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class="vector-toc-numb">3</span> <span>As a group</span> </div> </a> <button aria-controls="toc-As_a_group-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 As a group subsection</span> </button> <ul id="toc-As_a_group-sublist" class="vector-toc-list"> <li id="toc-Lattice_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lattice_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Lattice groups</span> </div> </a> <ul id="toc-Lattice_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Matrix_representation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matrix_representation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Matrix representation</span> </div> </a> <ul id="toc-Matrix_representation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Translation_of_axes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Translation_of_axes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Translation of axes</span> </div> </a> <ul id="toc-Translation_of_axes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Translational_symmetry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Translational_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Translational symmetry</span> </div> </a> <ul id="toc-Translational_symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Translations_of_a_graph" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Translations_of_a_graph"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Translations of a graph</span> </div> </a> <ul id="toc-Translations_of_a_graph-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-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">9</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">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Translation (geometry)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input 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 40 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-40" 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">40 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%A7%D9%86%D8%B3%D8%AD%D8%A7%D8%A8_(%D9%87%D9%86%D8%AF%D8%B3%D8%A9)" title="انسحاب (هندسة) – Arabic" lang="ar" hreflang="ar" data-title="انسحاب (هندسة)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D1%80%D0%B0%D0%BD%D1%81%D0%BB%D0%B0%D1%86%D0%B8%D1%8F" title="Транслация – Bulgarian" lang="bg" hreflang="bg" data-title="Транслация" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Translaci%C3%B3_(geometria)" title="Translació (geometria) – Catalan" lang="ca" hreflang="ca" data-title="Translació (geometria)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D0%BB%C4%95_%D0%BA%D1%83%C3%A7%D0%B0%D1%80%D1%83" title="Параллеллĕ куçару – Chuvash" lang="cv" hreflang="cv" data-title="Параллеллĕ куçару" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Posunut%C3%AD_(geometrie)" title="Posunutí (geometrie) – Czech" lang="cs" hreflang="cs" data-title="Posunutí (geometrie)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Translation_(geometri)" title="Translation (geometri) – Danish" lang="da" hreflang="da" data-title="Translation (geometri)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Parallelverschiebung" title="Parallelverschiebung – German" lang="de" hreflang="de" data-title="Parallelverschiebung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/R%C3%B6%C3%B6pl%C3%BCke" title="Rööplüke – Estonian" lang="et" hreflang="et" data-title="Rööplüke" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Traslaci%C3%B3n_(geometr%C3%ADa)" title="Traslación (geometría) – Spanish" lang="es" hreflang="es" data-title="Traslación (geometría)" 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/Translacio_(geometrio)" title="Translacio (geometrio) – Esperanto" lang="eo" hreflang="eo" data-title="Translacio (geometrio)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Translazio" title="Translazio – Basque" lang="eu" hreflang="eu" data-title="Translazio" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%86%D8%AA%D9%82%D8%A7%D9%84_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="انتقال (هندسه) – Persian" lang="fa" hreflang="fa" data-title="انتقال (هندسه)" 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/Translation" title="Translation – French" lang="fr" hreflang="fr" data-title="Translation" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Translaci%C3%B3n_(xeometr%C3%ADa)" title="Translación (xeometría) – Galician" lang="gl" hreflang="gl" data-title="Translación (xeometría)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8F%89%ED%96%89_%EC%9D%B4%EB%8F%99" title="평행 이동 – Korean" lang="ko" hreflang="ko" data-title="평행 이동" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Translasi_(geometri)" title="Translasi (geometri) – Indonesian" lang="id" hreflang="id" data-title="Translasi (geometri)" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Traslazione_(geometria)" title="Traslazione (geometria) – Italian" lang="it" hreflang="it" data-title="Traslazione (geometria)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%96%D7%96%D7%94_(%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94)" title="הזזה (גאומטריה) – Hebrew" lang="he" hreflang="he" data-title="הזזה (גאומטריה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Translatioun_(Mathematik)" title="Translatioun (Mathematik) – Luxembourgish" lang="lb" hreflang="lb" data-title="Translatioun (Mathematik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Eltol%C3%A1s" title="Eltolás – Hungarian" lang="hu" hreflang="hu" data-title="Eltolás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B0%D0%BD%D1%81%D0%BB%D0%B0%D1%86%D0%B8%D1%98%D0%B0_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0)" title="Транслација (геометрија) – Macedonian" lang="mk" hreflang="mk" data-title="Транслација (геометрија)" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Translatie_(meetkunde)" title="Translatie (meetkunde) – Dutch" lang="nl" hreflang="nl" data-title="Translatie (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B9%B3%E8%A1%8C%E7%A7%BB%E5%8B%95" title="平行移動 – Japanese" lang="ja" hreflang="ja" data-title="平行移動" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Translasjon_i_matematikk" title="Translasjon i matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Translasjon i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Translacion_(geometria)" title="Translacion (geometria) – Occitan" lang="oc" hreflang="oc" data-title="Translacion (geometria)" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Translacja_(matematyka)" title="Translacja (matematyka) – Polish" lang="pl" hreflang="pl" data-title="Translacja (matematyka)" 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/Transla%C3%A7%C3%A3o_(geometria)" title="Translação (geometria) – Portuguese" lang="pt" hreflang="pt" data-title="Translação (geometria)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Transla%C8%9Bie_(geometrie)" title="Translație (geometrie) – Romanian" lang="ro" hreflang="ro" data-title="Translație (geometrie)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%BF%D0%B5%D1%80%D0%B5%D0%BD%D0%BE%D1%81" title="Параллельный перенос – Russian" lang="ru" hreflang="ru" data-title="Параллельный перенос" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Translation_(geometry)" title="Translation (geometry) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Translation (geometry)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D1%80%D0%B0%D0%BD%D1%81%D0%BB%D0%B0%D1%86%D0%B8%D1%98%D0%B0_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0)" title="Транслација (геометрија) – Serbian" lang="sr" hreflang="sr" data-title="Транслација (геометрија)" 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/Translacija_(geometrija)" title="Translacija (geometrija) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Translacija (geometrija)" 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/Translaatio_(matematiikka)" title="Translaatio (matematiikka) – Finnish" lang="fi" hreflang="fi" data-title="Translaatio (matematiikka)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Translation_(matematik)" title="Translation (matematik) – Swedish" lang="sv" hreflang="sv" data-title="Translation (matematik)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%86%E0%AE%AF%E0%AE%B0%E0%AF%8D%E0%AE%9A%E0%AF%8D%E0%AE%9A%E0%AE%BF_(%E0%AE%B5%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="பெயர்ச்சி (வடிவவியல்) – Tamil" lang="ta" hreflang="ta" data-title="பெயர்ச்சி (வடிவவியல்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%80%E0%B8%A5%E0%B8%B7%E0%B9%88%E0%B8%AD%E0%B8%99%E0%B8%82%E0%B8%99%E0%B8%B2%E0%B8%99" title="การเลื่อนขนาน – Thai" lang="th" hreflang="th" data-title="การเลื่อนขนาน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%96teleme" title="Öteleme – Turkish" lang="tr" hreflang="tr" data-title="Öteleme" 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%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%B5%D0%BB%D1%8C%D0%BD%D0%B5_%D0%BF%D0%B5%D1%80%D0%B5%D0%BD%D0%B5%D1%81%D0%B5%D0%BD%D0%BD%D1%8F" title="Паралельне перенесення – Ukrainian" lang="uk" hreflang="uk" data-title="Паралельне перенесення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%A7%BB%E4%BD%8D" title="移位 – Cantonese" lang="yue" hreflang="yue" data-title="移位" 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/%E5%B9%B3%E7%A7%BB" title="平移 – Chinese" lang="zh" hreflang="zh" data-title="平移" data-language-autonym="中文" data-language-local-name="Chinese" 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Planar movement within a Euclidean space without rotation</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 <b>translation</b> is a <a href="/wiki/Geometric_transformation" title="Geometric transformation">geometric transformation</a> that moves every point of a figure, shape or space by the same <a href="/wiki/Distance_geometry" title="Distance geometry">distance</a> in a given <a href="/wiki/Direction_(geometry)" title="Direction (geometry)">direction</a>. 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>. 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>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="As_a_function">As a function</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Translation_(geometry)&action=edit&section=1" title="Edit section: As a function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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">See also: <a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">Displacement (geometry)</a></div> <p>If <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 \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> is a fixed vector, known as the <i>translation vector</i>, and <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 \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"></span> is the initial position of some object, then the translation function <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} }}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathbf {v} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9caf87fd0b979b2a34fabb83c43f78c3702d6d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.588ex; height:2.509ex;" alt="{\displaystyle T_{\mathbf {v} }}"></span> will work as <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} )=\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> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <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} )=\mathbf {p} +\mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ef190970526b4786ba110e009049bb934d3015" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.717ex; height:2.843ex;" alt="{\displaystyle T_{\mathbf {v} }(\mathbf {p} )=\mathbf {p} +\mathbf {v} }"></span>. </p><p>If <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> is a translation, then the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of a subset <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> under the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> is the <b>translate</b> of <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> by <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>. The translate of <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> by <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} }}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathbf {v} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9caf87fd0b979b2a34fabb83c43f78c3702d6d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.588ex; height:2.509ex;" alt="{\displaystyle T_{\mathbf {v} }}"></span> is often written as <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 A+\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+\mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1cbdd363a411f561d1df1d6ece16a792732ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.994ex; height:2.343ex;" alt="{\displaystyle A+\mathbf {v} }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Application_in_classical_physics">Application in classical physics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Translation_(geometry)&action=edit&section=2" title="Edit section: Application in classical physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Classical_physics" title="Classical physics">classical physics</a>, translational motion is movement that changes the <a href="/wiki/Position_(geometry)" title="Position (geometry)">position</a> of an object, as opposed to <a href="/wiki/Rotation" title="Rotation">rotation</a>. For example, according to Whittaker:<sup id="cite_ref-Whittaker_1-0" class="reference"><a href="#cite_note-Whittaker-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>If a body is moved from one position to another, and if the lines joining the initial and final points of each of the points of the body are a set of parallel straight lines of length <i>ℓ</i>, so that the orientation of the body in space is unaltered, the displacement is called a <i>translation parallel to the direction of the lines, through a distance ℓ</i>. </p><div class="templatequotecite">— <cite><a href="/wiki/E._T._Whittaker" title="E. T. Whittaker">E. T. Whittaker</a>, <i><a href="/wiki/A_Treatise_on_the_Analytical_Dynamics_of_Particles_and_Rigid_Bodies" class="mw-redirect" title="A Treatise on the Analytical Dynamics of Particles and Rigid Bodies">A Treatise on the Analytical Dynamics of Particles and Rigid Bodies</a></i>, p. 1</cite></div></blockquote> <p>A translation is the operation changing the positions of all points <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,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span> of an object according to the formula </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,y,z)\to (x+\Delta x,y+\Delta y,z+\Delta z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)\to (x+\Delta x,y+\Delta y,z+\Delta z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b242a68ba548eb06d7f4a6c0c96a63cc2270c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.417ex; height:2.843ex;" alt="{\displaystyle (x,y,z)\to (x+\Delta x,y+\Delta y,z+\Delta z)}"></span></dd></dl> <p>where <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 (\Delta x,\ \Delta y,\ \Delta z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Delta x,\ \Delta y,\ \Delta z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1e88155be8e90e86498eb270fef6fc6636d1c9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.419ex; height:2.843ex;" alt="{\displaystyle (\Delta x,\ \Delta y,\ \Delta z)}"></span> is the same <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> for each point of the object. The translation vector <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 (\Delta x,\ \Delta y,\ \Delta z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Delta x,\ \Delta y,\ \Delta z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1e88155be8e90e86498eb270fef6fc6636d1c9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.419ex; height:2.843ex;" alt="{\displaystyle (\Delta x,\ \Delta y,\ \Delta z)}"></span> common to all points of the object describes a particular type of <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a> of the object, usually called a <i>linear</i> displacement to distinguish it from displacements involving rotation, called <i>angular</i> displacements. </p><p>When considering <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>, a change of <a href="/wiki/Time" title="Time">time</a> coordinate is considered to be a translation. </p> <div class="mw-heading mw-heading2"><h2 id="As_an_operator">As an operator</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Translation_(geometry)&action=edit&section=3" title="Edit section: As an operator"><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">Main article: <a href="/wiki/Shift_operator" title="Shift operator">Shift operator</a></div> <p>The <a href="/wiki/Shift_operator" title="Shift