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Available in 29 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-29" 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">29 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 badge-Q70893996 mw-list-item" title=""><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%88%D9%8A%D9%84_(%D9%87%D9%86%D8%AF%D8%B3%D8%A9_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A9)" title="تحويل (هندسة رياضية) – Arabic" lang="ar" hreflang="ar" data-title="تحويل (هندسة رياضية)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/H%C9%99nd%C9%99si_%C3%A7evrilm%C9%99l%C9%99r" title="Həndəsi çevrilmələr – Azerbaijani" lang="az" hreflang="az" data-title="Həndəsi çevrilmələr" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Geometrijske_transformacije" title="Geometrijske transformacije – Bosnian" lang="bs" hreflang="bs" data-title="Geometrijske transformacije" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Transformaci%C3%B3_geom%C3%A8trica" title="Transformació geomètrica – Catalan" lang="ca" hreflang="ca" data-title="Transformació geomètrica" 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%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D1%81%D0%B5%D0%BD_%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D0%B9%C4%95" title="Координатсен трансформацийĕ – Chuvash" lang="cv" hreflang="cv" data-title="Координатсен трансформацийĕ" 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/Geometrick%C3%A9_zobrazen%C3%AD" title="Geometrické zobrazení – Czech" lang="cs" hreflang="cs" data-title="Geometrické zobrazení" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Trawsffurfiad_geometrig" title="Trawsffurfiad geometrig – Welsh" lang="cy" hreflang="cy" data-title="Trawsffurfiad geometrig" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Koordinatentransformation" title="Koordinatentransformation – German" lang="de" hreflang="de" data-title="Koordinatentransformation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CF%8C%CF%82_%CE%BC%CE%B5%CF%84%CE%B1%CF%83%CF%87%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CF%83%CE%BC%CF%8C%CF%82" title="Γεωμετρικός μετασχηματισμός – Greek" lang="el" hreflang="el" data-title="Γεωμετρικός μετασχηματισμός" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Transformaci%C3%B3n_geom%C3%A9trica" title="Transformación geométrica – Spanish" lang="es" hreflang="es" data-title="Transformación geométrica" 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/Geometria_transformado" title="Geometria transformado – Esperanto" lang="eo" hreflang="eo" data-title="Geometria transformado" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Transformation_g%C3%A9om%C3%A9trique" title="Transformation géométrique – French" lang="fr" hreflang="fr" data-title="Transformation géométrique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Transformasi_geometri" title="Transformasi geometri – Indonesian" lang="id" hreflang="id" data-title="Transformasi 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-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%AA" title="העתקה גאומטרית – Hebrew" lang="he" hreflang="he" data-title="העתקה גאומטרית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C4%A2eometrisk%C4%81_transform%C4%81cija" title="Ģeometriskā transformācija – Latvian" lang="lv" hreflang="lv" data-title="Ģeometriskā transformācija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Transzform%C3%A1ci%C3%B3_(matematika)" title="Transzformáció (matematika) – Hungarian" lang="hu" hreflang="hu" data-title="Transzformáció (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Co%C3%B6rdinatentransformatie" title="Coördinatentransformatie – Dutch" lang="nl" hreflang="nl" data-title="Coördinatentransformatie" 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%BE%E4%BD%95%E5%AD%A6%E7%9A%84%E5%A4%89%E6%8F%9B" title="幾何学的変換 – Japanese" lang="ja" hreflang="ja" data-title="幾何学的変換" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przekszta%C5%82cenie_geometryczne" title="Przekształcenie geometryczne – Polish" lang="pl" hreflang="pl" data-title="Przekształcenie geometryczne" 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/Transforma%C3%A7%C3%A3o_geom%C3%A9trica" title="Transformação geométrica – Portuguese" lang="pt" hreflang="pt" data-title="Transformação geométrica" 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/Transformare_geometric%C4%83" title="Transformare geometrică – Romanian" lang="ro" hreflang="ro" data-title="Transformare geometrică" 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%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="Преобразование координат – Russian" lang="ru" hreflang="ru" data-title="Преобразование координат" 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/Transformation" title="Transformation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Transformation" 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-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Koordinattransformation" title="Koordinattransformation – Swedish" lang="sv" hreflang="sv" data-title="Koordinattransformation" 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%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_%E0%AE%89%E0%AE%B0%E0%AF%81%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="வடிவவியல் உருமாற்றம் – Tamil" lang="ta" hreflang="ta" data-title="வடிவவியல் உருமாற்றம்" 