الأبعاد" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/2D_ra%C4%8Dunarska_grafika" title="2D računarska grafika – Bosnian" lang="bs" hreflang="bs" data-title="2D računarska grafika" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Computaci%C3%B3_gr%C3%A0fica_2D" title="Computació gràfica 2D – Catalan" lang="ca" hreflang="ca" data-title="Computació gràfica 2D" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Po%C4%8D%C3%ADta%C4%8Dov%C3%A1_2D_grafika" title="Počítačová 2D grafika – Czech" lang="cs" hreflang="cs" data-title="Počítačová 2D grafika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Computaci%C3%B3n_gr%C3%A1fica_2D" title="Computación gráfica 2D – Spanish" lang="es" hreflang="es" data-title="Computación gráfica 2D" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Dudimensia_komputila_grafiko" title="Dudimensia komputila grafiko – Esperanto" lang="eo" hreflang="eo" data-title="Dudimensia komputila grafiko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D8%A7%D9%81%DB%8C%DA%A9_%DA%A9%D8%A7%D9%85%D9%BE%DB%8C%D9%88%D8%AA%D8%B1%DB%8C_%D8%AF%D9%88%D8%A8%D8%B9%D8%AF%DB%8C" title="گرافیک کامپیوتری دوبعدی – Persian" lang="fa" hreflang="fa" data-title="گرافیک کامپیوتری دوبعدی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Graphisme_2D" title="Graphisme 2D – French" lang="fr" hreflang="fr" data-title="Graphisme 2D" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/2%EC%B0%A8%EC%9B%90_%EC%BB%B4%ED%93%A8%ED%84%B0_%EA%B7%B8%EB%9E%98%ED%94%BD%EC%8A%A4" title="2차원 컴퓨터 그래픽스 – Korean" lang="ko" hreflang="ko" data-title="2차원 컴퓨터 그래픽스" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D6%80%D5%AF%D5%B9%D5%A1%D6%83_%D5%B0%D5%A1%D5%B4%D5%A1%D5%AF%D5%A1%D6%80%D5%A3%D5%B9%D5%A1%D5%B5%D5%AB%D5%B6_%D5%A3%D6%80%D5%A1%D6%86%D5%AB%D5%AF%D5%A1" title="Երկչափ համակարգչային գրաֆիկա – Armenian" lang="hy" hreflang="hy" data-title="Երկչափ համակարգչային գրաֆիկա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Grafiko_dudimensiona" title="Grafiko dudimensiona – Ido" lang="io" hreflang="io" data-title="Grafiko dudimensiona" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Grafika_komputer_2D" title="Grafika komputer 2D – Indonesian" lang="id" hreflang="id" data-title="Grafika komputer 2D" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/2D_datorgrafika" title="2D datorgrafika – Latvian" lang="lv" hreflang="lv" data-title="2D datorgrafika" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/2%E6%AC%A1%E5%85%83%E3%82%B3%E3%83%B3%E3%83%94%E3%83%A5%E3%83%BC%E3%82%BF%E3%82%B0%E3%83%A9%E3%83%95%E3%82%A3%E3%83%83%E3%82%AF%E3%82%B9" title="2次元コンピュータグラフィックス – Japanese" lang="ja" hreflang="ja" data-title="2次元コンピュータグラフィックス" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/2D-grafikk" title="2D-grafikk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="2D-grafikk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grafika_dwuwymiarowa" title="Grafika dwuwymiarowa – Polish" lang="pl" hreflang="pl" data-title="Grafika dwuwymiarowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/2D_(computa%C3%A7%C3%A3o_gr%C3%A1fica)" title="2D (computação gráfica) – Portuguese" lang="pt" hreflang="pt" data-title="2D (computação gráfica)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Po%C4%8D%C3%ADta%C4%8Dov%C3%A1_2D_grafika" title="Počítačová 2D grafika – Slovak" lang="sk" hreflang="sk" data-title="Počítačová 2D grafika" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%88%DB%8E%D9%86%DB%95%DB%8C_%DA%A9%DB%86%D9%85%D9%BE%DB%8C%D9%88%D8%AA%DB%95%D8%B1%DB%8C%DB%8C%DB%8C_%D8%AF%D9%88%D9%88_%DA%95%DB%95%DA%BE%DB%95%D9%86%D8%AF%DB%8C" title="وێنەی کۆمپیوتەرییی دوو ڕەھەندی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="وێنەی کۆمپیوتەرییی دوو ڕەھەندی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/2D_ra%C4%8Dunarska_grafika" title="2D računarska grafika – Serbian" lang="sr" hreflang="sr" data-title="2D računarska grafika" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/2D_ra%C4%8Dunarska_grafika" title="2D računarska grafika – Serbo-Croatian" lang="sh" hreflang="sh" data-title="2D računarska grafika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/2D-grafiikka" title="2D-grafiikka – Finnish" lang="fi" hreflang="fi" data-title="2D-grafiikka" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/2_boyutlu_bilgisayar_grafikleri" title="2 boyutlu bilgisayar grafikleri – Turkish" lang="tr" hreflang="tr" data-title="2 boyutlu bilgisayar grafikleri" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%27%D1%8E%D1%82%D0%B5%D1%80%D0%BD%D0%B0_2D-%D0%B3%D1%80%D0%B0%D1%84%D1%96%D0%BA%D0%B0" title="Комп'ютерна 2D-графіка – Ukrainian" lang="uk" hreflang="uk" data-title="Комп'ютерна 2D-графіка" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AF%D9%88_%D8%AC%DB%81%D8%AA%DB%8C_%DA%A9%D9%85%D9%BE%DB%8C%D9%88%D9%B9%D8%B1_%DA%AF%D8%B1%D8%A7%D9%81%DA%A9%D8%B3" title="دو جہتی کمپیوٹر گرافکس – Urdu" lang="ur" hreflang="ur" data-title="دو جہتی کمپیوٹر گرافکس" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/2D_%E9%9B%BB%E8%85%A6%E5%9C%96%E5%83%8F" title="2D 電腦圖像 – Cantonese" lang="yue" hreflang="yue" data-title="2D 電腦圖像" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BA%8C%E7%BB%B4%E8%AE%A1%E7%AE%97%E6%9C%BA%E5%9B%BE%E5%BD%A2" title="二维计算机图形 – Chinese" lang="zh" hreflang="zh" data-title="二维计算机图形" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1194763#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a 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href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/2D_computer_graphics" title="Special:EditPage/2D computer graphics">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%222D+computer+graphics%22">"2D computer graphics"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%222D+computer+graphics%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%222D+computer+graphics%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%222D+computer+graphics%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%222D+computer+graphics%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%222D+computer+graphics%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">December 2009</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Original_research plainlinks metadata ambox ambox-content ambox-Original_research" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>possibly contains <a href="/wiki/Wikipedia:No_original_research" title="Wikipedia:No original research">original research</a></b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=2D_computer_graphics&action=edit">improve it</a> by <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verifying</a> the claims made and adding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a>. Statements consisting only of original research should be removed.</span> <span class="date-container"><i>(<span class="date">July 2019</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> </div> </div><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><b>2D computer graphics</b> is the <a href="/wiki/Computer-generated_imagery" title="Computer-generated imagery">computer-based</a> generation of <a href="/wiki/Digital_image" title="Digital image">digital images</a>—mostly from two-dimensional models (such as <a href="/wiki/2D_geometric_model" title="2D geometric model">2D geometric models</a>, text, and digital images) and by techniques specific to them. It may refer to the branch of <a href="/wiki/Computer_science" title="Computer science">computer science</a> that comprises such techniques or to the models themselves. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Blit_dot.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7c/Blit_dot.gif" decoding="async" width="96" height="144" class="mw-file-element" data-file-width="96" data-file-height="144" /></a><figcaption><a href="/wiki/Raster_graphics" title="Raster graphics">Raster graphic</a> <a href="/wiki/Sprite_(computer_graphics)" title="Sprite (computer graphics)">sprites</a> (left) and masks</figcaption></figure> <p>2D computer graphics are mainly used in applications that were originally developed upon traditional <a href="/wiki/Printing" title="Printing">printing</a> and <a href="/wiki/Drawing" title="Drawing">drawing</a> technologies, such as <a href="/wiki/Typography" title="Typography">typography</a>, <a href="/wiki/Cartography" title="Cartography">cartography</a>, <a href="/wiki/Technical_drawing" title="Technical drawing">technical drawing</a>, <a href="/wiki/Advertising" title="Advertising">advertising</a>, etc. In those applications, the two-dimensional <a href="/wiki/Image" title="Image">image</a> is not just a representation of a real-world object, but an independent artifact with added semantic value; two-dimensional models are therefore preferred, because they give more direct control of the image than <a href="/wiki/3D_computer_graphics" title="3D computer graphics">3D computer graphics</a> (whose approach is more akin to <a href="/wiki/Photography" title="Photography">photography</a> than to typography). </p><p>In many domains, such as <a href="/wiki/Desktop_publishing" title="Desktop publishing">desktop publishing</a>, <a href="/wiki/Engineering" title="Engineering">engineering</a>, and <a href="/wiki/Business" title="Business">business</a>, a description of a document based on 2D computer graphics techniques can be much smaller than the corresponding digital image—often by a factor of 1/1000 or more. This representation is also more flexible since it can be <a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">rendered</a> at different <a href="/wiki/Image_resolution" title="Image resolution">resolutions</a> to suit different <a href="/wiki/Output_device" title="Output device">output devices</a>. For these reasons, documents and illustrations are often stored or transmitted as <a href="/wiki/Graphics_file_format" class="mw-redirect" title="Graphics file format">2D graphic files</a>. </p><p>2D computer graphics started in the 1950s, based on <a href="/wiki/Vector_graphics" title="Vector graphics">vector graphics devices</a>. These were largely supplanted by <a href="/wiki/Raster_graphics" title="Raster graphics">raster-based devices</a> in the following decades. The <a href="/wiki/PostScript" title="PostScript">PostScript</a> language and the <a href="/wiki/X_Window_System" title="X Window System">X Window System</a> protocol were landmark developments in the field. </p><p>2D graphics models may combine <a href="/wiki/2D_geometric_model" title="2D geometric model">geometric models</a> (also called <a href="/wiki/Vector_graphics" title="Vector graphics">vector graphics</a>), digital images (also called <a href="/wiki/Raster_graphics" title="Raster graphics">raster graphics</a>), text to be <a href="/wiki/Typesetting" title="Typesetting">typeset</a> (defined by content, <a href="/wiki/Typeface" title="Typeface">font</a> style and size, color, position, and orientation), mathematical <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> and <a href="/wiki/Equation" title="Equation">equations</a>, and more. These components can be modified and manipulated by two-dimensional <a href="/wiki/Transformation_(geometry)" class="mw-redirect" title="Transformation (geometry)">geometric transformations</a> such as <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a>, <a href="/wiki/Rotation" title="Rotation">rotation</a>, and <a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">scaling</a>. In <a href="/wiki/Object-oriented_graphics" class="mw-redirect" title="Object-oriented graphics">object-oriented graphics</a>, the image is described indirectly by an <a href="/wiki/Object_(computer_science)" title="Object (computer science)">object</a> endowed with a self-<a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">rendering</a> <a href="/wiki/Method_(computer_science)" class="mw-redirect" title="Method (computer science)">method</a>—a procedure that assigns colors to the image <a href="/wiki/Pixel" title="Pixel">pixels</a> by an arbitrary algorithm. Complex models can be built by combining simpler objects, in the <a href="/wiki/Paradigms" class="mw-redirect" title="Paradigms">paradigms</a> of <a href="/wiki/Object-oriented_programming" title="Object-oriented programming">object-oriented programming</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Background_(geometry)"><span id="Background_.28geometry.29"></span>Background (geometry)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=1" title="Edit section: Background (geometry)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Duplication plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b><a href="/wiki/Wikipedia:CFORK" class="mw-redirect" title="Wikipedia:CFORK">duplicates</a> the scope of other articles</b>, specifically <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">Translation (geometry)</a> and <a href="/wiki/Rotation_(geometry)" class="mw-redirect" title="Rotation (geometry)">Rotation (geometry)</a>.