operator">translation operator</a> turns a function of the original position, <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 f(\mathbf {v} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathbf {v} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/387ffaa6b5e65940435d2ccc9228c471621a4bb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.499ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {v} )}"></span>, into a function of the final position, <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 f(\mathbf {v} +\mathbf {\delta } )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathbf {v} +\mathbf {\delta } )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/645c3deb2646de978c7b8d0c49c6955ae1e410d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.388ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {v} +\mathbf {\delta } )}"></span>. In other words, <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> is defined 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> This <a href="/wiki/Operator_(mathematics)" title="Operator (mathematics)">operator</a> is more abstract than a function, since <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> defines a relationship between two functions, rather than the underlying vectors themselves. The translation operator can act on many kinds of functions, such as when the <a href="/wiki/Translation_operator_(quantum_mechanics)" title="Translation operator (quantum mechanics)">translation operator acts on a wavefunction</a>, which is studied in the field of quantum mechanics. </p> <div class="mw-heading mw-heading2"><h2 id="As_a_group">As a group</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Translation_(geometry)&action=edit&section=4" title="Edit section: As a group"><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/Translation_operator_(quantum_mechanics)#Translation_group" title="Translation operator (quantum mechanics)">Translation operator (quantum mechanics) § Translation group</a></div> <p>The set of all translations forms the <b>translation group</b> <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 \mathbb {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c039979935c00b3b216cbb065999207872677f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} }"></span>, 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> <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 E(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b3b6ec0effe4cf5a6a54bd424fb04984dbe97d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.98ex; height:2.843ex;" alt="{\displaystyle E(n)}"></span>. The <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> of <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 E(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b3b6ec0effe4cf5a6a54bd424fb04984dbe97d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.98ex; height:2.843ex;" alt="{\displaystyle E(n)}"></span> by <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 \mathbb {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c039979935c00b3b216cbb065999207872677f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} }"></span> is isomorphic to the group of rigid motions which fix a particular origin point, the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</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 O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}"></span>: </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 E(n)/\mathbb {T} \cong O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo>≅</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(n)/\mathbb {T} \cong O(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f0cb95c20928b99fbfd6d9f8a34c08c559854d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.768ex; height:2.843ex;" alt="{\displaystyle E(n)/\mathbb {T} \cong O(n)}"></span></dd></dl> <p>Because translation is <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>, the translation group is <a href="/wiki/Abelian_group" title="Abelian group">abelian</a>. There are an infinite number of possible translations, so the translation group is an <a href="/wiki/Infinite_group" title="Infinite group">infinite group</a>. </p><p>In the <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a>, due to the treatment of space and time as a single <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>, translations can also refer to changes in the <a href="/wiki/Coordinate_time" title="Coordinate time">time coordinate</a>. For example, the <a href="/wiki/Galilean_group" class="mw-redirect" title="Galilean group">Galilean group</a> and the <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> include translations with respect to time. </p> <div class="mw-heading mw-heading3"><h3 id="Lattice_groups">Lattice groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Translation_(geometry)&action=edit&section=5" title="Edit section: Lattice groups"><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">Main article: <a href="/wiki/Lattice_(group)" title="Lattice (group)">Lattice (group)</a></div> <p>One kind of <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the three-dimensional translation group are the <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice groups</a>, which are <a href="/wiki/Infinite_group" title="Infinite group">infinite groups</a>, but unlike the translation groups, are <a href="/wiki/Finitely_generated_group" title="Finitely generated group">finitely generated</a>. That is, a finite <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generating set</a> generates the entire group. </p> <div class="mw-heading mw-heading2"><h2 