Cartesian coordinate system

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class="vector-toc-link" href="#Three_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Three dimensions</span> </div> </a> <ul id="toc-Three_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Higher_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Higher dimensions</span> </div> </a> <ul id="toc-Higher_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notations_and_conventions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notations_and_conventions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Notations and conventions</span> </div> </a> <button aria-controls="toc-Notations_and_conventions-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 Notations and conventions subsection</span> </button> <ul id="toc-Notations_and_conventions-sublist" class="vector-toc-list"> <li id="toc-Quadrants_and_octants" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quadrants_and_octants"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Quadrants and octants</span> </div> </a> <ul id="toc-Quadrants_and_octants-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cartesian_formulae_for_the_plane" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Cartesian_formulae_for_the_plane"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Cartesian formulae for the plane</span> </div> </a> <button aria-controls="toc-Cartesian_formulae_for_the_plane-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 Cartesian formulae for the plane subsection</span> </button> <ul id="toc-Cartesian_formulae_for_the_plane-sublist" class="vector-toc-list"> <li id="toc-Distance_between_two_points" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distance_between_two_points"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Distance between two points</span> </div> </a> <ul id="toc-Distance_between_two_points-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euclidean_transformations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euclidean_transformations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Euclidean transformations</span> </div> </a> <ul id="toc-Euclidean_transformations-sublist" class="vector-toc-list"> <li id="toc-Translation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Translation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.1</span> <span>Translation</span> </div> </a> <ul id="toc-Translation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rotation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Rotation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.2</span> <span>Rotation</span> </div> </a> <ul id="toc-Rotation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reflection" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Reflection"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.3</span> <span>Reflection</span> </div> </a> <ul id="toc-Reflection-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Glide_reflection" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Glide_reflection"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.4</span> <span>Glide reflection</span> </div> </a> <ul id="toc-Glide_reflection-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_matrix_form_of_the_transformations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#General_matrix_form_of_the_transformations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.5</span> <span>General matrix form of the transformations</span> </div> </a> <ul id="toc-General_matrix_form_of_the_transformations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Affine_transformation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Affine_transformation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Affine transformation</span> </div> </a> <ul id="toc-Affine_transformation-sublist" class="vector-toc-list"> <li id="toc-Scaling" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Scaling"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.1</span> <span>Scaling</span> </div> </a> <ul id="toc-Scaling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Shearing" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Shearing"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.2</span> <span>Shearing</span> </div> </a> <ul id="toc-Shearing-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Orientation_and_handedness" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Orientation_and_handedness"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Orientation and handedness</span> </div> </a> <button aria-controls="toc-Orientation_and_handedness-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 Orientation and handedness subsection</span> </button> <ul id="toc-Orientation_and_handedness-sublist" class="vector-toc-list"> <li id="toc-In_two_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_two_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>In two dimensions</span> </div> </a> <ul id="toc-In_two_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_three_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_three_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>In three dimensions</span> </div> </a> <ul id="toc-In_three_dimensions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Representing_a_vector_in_the_standard_basis" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Representing_a_vector_in_the_standard_basis"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Representing a vector in the standard basis</span> </div> </a> <ul id="toc-Representing_a_vector_in_the_standard_basis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_and_cited_references" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#General_and_cited_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>General and cited references</span> </div> </a> <ul id="toc-General_and_cited_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</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"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</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" title="Table of Contents" > <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">Cartesian coordinate system</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 74 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-74" 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">74 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Cartesiese_ko%C3%B6rdinatestelsel" title="Cartesiese koördinatestelsel – Afrikaans" lang="af" hreflang="af" data-title="Cartesiese koördinatestelsel" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%A7%D9%85_%D8%A5%D8%AD%D8%AF%D8%A7%D8%AB%D9%8A%D8%A7%D8%AA_%D8%AF%D9%8A%D9%83%D8%A7%D8%B1%D8%AA%D9%8A%D8%A9" title="نظام إحداثيات ديكارتية – Arabic" lang="ar" hreflang="ar" data-title="نظام إحداثيات ديكارتية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Coordenaes_cartesianes" title="Coordenaes cartesianes – Asturian" lang="ast" hreflang="ast" data-title="Coordenaes cartesianes" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Karteziyan_koordinat_sistemi" title="Karteziyan koordinat sistemi – Azerbaijani" lang="az" hreflang="az" data-title="Karteziyan koordinat sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A4%E0%A7%87%E0%A6%B8%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE%E0%A6%A8%E0%A6%BE%E0%A6%99%E0%A7%8D%E0%A6%95_%E0%A6%AC%E0%A7%8D%E0%A6%AF%E0%A6%AC%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE" title="কার্তেসীয় স্থানাঙ্ক ব্যবস্থা – Bangla" lang="bn" hreflang="bn" data-title="কার্তেসীয় স্থানাঙ্ক ব্যবস্থা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A2%D1%83%D1%80%D0%B0_%D0%BC%D3%A9%D0%B9%D3%A9%D1%88%D0%BB%D3%A9_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B0%D0%BB%D0%B0%D1%80_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0%D2%BB%D1%8B" title="Тура мөйөшлө координаталар системаһы – Bashkir" lang="ba" hreflang="ba" data-title="Тура