<span class="hide-when-compact"> Please <a href="/wiki/Talk:2D_computer_graphics#Duplication" title="Talk:2D computer graphics">discuss this issue</a> and help introduce a <a href="/wiki/Wikipedia:Summary_style" title="Wikipedia:Summary style">summary style</a> to the section by replacing the section with a link and a summary or by <a href="/wiki/Wikipedia:SPLIT" class="mw-redirect" title="Wikipedia:SPLIT">splitting the content</a> into a new article.</span> <span class="date-container"><i>(<span class="date">May 2022</span>)</i></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Rotations_and_reflections_in_two_dimensions" title="Rotations and reflections in two dimensions">Rotations and reflections in two dimensions</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Traslazione_OK.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Traslazione_OK.svg/220px-Traslazione_OK.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Traslazione_OK.svg/330px-Traslazione_OK.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/Traslazione_OK.svg/440px-Traslazione_OK.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>A translation moves every point of a figure or a space by the same amount in a given direction.</figcaption></figure> <p>In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, a <i><a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation (geometry)</a></i> moves every point a constant distance in a specified direction. A translation can be described as a <a href="/wiki/Euclidean_group" title="Euclidean group">rigid motion</a>: other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant <a href="/wiki/Vector_space" title="Vector space">vector</a> to every point, or as shifting the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> of the <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a>. A <i><a href="/wiki/Shift_operator" title="Shift operator">translation operator</a></i> is an <a href="/wiki/Operator_(mathematics)" title="Operator (mathematics)">operator</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathbf {\delta } }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathbf {\delta } }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a6730ac2e36afbbc5c42952ed336bb698195bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.331ex; height:2.509ex;" alt="{\displaystyle T_{\mathbf {\delta } }}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathbf {\delta } }f(\mathbf {v} )=f(\mathbf {v} +\mathbf {\delta } ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathbf {\delta } }f(\mathbf {v} )=f(\mathbf {v} +\mathbf {\delta } ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36a613b788721a990e94d2ab4bbfa34d304ff483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.963ex; height:2.843ex;" alt="{\displaystyle T_{\mathbf {\delta } }f(\mathbf {v} )=f(\mathbf {v} +\mathbf {\delta } ).}"></span> </p><p>If <b>v</b> is a fixed vector, then the translation <i>T</i><sub><b>v</b></sub> will work as <i>T</i><sub><b>v</b></sub>(<b>p</b>) = <b>p</b> + <b>v</b>. </p><p>If <i>T</i> is a translation, then the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of a subset <i>A</i> under the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <i>T</i> is the <b>translation</b> of <i>A</i> by <i>T</i>. The translation of <i>A</i> by <i>T</i><sub><b>v</b></sub> is often written <i>A</i> + <b>v</b>. </p><p>In a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, any translation is an <a href="/wiki/Isometry" title="Isometry">isometry</a>. The set of all translations forms the translation group <i>T</i>, which is isomorphic to the space itself, and a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> of <a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean group</a> <i>E</i>(<i>n</i> ). The <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> of <i>E</i>(<i>n</i> ) by <i>T</i> is isomorphic to the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> <i>O</i>(<i>n</i> ): </p> <dl><dd><i>E</i>(<i>n</i> ) <i>/ T</i> ≅ <i>O</i>(<i>n</i> ).</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Translation">Translation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=2" title="Edit section: Translation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since a translation is an <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a> but not a <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a>, <a href="/wiki/Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a> are normally used to represent the translation operator by a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> and thus to make it linear. Thus we write the 3-dimensional vector <b>w</b> = (<i>w</i><sub><i>x</i></sub>, <i>w</i><sub><i>y</i></sub>, <i>w</i><sub><i>z</i></sub>) using 4 homogeneous coordinates as <b>w</b> = (<i>w</i><sub><i>x</i></sub>, <i>w</i><sub><i>y</i></sub>, <i>w</i><sub><i>z</i></sub>, 1).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>To translate an object by a <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vector</a> <b>v</b>, each homogeneous vector <b>p</b> (written in homogeneous coordinates) would need to be multiplied by this <b>translation matrix</b>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathbf {v} }={\begin{bmatrix}1&0&0&v_{x}\\0&1&0&v_{y}\\0&0&1&v_{z}\\0&0&0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathbf {v} }={\begin{bmatrix}1&0&0&v_{x}\\0&1&0&v_{y}\\0&0&1&v_{z}\\0&0&0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8291f8c9e148126f931b49fa4fa9a96827aceeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:22.293ex; height:12.843ex;" alt="{\displaystyle T_{\mathbf {v} }={\begin{bmatrix}1&0&0&v_{x}\\0&1&0&v_{y}\\0&0&1&v_{z}\\0&0&0&1\end{bmatrix}}}"></span></dd></dl> <p>As shown below, the multiplication will give the expected result: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathbf {v} }\mathbf {p} ={\begin{bmatrix}1&0&0&v_{x}\\0&1&0&v_{y}\\0&0&1&v_{z}\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}+v_{x}\\p_{y}+v_{y}\\p_{z}+v_{z}\\1\end{bmatrix}}=\mathbf {p} +\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathbf {v} }\mathbf {p} ={\begin{bmatrix}1&0&0&v_{x}\\0&1&0&v_{y}\\0&0&1&v_{z}\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}+v_{x}\\p_{y}+v_{y}\\p_{z}+v_{z}\\1\end{bmatrix}}=\mathbf {p} +\mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7bb4edd686c84393b22e27b4507f2e1ebc03aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:53.241ex; height:12.843ex;" alt="{\displaystyle T_{\mathbf {v} }\mathbf {p} ={\begin{bmatrix}1&0&0&v_{x}\\0&1&0&v_{y}\\0&0&1&v_{z}\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}+v_{x}\\p_{y}+v_{y}\\p_{z}+v_{z}\\1\end{bmatrix}}=\mathbf {p} +\mathbf {v} }"></span></dd></dl> <p>The inverse of a translation matrix can be obtained by reversing the direction of the vector: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathbf {v} }^{-1}=T_{-\mathbf {v} }.\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </msub> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathbf {v} }^{-1}=T_{-\mathbf {v} }.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed2142aa1a48d83d35e67221cf40430956784ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:11.481ex; height:3.009ex;" alt="{\displaystyle T_{\mathbf {v} }^{-1}=T_{-\mathbf {v} }.\!}"></span></dd></dl> <p>Similarly, the product of translation matrices is given by adding the vectors: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathbf {u} }T_{\mathbf {v} }=T_{\mathbf {u} +\mathbf {v} }.\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </msub> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathbf {u} }T_{\mathbf {v} }=T_{\mathbf {u} +\mathbf {v} }.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a6ad015dc048e467ef0fd90a6af79b7b0375a3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:13.706ex; height:2.509ex;" alt="{\displaystyle T_{\mathbf {u} }T_{\mathbf {v} }=T_{\mathbf {u} +\mathbf {v} }.\!}"></span></dd></dl> <p>Because addition of vectors is <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices). </p> <div class="mw-heading mw-heading3"><h3 id="Rotation">Rotation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=3" title="Edit section: Rotation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, a <i><a href="/wiki/Rotation_matrix" title="Rotation matrix">rotation matrix</a></i> is a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> that is used to perform a <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a> in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4ee3f1ce8e028da5bd4219c9dc7fc2216543e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.509ex; height:6.176ex;" alt="{\displaystyle R={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}"></span></dd></dl> <p>rotates points in the <i>xy</i>-<a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian plane</a> counterclockwise through an angle <i>θ</i> about the origin of the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>. To perform the rotation using a rotation matrix <i>R</i>, the position of each point must be represented by a <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vector</a> <b>v</b>, containing the coordinates of the point. A rotated vector is obtained by using the <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> <i>R</i><b>v</b>. Since matrix multiplication has no effect on the zero vector (i.e., on the coordinates of the origin), rotation matrices can only be used to describe rotations about the origin of the coordinate system. </p><p>Rotation matrices provide a simple algebraic description of such rotations, and are used extensively for computations in <a href="/wiki/Geometry" title="Geometry">geometry</a>, <a href="/wiki/Physics" title="Physics">physics</a>, and <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>. In 2-dimensional space, a rotation can be simply described by an <a href="/wiki/Angle_of_rotation" class="mw-redirect" title="Angle of rotation">angle <i>θ</i> of rotation</a>, but it can be also represented by the 4 entries of a rotation matrix with 2 rows and 2 columns. In 3-dimensional space, every rotation can be interpreted as a rotation by a given angle about a single fixed axis of rotation (see <a href="/wiki/Euler%27s_rotation_theorem" title="Euler's rotation theorem">Euler's rotation theorem</a>), and hence it can be simply described by <a href="/wiki/Axis-angle_representation" class="mw-redirect" title="Axis-angle representation">an angle and a vector</a> with 3 entries. However, it can also be represented by the 9 entries of a rotation matrix with 3 rows and 3 columns. The notion of rotation is not commonly used in dimensions higher than 3; there is a notion of a <i><a href="/wiki/Rotational_displacement" class="mw-redirect" title="Rotational displacement">rotational displacement</a></i>, which can be represented by a matrix, but no associated single axis or angle. </p><p>Rotation matrices are <a href="/wiki/Square_matrix" title="Square matrix">square matrices</a>, with <a href="/wiki/Real_number" title="Real number">real</a> entries. More specifically they can be characterized as <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrices</a> with <a href="/wiki/Determinant" title="Determinant">determinant</a> 1: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{T}=R^{-1},\det R=1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mo movablelimits="true" form="prefix">det</mo> <mi>R</mi> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{T}=R^{-1},\det R=1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e505752de06e806363f2066a4bdefb75c83101af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.411ex; height:3.009ex;" alt="{\displaystyle R^{T}=R^{-1},\det R=1\,}"></span>.</dd></dl> <p>The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of all such matrices of size <i>n</i> forms a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, known as the <a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">special orthogonal group</a> <span class="texhtml">SO(<i>n</i>)</span>. </p> <div class="mw-heading mw-heading3"><h3 id="In_two_dimensions">In two dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=4" title="Edit section: In two dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Counterclockwise_rotation_SVG.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Counterclockwise_rotation_SVG.svg/220px-Counterclockwise_rotation_SVG.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Counterclockwise_rotation_SVG.svg/330px-Counterclockwise_rotation_SVG.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Counterclockwise_rotation_SVG.svg/440px-Counterclockwise_rotation_SVG.svg.png 2x" data-file-width="330" data-file-height="330" /></a><figcaption>A counterclockwise rotation of a vector through angle <i>θ</i>. The vector is initially aligned with the x-axis.