id="Matrix_representation">Matrix representation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Translation_(geometry)&action=edit&section=6" title="Edit section: Matrix representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A translation is an <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a> with <i>no</i> <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed points</a>. Matrix multiplications <i>always</i> have the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> as a fixed point. Nevertheless, there is a common <a href="/wiki/Workaround" title="Workaround">workaround</a> using <a href="/wiki/Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a> to represent a translation of a <a href="/wiki/Vector_space" title="Vector space">vector space</a> with <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>: Write the 3-dimensional vector <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 \mathbf {v} =(v_{x},v_{y},v_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0744af00f6b08dd2f26a2c76271389a3c25c5cb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.993ex; height:3.009ex;" alt="{\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z})}"></span> using 4 homogeneous coordinates as <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 \mathbf {v} =(v_{x},v_{y},v_{z},1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z},1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7aef15200ee6dc28cde107cdc09a604c362af7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.189ex; height:3.009ex;" alt="{\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z},1)}"></span>.<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>To translate an object by a <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vector</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 \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span>, each homogeneous vector <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 \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"></span> (written in homogeneous coordinates) can 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 {v} }T_{\mathbf {w} }=T_{\mathbf {v} +\mathbf {w} }.\!}"> <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> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mrow> </msub> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathbf {v} }T_{\mathbf {w} }=T_{\mathbf {v} +\mathbf {w} }.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794b2b94e0ae0d7fb594270c932d6c031b06c1e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:14.336ex; height:2.509ex;" alt="{\displaystyle T_{\mathbf {v} }T_{\mathbf {w} }=T_{\mathbf {v} +\mathbf {w} }.\!}"></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-heading2"><h2 id="Translation_of_axes">Translation of axes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Translation_(geometry)&action=edit&section=7" title="Edit section: Translation of axes"><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">Main article: <a href="/wiki/Translation_of_axes" title="Translation of axes">Translation of axes</a></div> <p>While geometric translation is often viewed as an <a href="/wiki/Active_transformation" class="mw-redirect" title="Active transformation">active transformation</a> that changes the position of a geometric object, a similar result can be achieved by a <a href="/wiki/Passive_transformation" class="mw-redirect" title="Passive transformation">passive transformation</a> that moves the coordinate system itself but leaves the object fixed. The passive version of an active geometric translation is known as a <i><a href="/wiki/Translation_of_axes" title="Translation of axes">translation of axes</a></i>. </p> <div class="mw-heading mw-heading2"><h2 id="Translational_symmetry">Translational symmetry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Translation_(geometry)&action=edit&section=8" title="Edit section: Translational symmetry"><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">Main article: <a href="/wiki/Translational_symmetry" title="Translational symmetry">Translational symmetry</a></div> <p>An object that looks the same before and after translation is said to have <a href="/wiki/Translational_symmetry" title="Translational symmetry">translational symmetry</a>. A common example is a <a href="/wiki/Periodic_function" title="Periodic function">periodic function</a>, which is an <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunction</a> of a translation operator. </p> <div class="mw-heading mw-heading2"><h2 id="Translations_of_a_graph">Translations of a graph <span class="anchor" id="Horizontal"></span><span class="anchor" id="Vertical"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Translation_(geometry)&action=edit&section=9" title="Edit section: Translations of a graph"><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">"Vertical translation" redirects here. For the concept in physics, see <a href="/wiki/Vertical_separation" class="mw-redirect" title="Vertical separation">Vertical separation</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Translated_graph_of_a_function.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Translated_graph_of_a_function.png/280px-Translated_graph_of_a_function.png" decoding="async" width="280" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Translated_graph_of_a_function.png/420px-Translated_graph_of_a_function.