мөйөшлө координаталар системаһы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B5%D0%BA%D0%B0%D1%80%D1%82%D0%BE%D0%B2%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Декартова координатна система – Bulgarian" lang="bg" hreflang="bg" data-title="Декартова координатна система" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Descartesov_koordinatni_sistem" title="Descartesov koordinatni sistem – Bosnian" lang="bs" hreflang="bs" data-title="Descartesov koordinatni sistem" 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/Sistema_de_coordenades_cartesianes" title="Sistema de coordenades cartesianes – Catalan" lang="ca" hreflang="ca" data-title="Sistema de coordenades cartesianes" 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%D3%B3%D1%80_%D0%BA%C4%95%D1%82%D0%B5%D1%81%D0%BB%D0%B5_%D1%82%D1%8B%D1%82%C4%83%D0%BC%C4%95" title="Координатсен тӳр кĕтесле тытăмĕ – Chuvash" lang="cv" hreflang="cv" data-title="Координатсен тӳр кĕтесле тытăмĕ" 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/Kart%C3%A9zsk%C3%A1_soustava_sou%C5%99adnic" title="Kartézská soustava souřadnic – Czech" lang="cs" hreflang="cs" data-title="Kartézská soustava souřadnic" 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/System_gyfesurynnol_Gartesaidd" title="System gyfesurynnol Gartesaidd – Welsh" lang="cy" hreflang="cy" data-title="System gyfesurynnol Gartesaidd" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kartesisk_koordinatsystem" title="Kartesisk koordinatsystem – Danish" lang="da" hreflang="da" data-title="Kartesisk koordinatsystem" 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/Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem – German" lang="de" hreflang="de" data-title="Kartesisches Koordinatensystem" 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/Descartesi_koordinaadid" title="Descartesi koordinaadid – Estonian" lang="et" hreflang="et" data-title="Descartesi koordinaadid" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B1%CF%81%CF%84%CE%B5%CF%83%CE%B9%CE%B1%CE%BD%CF%8C_%CF%83%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1_%CF%83%CF%85%CE%BD%CF%84%CE%B5%CF%84%CE%B1%CE%B3%CE%BC%CE%AD%CE%BD%CF%89%CE%BD" title="Καρτεσιανό σύστημα συντεταγμένων – Greek" lang="el" hreflang="el" data-title="Καρτεσιανό σύστημα συντεταγμένων" 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/Coordenadas_cartesianas" title="Coordenadas cartesianas – Spanish" lang="es" hreflang="es" data-title="Coordenadas cartesianas" 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/Kartezia_koordinato" title="Kartezia koordinato – Esperanto" lang="eo" hreflang="eo" data-title="Kartezia koordinato" 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/Koordenatu_kartesiar" title="Koordenatu kartesiar – Basque" lang="eu" hreflang="eu" data-title="Koordenatu kartesiar" 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%AF%D8%B3%D8%AA%DA%AF%D8%A7%D9%87_%D9%85%D8%AE%D8%AA%D8%B5%D8%A7%D8%AA_%D8%AF%DA%A9%D8%A7%D8%B1%D8%AA%DB%8C" title="دستگاه مختصات دکارتی – Persian" lang="fa" hreflang="fa" data-title="دستگاه مختصات دکارتی" 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/Coordonn%C3%A9es_cart%C3%A9siennes" title="Coordonnées cartésiennes – French" lang="fr" hreflang="fr" data-title="Coordonnées cartésiennes" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Comhordan%C3%A1id%C3%AD_Cairt%C3%A9iseacha" title="Comhordanáidí Cairtéiseacha – Irish" lang="ga" hreflang="ga" data-title="Comhordanáidí Cairtéiseacha" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Sistema_de_coordenadas_cartesianas" title="Sistema de coordenadas cartesianas – Galician" lang="gl" hreflang="gl" data-title="Sistema de coordenadas cartesianas" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/Cartesian_Coordinates" title="Cartesian Coordinates – Kikuyu" lang="ki" hreflang="ki" data-title="Cartesian Coordinates" data-language-autonym="Gĩkũyũ" data-language-local-name="Kikuyu" class="interlanguage-link-target"><span>Gĩkũyũ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8D%B0%EC%B9%B4%EB%A5%B4%ED%8A%B8_%EC%A2%8C%ED%91%9C%EA%B3%84" title="데카르트 좌표계 – Korean" lang="ko" hreflang="ko" data-title="데카르트 좌표계" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%A5%D5%AF%D5%A1%D6%80%D5%BF%D5%B5%D5%A1%D5%B6_%D5%AF%D5%B8%D5%B8%D6%80%D5%A4%D5%AB%D5%B6%D5%A1%D5%BF%D5%B6%D5%A5%D6%80%D5%AB_%D5%B0%D5%A1%D5%B4%D5%A1%D5%AF%D5%A1%D6%80%D5%A3" title="Դեկարտյան կոորդինատների համակարգ – Armenian" lang="hy" hreflang="hy" data-title="Դեկարտյան կոորդինատների համակարգ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%A8%E0%A4%BF%E0%A4%B0%E0%A5%8D%E0%A4%A6%E0%A5%87%E0%A4%B6%E0%A4%BE%E0%A4%82%E0%A4%95_%E0%A4%AA%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%A4%E0%A4%BF" title="कार्तीय निर्देशांक पद्धति – Hindi" lang="hi" hreflang="hi" data-title="कार्तीय निर्देशांक पद्धति" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kartezijev_koordinatni_sustav" title="Kartezijev koordinatni sustav – Croatian" lang="hr" hreflang="hr" data-title="Kartezijev koordinatni sustav" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Karteziana_sistemo_koordinatala" title="Karteziana sistemo koordinatala – Ido" lang="io" hreflang="io" data-title="Karteziana sistemo koordinatala" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sistem_koordinat_Cartesius" title="Sistem koordinat Cartesius – Indonesian" lang="id" hreflang="id" data-title="Sistem koordinat Cartesius" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Kartes%C3%ADusarhnitakerfi%C3%B0" title="Kartesíusarhnitakerfið – Icelandic" lang="is" hreflang="is" data-title="Kartesíusarhnitakerfið" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_di_riferimento_cartesiano" title="Sistema di riferimento cartesiano – Italian" lang="it" hreflang="it" data-title="Sistema di riferimento cartesiano" 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%9E%D7%A2%D7%A8%D7%9B%D7%AA_%D7%A6%D7%99%D7%A8%D7%99%D7%9D_%D7%A7%D7%A8%D7%98%D7%96%D7%99%D7%AA" title="מערכת צירים קרטזית – Hebrew" lang="he" hreflang="he" data-title="מערכת צירים קרטזית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%93%E1%83%94%E1%83%99%E1%83%90%E1%83%A0%E1%83%A2%E1%83%98%E1%83%A1_%E1%83%99%E1%83%9D%E1%83%9D%E1%83%A0%E1%83%93%E1%83%98%E1%83%9C%E1%83%90%E1%83%A2%E1%83%97%E1%83%90_%E1%83%A1%E1%83%98%E1%83%A1%E1%83%A2%E1%83%94%E1%83%9B%E1%83%90" title="დეკარტის კოორდინატთა სისტემა – Georgian" lang="ka" hreflang="ka" data-title="დეკარტის კოორდინატთა სისტემა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B5%D0%BA%D0%B0%D1%80%D1%82%D1%82%D1%8B%D2%9B_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D1%82%D0%B0%D1%80_%D0%B6%D2%AF%D0%B9%D0%B5%D1%81%D1%96" title="Декарттық координаттар жүйесі – Kazakh" lang="kk" hreflang="kk" data-title="Декарттық координаттар жүйесі" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Systema_Cartesianum_coordinatarum" title="Systema Cartesianum coordinatarum – Latin" lang="la" hreflang="la" data-title="Systema Cartesianum coordinatarum" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Dekarta_koordin%C4%81tu_sist%C4%93ma" title="Dekarta koordinātu sistēma – Latvian" lang="lv" hreflang="lv" data-title="Dekarta koordinātu sistēma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Dekarto_koordina%C4%8Di%C5%B3_sistema" title="Dekarto koordinačių sistema – Lithuanian" lang="lt" hreflang="lt" data-title="Dekarto koordinačių sistema" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Descartes-f%C3%A9le_koordin%C3%A1ta-rendszer" title="Descartes-féle koordináta-rendszer – Hungarian" lang="hu" hreflang="hu" data-title="Descartes-féle koordináta-rendszer" 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%94%D0%B5%D0%BA%D0%B0%D1%80%D1%82%D0%BE%D0%B2_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Декартов координатен систем – Macedonian" lang="mk" hreflang="mk" data-title="Декартов координатен систем" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%95%E0%B5%8D%E0%B4%B7%E0%B4%82" title="അക്ഷം – Malayalam" lang="ml" hreflang="ml" data-title="അക്ഷം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%9F%E0%A5%87%E0%A4%B6%E0%A4%BF%E0%A4%AF%E0%A4%A8_%E0%A4%B8%E0%A4%B9%E0%A4%A8%E0%A4%BF%E0%A4%B0%E0%A5%8D%E0%A4%A6%E0%A5%87%E0%A4%B6%E0%A4%95_%E0%A4%AA%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%A4%E0%A5%80" title="कार्टेशियन सहनिर्देशक पद्धती – Marathi" lang="mr" hreflang="mr" data-title="कार्टेशियन सहनिर्देशक पद्धती" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sistem_koordinat_Cartes" title="Sistem koordinat Cartes – Malay" lang="ms" hreflang="ms" data-title="Sistem koordinat Cartes" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%AC%E1%80%90%E1%80%80%E1%80%BA%E1%80%85%E1%80%AE%E1%80%B8%E1%80%9A%E1%80%94%E1%80%BA%E1%80%B8_%E1%80%80%E1%80%AD%E1%80%AF%E1%80%A9%E1%80%92%E1%80%AD%E1%80%94%E1%80%AD%E1%80%90%E1%80%BA%E1%80%85%E1%80%94%E1%80%85%E1%80%BA" title="ကာတက်စီးယန်း ကိုဩဒိနိတ်စနစ် – Burmese" lang="my" hreflang="my" data-title="ကာတက်စီးယန်း ကိုဩဒိနိတ်စနစ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Cartesisch_co%C3%B6rdinatenstelsel" title="Cartesisch coördinatenstelsel – Dutch" lang="nl" hreflang="nl" data-title="Cartesisch coördinatenstelsel" 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/%E7%9B%B4%E4%BA%A4%E5%BA%A7%E6%A8%99%E7%B3%BB" title="直交座標系 – Japanese" lang="ja" hreflang="ja" data-title="直交座標系" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kartesisk_koordinatsystem" title="Kartesisk koordinatsystem – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kartesisk koordinatsystem" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kartesisk_koordinatsystem" title="Kartesisk koordinatsystem – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kartesisk koordinatsystem" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Dekart_koordinatalar_tizimi" title="Dekart koordinatalar tizimi – Uzbek" lang="uz" hreflang="uz" data-title="Dekart koordinatalar tizimi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A8%BE%E0%A8%B0%E0%A8%9F%E0%A9%87%E0%A8%9C%E0%A8%BC%E0%A9%80_%E0%A8%97%E0%A9%81%E0%A8%A3%E0%A8%95_%E0%A8%AA%E0%A9%8D%E0%A8%B0%E0%A8%AC%E0%A9%B0%E0%A8%A7" title="ਕਾਰਟੇਜ਼ੀ ਗੁਣਕ ਪ੍ਰਬੰਧ – Punjabi" lang="pa" hreflang="pa" data-title="ਕਾਰਟੇਜ਼ੀ ਗੁਣਕ ਪ੍ਰਬੰਧ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Karteesch_Koordinatensystem" title="Karteesch Koordinatensystem – Low German" lang="nds" hreflang="nds" data-title="Karteesch Koordinatensystem" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Uk%C5%82ad_wsp%C3%B3%C5%82rz%C4%99dnych_kartezja%C5%84skich" title="Układ współrzędnych kartezjańskich – Polish" lang="pl" hreflang="pl" data-title="Układ współrzędnych kartezjańskich" 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/Sistema_de_coordenadas_cartesiano" title="Sistema de coordenadas cartesiano – Portuguese" lang="pt" hreflang="pt" data-title="Sistema de coordenadas cartesiano" 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/Coordonate_carteziene" title="Coordonate carteziene – Romanian" lang="ro" hreflang="ro" data-title="Coordonate carteziene" 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%D1%8F%D0%BC%D0%BE%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="Прямоугольная система координат – Russian" lang="ru" hreflang="ru" data-title="Прямоугольная система координат" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sistemi_koordinativ_kartezian" title="Sistemi koordinativ kartezian – Albanian" lang="sq" hreflang="sq" data-title="Sistemi koordinativ kartezian" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Sistema_di_rifirimentu_cartisianu" title="Sistema di rifirimentu cartisianu – Sicilian" lang="scn" hreflang="scn" data-title="Sistema di rifirimentu cartisianu" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system – Simple English" lang="en-simple" hreflang="en-simple" data-title="Cartesian coordinate system" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kartezi%C3%A1nska_s%C3%BAstava_s%C3%BAradn%C3%ADc_(v_naju%C5%BE%C5%A1om_zmysle)" title="Karteziánska sústava súradníc (v najužšom zmysle) – Slovak" 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searchaux" style="display:none">Most common coordinate system (geometry)</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cartesian-coordinate-system.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/220px-Cartesian-coordinate-system.svg.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/330px-Cartesian-coordinate-system.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/440px-Cartesian-coordinate-system.svg.png 2x" data-file-width="661" data-file-height="654" /></a><figcaption>Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: <span class="nowrap">(2, 3)</span> in green, <span class="nowrap">(−3, 1)</span> in red, <span class="nowrap">(−1.5, −2.5)</span> in blue, and the origin <span class="nowrap">(0, 0)</span> in purple.</figcaption></figure> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>Cartesian coordinate system</b> (<span class="rt-commentedText nowrap"><style data-mw-deduplicate="TemplateStyles:r1177148991">.mw-parser-output .IPA-label-small{font-size:85%}.mw-parser-output .references .IPA-label-small,.mw-parser-output .infobox .IPA-label-small,.mw-parser-output .navbox .IPA-label-small{font-size:100%}</style><span class="IPA-label IPA-label-small"><a href="/wiki/British_English" title="British English">UK</a>: </span><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="'k' in 'kind'">k</span><span title="/ɑːr/: 'ar' in 'far'">ɑːr</span><span title="/ˈ/: primary stress follows">ˈ</span><span title="'t' in 'tie'">t</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="/zj/: 'Z' in 'Zeus'">zj</span><span title="/ə/: 'a' in 'about'">ə</span><span title="'n' in 'nigh'">n</span></span>/</a></span></span>, <span class="rt-commentedText nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1177148991"><span class="IPA-label IPA-label-small"><a href="/wiki/American_English" title="American English">US</a>: </span><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="'k' in 'kind'">k</span><span title="/ɑːr/: 'ar' in 'far'">ɑːr</span><span title="/ˈ/: primary stress follows">ˈ</span><span title="'t' in 'tie'">t</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="/ʒ/: 's' in 'pleasure'">ʒ</span><span title="/ə/: 'a' in 'about'">ə</span><span title="'n' in 'nigh'">n</span></span>/</a></span></span>) in a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> is a <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> that specifies each <a href="/wiki/Point_(geometry)" title="Point (geometry)">point</a> uniquely by a pair of <a href="/wiki/Real_number" title="Real number">real numbers</a> called <i>coordinates</i>, which are the <a href="/wiki/Positive_and_negative_numbers" class="mw-redirect" title="Positive and negative numbers">signed</a> distances to the point from two fixed <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> <a href="/wiki/Oriented_line" class="mw-redirect" title="Oriented line">oriented lines</a>, called <i><a href="/wiki/Coordinate_line" class="mw-redirect" title="Coordinate line">coordinate lines</a></i>, <i>coordinate axes</i> or just <i>axes</i> (plural of <i>axis</i>) of the system. The point where the axes meet is called the <i><a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a></i> and has <span class="texhtml">(0, 0)</span> as coordinates. The axes <a href="/wiki/Direction_(geometry)" title="Direction (geometry)">directions</a> represent an <a href="/wiki/Orthogonal_basis" title="Orthogonal basis">orthogonal basis</a>. The combination of origin and basis forms a <a href="/wiki/Coordinate_frame" class="mw-redirect" title="Coordinate frame">coordinate frame</a> called the <b>Cartesian frame</b>. </p><p>Similarly, the position of any point in <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a> can be specified by three <i>Cartesian coordinates</i>, which are the signed distances from the point to three mutually perpendicular planes. More generally, <span class="texhtml"><i>n</i></span> Cartesian coordinates specify the point in an <span class="texhtml"><i>n</i></span>-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> for any <a href="/wiki/Dimension" title="Dimension">dimension</a> <span class="texhtml"><i>n</i></span>. These coordinates are the signed distances from the point to <span class="texhtml"><i>n</i></span> mutually perpendicular fixed <a href="/wiki/Hyperplane" title="Hyperplane">hyperplanes</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cartesian-coordinate-system-with-circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Cartesian-coordinate-system-with-circle.svg/220px-Cartesian-coordinate-system-with-circle.svg.png" decoding="async" width="220" height="226" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Cartesian-coordinate-system-with-circle.svg/330px-Cartesian-coordinate-system-with-circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Cartesian-coordinate-system-with-circle.svg/440px-Cartesian-coordinate-system-with-circle.svg.png 2x" data-file-width="768" data-file-height="790" /></a><figcaption>Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is <span class="nowrap">(<i>x</i> − <i>a</i>)<sup>2</sup> + (<i>y</i> − <i>b</i>)<sup>2</sup> = <i>r</i><sup>2</sup></span> where <i>a</i> and <i>b</i> are the coordinates of the center <span class="nowrap">(<i>a</i>, <i>b</i>)</span> and <i>r</i> is the radius.</figcaption></figure> <p>Cartesian coordinates are named for <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>, whose invention of them in the 17th century revolutionized mathematics by allowing the expression of problems of geometry in terms of <a href="/wiki/Algebra" title="Algebra">algebra</a> and <a href="/wiki/Calculus" title="Calculus">calculus</a>. Using the Cartesian coordinate system, geometric shapes (such as <a href="/wiki/Curve" title="Curve">curves</a>) can be described by <a href="/wiki/Equation" title="Equation">equations</a> involving the coordinates of points of the shape. For example, a <a href="/wiki/Circle" title="Circle">circle</a> of radius 2, centered at the origin of the plane, may be described as the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of all points whose coordinates <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>y</i></span> satisfy the equation <span class="texhtml"><i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 4</span>; the <a href="/wiki/Area" title="Area">area</a>, the <a href="/wiki/Perimeter" title="Perimeter">perimeter</a> and the <a href="/wiki/Tangent_line" class="mw-redirect" title="Tangent line">tangent line</a> at any point can be computed from this equation by using <a href="/wiki/Integral" title="Integral">integrals</a> and <a href="/wiki/Derivative" title="Derivative">derivatives</a>, in a way that can be applied to any curve. </p><p>Cartesian coordinates are the foundation of <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, and provide enlightening geometric interpretations for many other branches of mathematics, such as <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, multivariate <a href="/wiki/Calculus" title="Calculus">calculus</a>, <a href="/wiki/Group_theory" title="Group theory">group theory</a> and more. A familiar example is the concept of the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph of a function</a>. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Engineering" title="Engineering">engineering</a> and many more. They are the most common coordinate system used in <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>, <a href="/wiki/Computer-aided_geometric_design" class="mw-redirect" title="Computer-aided geometric design">computer-aided geometric design</a> and other <a href="/wiki/Computational_geometry" title="Computational geometry">geometry-related data processing</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The adjective <i>Cartesian</i> refers to the French <a href="/wiki/Mathematician" title="Mathematician">mathematician</a> and <a href="/wiki/Philosopher" class="mw-redirect" title="Philosopher">philosopher</a> <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>, who published this idea in 1637 while he was resident in the Netherlands. It was independently discovered by <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a>, who also worked in three dimensions, although Fermat did not publish the discovery.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The French cleric <a href="/wiki/Nicole_Oresme#Mathematics" title="Nicole Oresme">Nicole Oresme</a> used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.<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>Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis.<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> The concept of using a pair of axes was introduced later, after Descartes' <i><a href="/wiki/La_G%C3%A9om%C3%A9trie" title="La Géométrie">La Géométrie</a></i> was translated into Latin in 1649 by <a href="/wiki/Frans_van_Schooten" title="Frans van Schooten">Frans van Schooten</a> and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes's work.<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><p>The development of the Cartesian coordinate system would play a fundamental role in the development of the <a href="/wiki/Calculus" title="Calculus">calculus</a> by <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> and <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The two-coordinate description of the plane was later generalized into the concept of <a href="/wiki/Vector_spaces" class="mw-redirect" title="Vector spaces">vector spaces</a>.<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> </p><p>Many other coordinate systems have been developed since Descartes, such as the <a href="/wiki/Polar_coordinate_system" title="Polar coordinate system">polar coordinates</a> for the plane, and the <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical</a> and <a href="/wiki/Cylindrical_coordinate_system" title="Cylindrical coordinate system">cylindrical coordinates</a> for three-dimensional space. </p> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=2" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Cartesian_coordinates_in_one_dimension"></span> </p> <div class="mw-heading mw-heading3"><h3 id="One_dimension">One dimension</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=3" title="Edit section: One dimension"><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">Main article: <a href="/wiki/Number_line" title="Number line">Number line</a></div> <p>An <a href="/wiki/Affine_line" class="mw-redirect" title="Affine line">affine line</a> with a chosen Cartesian coordinate system is called a <i>number line</i>. Every point on the line has a real-number coordinate, and every real number represents some point on the line. </p><p>There are two <a href="/wiki/Degree_of_freedom" class="mw-redirect" title="Degree of freedom">degrees of freedom</a> in the choice of Cartesian coordinate system for a line, which can be specified by choosing two distinct points along the line and assigning them to two distinct <a href="/wiki/Real_number" title="Real number">real numbers</a> (most commonly zero and one). Other points can then be uniquely assigned to numbers by <a href="/wiki/Linear_interpolation" title="Linear interpolation">linear interpolation</a>. Equivalently, one point can be assigned to a specific real number, for instance an <i>origin</i> point corresponding to zero, and an <a href="/wiki/Curve_orientation" title="Curve orientation">oriented</a> length along the line can be chosen as a unit, with the orientation indicating the correspondence between directions along the line and positive or negative numbers.