</figcaption></figure> <p>In two dimensions every rotation matrix has the following form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(\theta )={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(\theta )={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0166e674df67cf24314537211848adec91813945" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.408ex; height:6.176ex;" alt="{\displaystyle R(\theta )={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}"></span>.</dd></dl> <p>This rotates <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vectors</a> by means of the following <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}x'\\y'\\\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}{\begin{bmatrix}x\\y\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}x'\\y'\\\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}{\begin{bmatrix}x\\y\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50622f9a4a7ba2961f5df5f7e0882983cf2f1d2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.502ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}x'\\y'\\\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}{\begin{bmatrix}x\\y\\\end{bmatrix}}}"></span>.</dd></dl> <p>So the coordinates (x',y') of the point (x,y) after rotation are: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'=x\cos \theta -y\sin \theta \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>y</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'=x\cos \theta -y\sin \theta \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/657b520ec337f95a996bc9e77f07401778d272af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.522ex; height:2.843ex;" alt="{\displaystyle x'=x\cos \theta -y\sin \theta \,}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'=x\sin \theta +y\cos \theta \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>y</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'=x\sin \theta +y\cos \theta \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddafa97cf937c752708b51b3ba65d9e4e797e6c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.352ex; height:2.843ex;" alt="{\displaystyle y'=x\sin \theta +y\cos \theta \,}"></span>.</dd></dl> <p>The direction of vector rotation is counterclockwise if θ is positive (e.g. 90°), and clockwise if θ is negative (e.g. -90°). </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(-\theta )={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{bmatrix}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(-\theta )={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{bmatrix}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/177a87efda16575628c480a24d6556e1fed86eff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.604ex; height:6.176ex;" alt="{\displaystyle R(-\theta )={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{bmatrix}}\,}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Non-standard_orientation_of_the_coordinate_system">Non-standard orientation of the coordinate system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=5" title="Edit section: Non-standard orientation of the coordinate system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Clockwise_rotation_SVG.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Clockwise_rotation_SVG.svg/220px-Clockwise_rotation_SVG.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Clockwise_rotation_SVG.svg/330px-Clockwise_rotation_SVG.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Clockwise_rotation_SVG.svg/440px-Clockwise_rotation_SVG.svg.png 2x" data-file-width="330" data-file-height="330" /></a><figcaption>A rotation through angle <i>θ</i> with non-standard axes</figcaption></figure> <p>If a standard <a href="/wiki/Orientation_(space)" class="mw-redirect" title="Orientation (space)">right-handed</a> <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a> is used, with the <i>x</i> axis to the right and the <i>y</i> axis up, the rotation R(<i>θ</i>) is counterclockwise. If a left-handed Cartesian coordinate system is used, with <i>x</i> directed to the right but <i>y</i> directed down, R(<i>θ</i>) is clockwise. Such non-standard orientations are rarely used in mathematics but are common in 2D computer graphics, which often have the origin in the top left corner and the <i>y</i>-axis down the screen or page.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>See <a href="/wiki/Rotation_matrix#Ambiguities" title="Rotation matrix">below</a> for other alternative conventions which may change the sense of the rotation produced by a <a href="/wiki/Rotation_matrix" title="Rotation matrix">rotation matrix</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Common_rotations">Common rotations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=6" title="Edit section: Common rotations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Particularly useful are the matrices for 90° and 180° rotations: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(90^{\circ })={\begin{bmatrix}0&-1\\[3pt]1&0\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="0.7em 0.4em" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(90^{\circ })={\begin{bmatrix}0&-1\\[3pt]1&0\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e500c61fb00503f908125a1347df16aaa30e8080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.968ex; height:7.509ex;" alt="{\displaystyle R(90^{\circ })={\begin{bmatrix}0&-1\\[3pt]1&0\\\end{bmatrix}}}"></span> (90° counterclockwise rotation)</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(180^{\circ })={\begin{bmatrix}-1&0\\[3pt]0&-1\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="0.7em 0.4em" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(180^{\circ })={\begin{bmatrix}-1&0\\[3pt]0&-1\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214804eae9900efc7c423dd9d80402923051365e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.939ex; height:7.509ex;" alt="{\displaystyle R(180^{\circ })={\begin{bmatrix}-1&0\\[3pt]0&-1\\\end{bmatrix}}}"></span> (180° rotation in either direction – a half-turn)</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(270^{\circ })={\begin{bmatrix}0&1\\[3pt]-1&0\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <msup> <mn>270</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="0.7em 0.4em" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(270^{\circ })={\begin{bmatrix}0&1\\[3pt]-1&0\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47626397b5ce1380d23373ea3efbb6133a054a75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.131ex; height:7.509ex;" alt="{\displaystyle R(270^{\circ })={\begin{bmatrix}0&1\\[3pt]-1&0\\\end{bmatrix}}}"></span> (270° counterclockwise rotation, the same as a 90° clockwise rotation)</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Scaling">Scaling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=7" title="Edit section: Scaling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/2D_computer_graphics" title="Special:EditPage/2D computer graphics">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%222D+computer+graphics%22">"2D computer graphics"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%222D+computer+graphics%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%222D+computer+graphics%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%222D+computer+graphics%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%222D+computer+graphics%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%222D+computer+graphics%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">April 2008</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, <b>uniform scaling</b> (<b><a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a> scaling</b>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <b>homogeneous dilation</b>, <a href="/wiki/Homothetic_transformation" class="mw-redirect" title="Homothetic transformation">homothety</a>) is a <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> that enlarges (increases) or shrinks (diminishes) objects by a <a href="/wiki/Scale_factor" class="mw-redirect" title="Scale factor">scale factor</a> that is the same in all directions. The result of uniform scaling is <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a> (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. (Some school text books specifically exclude this possibility, just as some exclude squares from being rectangles or circles from being ellipses.) </p><p>More general is <b>scaling</b> with a separate scale factor for each axis direction. <b>Non-uniform scaling</b> (<b><a href="/wiki/Anisotropic" class="mw-redirect" title="Anisotropic">anisotropic</a> scaling</b>, <b>inhomogeneous dilation</b>) is obtained when at least one of the scaling factors is different from the others; a special case is <b>directional scaling</b> or <b>stretching</b> (in one direction). Non-uniform scaling changes the <a href="/wiki/Shape" title="Shape">shape</a> of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to the scaling axes (the angles between lines parallel to the axes are preserved, but not all angles). </p><p>A scaling can be represented by a scaling matrix. To scale an object by a <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vector</a> <i>v</i> = (<i>v<sub>x</sub>, v<sub>y</sub>, v<sub>z</sub></i>), each point <i>p</i> = (<i>p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub></i>) would need to be multiplied with this <a href="/wiki/Scaling_matrix" class="mw-redirect" title="Scaling matrix">scaling matrix</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4561ec79e4ad2fb898cf7b2f4ec0873a01fb31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:21.303ex; height:9.509ex;" alt="{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}.}"></span></dd></dl> <p>As shown below, the multiplication will give the expected result: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b981072c2d00c1ce373cd483c00fd6d927f668" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:40.26ex; height:9.509ex;" alt="{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\end{bmatrix}}.}"></span></dd></dl> <p>Such a scaling changes the <a href="/wiki/Diameter" title="Diameter">diameter</a> of an object by a factor between the scale factors, the <a href="/wiki/Area" title="Area">area</a> by a factor between the smallest and the largest product of two scale factors, and the <a href="/wiki/Volume" title="Volume">volume</a> by the product of all three. </p><p>The scaling is uniform <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the scaling factors are equal (<i>v<sub>x</sub> = v<sub>y</sub> = v<sub>z</sub></i>). If all except one of the scale factors are equal to 1, we have directional scaling. </p><p>In the case where <i>v<sub>x</sub> = v<sub>y</sub> = v<sub>z</sub> = k</i>, the scaling is also called an <b>enlargement</b> or <b><a href="/wiki/Dilation_(metric_space)" title="Dilation (metric space)">dilation</a></b> by a factor k, increasing the area by a factor of k<sup>2</sup> and the volume by a factor of k<sup>3</sup>. </p><p>Scaling in the most general sense is any <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a> with a <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizable matrix</a>. It includes the case that the three directions of scaling are not perpendicular. It includes also the case that one or more scale factors are equal to zero (<a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">projection</a>), and the case of one or more negative scale factors. The latter corresponds to a combination of scaling proper and a kind of reflection: along lines in a particular direction we take the reflection in the point of intersection with a plane that need not be perpendicular; therefore it is more general than ordinary reflection in the plane. </p> <div class="mw-heading mw-heading3"><h3 id="Using_homogeneous_coordinates">Using homogeneous coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=8" title="Edit section: Using homogeneous coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>, often used in <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>, points are represented using <a href="/wiki/Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a>. To scale an object by a <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vector</a> <i>v</i> = (<i>v<sub>x</sub>, v<sub>y</sub>, v<sub>z</sub></i>), each homogeneous coordinate vector <i>p</i> = (<i>p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub></i>, 1) would need to be multiplied with this <a href="/wiki/Projective_transformation" class="mw-redirect" title="Projective transformation">projective transformation</a> matrix: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba17744c20c70ba66748f68725966e236de30eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:24.789ex; height:12.843ex;" alt="{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}.}"></span></dd></dl> <p>As shown below, the multiplication will give the expected result: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\\1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\\1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/539be147e9c632c58c8a87e4ecf6c2f3d4f857db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:43.745ex; height:12.843ex;" alt="{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\\1\end{bmatrix}}.