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%81%E0%B8%9B%E0%B8%A5%E0%B8%87%E0%B8%97%E0%B8%B2%E0%B8%87%E0%B9%80%E0%B8%A3%E0%B8%82%E0%B8%B2%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95" title="การแปลงทางเรขาคณิต – Thai" lang="th" hreflang="th" data-title="การแปลงทางเรขาคณิต" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B5_%D0%BF%D0%B5%D1%80%D0%B5%D1%82%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F" title="Геометричне перетворення – Ukrainian" lang="uk" hreflang="uk" data-title="Геометричне перетворення" 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/%E5%B9%BE%E4%BD%95%E8%AE%8A%E6%8F%9B" title="幾何變換 – Cantonese" lang="yue" hreflang="yue" data-title="幾何變換" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link 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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">For broader coverage of this topic, see <a href="/wiki/Transformation_(mathematics)" class="mw-redirect" title="Transformation (mathematics)">Transformation (mathematics)</a>.</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>geometric transformation</b> is any <a href="/wiki/Bijection" title="Bijection">bijection</a> of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> to itself (or to another such set) with some salient <a href="/wiki/Geometry" title="Geometry">geometrical</a> underpinning, such as preserving distances, angles, or ratios (scale). More specifically, it is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> whose <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> and <a href="/wiki/Range_of_a_function" title="Range of a function">range</a> are sets of points — most often both <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 {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> or both <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 {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> — such that the function is <a href="/wiki/Bijective_function" class="mw-redirect" title="Bijective function">bijective</a> so that its <a href="/wiki/Inverse_function" title="Inverse function">inverse</a> exists.<sup id="cite_ref-Usiskin_2003_p._84_1-0" class="reference"><a href="#cite_note-Usiskin_2003_p._84-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The study of geometry may be approached by the study of these transformations, such as in <a href="/wiki/Transformation_geometry" title="Transformation geometry">transformation geometry</a>.<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> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Classifications">Classifications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_transformation&action=edit&section=1" title="Edit section: Classifications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between, say, planar transformations and spatial transformations). They can also be classified according to the properties they preserve: </p> <ul><li><a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">Displacements</a> preserve <a href="/wiki/Distance" title="Distance">distances</a> and <a href="/wiki/Angle" title="Angle">oriented angles</a> (e.g., <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a>);<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></li> <li><a href="/wiki/Isometry" title="Isometry">Isometries</a> preserve angles and distances (e.g., <a href="/wiki/Euclidean_transformation" class="mw-redirect" title="Euclidean transformation">Euclidean transformations</a>);<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><sup id="cite_ref-Berger_5-0" class="reference"><a href="#cite_note-Berger-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">Similarities</a> preserve angles and ratios between distances (e.g., resizing);<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Affine_transformation" title="Affine transformation">Affine transformations</a> preserve <a href="/wiki/Parallelism_(geometry)" class="mw-redirect" title="Parallelism (geometry)">parallelism</a> (e.g., <a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">scaling</a>, <a href="/wiki/Shear_mapping" title="Shear mapping">shear</a>);<sup id="cite_ref-Berger_5-1" class="reference"><a href="#cite_note-Berger-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Projective_transformation" class="mw-redirect" title="Projective transformation">Projective transformations</a> preserve <a href="/wiki/Collinearity_(geometry)" class="mw-redirect" title="Collinearity (geometry)">collinearity</a>;<sup id="cite_ref-Wilkinson_8-0" class="reference"><a href="#cite_note-Wilkinson-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul> <p>Each of these classes contains the previous one.<sup id="cite_ref-Wilkinson_8-1" class="reference"><a href="#cite_note-Wilkinson-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <ul><li><a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a> using complex coordinates on the plane (as well as <a href="/wiki/Circle_inversion" class="mw-redirect" title="Circle inversion">circle inversion</a>) preserve the set of all lines and circles, but may interchange lines and circles.