<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> Each point corresponds to its signed distance from the origin (a number with an absolute value equal to the distance and a <span class="texhtml">+</span> or <span class="texhtml">−</span> sign chosen based on direction). </p><p>A <a href="/wiki/Geometric_transformation" title="Geometric transformation">geometric transformation</a> of the line can be represented by a <a href="/wiki/Function_of_a_real_variable" title="Function of a real variable">function of a real variable</a>, for example <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> of the line corresponds to addition, and <a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">scaling</a> the line corresponds to multiplication. Any two Cartesian coordinate systems on the line can be related to each-other by a <a href="/wiki/Linear_function" title="Linear function">linear function</a> (function of the form <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 ax+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto ax+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae7527719a79cc12409d36c3dfef7323548fbdd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.341ex; height:2.343ex;" alt="{\displaystyle x\mapsto ax+b}"></span>)</span> taking a specific point's coordinate in one system to its coordinate in the other system. Choosing a coordinate system for each of two different lines establishes an <a href="/wiki/Affine_transformation" title="Affine transformation">affine map</a> from one line to the other taking each point on one line to the point on the other line with the same coordinate. </p><p><span class="anchor" id="Cartesian_coordinates_in_two_dimensions"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Two_dimensions">Two dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=4" title="Edit section: Two dimensions"><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">Further information: <a href="/wiki/Two-dimensional_space" title="Two-dimensional space">Two-dimensional space</a></div> <p>A Cartesian coordinate system in two dimensions (also called a <b>rectangular coordinate system</b> or an <b>orthogonal coordinate system</b><sup id="cite_ref-:0_8-0" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>) is defined by an <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a> of <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> lines (axes), a single <a href="/wiki/Unit_of_length" title="Unit of length">unit of length</a> for both axes, and an orientation for each axis. The point where the axes meet is taken as the origin for both, thus turning each axis into a number line. For any point <i>P</i>, a line is drawn through <i>P</i> perpendicular to each axis, and the position where it meets the axis is interpreted as a number. The two numbers, in that chosen order, are the <i>Cartesian coordinates</i> of <i>P</i>. The reverse construction allows one to determine the point <i>P</i> given its coordinates. </p><p>The first and second coordinates are called the <i><a href="/wiki/Abscissa" class="mw-redirect" title="Abscissa">abscissa</a></i> and the <i><a href="/wiki/Ordinate" class="mw-redirect" title="Ordinate">ordinate</a></i> of <i>P</i>, respectively; and the point where the axes meet is called the <i>origin</i> of the coordinate system. The coordinates are usually written as two numbers in parentheses, in that order, separated by a comma, as in <span class="nowrap">(3, −10.5)</span>. Thus the origin has coordinates <span class="nowrap">(0, 0)</span>, and the points on the positive half-axes, one unit away from the origin, have coordinates <span class="nowrap">(1, 0)</span> and <span class="nowrap">(0, 1)</span>. </p><p>In mathematics, physics, and engineering, the first axis is usually defined or depicted as horizontal and oriented to the right, and the second axis is vertical and oriented upwards. (However, in some <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a> contexts, the ordinate axis may be oriented downwards.) The origin is often labeled <i>O</i>, and the two coordinates are often denoted by the letters <i>X</i> and <i>Y</i>, or <i>x</i> and <i>y</i>. The axes may then be referred to as the <i>X</i>-axis and <i>Y</i>-axis. The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values. </p><p>A <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a> with a chosen Cartesian coordinate system is called a <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="Cartesian_plane"></span><span class="vanchor-text">Cartesian plane</span></span></b>. In a Cartesian plane, one can define canonical representatives of certain geometric figures, such as the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> (with radius equal to the length unit, and center at the origin), the <a href="/wiki/Unit_square" title="Unit square">unit square</a> (whose diagonal has endpoints at <span class="nowrap">(0, 0)</span> and <span class="nowrap">(1, 1)</span>), the <a href="/wiki/Unit_hyperbola" title="Unit hyperbola">unit hyperbola</a>, and so on. </p><p>The two axes divide the plane into four <a href="/wiki/Right_angle" title="Right angle">right angles</a>, called <i>quadrants</i>. The quadrants may be named or numbered in various ways, but the quadrant where all coordinates are positive is usually called the <i>first quadrant</i>. </p><p>If the coordinates of a point are <span class="nowrap">(<i>x</i>, <i>y</i>)</span>, then its <a href="/wiki/Distance_from_a_point_to_a_line" title="Distance from a point to a line">distances</a> from the <i>X</i>-axis and from the <i>Y</i>-axis are |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>y</i></span>| and |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>|, respectively; where | · | denotes the <a href="/wiki/Absolute_value_(algebra)" title="Absolute value (algebra)">absolute value</a> of a number. </p><p><span class="anchor" id="Cartesian_coordinates_in_three_dimensions"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Three_dimensions">Three dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=5" title="Edit section: Three dimensions"><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">Further information: <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">Three-dimensional space</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Coord_system_CA_0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/220px-Coord_system_CA_0.svg.png" decoding="async" width="220" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/330px-Coord_system_CA_0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/440px-Coord_system_CA_0.svg.png 2x" data-file-width="620" data-file-height="600" /></a><figcaption>A three dimensional Cartesian coordinate system, with origin <i>O</i> and axis lines <i>X</i>, <i>Y</i> and <i>Z</i>, oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinates <span class="texhtml"><i>x</i> = 2</span>, <span class="texhtml"><i>y</i> = 3</span>, and <span class="texhtml"><i>z</i> = 4</span>, or <span class="texhtml">(2, 3, 4)</span>.