}"></span></dd></dl> <p>Since the last component of a homogeneous coordinate can be viewed as the denominator of the other three components, a uniform scaling by a common factor <i>s</i> (uniform scaling) can be accomplished by using this scaling matrix: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{v}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2ac4cca4d86727353d98875c9208b7e434bca8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:22.165ex; height:13.176ex;" alt="{\displaystyle S_{v}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}.}"></span></dd></dl> <p>For each vector <i>p</i> = (<i>p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub></i>, 1) we would have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{v}p={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\{\frac {1}{s}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{v}p={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\{\frac {1}{s}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9175fd0f0653a12ae2dcd59bbe45a7e0c33013bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:38.174ex; height:13.509ex;" alt="{\displaystyle S_{v}p={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\{\frac {1}{s}}\end{bmatrix}}}"></span></dd></dl> <p>which would be homogenized to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}sp_{x}\\sp_{y}\\sp_{z}\\1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>s</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}sp_{x}\\sp_{y}\\sp_{z}\\1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5117bb588482f8e8ceb7a94ba373faa08bb615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:7.931ex; height:12.843ex;" alt="{\displaystyle {\begin{bmatrix}sp_{x}\\sp_{y}\\sp_{z}\\1\end{bmatrix}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Techniques">Techniques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=9" title="Edit section: Techniques"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Direct_painting">Direct painting</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=10" title="Edit section: Direct painting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A convenient way to create a complex image is to start with a blank "canvas" <a href="/wiki/Raster_graphics" title="Raster graphics">raster map</a> (an array of <a href="/wiki/Pixel" title="Pixel">pixels</a>, also known as a <a href="/wiki/Bitmap" title="Bitmap">bitmap</a>) filled with some uniform <a href="/wiki/Background_color" class="mw-redirect" title="Background color">background color</a> and then "draw", "paint" or "paste" simple patches of color onto it, in an appropriate order. In particular the canvas may be the <a href="/wiki/Frame_buffer" class="mw-redirect" title="Frame buffer">frame buffer</a> for a <a href="/wiki/Computer_display" class="mw-redirect" title="Computer display">computer display</a>. </p><p>Some programs will set the pixel colors directly, but most will rely on some 2D <a href="/wiki/Graphics_library" title="Graphics library">graphics library</a> or the machine's <a href="/wiki/Graphics_card" title="Graphics card">graphics card</a>, which usually implement the following operations: </p> <ul><li>paste a given <a href="/wiki/Digital_image" title="Digital image">image</a> at a specified offset onto the canvas;</li> <li>write a string of characters with a specified font, at a given position and angle;</li> <li>paint a simple <a href="/wiki/Geometric_shape" class="mw-redirect" title="Geometric shape">geometric shape</a>, such as a <a href="/wiki/Triangle" title="Triangle">triangle</a> defined by three corners, or a <a href="/wiki/Circle" title="Circle">circle</a> with given center and radius;</li> <li>draw a <a href="/wiki/Line_segment" title="Line segment">line segment</a>, <a href="/wiki/Arc_(geometry)" class="mw-redirect" title="Arc (geometry)">arc</a>, or simple curve with a <i>virtual pen</i> of given width.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Extended_color_models">Extended color models</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=11" title="Edit section: Extended color models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Text, shapes and lines are rendered with a client-specified color. Many libraries and cards provide <a href="/wiki/Color_gradient" title="Color gradient">color gradients</a>, which are handy for the generation of smoothly-varying backgrounds, shadow effects, etc. (See also <a href="/wiki/Gouraud_shading" title="Gouraud shading">Gouraud shading</a>). The pixel colors can also be taken from a texture, e.g. a <a href="/wiki/Digital_image" title="Digital image">digital image</a> (thus emulating rub-on <a href="/wiki/Screentone" title="Screentone">screentones</a> and the fabled <i>checker paint</i> which used to be available only in <a href="/wiki/Cartoon" title="Cartoon">cartoons</a>). </p><p>Painting a <a href="/wiki/Pixel" title="Pixel">pixel</a> with a given color usually replaces its previous color. However, many systems support painting with <a href="/wiki/Transparency_(graphic)" title="Transparency (graphic)">transparent</a> and <a href="/wiki/Translucent" class="mw-redirect" title="Translucent">translucent</a> colors, which only modify the previous pixel values. The two colors may also be combined in more complex ways, e.g. by computing their <a href="/wiki/Bitwise_operation" title="Bitwise operation">bitwise</a> <a href="/wiki/Exclusive_or" title="Exclusive or">exclusive or</a>. This technique is known as <i>inverting color</i> or <i>color inversion</i>, and is often used in <a href="/wiki/Graphical_user_interface" title="Graphical user interface">graphical user interfaces</a> for highlighting, rubber-band drawing, and other volatile painting—since re-painting the same shapes with the same color will restore the original pixel values. </p> <div class="mw-heading mw-heading4"><h4 id="Layers">Layers</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=12" title="Edit section: Layers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:BabyAnimation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/BabyAnimation.png/220px-BabyAnimation.png" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/BabyAnimation.png/330px-BabyAnimation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/BabyAnimation.png/440px-BabyAnimation.png 2x" data-file-width="1920" data-file-height="1080" /></a><figcaption>A 2D animated character <a href="/wiki/Compositing" title="Compositing">composited</a> with 3D backgrounds using layers</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Layers_(digital_image_editing)" title="Layers (digital image editing)">Layers (digital image editing)</a></div> <p>The models used in 2D computer graphics usually do not provide for three-dimensional shapes, or three-dimensional optical phenomena such as lighting, <a href="/wiki/Shadow" title="Shadow">shadows</a>, <a href="/wiki/Reflection_(physics)" title="Reflection (physics)">reflection</a>, <a href="/wiki/Refraction" title="Refraction">refraction</a>, etc. However, they usually can model multiple <i>layers</i> (conceptually of ink, paper, or film; opaque, <a href="/wiki/Translucent" class="mw-redirect" title="Translucent">translucent</a>, or <a href="/wiki/Transparency_(graphic)" title="Transparency (graphic)">transparent</a>—stacked in a specific order. The ordering is usually defined by a single number (the layer's <i>depth</i>, or distance from the viewer). </p><p>Layered models are sometimes called "2<style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>-D computer graphics". They make it possible to mimic traditional drafting and printing techniques based on film and paper, such as cutting and pasting; and allow the user to edit any layer without affecting the others. For these reasons, they are used in most <a href="/wiki/Graphics_editor" class="mw-redirect" title="Graphics editor">graphics editors</a>. Layered models also allow better <a href="/wiki/Spatial_anti-aliasing" title="Spatial anti-aliasing">spatial anti-aliasing</a> of complex drawings and provide a sound model for certain techniques such as <i>mitered joints</i> and the <a href="/wiki/Even%E2%80%93odd_rule" title="Even–odd rule">even–odd rule</a>. </p><p>Layered models are also used to allow the user to suppress unwanted information when viewing or printing a document, e.g. roads or railways from a <a href="/wiki/Map" title="Map">map</a>, certain process layers from an <a href="/wiki/Integrated_circuit" title="Integrated circuit">integrated circuit</a> diagram, or hand annotations from a business letter. </p><p>In a layer-based model, the target image is produced by "painting" or "pasting" each layer, in order of decreasing depth, on the virtual canvas. Conceptually, each layer is first <a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">rendered</a> on its own, yielding a <a href="/wiki/Digital_image" title="Digital image">digital image</a> with the desired <a href="/wiki/Image_resolution" title="Image resolution">resolution</a> which is then painted over the canvas, pixel by pixel. Fully transparent parts of a layer need not be rendered, of course. The rendering and painting may be done in parallel, i.e., each layer pixel may be painted on the canvas as soon as it is produced by the rendering procedure. </p><p>Layers that consist of complex geometric objects (such as <a href="/wiki/String_(computer_science)" title="String (computer science)">text</a> or <a href="/wiki/Polyline" class="mw-redirect" title="Polyline">polylines</a>) may be broken down into simpler elements (<a href="/wiki/Character_(computing)" title="Character (computing)">characters</a> or <a href="/wiki/Line_segment" title="Line segment">line segments</a>, respectively), which are then painted as separate layers, in some order. However, this solution may create undesirable <a href="/wiki/Aliasing" title="Aliasing">aliasing</a> artifacts wherever two elements overlap the same pixel. </p><p>See also <a href="/wiki/Portable_Document_Format#Layers" class="mw-redirect" title="Portable Document Format">Portable Document Format#Layers</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Hardware">Hardware</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=13" title="Edit section: Hardware"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Video_display_controller" title="Video display controller">Video display controller</a>, <a href="/wiki/List_of_home_computers_by_video_hardware" title="List of home computers by video hardware">List of home computers by video hardware</a>, and <a href="/wiki/Sprite_(computer_graphics)" title="Sprite (computer graphics)">Sprite (computer graphics)</a></div> <p>Modern computer <a href="/wiki/Graphics_card" title="Graphics card">graphics card</a> displays almost overwhelmingly use raster techniques, dividing the screen into a rectangular grid of <a href="/wiki/Pixel" title="Pixel">pixels</a>, due to the relatively low cost of raster-based video hardware as compared with vector graphic hardware. Most graphic hardware has internal support for <a href="/wiki/BitBLT" class="mw-redirect" title="BitBLT">blitting</a> operations or <a href="/wiki/Sprite_(computer_graphics)" title="Sprite (computer graphics)">sprite</a> drawing. A <a href="/wiki/Co-processor" class="mw-redirect" title="Co-processor">co-processor</a> dedicated to <a href="/wiki/BitBLT" class="mw-redirect" title="BitBLT">blitting</a> is known as a <i><a href="/wiki/Blitter" title="Blitter">Blitter</a> chip</i>. </p><p>Classic 2D <a href="/wiki/Graphics_chip" class="mw-redirect" title="Graphics chip">graphics chips</a> and <a href="/wiki/Graphics_processing_unit" title="Graphics processing unit">graphics processing units</a> of the late 1970s to 1980s, used in <a href="/wiki/8-bit_computing" title="8-bit computing">8-bit</a> to early <a href="/wiki/16-bit_computing" title="16-bit computing">16-bit</a>, <a href="/wiki/Arcade_game" title="Arcade game">arcade games</a>, <a href="/wiki/Video_game_console" title="Video game console">video game consoles</a>, and <a href="/wiki/Home_computer" title="Home computer">home computers</a>, include: </p> <ul><li><a href="/wiki/Atari,_Inc." title="Atari, Inc.">Atari, Inc.