</li></ul> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:France_identique.gif" class="mw-file-description" title="Original image (based on the map of France)"><img alt="Original image (based on the map of France)" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/France_identique.gif/108px-France_identique.gif" decoding="async" width="108" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/ef/France_identique.gif 1.5x" data-file-width="161" data-file-height="178" /></a></span></div> <div class="gallerytext"> Original image (based on the map of France)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:France_par_rotation_180deg.gif" class="mw-file-description" title="Isometry"><img alt="Isometry" src="//upload.wikimedia.org/wikipedia/commons/6/64/France_par_rotation_180deg.gif" decoding="async" width="109" height="120" class="mw-file-element" data-file-width="108" data-file-height="119" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Isometry" title="Isometry">Isometry</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:France_par_similitude.gif" class="mw-file-description" title="Similarity"><img alt="Similarity" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/France_par_similitude.gif/120px-France_par_similitude.gif" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/France_par_similitude.gif/180px-France_par_similitude.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/4/47/France_par_similitude.gif 2x" data-file-width="200" data-file-height="200" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">Similarity</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:France_affine_(1).gif" class="mw-file-description" title="Affine transformation"><img alt="Affine transformation" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/France_affine_%281%29.gif/120px-France_affine_%281%29.gif" decoding="async" width="120" height="53" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/France_affine_%281%29.gif/180px-France_affine_%281%29.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/France_affine_%281%29.gif/240px-France_affine_%281%29.gif 2x" data-file-width="283" data-file-height="126" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Affine_transformation" title="Affine transformation">Affine transformation</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:France_homographie.gif" class="mw-file-description" title="Projective transformation"><img alt="Projective transformation" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/France_homographie.gif/112px-France_homographie.gif" decoding="async" width="112" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/France_homographie.gif/169px-France_homographie.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/5/5a/France_homographie.gif 2x" data-file-width="224" data-file-height="239" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Projective_transformation" class="mw-redirect" title="Projective transformation">Projective transformation</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:France_circ.gif" class="mw-file-description" title="Inversion"><img alt="Inversion" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/France_circ.gif/120px-France_circ.gif" decoding="async" width="120" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/France_circ.gif/180px-France_circ.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/4/4b/France_circ.gif 2x" data-file-width="197" data-file-height="191" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Circle_inversion" class="mw-redirect" title="Circle inversion">Inversion</a></div> </li> </ul> <ul><li><a href="/wiki/Conformal_transformation" class="mw-redirect" title="Conformal transformation">Conformal transformations</a> preserve angles, and are, in the first order, similarities.</li> <li><a href="/wiki/Equiareal_map" title="Equiareal map">Equiareal transformations</a>, preserve areas in the planar case or volumes in the three dimensional case.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and are, in the first order, affine transformations of <a href="/wiki/Determinant" title="Determinant">determinant</a> 1.</li> <li><a href="/wiki/Homeomorphism" title="Homeomorphism">Homeomorphisms</a> (bicontinuous transformations) preserve the neighborhoods of points.</li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphisms</a> (bidifferentiable transformations) are the transformations that are affine in the first order; they contain the preceding ones as special cases, and can be further refined.</li></ul> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Fconf.gif" class="mw-file-description" title="Conformal transformation"><img alt="Conformal transformation" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Fconf.gif/106px-Fconf.gif" decoding="async" width="106" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Fconf.gif/159px-Fconf.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/9/94/Fconf.gif 2x" data-file-width="164" data-file-height="186" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Conformal_transformation" class="mw-redirect" title="Conformal transformation">Conformal transformation</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:France_aire.gif" class="mw-file-description" title="Equiareal transformation"><img alt="Equiareal transformation" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/France_aire.gif/120px-France_aire.gif" decoding="async" width="120" height="84" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/France_aire.gif/180px-France_aire.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0a/France_aire.gif 2x" data-file-width="217" data-file-height="152" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Equiareal_map" title="Equiareal map">Equiareal transformation</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:France_homothetie.gif" class="mw-file-description" title="Homeomorphism"><img alt="Homeomorphism" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/France_homothetie.gif/120px-France_homothetie.gif" decoding="async" width="120" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/France_homothetie.gif/180px-France_homothetie.