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/Translated_graph_of_a_function.png/560px-Translated_graph_of_a_function.png 2x" data-file-width="1430" data-file-height="1130" /></a><figcaption>Compared to the graph <span class="texhtml"><i>y</i> = <i>f</i>(<i>x</i>)</span>, the graph <span class="texhtml"><i>y</i> = <i>f</i>(<i>x</i> − <i>a</i>)</span> has been translated horizontally by <span class="texhtml mvar" style="font-style:italic;">a</span>, while the graph <span class="texhtml"><i>y</i> = <i>f</i>(<i>x</i>) + b</span> has been translated vertically by <span class="texhtml mvar" style="font-style:italic;">b</span>.</figcaption></figure> <p>The <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of a <a href="/wiki/Real_function" class="mw-redirect" title="Real function">real function</a> <span class="texhtml mvar" style="font-style:italic;">f</span>, the set of points <span class="nowrap">⁠<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,f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,f(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21dd0c5c5815bc0516f679f631fd588ceb458d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.59ex; height:2.843ex;" alt="{\displaystyle (x,f(x))}"></span>⁠</span>, is often pictured in the <a href="/wiki/Real_coordinate_plane" class="mw-redirect" title="Real coordinate plane">real coordinate plane</a> with <span class="texhtml mvar" style="font-style:italic;">x</span> as the horizontal coordinate and <span class="nowrap">⁠<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=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"></span>⁠</span> as the vertical coordinate. </p><p>Starting from the graph of <span class="texhtml mvar" style="font-style:italic;">f</span>, a <b>horizontal translation</b> means <a href="/wiki/Function_composition" title="Function composition">composing</a> <span class="texhtml mvar" style="font-style:italic;">f</span> with a function <span class="nowrap">⁠<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\mapsto x-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x-a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb1a42aba777da68a7af2ecb9679a0226b4d5fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.344ex; height:2.176ex;" alt="{\displaystyle x\mapsto x-a}"></span>⁠</span>, for some constant number <span class="texhtml mvar" style="font-style:italic;">a</span>, resulting in a graph consisting of points <span class="nowrap">⁠<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,f(x-a))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,f(x-a))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beb4406b0b71bc6c906c1c9a7e0cb9714e1cafc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.661ex; height:2.843ex;" alt="{\displaystyle (x,f(x-a))}"></span>⁠</span>. Each point <span class="nowrap">⁠<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,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span>⁠</span> of the original graph corresponds to the point <span class="nowrap">⁠<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+a,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+a,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd627858d06c17459b4d8872bdfc36073bbbb3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.399ex; height:2.843ex;" alt="{\displaystyle (x+a,y)}"></span>⁠</span> in the new graph, which pictorially results in a horizontal shift. </p><p>A <b>vertical translation</b> means composing the function <span class="nowrap">⁠<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\mapsto y+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">↦</mo> <mi>y</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\mapsto y+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb5f7d080dd0d2e259e958cf8639c98083f7f867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.763ex; height:2.509ex;" alt="{\displaystyle y\mapsto y+b}"></span>⁠</span> with <span class="texhtml mvar" style="font-style:italic;">f</span>, for some constant <span class="texhtml mvar" style="font-style:italic;">b</span>, resulting in a graph consisting of the points <span class="nowrap">⁠<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 {\bigl (}x,f(x)+b{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}x,f(x)+b{\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d143bba84225ddebe9bc64cf4e8c4c088a59159d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.749ex; height:3.176ex;" alt="{\displaystyle {\bigl (}x,f(x)+b{\bigr )}}"></span>⁠</span>. Each point <span class="nowrap">⁠<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,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span>⁠</span> of the original graph corresponds to the point <span class="nowrap">⁠<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,y+b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y+b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83b67b30a359932520ea7a581fa95eb26e8eb97e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.166ex; height:2.843ex;" alt="{\displaystyle (x,y+b)}"></span>⁠</span> in the new graph, which pictorially results in a vertical shift.