</a>'s <a href="/wiki/Television_Interface_Adapter" class="mw-redirect" title="Television Interface Adapter">TIA</a>, <a href="/wiki/ANTIC" title="ANTIC">ANTIC</a>, <a href="/wiki/CTIA_and_GTIA" title="CTIA and GTIA">CTIA and GTIA</a></li> <li><a href="/wiki/Capcom" title="Capcom">Capcom</a>'s <a href="/wiki/CP_System" title="CP System">CPS-A and CPS-B</a></li> <li><a href="/wiki/Commodore_International" title="Commodore International">Commodore</a>'s <a href="/wiki/Original_Chip_Set" class="mw-redirect" title="Original Chip Set">OCS</a></li> <li><a href="/wiki/MOS_Technology" title="MOS Technology">MOS Technology</a>'s <a href="/wiki/MOS_Technology_VIC" title="MOS Technology VIC">VIC</a> and <a href="/wiki/MOS_Technology_VIC-II" title="MOS Technology VIC-II">VIC-II</a></li> <li><a href="/wiki/Hudson_Soft" title="Hudson Soft">Hudson Soft</a>'s <a href="/wiki/X68000#Technical_specifications" title="X68000">Cynthia</a> and <a href="/wiki/Hudson_Soft_HuC6270" title="Hudson Soft HuC6270">HuC6270</a></li> <li><a href="/wiki/NEC" title="NEC">NEC</a>'s <a href="/wiki/NEC_%CE%BCPD7220" title="NEC μPD7220">μPD7220 and μPD72120</a></li> <li><a href="/wiki/Ricoh" title="Ricoh">Ricoh</a>'s <a href="/wiki/Picture_Processing_Unit" class="mw-redirect" title="Picture Processing Unit">PPU</a> and <a href="/wiki/Super_Nintendo_Entertainment_System_technical_specifications" class="mw-redirect" title="Super Nintendo Entertainment System technical specifications">S-PPU</a></li> <li><a href="/wiki/Sega" title="Sega">Sega</a>'s <a href="/wiki/Video_display_controller" title="Video display controller">VDP</a>, <a href="/wiki/Sega_Super_Scaler" class="mw-redirect" title="Sega Super Scaler">Super Scaler</a>, <a href="/wiki/Sega_System_16" class="mw-redirect" title="Sega System 16">315-5011/315-5012</a> and <a href="/wiki/Sega_System_16" class="mw-redirect" title="Sega System 16">315-5196/315-5197</a></li> <li><a href="/wiki/Texas_Instruments" title="Texas Instruments">Texas Instruments</a>' <a href="/wiki/TMS9918" title="TMS9918">TMS9918</a></li> <li><a href="/wiki/Yamaha_Corporation" title="Yamaha Corporation">Yamaha</a>'s <a href="/wiki/Yamaha_V9938" title="Yamaha V9938">V9938</a>, <a href="/wiki/Yamaha_V9958" title="Yamaha V9958">V9958</a> and <a href="/wiki/Mega_Drive#Technical_specifications" class="mw-redirect" title="Mega Drive">YM7101 VDP</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Software">Software</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=14" title="Edit section: Software"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many <a href="/wiki/Graphical_user_interface" title="Graphical user interface">graphical user interfaces</a> (GUIs), including <a href="/wiki/MacOS" title="MacOS">macOS</a>, <a href="/wiki/Microsoft_Windows" title="Microsoft Windows">Microsoft Windows</a>, or the <a href="/wiki/X_Window_System" title="X Window System">X Window System</a>, are primarily based on 2D graphical concepts. Such software provides a visual environment for interacting with the computer, and commonly includes some form of <a href="/wiki/Window_manager" title="Window manager">window manager</a> to aid the user in conceptually distinguishing between different applications. The user interface within individual software applications is typically 2D in nature as well, due in part to the fact that most common <a href="/wiki/Input_device" title="Input device">input devices</a>, such as the <a href="/wiki/Computer_mouse" title="Computer mouse">mouse</a>, are constrained to two dimensions of movement. </p><p>2D graphics are very important in the control peripherals such as printers, plotters, sheet cutting machines, etc. They were also used in most early <a href="/wiki/Video_game" title="Video game">video games</a>; and are still used for card and board games such as <a href="/wiki/Solitaire_(game)" class="mw-redirect" title="Solitaire (game)">solitaire</a>, <a href="/wiki/Chess" title="Chess">chess</a>, <a href="/wiki/Mahjongg" class="mw-redirect" title="Mahjongg">mahjongg</a>, etc. </p><p>2D graphics editors or <i>drawing programs</i> are application-level software for the creation of images, diagrams and illustrations by direct manipulation (through the mouse, <a href="/wiki/Graphics_tablet" title="Graphics tablet">graphics tablet</a>, or similar device) of 2D computer graphics primitives. These editors generally provide <a href="/wiki/2D_geometric_primitive" class="mw-redirect" title="2D geometric primitive">geometric primitives</a> as well as <a href="/wiki/Digital_image" title="Digital image">digital images</a>; and some even support procedural models. The illustration is usually represented internally as a layered model, often with a hierarchical structure to make editing more convenient. These editors generally output <a href="/wiki/Graphics_file_format" class="mw-redirect" title="Graphics file format">graphics files</a> where the layers and primitives are separately preserved in their original form. <a href="/wiki/MacDraw" title="MacDraw">MacDraw</a>, introduced in 1984 with the <a href="/wiki/Apple_Macintosh" class="mw-redirect" title="Apple Macintosh">Macintosh</a> line of computers, was an early example of this class; recent examples are the commercial products <a href="/wiki/Adobe_Illustrator" title="Adobe Illustrator">Adobe Illustrator</a> and <a href="/wiki/CorelDRAW" title="CorelDRAW">CorelDRAW</a>, and the free editors such as <a href="/wiki/Xfig" title="Xfig">xfig</a> or <a href="/wiki/Inkscape" title="Inkscape">Inkscape</a>. There are also many 2D graphics editors specialized for certain types of drawings such as electrical, electronic and VLSI diagrams, topographic maps, computer fonts, etc. </p><p><a href="/wiki/Digital_image_processing" title="Digital image processing">Image editors</a> are specialized for the manipulation of <a href="/wiki/Digital_image" title="Digital image">digital images</a>, mainly by means of free-hand drawing/painting and <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a> operations. They typically use a direct-painting paradigm, where the user controls virtual pens, brushes, and other free-hand artistic instruments to apply paint to a virtual canvas. Some image editors support a multiple-layer model; however, in order to support signal-processing operations like blurring each layer is normally represented as a digital image. Therefore, any geometric primitives that are provided by the editor are immediately converted to pixels and painted onto the canvas. The name <i>raster graphics editor</i> is sometimes used to contrast this approach to that of general editors which also handle <i>vector graphics</i>. One of the first popular image editors was <a href="/wiki/Apple_Computer" class="mw-redirect" title="Apple Computer">Apple</a>'s <a href="/wiki/MacPaint" title="MacPaint">MacPaint</a>, companion to <a href="/wiki/MacDraw" title="MacDraw">MacDraw</a>. Modern examples are the free <a href="/wiki/GIMP" title="GIMP">GIMP</a> editor, and the commercial products <a href="/wiki/Photoshop" class="mw-redirect" title="Photoshop">Photoshop</a> and <a href="/wiki/Paint_Shop_Pro" class="mw-redirect" title="Paint Shop Pro">Paint Shop Pro</a>. This class too includes many specialized editors—for medicine, remote sensing, <a href="/wiki/Digital_photography" title="Digital photography">digital photography</a>, etc. </p> <div class="mw-heading mw-heading2"><h2 id="Developmental_animation">Developmental animation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=15" title="Edit section: Developmental animation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>With the resurgence<sup id="cite_ref-Pile_4-0" class="reference"><a href="#cite_note-Pile-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 8">: 8 </span></sup> of 2D animation, free and proprietary software packages have become widely available for amateurs and professional animators. With software like <a href="/wiki/RETAS" title="RETAS">RETAS</a> <a href="/wiki/UbiArt_Framework" title="UbiArt Framework">UbiArt Framework</a> and <a href="/wiki/Adobe_After_Effects" title="Adobe After Effects">Adobe After Effects</a>, coloring and compositing can be done in less time.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2013)">citation needed</span></a></i>]</sup> </p><p>Various approaches have been developed<sup id="cite_ref-Pile_4-1" class="reference"><a href="#cite_note-Pile-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 38">: 38 </span></sup> to aid and speed up the process of digital 2D animation. For example, by <a href="/wiki/Vector_graphics_editor" title="Vector graphics editor">generating vector artwork</a> in a tool like <a href="/wiki/Adobe_Flash" title="Adobe Flash">Adobe Flash</a> an artist may employ software-driven automatic coloring and <a href="/wiki/Tweening" class="mw-redirect" title="Tweening">in-betweening</a>. </p><p>Programs like <a href="/wiki/Blender_(software)" title="Blender (software)">Blender</a> or <a href="/w/index.php?title=Adobe_Substance&action=edit&redlink=1" class="new" title="Adobe Substance (page does not exist)">Adobe Substance</a> allow the user to do either 3D animation, 2D animation or combine both in its software allowing experimentation with multiple forms of animation.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:2D" class="extiw" title="commons:Category:2D">2D</a></span>.</div></div> </div> <ul><li><a href="/wiki/2.5D" title="2.5D">2.5D</a></li> <li><a href="/wiki/3D_computer_graphics" title="3D computer graphics">3D computer graphics</a></li> <li><a href="/wiki/Computer_animation" title="Computer animation">Computer animation</a></li> <li><a href="/wiki/CGI_animation" class="mw-redirect" title="CGI animation">CGI</a></li> <li><a href="/wiki/Bit_blit" title="Bit blit">Bit blit</a></li> <li><a href="/wiki/Computer_graphics" title="Computer graphics">Computer graphics</a></li> <li><a href="/wiki/Graphic_art_software" title="Graphic art software">Graphic art software</a></li> <li><a href="/wiki/Graphics" title="Graphics">Graphics</a></li> <li><a href="/wiki/Image_scaling" title="Image scaling">Image scaling</a></li> <li><a href="/wiki/List_of_home_computers_by_video_hardware" title="List of home computers by video hardware">List of home computers by video hardware</a></li> <li><a href="/wiki/Turtle_graphics" title="Turtle graphics">Turtle graphics</a></li> <li><a href="/wiki/Transparency_(graphic)" title="Transparency (graphic)">Transparency in graphics</a></li> <li><a href="/wiki/Palette_(computing)" title="Palette (computing)">Palette (computing)</a></li> <li><a href="/wiki/Pixel_art" title="Pixel art">Pixel art</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=2D_computer_graphics&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Richard Paul, 1981, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UzZ3LAYqvRkC">Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators</a>, MIT Press, Cambridge, MA</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://www.w3.org/TR/SVG/coords.html#InitialCoordinateSystem">"Scalable Vector Graphics -- the initial coordinate system"</a>, <i>w3.org</i>, 2003</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=w3.org&rft.atitle=Scalable+Vector+Graphics+--+the+initial+coordinate+system&rft.date=2003&rft_id=http%3A%2F%2Fwww.w3.org%2FTR%2FSVG%2Fcoords.html%23InitialCoordinateSystem&rfr_id=info%3Asid%2Fen.wikipedia.org%3A2D+computer+graphics" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDurandCutler" class="citation web cs1">Durand; Cutler. <a 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Massachusetts Institute of Technology<span class="reference-accessdate">. Retrieved <span class="nowrap">12 September</span> 2008</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Transformations&rft.pub=Massachusetts+Institute+of+Technology&rft.au=Durand&rft.au=Cutler&rft_id=http%3A%2F%2Fgroups.csail.mit.edu%2Fgraphics%2Fclasses%2F6.837%2FF03%2Flectures%2F04_transformations.ppt&rfr_id=info%3Asid%2Fen.wikipedia.org%3A2D+computer+graphics" class="Z3988"></span></span> </li> <li id="cite_note-Pile-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Pile_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Pile_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPile2013" class="citation book cs1"><a href="/w/index.php?title=John_Pile_Jr&action=edit&redlink=1" class="new" title="John Pile Jr (page does not exist)">Pile, John Jr.</a> (May 2013). <a rel="nofollow" class="external text" href="http://www.crcpress.com/product/isbn/9781466501898"><i>2D Graphics Programming for Games</i></a>. New York, NY: CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1466501898" title="Special:BookSources/978-1466501898"><bdi>978-1466501898</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=2D+Graphics+Programming+for+Games&rft.place=New+York%2C+NY&rft.pub=CRC+Press&rft.date=2013-05&rft.isbn=978-1466501898&rft.aulast=Pile&rft.aufirst=John+Jr.&rft_id=http%3A%2F%2Fwww.crcpress.com%2Fproduct%2Fisbn%2F9781466501898&rfr_id=info%3Asid%2Fen.wikipedia.org%3A2D+computer+graphics" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.blender.org/">"blender.org - Home of the Blender project - Free and Open 3D Creation Software"</a>. <i>blender.org</i><span class="reference-accessdate">. 