</figcaption></figure> <p>A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (the <i>axes</i>) that go through a common point (the <i>origin</i>), and are pair-wise perpendicular; an orientation for each axis; and a single unit of length for all three axes. As in the two-dimensional case, each axis becomes a number line. For any point <i>P</i> of space, one considers a plane through <i>P</i> perpendicular to each coordinate axis, and interprets the point where that plane cuts the axis as a number. The Cartesian coordinates of <i>P</i> are those three numbers, in the chosen order. The reverse construction determines the point <i>P</i> given its three coordinates. </p><p>Alternatively, each coordinate of a point <i>P</i> can be taken as the distance from <i>P</i> to the plane defined by the other two axes, with the sign determined by the orientation of the corresponding axis. </p><p>Each pair of axes defines a <i>coordinate plane</i>. These planes divide space into eight <i><a href="/wiki/Octant_(solid_geometry)" title="Octant (solid geometry)">octants</a></i>. The octants are: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(+x,+y,+z)&&(-x,+y,+z)&&(+x,-y,+z)&&(+x,+y,-z)\\(+x,-y,-z)&&(-x,+y,-z)&&(-x,-y,+z)&&(-x,-y,-z)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mo>+</mo> <mi>x</mi> <mo>,</mo> <mo>+</mo> <mi>y</mi> <mo>,</mo> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo>,</mo> <mo>+</mo> <mi>y</mi> <mo>,</mo> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mo stretchy="false">(</mo> <mo>+</mo> <mi>x</mi> <mo>,</mo> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mo stretchy="false">(</mo> <mo>+</mo> <mi>x</mi> <mo>,</mo> <mo>+</mo> <mi>y</mi> <mo>,</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mo>+</mo> <mi>x</mi> <mo>,</mo> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo>,</mo> <mo>+</mo> <mi>y</mi> <mo>,</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo>,</mo> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo>,</mo> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(+x,+y,+z)&&(-x,+y,+z)&&(+x,-y,+z)&&(+x,+y,-z)\\(+x,-y,-z)&&(-x,+y,-z)&&(-x,-y,+z)&&(-x,-y,-z)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b75c6ee0f0d78eb6a7dee3bc93587ebb5ee87fb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.187ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}(+x,+y,+z)&&(-x,+y,+z)&&(+x,-y,+z)&&(+x,+y,-z)\\(+x,-y,-z)&&(-x,+y,-z)&&(-x,-y,+z)&&(-x,-y,-z)\end{aligned}}}"></span> </p><p>The coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and separated by commas, as in <span class="texhtml">(3, −2.5, 1)</span> or <span class="texhtml">(<i>t</i>, <i>u</i> + <i>v</i>, <i>π</i>/2)</span>. Thus, the origin has coordinates <span class="texhtml">(0, 0, 0)</span>, and the unit points on the three axes are <span class="texhtml">(1, 0, 0)</span>, <span class="texhtml">(0, 1, 0)</span>, and <span class="texhtml">(0, 0, 1)</span>. </p><p>Standard names for the coordinates in the three axes are <i>abscissa</i>, <i>ordinate</i> and <i>applicate</i>.<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> The coordinates are often denoted by the letters <i>x</i>, <i>y</i>, and <i>z</i>. The axes may then be referred to as the <i>x</i>-axis, <i>y</i>-axis, and <i>z</i>-axis, respectively. Then the coordinate planes can be referred to as the <i>xy</i>-plane, <i>yz</i>-plane, and <i>xz</i>-plane. </p><p>In mathematics, physics, and engineering contexts, the first two axes are often defined or depicted as horizontal, with the third axis pointing up. In that case the third coordinate may be called <i>height</i> or <i>altitude</i>. The orientation is usually chosen so that the 90-degree angle from the first axis to the second axis looks counter-clockwise when seen from the point <span class="texhtml">(0, 0, 1)</span>; a convention that is commonly called <i>the <a href="/wiki/Right-hand_rule" title="Right-hand rule">right-hand rule</a></i>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cartesian_coordinate_surfaces.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Cartesian_coordinate_surfaces.png/220px-Cartesian_coordinate_surfaces.png" decoding="async" width="220" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Cartesian_coordinate_surfaces.png/330px-Cartesian_coordinate_surfaces.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Cartesian_coordinate_surfaces.png/440px-Cartesian_coordinate_surfaces.png 2x" data-file-width="560" data-file-height="540" /></a><figcaption> The <a href="/wiki/Coordinate_system#Coordinate_surface" title="Coordinate system">coordinate surfaces</a> of the Cartesian coordinates <span class="texhtml">(<i>x</i>, <i>y</i>, <i>z</i>)</span>. The <i>z</i>-axis is vertical and the <i>x</i>-axis is highlighted in green. Thus, the red plane shows the points with <span class="texhtml"><i>x</i> = 1</span>, the blue plane shows the points with <span class="texhtml"><i>z</i> = 1</span>, and the yellow plane shows the points with <span class="texhtml"><i>y</i> = −1</span>. The three surfaces intersect at the point <i>P</i> (shown as a black sphere) with the Cartesian coordinates <span class="texhtml">(1, −1, 1</span>).</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Higher_dimensions">Higher dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=6" title="Edit section: Higher dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of <a href="/wiki/Real_number" title="Real number">real numbers</a>; that is, with the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</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 \mathbb {R} ^{2}=\mathbb {R} \times \mathbb {R} }"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}=\mathbb {R} \times \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1ccce2a9e51985a297fe9cb76a94c9afa24e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.027ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}=\mathbb {R} \times \mathbb {R} }"></span>, 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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> is the set of all real numbers. In the same way, the points in any <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> of dimension <i>n</i> be identified with the <a href="/wiki/Tuple" title="Tuple">tuples</a> (lists) of <i>n</i> real numbers; that is, with the Cartesian product <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} ^{n}}"> <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"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Generalizations">Generalizations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=7" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept of Cartesian coordinates generalizes to allow axes that are not perpendicular to each other, and/or different units along each axis. In that case, each coordinate is obtained by projecting the point onto one axis along a direction that is parallel to the other axis (or, in general, to the <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a> defined by all the other axes). In such an <i><a href="/wiki/Oblique_coordinate_system" class="mw-redirect" title="Oblique coordinate system">oblique coordinate system</a></i> the computations of distances and angles must be modified from that in standard Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not hold (see <a href="/wiki/Affine_plane" title="Affine plane">affine plane</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Notations_and_conventions">Notations and conventions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=8" title="Edit section: Notations and conventions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Cartesian coordinates of a point are usually written in <a href="/wiki/Parentheses" class="mw-redirect" title="Parentheses">parentheses</a> and separated by commas, as in <span class="nowrap">(10, 5)</span> or <span class="nowrap">(3, 5, 7)</span>. The origin is often labelled with the capital letter <i>O</i>. In analytic geometry, unknown or generic coordinates are often denoted by the letters (<i>x</i>, <i>y</i>) in the plane, and (<i>x</i>, <i>y</i>, <i>z</i>) in three-dimensional space. This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as the coordinates of points in many geometric problems), and letters near the beginning for given quantities. </p><p>These conventional names are often used in other domains, such as physics and engineering, although other letters may be used. For example, in a graph showing how a <a href="/wiki/Pressure" title="Pressure">pressure</a> varies with <a href="/wiki/Time" title="Time">time</a>, the graph coordinates may be denoted <i>p</i> and <i>t</i>. Each axis is usually named after the coordinate which is measured along it; so one says the <i>x-axis</i>, the <i>y-axis</i>, the <i>t-axis</i>, etc. </p><p>Another common