<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> </p><p>For example, taking the <a href="/wiki/Quadratic_function" title="Quadratic function">quadratic function</a> <span class="nowrap">⁠<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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1108c4c9ee8ac7de90b77f9bd27415b13b6bf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.638ex; height:3.009ex;" alt="{\displaystyle y=x^{2}}"></span>⁠</span>, whose graph is a <a href="/wiki/Parabola" title="Parabola">parabola</a> with vertex at <span class="nowrap">⁠<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 (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"></span>⁠</span>, a horizontal translation 5 units to the right would be the new function <span class="nowrap">⁠<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-5)^{2}=x^{2}-10x+25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>5</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>10</mn> <mi>x</mi> <mo>+</mo> <mn>25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=(x-5)^{2}=x^{2}-10x+25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4db1642234a168bc4f0611f7bcd52c8d19bdecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.593ex; height:3.176ex;" alt="{\displaystyle y=(x-5)^{2}=x^{2}-10x+25}"></span>⁠</span> whose vertex has coordinates <span class="nowrap">⁠<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 (5,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>5</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (5,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a6520c29d84cc00fbe6235cbdcb0a8e79b95e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (5,0)}"></span>⁠</span>. A vertical translation 3 units upward would be the new function <span class="nowrap">⁠<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^{2}+3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x^{2}+3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/931e4a245570a2590951a53eec38296778a172b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.641ex; height:3.009ex;" alt="{\displaystyle y=x^{2}+3}"></span>⁠</span> whose vertex has coordinates <span class="nowrap">⁠<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 (0,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/facd8877b12aa8477bf175fbfd33ce75a93955cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,3)}"></span>⁠</span>. </p><p>The <a href="/wiki/Antiderivative" title="Antiderivative">antiderivatives</a> of a function all differ from each other by a <a href="/wiki/Constant_of_integration" title="Constant of integration">constant of integration</a> and are therefore vertical translates of each other.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Translation_(geometry)&action=edit&section=10" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For describing <a href="/wiki/Vehicle_dynamics" title="Vehicle dynamics">vehicle dynamics</a> (or movement of any <a href="/wiki/Rigid_body" title="Rigid body">rigid body</a>), including <a href="/wiki/Ship_motions" title="Ship motions">ship dynamics</a> and <a href="/wiki/Aircraft_principal_axes" title="Aircraft principal axes">aircraft dynamics</a>, it is common to use a mechanical model consisting of six <a href="/wiki/Degrees_of_freedom_(mechanics)" title="Degrees of freedom (mechanics)">degrees of freedom</a>, which includes translations along three reference axes (as well as rotations about those three axes). These translations are often called <a href="/wiki/Ship_motions#Surge" title="Ship motions"><i>surge</i></a>, <a href="/wiki/Ship_motions#Sway" title="Ship motions"><i>sway</i></a>, and <a href="/wiki/Ship_motions#Heave" title="Ship motions"><i>heave</i></a>. </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=Translation_(geometry)&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 40em;"> <ul><li><a href="/wiki/2D_computer_graphics#Translation" title="2D computer graphics">2D computer graphics#Translation</a></li> <li><a href="/wiki/Advection" title="Advection">Advection</a></li> <li><a href="/wiki/Change_of_basis" title="Change of basis">Change of basis</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li> <li><a href="/wiki/Rotation_matrix" title="Rotation matrix">Rotation matrix</a></li> <li><a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">Scaling (geometry)</a></li> <li><a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation matrix</a></li> <li><a href="/wiki/Translational_symmetry" title="Translational symmetry">Translational symmetry</a></li></ul> </div> <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=Translation_(geometry)&action=edit&section=12" 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-Whittaker-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Whittaker_1-0">^</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 id="CITEREFEdmund_Taylor_Whittaker1988" class="citation book cs1"><a href="/wiki/E._T._Whittaker" title="E. T. Whittaker">Edmund Taylor Whittaker</a> (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=epH1hCB7N2MC&q=rigid+bodies+translation&pg=PA4"><i>A Treatise on the Analytical Dynamics of Particles and Rigid Bodies</i></a> (Reprint of fourth edition of 1936 with foreword by William McCrea ed.). Cambridge University Press. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-35883-3" title="Special:BookSources/0-521-35883-3"><bdi>0-521-35883-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+on+the+Analytical+Dynamics+of+Particles+and+Rigid+Bodies&rft.pages=1&rft.edition=Reprint+of+fourth+edition+of+1936+with+foreword+by+William+McCrea&rft.pub=Cambridge+University+Press&rft.date=1988&rft.isbn=0-521-35883-3&rft.au=Edmund+Taylor+Whittaker&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DepH1hCB7N2MC%26q%3Drigid%2Bbodies%2Btranslation%26pg%3DPA4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATranslation+%28geometry%29" class="Z3988"></span></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">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-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="CITEREFDoughertyAstol1999" class="citation cs2">Dougherty, Edward R.; Astol, Jaakko (1999), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4PV-sTF6qJQC&pg=PA169"><i>Nonlinear Filters for Image Processing</i></a>, SPIE/IEEE series on imaging science & engineering, vol. 59, SPIE Press, p. 169, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780819430335" title="Special:BookSources/9780819430335"><bdi>9780819430335</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nonlinear+Filters+for+Image+Processing&rft.series=SPIE%2FIEEE+series+on+imaging+science+%26+engineering&rft.pages=169&rft.pub=SPIE+Press&rft.date=1999&rft.isbn=9780819430335&rft.aulast=Dougherty&rft.aufirst=Edward+R.