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/e/ef/France_homothetie.gif 2x" data-file-width="182" data-file-height="159" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Homeomorphism" title="Homeomorphism">Homeomorphism</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:France_diff.gif" class="mw-file-description" title="Diffeomorphism"><img alt="Diffeomorphism" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/France_diff.gif/120px-France_diff.gif" decoding="async" width="120" height="112" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/France_diff.gif/180px-France_diff.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/7/79/France_diff.gif 2x" data-file-width="188" data-file-height="175" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></div> </li> </ul> <p>Transformations of the same type form <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> that may be sub-groups of other transformation groups. </p> <div class="mw-heading mw-heading2"><h2 id="Opposite_group_actions">Opposite group actions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_transformation&action=edit&section=2" title="Edit section: Opposite group actions"><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 articles: <a href="/wiki/Group_action" title="Group action">Group action</a> and <a href="/wiki/Opposite_group" title="Opposite group">Opposite group</a></div> <p>Many geometric transformations are expressed with linear algebra. The bijective linear transformations are elements of a <a href="/wiki/General_linear_group" title="General linear group">general linear group</a>. The <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> <i>A</i> is non-singular. For a <a href="/wiki/Row_vector" class="mw-redirect" title="Row vector">row vector</a> <i>v</i>, the <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">matrix product</a> <i>vA</i> gives another row vector <i>w</i> = <i>vA</i>. </p><p>The <a href="/wiki/Transpose" title="Transpose">transpose</a> of a row vector <i>v</i> is a column vector <i>v</i><sup>T</sup>, and the transpose of the above equality is <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 w^{T}=(vA)^{T}=A^{T}v^{T}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mi>A</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w^{T}=(vA)^{T}=A^{T}v^{T}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6dacece82e5ce232dee41305165d709f8b09d3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.616ex; height:3.176ex;" alt="{\displaystyle w^{T}=(vA)^{T}=A^{T}v^{T}.}"></span> Here <i>A</i><sup>T</sup> provides a left action on column vectors. </p><p>In transformation geometry there are <a href="/wiki/Composition_of_relations" title="Composition of relations">compositions</a> <i>AB</i>. Starting with a row vector <i>v</i>, the right action of the composed transformation is <i>w</i> = <i>vAB</i>. After transposition, </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 w^{T}=(vAB)^{T}=(AB)^{T}v^{T}=B^{T}A^{T}v^{T}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mi>A</mi> <mi>B</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w^{T}=(vAB)^{T}=(AB)^{T}v^{T}=B^{T}A^{T}v^{T}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec14a0f8b37041706737aaa9646badfd7414a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.854ex; height:3.176ex;" alt="{\displaystyle w^{T}=(vAB)^{T}=(AB)^{T}v^{T}=B^{T}A^{T}v^{T}.}"></span></dd></dl> <p>Thus for <i>AB</i> the associated left <a href="/wiki/Group_action" title="Group action">group action</a> is <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 B^{T}A^{T}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B^{T}A^{T}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5517ef36ce6a5fd93190d40c31690b92c2367f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.932ex; height:2.676ex;" alt="{\displaystyle B^{T}A^{T}.}"></span> In the study of <a href="/wiki/Opposite_group" title="Opposite group">opposite groups</a>, the distinction is made between opposite group actions because <a href="/wiki/Commutative_group" class="mw-redirect" title="Commutative group">commutative groups</a> are the only groups for which these opposites are equal. </p> <div class="mw-heading mw-heading2"><h2 id="Active_and_passive_transformations">Active and passive transformations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_transformation&action=edit&section=3" title="Edit section: Active and passive transformations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Active_and_passive_transformation" title="Active and passive transformation">Active and passive transformation</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Active_and_passive_transformation&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure typeof="mw:File/Thumb"><a href="/wiki/File:PassiveActive.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/PassiveActive.JPG/310px-PassiveActive.JPG" decoding="async" width="310" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/PassiveActive.JPG/465px-PassiveActive.