convention for coordinate naming is to use subscripts, as (<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, ..., <i>x</i><sub><i>n</i></sub>) for the <i>n</i> coordinates in an <i>n</i>-dimensional space, especially when <i>n</i> is greater than 3 or unspecified. Some authors prefer the numbering (<i>x</i><sub>0</sub>, <i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>n</i>−1</sub>). These notations are especially advantageous in <a href="/wiki/Computer_programming" title="Computer programming">computer programming</a>: by storing the coordinates of a point as an <a href="/wiki/Array_data_type" class="mw-redirect" title="Array data type">array</a>, instead of a <a href="/wiki/Record_(computer_science)" title="Record (computer science)">record</a>, the <a href="/wiki/Subscript" class="mw-redirect" title="Subscript">subscript</a> can serve to index the coordinates. </p><p>In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called the <a href="/wiki/Abscissa" class="mw-redirect" title="Abscissa">abscissa</a>) is measured along a <a href="/wiki/Horizontal_plane" class="mw-redirect" title="Horizontal plane">horizontal</a> axis, oriented from left to right. The second coordinate (the <a href="/wiki/Ordinate" class="mw-redirect" title="Ordinate">ordinate</a>) is then measured along a <a href="/wiki/Vertical_direction" class="mw-redirect" title="Vertical direction">vertical</a> axis, usually oriented from bottom to top. Young children learning the Cartesian system, commonly learn the order to read the values before cementing the <i>x</i>-, <i>y</i>-, and <i>z</i>-axis concepts, by starting with 2D mnemonics (for example, 'Walk along the hall then up the stairs' akin to straight across the <i>x</i>-axis then up vertically along the <i>y</i>-axis). </p><p>Computer graphics and <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a>, however, often use a coordinate system with the <i>y</i>-axis oriented downwards on the computer display. This convention developed in the 1960s (or earlier) from the way that images were originally stored in <a href="/wiki/Framebuffer" title="Framebuffer">display buffers</a>. </p><p>For three-dimensional systems, a convention is to portray the <i>xy</i>-plane horizontally, with the <i>z</i>-axis added to represent height (positive up). Furthermore, there is a convention to orient the <i>x</i>-axis toward the viewer, biased either to the right or left. If a diagram (<a href="/wiki/3D_projection" title="3D projection">3D projection</a> or <a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">2D perspective drawing</a>) shows the <i>x</i>- and <i>y</i>-axis horizontally and vertically, respectively, then the <i>z</i>-axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the <i>z</i>-axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera <a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">perspective</a>. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the <a href="/wiki/Right-hand_rule" title="Right-hand rule">right-hand rule</a>, unless specifically stated otherwise. All laws of physics and math assume this <a href="#Orientation_and_handedness">right-handedness</a>, which ensures consistency. </p><p>For 3D diagrams, the names "abscissa" and "ordinate" are rarely used for <i>x</i> and <i>y</i>, respectively. When they are, the <i>z</i>-coordinate is sometimes called the <b>applicate</b>. The words <i>abscissa</i>, <i>ordinate</i> and <i>applicate</i> are sometimes used to refer to coordinate axes rather than the coordinate values.<sup id="cite_ref-:0_8-1" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Quadrants_and_octants">Quadrants and octants</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=9" title="Edit section: Quadrants and octants"><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/Octant_(solid_geometry)" title="Octant (solid geometry)">Octant (solid geometry)</a> and <a href="/wiki/Quadrant_(plane_geometry)" title="Quadrant (plane geometry)">Quadrant (plane geometry)</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cartesian_coordinates_2D.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Cartesian_coordinates_2D.svg/220px-Cartesian_coordinates_2D.svg.png" decoding="async" width="220" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Cartesian_coordinates_2D.svg/330px-Cartesian_coordinates_2D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Cartesian_coordinates_2D.svg/440px-Cartesian_coordinates_2D.svg.png 2x" data-file-width="800" data-file-height="742" /></a><figcaption>The four quadrants of a Cartesian coordinate system</figcaption></figure> <p>The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called <i>quadrants</i>,<sup id="cite_ref-:0_8-2" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by <a href="/wiki/Roman_numeral" class="mw-redirect" title="Roman numeral">Roman numerals</a>: I (where the coordinates both have positive signs), II (where the abscissa is negative − and the ordinate is positive +), III (where both the abscissa and the ordinate are −), and IV (abscissa +, ordinate −). When the axes are drawn according to the mathematical custom, the numbering goes <a href="/wiki/Clockwise" title="Clockwise">counter-clockwise</a> starting from the upper right ("north-east") quadrant. </p><p>Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or <b>octants</b>,<sup id="cite_ref-:0_8-3" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> according to the signs of the coordinates of the points. The convention used for naming a specific octant is to list its signs; for example, <span class="nowrap">(+ + +)</span> or <span class="nowrap">(− + −)</span>. The generalization of the quadrant and octant to an arbitrary number of dimensions is the <b><a href="/wiki/Orthant" title="Orthant">orthant</a></b>, and a similar naming system applies. </p> <div class="mw-heading mw-heading2"><h2 id="Cartesian_formulae_for_the_plane">Cartesian formulae for the plane</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=10" title="Edit section: Cartesian formulae for the plane"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Distance_between_two_points">Distance between two points</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=11" title="Edit section: Distance between two points"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a> between two points of the plane with Cartesian coordinates <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_{1},y_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc74086e56542bd28b46a84faaee3cebdd4a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1})}"></span> 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 (x_{2},y_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{2},y_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52d44e16a796acee486af49af05f678566d181a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{2},y_{2})}"></span> is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba52e38f3619c2c8ff3aa5255c92920174e1d95" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:30.688ex; height:4.843ex;" alt="{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}"></span> </p><p>This is the Cartesian version of <a href="/wiki/Pythagoras%27s_theorem" class="mw-redirect" title="Pythagoras's theorem">Pythagoras's theorem</a>. In three-dimensional space, the distance between 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_{1},y_{1},z_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1},z_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/781d7569148878d5fab8e65498162edcc5430791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1},z_{1})}"></span> 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 (x_{2},y_{2},z_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{2},y_{2},z_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c761bb04bd0bf983b9b8e83310e5407b7426ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{2},y_{2},z_{2})}"></span> is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d6cd0e23208033c817be89edb353f500eb8d04" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:43.503ex; height:4.843ex;" alt="{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},}"></span> </p><p>which can be obtained by two consecutive applications of Pythagoras' theorem.