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casting</a></li> <li><a href="/wiki/Skybox_(video_games)" title="Skybox (video games)">Skybox</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/3D_computer_graphics" title="3D computer graphics">3D graphics</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/3D_projection" title="3D projection">3D projection</a></li> <li><a href="/wiki/3D_rendering" title="3D rendering">3D rendering</a></li> <li>(<a href="/wiki/Image-based_modeling_and_rendering" title="Image-based modeling and rendering">Image-based</a></li> <li><a href="/wiki/Spectral_rendering" title="Spectral rendering">Spectral</a></li> <li><a href="/wiki/Unbiased_rendering" title="Unbiased rendering">Unbiased</a>)</li> <li><a href="/wiki/Aliasing" title="Aliasing">Aliasing</a></li> <li><a href="/wiki/Anisotropic_filtering" title="Anisotropic filtering">Anisotropic filtering</a></li> <li><a href="/wiki/Cel_shading" title="Cel shading">Cel shading</a></li> <li><a href="/wiki/Fluid_animation" title="Fluid animation">Fluid animation</a></li> <li><a href="/wiki/Computer_graphics_lighting" title="Computer graphics lighting">Lighting</a> <ul><li><a href="/wiki/Global_illumination" title="Global illumination">Global illumination</a></li></ul></li> <li><a href="/wiki/Hidden-surface_determination" title="Hidden-surface determination">Hidden-surface determination</a></li> <li><a href="/wiki/Polygon_mesh" title="Polygon mesh">Polygon mesh</a></li> <li>(<a href="/wiki/Triangle_mesh" title="Triangle mesh">Triangle mesh</a>)</li> <li><a href="/wiki/Shading" title="Shading">Shading</a> <ul><li><a href="/wiki/Deferred_shading" title="Deferred shading">Deferred</a></li></ul></li> <li><a href="/wiki/Surface_triangulation" title="Surface triangulation">Surface triangulation</a></li> <li><a href="/wiki/Wire-frame_model" title="Wire-frame model">Wire-frame model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_transformation" title="Affine transformation">Affine transformation</a></li> <li><a href="/wiki/Back-face_culling" title="Back-face culling">Back-face culling</a></li> <li><a href="/wiki/Clipping_(computer_graphics)" title="Clipping (computer graphics)">Clipping</a></li> <li><a href="/wiki/Collision_detection" title="Collision detection">Collision detection</a></li> <li><a href="/wiki/Planar_projection" title="Planar projection">Planar projection</a></li> <li><a href="/wiki/Reflection_(computer_graphics)" title="Reflection (computer graphics)">Reflection</a></li> <li><a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">Rendering</a> <ul><li><a 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matrix">Shear matrix</a></li> <li><a href="/wiki/Shader" title="Shader">Shader</a></li> <li><a href="/wiki/Texel_(graphics)" title="Texel (graphics)">Texel</a></li> <li><a href="/wiki/Translation_(geometry)" title="Translation (geometry)">Translation</a></li> <li><a href="/wiki/Volume_rendering" title="Volume rendering">Volume rendering</a></li> <li><a href="/wiki/Voxel" title="Voxel">Voxel</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Graphics_software" title="Graphics software">Graphics software</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_3D_computer_graphics_software" title="List of 3D computer graphics software">3D computer graphics software</a> <ul><li><a href="/wiki/List_of_3D_animation_software" title="List of 3D animation software">animation</a></li> <li><a href="/wiki/List_of_3D_modeling_software" title="List of 3D modeling software">modeling</a></li> <li><a href="/wiki/List_of_3D_rendering_software" title="List of 3D rendering software">rendering</a></li></ul></li> <li><a href="/wiki/Raster_graphics_editor" title="Raster graphics editor">Raster graphics editors</a></li> <li><a href="/wiki/Comparison_of_vector_graphics_editors" title="Comparison of vector graphics editors">Vector graphics editors</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algorithms</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_computer_graphics_algorithms" class="mw-redirect" title="List of computer graphics algorithms">List of computer graphics algorithms</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Visualization_of_technical_information" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Visualization" title="Template:Visualization"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Visualization" title="Template talk:Visualization"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Visualization" title="Special:EditPage/Template:Visualization"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Visualization_of_technical_information" style="font-size:114%;margin:0 4em"><a href="/wiki/Visualization_(graphics)" title="Visualization (graphics)">Visualization</a> of technical information</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Biological_data_visualization" title="Biological data visualization">Biological data visualization</a></li> <li><a href="/wiki/Chemical_imaging" title="Chemical imaging">Chemical imaging</a></li> <li><a href="/wiki/Crime_mapping" title="Crime mapping">Crime mapping</a></li> <li><a href="/wiki/Data_visualization" class="mw-redirect" title="Data visualization">Data visualization</a></li> <li><a href="/wiki/Visualization_(graphics)" title="Visualization (graphics)">Educational visualization</a></li> <li><a href="/wiki/Flow_visualization" title="Flow visualization">Flow visualization</a></li> <li><a href="/wiki/Geovisualization" title="Geovisualization">Geovisualization</a></li> <li><a href="/wiki/Information_visualization" class="mw-redirect" title="Information visualization">Information visualization</a></li> <li><a href="/wiki/Mathematical_diagram" title="Mathematical diagram">Mathematical visualization</a></li> <li><a href="/wiki/Medical_imaging" title="Medical imaging">Medical imaging</a></li> <li><a href="/wiki/Molecular_graphics" title="Molecular graphics">Molecular graphics</a></li> <li><a href="/wiki/Visualization_(graphics)" title="Visualization (graphics)">Product visualization</a></li> <li><a href="/wiki/Scientific_visualization" title="Scientific visualization">Scientific visualization</a></li> <li><a href="/wiki/Social_visualization" title="Social visualization">Social visualization</a></li> <li><a href="/wiki/Software_visualization" title="Software visualization">Software visualization</a></li> <li><a href="/wiki/Technical_drawing" title="Technical drawing">Technical drawing</a></li> <li><a href="/wiki/User_interface_design" title="User interface design">User interface design</a></li> <li><a href="/wiki/Visual_culture" title="Visual culture">Visual culture</a></li> <li><a href="/wiki/Volume_rendering" title="Volume rendering">Volume visualization</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Image <br />types</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chart" title="Chart">Chart</a></li> <li><a href="/wiki/Diagram" title="Diagram">Diagram</a></li> <li><a href="/wiki/Engineering_drawing" title="Engineering drawing">Engineering drawing</a></li> <li><a href="/wiki/Graph_of_a_function" title="Graph of a function">Graph of a function</a></li> <li><a href="/wiki/Ideogram" title="Ideogram">Ideogram</a></li> <li><a href="/wiki/Map" title="Map">Map</a></li> <li><a href="/wiki/Photograph" title="Photograph">Photograph</a></li> <li><a href="/wiki/Pictogram" title="Pictogram">Pictogram</a></li> <li><a href="/wiki/Plot_(graphics)" title="Plot (graphics)">Plot</a></li> <li><a href="/wiki/Sankey_diagram" title="Sankey diagram">Sankey diagram</a></li> <li><a href="/wiki/Schematic" title="Schematic">Schematic</a></li> <li><a href="/wiki/Skeletal_formula" title="Skeletal formula">Skeletal formula</a></li> <li><a href="/wiki/Statistical_graphics" title="Statistical graphics">Statistical graphics</a></li> <li><a href="/wiki/Table_(information)" title="Table (information)">Table</a></li> <li><a href="/wiki/Technical_drawing" title="Technical drawing">Technical drawings</a></li> <li><a href="/wiki/Technical_illustration" title="Technical illustration">Technical illustration</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Pre-19th century</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Edmond_Halley" title="Edmond Halley">Edmond Halley</a></li> <li><a href="/wiki/Charles-Ren%C3%A9_de_Fourcroy" title="Charles-René de Fourcroy">Charles-René de Fourcroy</a></li> <li><a href="/wiki/Joseph_Priestley" title="Joseph Priestley">Joseph Priestley</a></li> <li><a href="/wiki/Gaspard_Monge" title="Gaspard Monge">Gaspard Monge</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">19th century</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Charles_Dupin" title="Charles Dupin">Charles Dupin</a></li> <li><a href="/wiki/Adolphe_Quetelet" title="Adolphe Quetelet">Adolphe Quetelet</a></li> <li><a href="/wiki/Andr%C3%A9-Michel_Guerry" title="André-Michel Guerry">André-Michel Guerry</a></li> <li><a href="/wiki/William_Playfair" title="William Playfair">William Playfair</a></li> <li><a href="/wiki/August_Kekul%C3%A9" title="August Kekulé">August Kekulé</a></li> <li><a href="/wiki/Charles_Joseph_Minard" title="Charles Joseph Minard">Charles Joseph Minard</a></li> <li><a href="/w/index.php?title=Luigi_Perozzo&action=edit&redlink=1" class="new" title="Luigi Perozzo (page does not exist)">Luigi Perozzo</a></li> <li><a href="/wiki/Francis_Amasa_Walker" title="Francis Amasa Walker">Francis Amasa Walker</a></li> <li><a href="/wiki/John_Venn" title="John Venn">John Venn</a></li> <li><a href="/wiki/Oliver_Byrne_(mathematician)" title="Oliver Byrne (mathematician)">Oliver Byrne</a></li> <li><a href="/wiki/Matthew_Henry_Phineas_Riall_Sankey" title="Matthew Henry Phineas Riall Sankey">Matthew Sankey</a></li> <li><a href="/wiki/Charles_Booth_(social_reformer)" title="Charles Booth (social reformer)">Charles Booth</a></li> <li><a href="/w/index.php?title=Georg_von_Mayr&action=edit&redlink=1" class="new" title="Georg von Mayr (page does not exist)">Georg von Mayr</a></li> <li><a href="/wiki/John_Snow" title="John Snow">John Snow</a></li> <li><a href="/wiki/Florence_Nightingale" title="Florence Nightingale">Florence Nightingale</a></li> <li><a href="/wiki/Karl_Wilhelm_Pohlke" title="Karl Wilhelm Pohlke">Karl Wilhelm Pohlke</a></li> <li><a href="/wiki/Toussaint_Loua" title="Toussaint Loua">Toussaint Loua</a></li> <li><a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Early 20th century</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Edward_Walter_Maunder" title="Edward Walter Maunder">Edward Walter Maunder</a></li> <li><a href="/wiki/Otto_Neurath" title="Otto Neurath">Otto Neurath</a></li> <li><a href="/wiki/W._E._B._Du_Bois" title="W. E. B. Du Bois">W. E. B. Du Bois</a></li> <li><a href="/wiki/Henry_Gantt" title="Henry Gantt">Henry Gantt</a></li> <li><a href="/wiki/Arthur_Lyon_Bowley" title="Arthur Lyon Bowley">Arthur Lyon Bowley</a></li> <li><a href="/wiki/Howard_G._Funkhouser" title="Howard G. Funkhouser">Howard G. Funkhouser</a></li> <li><a href="/wiki/John_B._Peddle" title="John B. Peddle">John B. Peddle</a></li> <li><a href="/wiki/Ejnar_Hertzsprung" title="Ejnar Hertzsprung">Ejnar Hertzsprung</a></li> <li><a href="/wiki/Henry_Norris_Russell" title="Henry Norris Russell">Henry Norris Russell</a></li> <li><a href="/wiki/Max_O._Lorenz" title="Max O. Lorenz">Max O. Lorenz</a></li> <li><a href="/wiki/Fritz_Kahn" title="Fritz Kahn">Fritz Kahn</a></li> <li><a href="/wiki/Harry_Beck" title="Harry Beck">Harry Beck</a></li> <li><a href="/wiki/Erwin_Raisz" title="Erwin Raisz">Erwin Raisz</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mid 20th century</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Jacques_Bertin" title="Jacques Bertin">Jacques Bertin</a></li> <li><a href="/wiki/Rudolf_Modley" title="Rudolf Modley">Rudolf Modley</a></li> <li><a href="/wiki/Arthur_H._Robinson" title="Arthur H. Robinson">Arthur H. Robinson</a></li> <li><a href="/wiki/John_Tukey" title="John Tukey">John Tukey</a></li> <li><a href="/wiki/Mary_Eleanor_Spear" title="Mary Eleanor Spear">Mary Eleanor Spear</a></li> <li><a href="/wiki/Edgar_Anderson" title="Edgar Anderson">Edgar Anderson</a></li> <li><a href="/wiki/Howard_T._Fisher" title="Howard T. Fisher">Howard T. Fisher</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Late 20th century</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Borden_Dent" title="Borden Dent">Borden Dent</a></li> <li><a href="/wiki/Nigel_Holmes" title="Nigel Holmes">Nigel Holmes</a></li> <li><a href="/wiki/William_S._Cleveland" title="William S. Cleveland">William S. Cleveland</a></li> <li><a href="/wiki/George_G._Robertson" title="George G. Robertson">George G. Robertson</a></li> <li><a href="/wiki/Bruce_H._McCormick" title="Bruce H. McCormick">Bruce H. McCormick</a></li> <li><a href="/wiki/Catherine_Plaisant" title="Catherine Plaisant">Catherine Plaisant</a></li> <li><a href="/wiki/Stuart_Card" title="Stuart Card">Stuart Card</a></li> <li><a href="/wiki/Pat_Hanrahan" title="Pat Hanrahan">Pat Hanrahan</a></li> <li><a href="/wiki/Edward_Tufte" title="Edward Tufte">Edward Tufte</a></li> <li><a href="/wiki/Ben_Shneiderman" title="Ben Shneiderman">Ben Shneiderman</a></li> <li><a href="/wiki/Michael_Friendly" title="Michael Friendly">Michael Friendly</a></li> <li><a href="/wiki/Howard_Wainer" title="Howard Wainer">Howard Wainer</a></li> <li><a href="/wiki/Clifford_A._Pickover" title="Clifford A. Pickover">Clifford A. Pickover</a></li> <li><a