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/PassiveActive.JPG/620px-PassiveActive.JPG 2x" data-file-width="624" data-file-height="273" /></a><figcaption>In the active transformation (left), a point <span class="texhtml mvar" style="font-style:italic;">P</span> is transformed to point <span class="texhtml mvar" style="font-style:italic;">P<span class="nowrap" style="padding-left:0.05em;">′</span></span> by rotating clockwise by <a href="/wiki/Angle" title="Angle">angle</a> <span class="texhtml mvar" style="font-style:italic;">θ</span> about the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> of a fixed coordinate system. In the passive transformation (right), point <span class="texhtml mvar" style="font-style:italic;">P</span> stays fixed, while the coordinate system rotates counterclockwise by an angle <span class="texhtml mvar" style="font-style:italic;">θ</span> about its origin. The coordinates of <span class="texhtml mvar" style="font-style:italic;">P<span class="nowrap" style="padding-left:0.05em;">′</span></span> after the active transformation relative to the original coordinate system are the same as the coordinates of <span class="texhtml mvar" style="font-style:italic;">P</span> relative to the rotated coordinate system.</figcaption></figure> <p>Geometric transformations can be distinguished into two types: <a href="/wiki/Active_and_passive_transformation" title="Active and passive transformation">active</a> or alibi transformations which change the physical position of a set of <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a> relative to a fixed <a href="/wiki/Frame_of_reference" title="Frame of reference">frame of reference</a> or <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> (<i><a href="/wiki/Alibi" title="Alibi">alibi</a></i> meaning "being somewhere else at the same time"); and passive or alias transformations which leave points fixed but change the frame of reference or coordinate system relative to which they are described (<i><a href="/wiki/Pseudonym" title="Pseudonym">alias</a></i> meaning "going under a different name").<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Active_and_passive_transformation_Davidson_11-0" class="reference"><a href="#cite_note-Active_and_passive_transformation_Davidson-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> By <i>transformation</i>, <a href="/wiki/Mathematician" title="Mathematician">mathematicians</a> usually refer to active transformations, while <a href="/wiki/Physicist" title="Physicist">physicists</a> and <a href="/wiki/Engineer" title="Engineer">engineers</a> could mean either.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2023)">citation needed</span></a></i>]</sup> </p><p>For instance, active transformations are useful to describe successive positions of a <a href="/wiki/Rigid_body" title="Rigid body">rigid body</a>. On the other hand, passive transformations may be useful in human motion analysis to observe the motion of the <a href="/wiki/Tibia" title="Tibia">tibia</a> relative to the <a href="/wiki/Femur" title="Femur">femur</a>, that is, its motion relative to a (<i>local</i>) coordinate system which moves together with the femur, rather than a (<i>global</i>) coordinate system which is fixed to the floor.<sup id="cite_ref-Active_and_passive_transformation_Davidson_11-1" class="reference"><a href="#cite_note-Active_and_passive_transformation_Davidson-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Three-dimensional_Euclidean_space" class="mw-redirect" title="Three-dimensional Euclidean space">three-dimensional Euclidean space</a>, any <a href="/wiki/Rigid_transformation" title="Rigid transformation">proper rigid transformation</a>, whether active or passive, can be represented as a <a href="/wiki/Screw_displacement" class="mw-redirect" title="Screw displacement">screw displacement</a>, the composition of a <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> along an axis and a <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a> about that axis. </p> The terms <i>active transformation</i> and <i>passive transformation</i> were first introduced in 1957 by <a href="/wiki/Valentine_Bargmann" title="Valentine Bargmann">Valentine Bargmann</a> for describing <a href="/wiki/Lorentz_transformations" class="mw-redirect" title="Lorentz transformations">Lorentz transformations</a> in <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></div></div> <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=Geometric_transformation&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Coordinate_transformation" class="mw-redirect" title="Coordinate transformation">Coordinate transformation</a></li> <li><a href="/wiki/Erlangen_program" title="Erlangen program">Erlangen program</a></li> <li><a href="/wiki/Symmetry_(geometry)" title="Symmetry (geometry)">Symmetry (geometry)</a></li> <li><a href="/wiki/Motion_(geometry)" title="Motion (geometry)">Motion</a></li> <li><a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">Reflection</a></li> <li><a href="/wiki/Rigid_transformation" title="Rigid transformation">Rigid transformation</a></li> <li><a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">Rotation</a></li> <li><a href="/wiki/Topology" title="Topology">Topology</a></li> <li><a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation matrix</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometric_transformation&action=edit&section=5" 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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Usiskin_2003_p._84-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Usiskin_2003_p._84_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="CITEREFUsiskinPeressiniMarchisottoStanley2003" class="citation book cs1"><a href="/wiki/Zalman_Usiskin" title="Zalman Usiskin">Usiskin, Zalman</a>; Peressini, Anthony L.; <a href="/wiki/Elena_Marchisotto" title="Elena Marchisotto">Marchisotto, Elena</a>; Stanley, Dick (2003). <i>Mathematics for High School Teachers: An Advanced Perspective</i>. Pearson Education. p. 84. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-044941-5" title="Special:BookSources/0-13-044941-5"><bdi>0-13-044941-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/50004269">50004269</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+High+School+Teachers%3A+An+Advanced+Perspective&rft.pages=84&rft.pub=Pearson+Education&rft.date=2003&rft_id=info%3Aoclcnum%2F50004269&rft.isbn=0-13-044941-5&rft.aulast=Usiskin&rft.aufirst=Zalman&rft.au=Peressini%2C+Anthony+L.&rft.au=Marchisotto%2C+Elena&rft.au=Stanley%2C+Dick&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometric+transformation" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVenema2006" class="citation cs2">Venema, Gerard A. 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Retrieved <span class="nowrap">2020-05-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Geometry+Translation&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Fgeometry%2Ftranslation.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometric+transformation" 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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html#euclidean">"Geometric Transformations — Euclidean Transformations"</a>. <i>pages.mtu.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-05-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=pages.mtu.edu&rft.atitle=Geometric+Transformations+%E2%80%94+Euclidean+Transformations&rft_id=https%3A%2F%2Fpages.mtu.edu%2F~shene%2FCOURSES%2Fcs3621%2FNOTES%2Fgeometry%2Fgeo-tran.html%23euclidean&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometric+transformation" class="Z3988"></span></span> </li> <li id="cite_note-Berger-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Berger_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Berger_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=pN0iAVavPR8C&pg=PA131">Geometric transformation</a></i>, p. 131, at <a href="/wiki/Google_Books" title="Google Books">Google Books</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/geometry/transformations.html">"Transformations"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-05-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Transformations&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Fgeometry%2Ftransformations.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometric+transformation" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html#affine">"Geometric Transformations — Affine Transformations"</a>. <i>pages.mtu.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-05-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=pages.mtu.edu&rft.atitle=Geometric+Transformations+%E2%80%94+Affine+Transformations&rft_id=https%3A%2F%2Fpages.mtu.edu%2F~shene%2FCOURSES%2Fcs3621%2FNOTES%2Fgeometry%2Fgeo-tran.html%23affine&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometric+transformation" class="Z3988"></span></span> </li> <li id="cite_note-Wilkinson-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Wilkinson_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Wilkinson_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Leland Wilkinson, D. Wills, D. Rope, A. Norton, R. Dubbs – '<b><a rel="nofollow" class="external text" href="https://books.google.com/books?id=NRyGnjeNKJIC&pg=PA182">Geometric transformation</a><i>, p. 182, at <a href="/wiki/Google_Books" title="Google Books">Google Books</a></i></b></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=Y6jDAgAAQBAJ&pg=PA191">Geometric transformation</a></i>, p. 191, at <a href="/wiki/Google_Books" title="Google Books">Google Books</a> Bruce E. Meserve – Fundamental Concepts of Geometry, page 191.]</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrampinPirani1986" class="citation book cs1">Crampin, M.; Pirani, F.A.E. (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iDfk7bjI5qAC&pg=PA22"><i>Applicable Differential Geometry</i></a>. Cambridge University Press. p. 22. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-23190-9" title="Special:BookSources/978-0-521-23190-9"><bdi>978-0-521-23190-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applicable+Differential+Geometry&rft.pages=22&rft.pub=Cambridge+University+Press&rft.date=1986&rft.isbn=978-0-521-23190-9&rft.aulast=Crampin&rft.aufirst=M.&rft.au=Pirani%2C+F.A.E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DiDfk7bjI5qAC%26pg%3DPA22&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometric+transformation" class="Z3988"></span></span> </li> <li id="cite_note-Active_and_passive_transformation_Davidson-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-Active_and_passive_transformation_Davidson_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Active_and_passive_transformation_Davidson_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoseph_K._Davidson,_Kenneth_Henderson_Hunt2004" class="citation book cs1">Joseph K. Davidson, Kenneth Henderson Hunt (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OQq67Tr7D0cC&pg=PA74">"§4.4.1 The active interpretation and the active transformation"</a>. <i>Robots and screw theory: applications of kinematics and statics to robotics</i>. Oxford University Press. p. 74 <i>ff</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-856245-4" title="Special:BookSources/0-19-856245-4"><bdi>0-19-856245-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A74.4.1+The+active+interpretation+and+the+active+transformation&rft.btitle=Robots+and+screw+theory%3A+applications+of+kinematics+and+statics+to+robotics&rft.pages=74+%27%27ff%27%27&rft.pub=Oxford+University+Press&rft.date=2004&rft.isbn=0-19-856245-4&rft.au=Joseph+K.