&rft.au=Astol%2C+Jaakko&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4PV-sTF6qJQC%26pg%3DPA169&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATranslation+%28geometry%29" class="Z3988"></span>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZillWright2009" class="citation cs2">Zill, Dennis; Wright, Warren S. (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0n0iPYKLo74C&pg=PA269"><i>Single Variable Calculus: Early Transcendentals</i></a>, Jones & Bartlett Learning, p. 269, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780763749651" title="Special:BookSources/9780763749651"><bdi>9780763749651</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Single+Variable+Calculus%3A+Early+Transcendentals&rft.pages=269&rft.pub=Jones+%26+Bartlett+Learning&rft.date=2009&rft.isbn=9780763749651&rft.aulast=Zill&rft.aufirst=Dennis&rft.au=Wright%2C+Warren+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0n0iPYKLo74C%26pg%3DPA269&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATranslation+%28geometry%29" class="Z3988"></span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Translation_(geometry)&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Zazkis, R., Liljedahl, P., & Gadowsky, K. Conceptions of function translation: obstacles, intuitions, and rerouting. Journal of Mathematical Behavior, 22, 437-450. Retrieved April 29, 2014, from www.elsevier.com/locate/jmathb</li> <li><a rel="nofollow" class="external text" href="http://www.biology.arizona.edu/biomath/tutorials/transformations/horizontaltranslations.html">Transformations of Graphs: Horizontal Translations</a>. (2006, January 1). BioMath: Transformation of Graphs. Retrieved April 29, 2014</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Translation_(geometry)&action=edit&section=14" title="Edit section: External links"><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 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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:Translation_(geometry)" class="extiw" title="commons:Category:Translation (geometry)">Translation (geometry)</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Geometry/Translation.shtml">Translation Transform</a> at <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li> <li><a rel="nofollow" class="external text" href="http://www.mathsisfun.com/geometry/translation.html">Geometric Translation (Interactive Animation)</a> at Math Is Fun</li> <li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/Understanding2DTranslation/">Understanding 2D Translation</a> and <a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/Understanding3DTranslation/">Understanding 3D Translation</a> by Roger Germundsson, <a href="/wiki/The_Wolfram_Demonstrations_Project" class="mw-redirect" title="The Wolfram Demonstrations Project">The Wolfram Demonstrations Project</a>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist 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graphics</a></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/Diffusion_curve" title="Diffusion curve">Diffusion curve</a></li> <li><a href="/wiki/Pixel" title="Pixel">Pixel</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/2D_computer_graphics" title="2D computer graphics">2D 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/Alpha_compositing" title="Alpha compositing">Alpha compositing</a></li> <li><a href="/wiki/Layers_(digital_image_editing)" title="Layers (digital image editing)">Layers</a></li> <li><a href="/wiki/Text-to-image_model" title="Text-to-image model">Text-to-image</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="2.5D" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/2.5D" title="2.5D">2.5D</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/Isometric_video_game_graphics" title="Isometric video game graphics">Isometric graphics</a></li> <li><a href="/wiki/Mode_7" title="Mode 7">Mode 7</a></li> <li><a href="/wiki/Parallax_scrolling" title="Parallax scrolling">Parallax scrolling</a></li> <li><a href="/wiki/Ray_casting" title="Ray casting">Ray 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 href="/wiki/Beam_tracing" title="Beam tracing">Beam tracing</a></li> <li><a href="/wiki/Cone_tracing" title="Cone tracing">Cone tracing</a></li> <li><a href="/wiki/Checkerboard_rendering" title="Checkerboard rendering">Checkerboard rendering</a></li> <li><a href="/wiki/Ray_tracing_(graphics)" title="Ray tracing (graphics)">Ray tracing</a></li> <li><a href="/wiki/Path_tracing" title="Path tracing">Path tracing</a></li> <li><a href="/wiki/Ray_casting" title="Ray casting">Ray casting</a></li> <li><a href="/wiki/Scanline_rendering" title="Scanline rendering">Scanline rendering</a></li></ul></li> <li><a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">Rotation</a></li> <li><a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">Scaling</a></li> <li><a href="/wiki/Shadow_mapping" title="Shadow mapping">Shadow mapping</a></li> <li><a href="/wiki/Shadow_volume" title="Shadow volume">Shadow volume</a></li> <li><a href="/wiki/Shear_matrix" class="mw-redirect" title="Shear 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 class="mw-selflink selflink">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 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