href="/wiki/Lawrence_J._Rosenblum" title="Lawrence J. Rosenblum">Lawrence J. Rosenblum</a></li> <li><a href="/wiki/Thomas_A._DeFanti" title="Thomas A. DeFanti">Thomas A. DeFanti</a></li> <li><a href="/wiki/George_Furnas" title="George Furnas">George Furnas</a></li> <li><a href="/wiki/Sheelagh_Carpendale" title="Sheelagh Carpendale">Sheelagh Carpendale</a></li> <li><a href="/wiki/Cynthia_Brewer" title="Cynthia Brewer">Cynthia Brewer</a></li> <li><a href="/wiki/Jock_D._Mackinlay" title="Jock D. Mackinlay">Jock D. Mackinlay</a></li> <li><a href="/wiki/Alan_MacEachren" title="Alan MacEachren">Alan MacEachren</a></li> <li><a href="/wiki/David_Goodsell" title="David Goodsell">David Goodsell</a></li> <li><a href="/wiki/Kwan-Liu_Ma" title="Kwan-Liu Ma">Kwan-Liu Ma</a></li> <li><a href="/wiki/Michael_Maltz" class="mw-redirect" title="Michael Maltz">Michael Maltz</a></li> <li><a href="/wiki/Leland_Wilkinson" title="Leland Wilkinson">Leland Wilkinson</a></li> <li><a href="/wiki/Alfred_Inselberg" title="Alfred Inselberg">Alfred Inselberg</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Early 21st century</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=Polo_Chau&action=edit&redlink=1" class="new" title="Polo Chau (page does not exist)">Polo Chau</a></li> <li><a href="/wiki/Ben_Fry" title="Ben Fry">Ben Fry</a></li> <li><a href="/wiki/Jeffrey_Heer" title="Jeffrey Heer">Jeffrey Heer</a></li> <li><a href="/wiki/Jessica_Hullman" title="Jessica Hullman">Jessica Hullman</a></li> <li><a href="/wiki/Gordon_Kindlmann" title="Gordon Kindlmann">Gordon Kindlmann</a></li> <li><a href="/wiki/Aaron_Koblin" title="Aaron Koblin">Aaron Koblin</a></li> <li><a href="/w/index.php?title=Martin_Krzywinski&action=edit&redlink=1" class="new" title="Martin Krzywinski (page does not exist)">Martin Krzywinski</a></li> <li><a href="/wiki/Christopher_R._Johnson" title="Christopher R. Johnson">Christopher R. Johnson</a></li> <li><a href="/wiki/Manuel_Lima" title="Manuel Lima">Manuel Lima</a></li> <li><a href="/wiki/David_McCandless" title="David McCandless">David McCandless</a></li> <li><a href="/wiki/Mauro_Martino" title="Mauro Martino">Mauro Martino</a></li> <li><a href="/wiki/John_Maeda" title="John Maeda">John Maeda</a></li> <li><a href="/wiki/Miriah_Meyer" title="Miriah Meyer">Miriah Meyer</a></li> <li><a href="/wiki/Tamara_Munzner" title="Tamara Munzner">Tamara Munzner</a></li> <li><a href="/wiki/Ade_Olufeko" class="mw-redirect" title="Ade Olufeko">Ade Olufeko</a></li> <li><a href="/wiki/Hanspeter_Pfister" title="Hanspeter Pfister">Hanspeter Pfister</a></li> <li><a href="/wiki/Hans_Rosling" title="Hans Rosling">Hans Rosling</a></li> <li><a href="/wiki/Claudio_Silva_(computer_scientist)" title="Claudio Silva (computer scientist)">Claudio Silva</a></li> <li><a href="/wiki/Moritz_Stefaner" title="Moritz Stefaner">Moritz Stefaner</a></li> <li><a href="/wiki/Fernanda_Vi%C3%A9gas" title="Fernanda Viégas">Fernanda Viégas</a></li> <li><a href="/wiki/Martin_M._Wattenberg" title="Martin M. Wattenberg">Martin Wattenberg</a></li> <li><a href="/wiki/Bang_Wong" title="Bang Wong">Bang Wong</a></li> <li><a href="/wiki/Hadley_Wickham" title="Hadley Wickham">Hadley Wickham</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related <br />topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cartography" title="Cartography">Cartography</a></li> <li><a href="/wiki/Chartjunk" title="Chartjunk">Chartjunk</a></li> <li><a href="/wiki/Color_coding_in_data_visualization" class="mw-redirect" title="Color coding in data visualization">Color coding</a></li> <li><a href="/wiki/Computer_graphics" title="Computer graphics">Computer graphics</a> <ul><li><a href="/wiki/Computer_graphics_(computer_science)" title="Computer graphics (computer science)">in computer science</a></li></ul></li> <li><a href="/wiki/CPK_coloring" title="CPK coloring">CPK coloring</a></li> <li><a href="/wiki/Graph_drawing" title="Graph drawing">Graph drawing</a></li> <li><a href="/wiki/Graphic_design" title="Graphic design">Graphic design</a></li> <li><a href="/wiki/Graphic_organizer" title="Graphic organizer">Graphic organizer</a></li> <li><a href="/wiki/Imaging_science" class="mw-redirect" title="Imaging science">Imaging science</a></li> <li><a href="/wiki/Infographic" title="Infographic">Information graphics</a></li> <li><a href="/wiki/Information_science" title="Information science">Information science</a></li> <li><a href="/wiki/Misleading_graph" title="Misleading graph">Misleading graph</a></li> <li><a href="/wiki/Neuroimaging" title="Neuroimaging">Neuroimaging</a></li> <li><a href="/wiki/Patent_drawing" title="Patent drawing">Patent drawing</a></li> <li><a href="/wiki/Scientific_modelling" title="Scientific modelling">Scientific modelling</a></li> <li><a href="/wiki/Spatial_analysis" title="Spatial analysis">Spatial analysis</a></li> <li><a href="/wiki/Visual_analytics" title="Visual analytics">Visual analytics</a></li> <li><a href="/wiki/Visual_perception" title="Visual perception">Visual perception</a></li> <li><a href="/wiki/Volume_cartography" title="Volume cartography">Volume cartography</a></li> <li><a href="/wiki/Volume_rendering" title="Volume rendering">Volume rendering</a></li> <li><a href="/wiki/Information_art" title="Information art">Information art</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Animation_topics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Animation" title="Template:Animation"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Animation" title="Template talk:Animation"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Animation" title="Special:EditPage/Template:Animation"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Animation_topics" style="font-size:114%;margin:0 4em"><a href="/wiki/Animation" title="Animation">Animation</a> topics</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By country</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bangladeshi_animation" title="Bangladeshi animation">Bangladesh</a></li> <li><a href="/wiki/Bhutanese_animation" title="Bhutanese animation">Bhutan</a></li> <li><a href="/wiki/Chinese_animation" title="Chinese animation">China</a></li> <li><a href="/wiki/Czech_animation" title="Czech animation">Czechia</a></li> <li><a href="/wiki/Estonian_animation" title="Estonian animation">Estonia</a></li> <li><a href="/wiki/Indian_animation_industry" title="Indian animation industry">India</a></li> <li><a href="/wiki/Anime" title="Anime">Japan</a></li> <li><a href="/wiki/Malaysian_animation" title="Malaysian animation">Malaysia</a></li> <li><a href="/wiki/Mexican_animation" title="Mexican animation">Mexico</a></li> <li><a href="/wiki/North_Korean_animation" title="North Korean animation">North Korea</a></li> <li><a href="/wiki/Filipino_animation" class="mw-redirect" title="Filipino animation">Philippines</a></li> <li><a href="/wiki/Portuguese_animation" title="Portuguese animation">Portugal</a></li> <li><a href="/wiki/Romanian_animation" title="Romanian animation">Romania</a></li> <li><a href="/wiki/Animation_industry_in_South_Africa" title="Animation industry in South Africa">South Africa</a></li> <li><a href="/wiki/South_Korean_animation" title="South Korean animation">South Korea</a></li> <li><a href="/wiki/Spanish_animation" title="Spanish animation">Spain</a></li> <li><a href="/wiki/Taiwanese_animation" title="Taiwanese animation">Taiwan</a></li> <li><a href="/wiki/Thai_animation" title="Thai animation">Thailand</a></li> <li><a href="/wiki/American_animation" title="American animation">United States</a></li> <li><a href="/wiki/Vietnamese_animation" title="Vietnamese animation">Vietnam</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_animation" title="History of animation">History</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_Azerbaijani_animation" class="mw-redirect" title="History of Azerbaijani animation">Azerbaijan</a></li> <li><a href="/wiki/Bangladeshi_animation#History" title="Bangladeshi animation">Bangladesh</a></li> <li><a href="/wiki/History_of_Brazilian_animation" class="mw-redirect" title="History of Brazilian animation">Brazil</a></li> <li><a href="/wiki/History_of_Canadian_animation" title="History of Canadian animation">Canada</a></li> <li><a href="/wiki/History_of_Chinese_animation" title="History of Chinese animation">China</a></li> <li><a href="/wiki/History_of_French_animation" title="History of French animation">France</a></li> <li><a href="/wiki/History_of_Hungarian_animation" title="History of Hungarian animation">Hungary</a></li> <li><a href="/wiki/History_of_animation_in_India" title="History of animation in India">India</a></li> <li><a href="/wiki/History_of_Iranian_animation" class="mw-redirect" title="History of Iranian animation">Iran</a></li> <li><a href="/wiki/History_of_anime" title="History of anime">Japan</a></li> <li><a href="/wiki/History_of_Korean_animation" title="History of Korean animation">Korea</a></li> <li><a href="/wiki/History_of_Russian_animation" title="History of Russian animation">Russia</a></li> <li><a href="/wiki/History_of_Ukrainian_animation" class="mw-redirect" title="History of Ukrainian animation">Ukraine</a></li> <li><a href="/wiki/History_of_British_animation" class="mw-redirect" title="History of British animation">United Kingdom</a></li> <li>United States <ul><li><a href="/wiki/Animation_in_the_United_States_during_the_silent_era" title="Animation in the United States during the silent era">Silent Era</a></li> <li><a href="/wiki/Golden_age_of_American_animation" title="Golden age of American animation">The Golden Age</a></li> <li><a href="/wiki/World_War_II_and_American_animation" title="World War II and American animation">World War II</a></li> <li><a href="/wiki/Animation_in_the_United_States_in_the_television_era" title="Animation in the United States in the television era">Early TV broadcast era</a></li> <li><a href="/wiki/Modern_animation_in_the_United_States" title="Modern animation in the United States">Modern TV cable and streaming era</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Industry</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Animator" title="Animator">Animator</a> <ul><li><a href="/wiki/List_of_animators" title="List of animators">List</a></li></ul></li> <li><a href="/wiki/Animation_department" title="Animation department">Animation department</a></li> <li><a href="/wiki/Animation_director" title="Animation director">Animation director</a></li> <li><a href="/wiki/Story_artist" title="Story artist">Story artist</a></li> <li><a href="/wiki/Animation_studio" title="Animation studio">Animation studios</a> <ul><li><a href="/wiki/List_of_animation_studios" title="List of animation studios">List</a></li></ul></li> <li><a href="/wiki/Animation_database" title="Animation database">Animation database</a></li> <li><a href="/wiki/Art_pipeline" title="Art pipeline">Art pipeline</a></li> <li><a href="/wiki/Category:Anatomical_simulation" title="Category:Anatomical simulation">Biologist simulators</a></li> <li><a href="/wiki/Category:Animation_film_festivals" title="Category:Animation film festivals">Animation film festivals</a> <ul><li><a href="/wiki/List_of_international_animation_festivals" title="List of international animation festivals">international</a></li> <li><a href="/wiki/List_of_regional_animation_festivals" title="List of regional animation festivals">regional</a></li></ul></li> <li><a href="/wiki/List_of_highest-grossing_animated_films" title="List of highest-grossing animated films">Highest-grossing films</a> (<a href="/wiki/List_of_highest-grossing_openings_for_animated_films" title="List of highest-grossing openings for animated films">Opening weekends</a>)</li> <li><a href="/wiki/Outsourcing_of_animation" title="Outsourcing of animation">Outsourcing</a></li> <li><a href="/wiki/International_Animation_Day" title="International Animation Day">International Animation Day</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Works</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Films <ul><li><a href="/wiki/List_of_computer-animated_films" title="List of computer-animated films">Computer-animated</a></li> <li><a href="/wiki/Lists_of_animated_films" title="Lists of animated films">Feature-length</a></li> <li><a href="/wiki/List_of_lost_or_unfinished_animated_films" title="List of lost or unfinished animated films">Lost or unfinished</a></li> <li><a href="/wiki/List_of_animated_package_films" title="List of animated package films">Package</a></li> <li><a href="/wiki/List_of_animated_short_films" title="List of animated short films">Short</a></li> <li><a href="/wiki/List_of_animated_short_series" class="mw-redirect" title="List of animated short series">Short series</a></li> <li><a href="/wiki/List_of_stop_motion_films" title="List of stop motion films">Stop-motion</a></li> <li><a href="/wiki/List_of_adult_animated_feature_films" title="List of adult animated feature films">Adult animated films</a></li> <li><a href="/wiki/List_of_children%27s_animated_films" title="List