+Davidson%2C+Kenneth+Henderson+Hunt&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOQq67Tr7D0cC%26pg%3DPA74&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometric+transformation" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBargmann1957" class="citation journal cs1">Bargmann, Valentine (1957). "Relativity". <i>Reviews of Modern Physics</i>. <b>29</b> (2): 161–174. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1957RvMP...29..161B">1957RvMP...29..161B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.29.161">10.1103/RevModPhys.29.161</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Relativity&rft.volume=29&rft.issue=2&rft.pages=161-174&rft.date=1957&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.29.161&rft_id=info%3Abibcode%2F1957RvMP...29..161B&rft.aulast=Bargmann&rft.aufirst=Valentine&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometric+transformation" class="Z3988"></span></span> </li> </ol></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=Geometric_transformation&action=edit&section=6" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Transformations_(geometry)" class="extiw" title="commons:Category:Transformations (geometry)">Transformations (geometry)</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdler2012" class="citation cs2"><a href="/wiki/Irving_Adler" title="Irving Adler">Adler, Irving</a> (2012) [1966], <i>A New Look at Geometry</i>, Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-49851-5" title="Special:BookSources/978-0-486-49851-5"><bdi>978-0-486-49851-5</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+New+Look+at+Geometry&rft.pub=Dover&rft.date=2012&rft.isbn=978-0-486-49851-5&rft.aulast=Adler&rft.aufirst=Irving&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometric+transformation" class="Z3988"></span></li> <li><a href="/wiki/Zolt%C3%A1n_P%C3%A1l_Dienes" title="Zoltán Pál Dienes">Dienes, Z. P.</a>; Golding, E. W. (1967) . <i>Geometry Through Transformations</i> (3 vols.): <i>Geometry of Distortion</i>, <i>Geometry of Congruence</i>, and <i>Groups and Coordinates</i>. New York: Herder and Herder.</li> <li><a href="/wiki/David_Gans" title="David Gans">David Gans</a> – <i>Transformations and geometries</i>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbertCohn-Vossen1952" class="citation book cs1"><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert, David</a>; <a href="/wiki/Stephan_Cohn-Vossen" class="mw-redirect" title="Stephan Cohn-Vossen">Cohn-Vossen, Stephan</a> (1952). <i>Geometry and the Imagination</i> (2nd ed.). Chelsea. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8284-1087-9" title="Special:BookSources/0-8284-1087-9"><bdi>0-8284-1087-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+and+the+Imagination&rft.edition=2nd&rft.pub=Chelsea&rft.date=1952&rft.isbn=0-8284-1087-9&rft.aulast=Hilbert&rft.aufirst=David&rft.au=Cohn-Vossen%2C+Stephan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometric+transformation" class="Z3988"></span></li> <li>John McCleary (2013) <i>Geometry from a Differentiable Viewpoint</i>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-11607-7" title="Special:BookSources/978-0-521-11607-7">978-0-521-11607-7</a></li> <li>Modenov, P. S.; Parkhomenko, A. S. (1965) . <i>Geometric Transformations</i> (2 vols.): <i>Euclidean and Affine Transformations</i>, and <i>Projective Transformations</i>. New York: Academic Press.</li> <li>A. N. Pressley – <i>Elementary Differential Geometry</i>.</li> <li><a href="/wiki/Isaak_Yaglom" title="Isaak Yaglom">Yaglom, I. M.</a> (1962, 1968, 1973, 2009) . <i>Geometric Transformations</i> (4 vols.). <a href="/wiki/Random_House" title="Random House">Random House</a> (I, II & III), <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">MAA</a> (I, II, III & IV).</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Geometric_transformation&oldid=1251333483">https://en.wikipedia.org/w/index.php?title=Geometric_transformation&oldid=1251333483</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Geometry" title="Category:Geometry">Geometry</a></li><li><a href="/wiki/Category:Functions_and_mappings" title="Category:Functions and mappings">Functions and mappings</a></li><li><a href="/wiki/Category:Symmetry" title="Category:Symmetry">Symmetry</a></li><li><a href="/wiki/Category:Transformation_(function)" title="Category:Transformation (function)">Transformation (function)</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_with_excerpts" title="Category:Articles with excerpts">Articles with excerpts</a></li><li><a href="/wiki/Category:Commons_category_link_is_on_Wikidata" title="Category:Commons category link is on Wikidata">Commons category link is on Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 15 October 2024, at 16:49<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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