of children's animated films">Children's animated films</a></li></ul></li> <li><a href="/wiki/Animated_series" title="Animated series">Series</a> <ul><li><a href="/wiki/List_of_adult_animated_television_series" title="List of adult animated television series">Adult animated</a></li> <li><a href="/wiki/List_of_children%27s_animated_television_series" title="List of children's animated television series">Children's animated</a></li> <li><a href="/wiki/List_of_computer-animated_television_series" title="List of computer-animated television series">Computer-animated</a></li> <li><a href="/wiki/List_of_animated_direct-to-video_series" title="List of animated direct-to-video series">Direct-to-video</a></li> <li><a href="/wiki/List_of_Flash_animated_television_series" title="List of Flash animated television series">Flash</a></li> <li><a href="/wiki/List_of_animated_Internet_series" title="List of animated Internet series">Internet</a></li> <li><a href="/wiki/Lists_of_animated_television_series" title="Lists of animated television series">Television</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Techniques</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Traditional_animation" title="Traditional animation">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrier-grid_animation_and_stereography" title="Barrier-grid animation and stereography">Barrier-grid and stereography</a></li> <li><a href="/wiki/Flip_book" title="Flip book">Flip book</a></li> <li><a href="/wiki/Limited_animation" title="Limited animation">Limited animation</a></li> <li><a href="/wiki/Masking_(art)" title="Masking (art)">Masking</a></li> <li><a href="/wiki/Rotoscoping" title="Rotoscoping">Rotoscoping</a></li> <li><a href="/wiki/Exposure_sheet" title="Exposure sheet">Exposure sheet</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Stop_motion" title="Stop motion">Stop motion</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Claymation" title="Claymation">Claymation</a> <ul><li><a href="/wiki/Clay_painting" title="Clay painting">clay painting</a>, <a href="/wiki/Strata-cut_animation" title="Strata-cut animation">strata-cut</a></li></ul></li> <li><a href="/wiki/Cutout_animation" title="Cutout animation">Cutout</a> (<a href="/wiki/Silhouette_animation" title="Silhouette animation">silhouette</a>)</li> <li><a href="/wiki/Graphic_animation" class="mw-redirect" title="Graphic animation">Graphic</a></li> <li><a href="/wiki/Model_animation" title="Model animation">Model</a> <ul><li><a href="/wiki/Go_motion" title="Go motion">go motion</a></li></ul></li> <li><a href="/wiki/Object_animation" title="Object animation">Object</a> <ul><li><a href="/wiki/Brickfilm" title="Brickfilm">Brickfilm</a></li></ul></li> <li><a href="/wiki/Pixilation" title="Pixilation">Pixilation</a></li> <li><a href="/wiki/Puppetoons" title="Puppetoons">Puppetoons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_animation" title="Computer animation">Computer</a> <br />(<a href="/wiki/History_of_computer_animation" title="History of computer animation">history</a>, <a href="/wiki/Timeline_of_computer_animation_in_film_and_television" title="Timeline of computer animation in film and television">timeline</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">2D</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/2.5D" title="2.5D">2.5D</a></li> <li><a href="/wiki/Flash_animation" title="Flash animation">Flash</a></li> <li><a href="/wiki/PowerPoint_animation" title="PowerPoint animation">PowerPoint</a></li> <li><a href="/wiki/SVG_animation" title="SVG animation">SVG</a></li> <li><a href="/wiki/CSS_animations" title="CSS animations">CSS</a></li> <li><a href="/wiki/Multi-sketch" title="Multi-sketch">Multi-sketch</a></li> <li><a href="/wiki/Onion_skinning" title="Onion skinning">Onion skinning</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/3D_computer_graphics" title="3D computer graphics">3D</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/T-pose" title="T-pose">T-pose</a></li> <li><a href="/wiki/Cel_shading" title="Cel shading">Cel shading</a></li> <li><a href="/wiki/Computer-generated_imagery" title="Computer-generated imagery">CGI</a></li> <li><a href="/wiki/Crowd_simulation" title="Crowd simulation">Crowd</a></li> <li><a href="/wiki/Computer_facial_animation" title="Computer facial animation">Facial animation</a></li> <li><a href="/wiki/Morph_target_animation" title="Morph target animation">Morph target</a></li> <li><a href="/wiki/Motion_capture" title="Motion capture">Motion capture</a> <ul><li><a href="/wiki/Facial_motion_capture" title="Facial motion capture">facial</a></li> <li><a href="/wiki/Finger_tracking" title="Finger tracking">hand tracking</a></li> <li><a href="/wiki/Eye_tracking" title="Eye tracking">eye tracking</a></li></ul></li> <li><a href="/wiki/Non-photorealistic_rendering" title="Non-photorealistic rendering">Non-photorealistic rendering</a></li> <li><a href="/wiki/Physically_based_animation" title="Physically based animation">Physically based animation</a></li> <li><a href="/wiki/Procedural_animation" title="Procedural animation">Procedural</a></li> <li><a href="/wiki/Skeletal_animation" title="Skeletal animation">Skeletal</a></li> <li><a href="/wiki/Virtual_cinematography" title="Virtual cinematography">Virtual cinematography</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Puppetry" title="Puppetry">Puppetry</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:transparent; font-weight:normal;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Puppet" title="Puppet">Traditional puppetry</a></li> <li><a href="/wiki/Digital_puppetry" title="Digital puppetry">Digital puppetry</a> <ul><li><a href="/wiki/Machinima" title="Machinima">Machinima</a></li> <li><a href="/wiki/Aniforms" title="Aniforms">Aniforms</a></li> <li><a href="/wiki/Virtual_human" title="Virtual human">Virtual human</a></li> <li><a href="/wiki/Live2D" title="Live2D">Live2D</a></li></ul></li> <li><a href="/wiki/Supermarionation" title="Supermarionation">Supermarionation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mechanics" title="Mechanics">Mechanical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Animatronics" title="Animatronics">Animatronics</a> <ul><li><a href="/wiki/Audio-Animatronics" title="Audio-Animatronics">Audio-Animatronics</a></li> <li><a href="/wiki/Linear_alternator" title="Linear alternator">Linear Animation Generator</a></li></ul></li> <li><a href="/wiki/Direct_manipulation_animation" title="Direct manipulation animation">Direct manipulation animation</a></li> <li><a href="/wiki/Humanoid_animation" title="Humanoid animation">Humanoid animation</a></li> <li><a href="/wiki/Idle_animation" title="Idle animation">Idle animation</a></li> <li><a href="/wiki/Ink-wash_animation" class="mw-redirect" title="Ink-wash animation">Ink-wash animation</a></li> <li><a href="/wiki/Magic_lantern" title="Magic lantern">Magic Lantern</a></li> <li><a href="/wiki/Scanimate" title="Scanimate">Scanimate</a></li> <li><a href="/wiki/Shadowmation" title="Shadowmation">Shadowmation</a></li> <li><a href="/wiki/Squigglevision" title="Squigglevision">Squigglevision</a></li> <li><a href="/wiki/Whiteboard_animation" title="Whiteboard animation">Whiteboard animation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other methods</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:transparent; font-weight:normal;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blocking_(animation)" title="Blocking (animation)">Blocking</a></li> <li><a href="/wiki/Character_animation" title="Character animation">Character animation</a> <ul><li><a href="/wiki/Model_sheet" title="Model sheet">model sheet</a></li> <li><a href="/wiki/Walk_cycle" title="Walk cycle">walk cycle</a></li> <li><a href="/wiki/Lip_sync" title="Lip sync">lip sync</a></li> <li><a href="/wiki/Off-model" title="Off-model">off-model</a></li></ul></li> <li><a href="/wiki/Chuckimation" class="mw-redirect" title="Chuckimation">Chuckimation</a></li> <li><a href="/wiki/Drawn-on-film_animation" title="Drawn-on-film animation">Drawn-on-film</a></li> <li><a href="/wiki/Charcoal" title="Charcoal">Erasure animation</a></li> <li><a href="/wiki/Hydraulic_engineering" title="Hydraulic engineering">Hydrotechnics</a></li> <li><a href="/wiki/Inbetweening" title="Inbetweening">Inbetweening</a></li> <li><a href="/wiki/Morphing" title="Morphing">Morphing</a></li> <li><a href="/wiki/Paint-on-glass_animation" title="Paint-on-glass animation">Paint-on-glass</a></li> <li><a href="/wiki/Pinscreen_animation" title="Pinscreen animation">Pinscreen</a></li> <li><a href="/wiki/Pixel_artist" class="mw-redirect" title="Pixel artist">Pixel art</a></li> <li><a href="/wiki/Pose_to_pose_animation" title="Pose to pose animation">Pose to pose</a></li> <li><a href="/wiki/Straight_ahead_animation" title="Straight ahead animation">Straight ahead</a></li> <li><a href="/wiki/Rubber_hose_animation" title="Rubber hose animation">Rubber hose</a></li> <li><a href="/wiki/Special_effects" class="mw-redirect" title="Special effects">Special effects</a></li> <li><a href="/wiki/Sand_animation" title="Sand animation">Sand</a></li> <li><a href="/wiki/Syncro-Vox" title="Syncro-Vox">Syncro-Vox</a></li> <li><a href="/wiki/Zoetrope" title="Zoetrope">Zoetrope</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Variants</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_animation" class="mw-redirect" title="Abstract animation">Abstract animation</a> (<a href="/wiki/Visual_music" title="Visual music">visual music</a>)</li> <li><a href="/wiki/Adult_animation" title="Adult animation">Adult animation</a></li> <li><a href="/wiki/Animated_cartoon" class="mw-redirect" title="Animated cartoon">Animated cartoon</a></li> <li><a href="/wiki/Animated_sitcom" title="Animated sitcom">Animated sitcom</a></li> <li><a href="/wiki/Animated_documentary" title="Animated documentary">Animated documentary</a></li> <li><a href="/wiki/Anime" title="Anime">Anime</a></li> <li><a href="/wiki/Educational_animation" title="Educational animation">Educational animation</a></li> <li><a href="/wiki/Erotic_animation" class="mw-redirect" title="Erotic animation">Erotic animation</a></li> <li><a href="/wiki/Independent_animation" title="Independent animation">Independent animation</a></li> <li><a href="/wiki/Instructional_animation" title="Instructional animation">Instructional animation</a></li> <li><a href="/wiki/Virtual_newscaster" title="Virtual newscaster">Virtual newscaster</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Animation_music" title="Animation music">Animation music</a> <ul><li><a href="/wiki/Bouncing_ball_(music)" title="Bouncing ball (music)">Bouncing ball</a></li> <li><a href="/wiki/Mickey_Mousing" title="Mickey Mousing">Mickey Mousing</a></li></ul></li> <li><a href="/wiki/Key_frame" title="Key frame">Key frame</a></li> <li><a href="/wiki/Cel" title="Cel">Cel</a></li> <li><a href="/wiki/Character_animation" title="Character animation">Character animation</a> <ul><li><a href="/wiki/Model_sheet" title="Model sheet">model sheet</a></li> <li><a href="/wiki/Walk_cycle" title="Walk cycle">walk cycle</a></li> <li><a href="/wiki/Lip_sync" title="Lip sync">lip sync</a></li> <li><a href="/wiki/Off-model" title="Off-model">off-model</a></li></ul></li> <li><a href="/wiki/Creature_animation" title="Creature animation">Creature animation</a></li> <li><a href="/wiki/Twelve_basic_principles_of_animation" title="Twelve basic principles of animation">Twelve basic principles</a></li> <li><a href="/wiki/Motion_comic" title="Motion comic">Motion comic</a></li> <li><a href="/wiki/List_of_films_with_live_action_and_animation" title="List of films with live action and animation">Films with live action and animation</a> <ul><li><a href="/wiki/List_of_highest-grossing_live-action/animated_films" title="List of highest-grossing live-action/animated films">highest grossing</a></li></ul></li> <li><a href="/wiki/Cartoon_physics" title="Cartoon physics">Cartoon physics</a></li> <li><a href="/wiki/Cartoon_violence" title="Cartoon violence">Cartoon violence</a></li> <li><a href="/wiki/List_of_most_expensive_animated_films" title="List of most expensive animated films">Most expensive animated films</a></li> <li><a href="/wiki/List_of_animated_films_by_box_office_admissions" title="List of animated films by box office admissions">List of animated films by box office admissions</a></li> <li><a href="/wiki/List_of_animated_television_series_by_episode_count" title="List of animated television series by episode count">List of animated television series by episode count</a> <ul><li><a href="/wiki/List_of_anime_series_by_episode_count" title="List of anime series by episode count">anime series</a> <ul><li><a href="/wiki/List_of_anime_franchises_by_episode_count" title="List of anime franchises by episode count">anime franchises</a></li></ul></li></ul></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" 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2D computer graphics - Wikipedia