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Euclidean_transformations">Euclidean transformations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=12" title="Edit section: Euclidean transformations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Euclidean_plane_isometry" title="Euclidean plane isometry">Euclidean transformations</a> or <b>Euclidean motions</b> are the (<a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a>) mappings of points of the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a> to themselves which preserve distances between points. There are four types of these mappings (also called isometries): <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a>, <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a>, <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflections</a> and <a href="/wiki/Glide_reflection" title="Glide reflection">glide reflections</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Translation">Translation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=13" title="Edit section: Translation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Translation_(geometry)" title="Translation (geometry)">Translating</a> a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers <span class="nowrap">(<i>a</i>, <i>b</i>)</span> to the Cartesian coordinates of every point in the set. That is, if the original coordinates of a point are <span class="nowrap">(<i>x</i>, <i>y</i>)</span>, after the translation they will be </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x',y')=(x+a,y+b).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x',y')=(x+a,y+b).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d55e7205584f1956fbda574fd7ae9d7e71a80ae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.685ex; height:3.009ex;" alt="{\displaystyle (x',y')=(x+a,y+b).}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Rotation">Rotation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=14" title="Edit section: Rotation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To <a href="/wiki/Rotation_(geometry)" class="mw-redirect" title="Rotation (geometry)">rotate</a> a figure <a href="/wiki/Clockwise" title="Clockwise">counterclockwise</a> around the origin by some angle <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> is equivalent to replacing every point with coordinates (<i>x</i>,<i>y</i>) by the point with coordinates (<i>x'</i>,<i>y'</i>), where </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x'&=x\cos \theta -y\sin \theta \\y'&=x\sin \theta +y\cos \theta .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>y</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>y</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x'&=x\cos \theta -y\sin \theta \\y'&=x\sin \theta +y\cos \theta .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e8193cc301ba228063af7ecdf292c2b8c7e76d3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.533ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}x'&=x\cos \theta -y\sin \theta \\y'&=x\sin \theta +y\cos \theta .\end{aligned}}}"></span> </p><p>Thus: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x',y')=((x\cos \theta -y\sin \theta \,),(x\sin \theta +y\cos \theta \,)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>y</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>y</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x',y')=((x\cos \theta -y\sin \theta \,),(x\sin \theta +y\cos \theta \,)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce9f7cd7ae4908e5a226f687c4dcfaa9c8995fb6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.727ex; height:3.009ex;" alt="{\displaystyle (x',y')=((x\cos \theta -y\sin \theta \,),(x\sin \theta +y\cos \theta \,)).}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Reflection">Reflection</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=15" title="Edit section: Reflection"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="nowrap">(<i>x</i>, <i>y</i>)</span> are the Cartesian coordinates of a point, then <span class="nowrap">(−<i>x</i>, <i>y</i>)</span> are the coordinates of its <a href="/wiki/Coordinate_rotations_and_reflections" class="mw-redirect" title="Coordinate rotations and reflections">reflection</a> across the second coordinate axis (the y-axis), as if that line were a mirror. Likewise, <span class="nowrap">(<i>x</i>, −<i>y</i>)</span> are the coordinates of its reflection across the first coordinate axis (the x-axis). In more generality, reflection across a line through the origin making an angle <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> with the x-axis, is equivalent to replacing every point with coordinates <span class="nowrap">(<i>x</i>, <i>y</i>)</span> by the point with coordinates <span class="nowrap">(<i>x</i>′,<i>y</i>′)</span>, where </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x'&=x\cos 2\theta +y\sin 2\theta \\y'&=x\sin 2\theta -y\cos 2\theta .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mo>+</mo> <mi>y</mi> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mo>−</mo> <mi>y</mi> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x'&=x\cos 2\theta +y\sin 2\theta \\y'&=x\sin 2\theta -y\cos 2\theta .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b50fb88510752090d6b3256d99672b01a5b77ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.858ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}x'&=x\cos 2\theta +y\sin 2\theta \\y'&=x\sin 2\theta -y\cos 2\theta .\end{aligned}}}"></span> </p><p>Thus: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x',y')=((x\cos 2\theta +y\sin 2\theta \,),(x\sin 2\theta -y\cos 2\theta \,)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mo>+</mo> <mi>y</mi> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mspace width="thinmathspace" /> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mo>−</mo> <mi>y</mi> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mspace width="thinmathspace" /> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x',y')=((x\cos 2\theta +y\sin 2\theta \,),(x\sin 2\theta -y\cos 2\theta \,)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe2674eddd7ed77d14b8b496ce4d81692fcd3cb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.377ex; height:3.009ex;" alt="{\displaystyle (x',y')=((x\cos 2\theta +y\sin 2\theta \,),(x\sin 2\theta -y\cos 2\theta \,)).}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Glide_reflection">Glide reflection</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=16" title="Edit section: Glide reflection"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A glide reflection is the composition of a reflection across a line followed by a translation in the direction of that line. It can be seen that the order of these operations does not matter (the translation can come first, followed by the reflection). </p> <div class="mw-heading mw-heading4"><h4 id="General_matrix_form_of_the_transformations">General matrix form of the transformations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=17" title="Edit section: General matrix form of the transformations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformations</a> of the plane can be described in a uniform way by using matrices. For this purpose, the coordinates <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> of a point are commonly represented as the <a href="/wiki/Column_matrix" class="mw-redirect" title="Column matrix">column matrix</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 {\begin{pmatrix}x\\y\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3abb48a7f7870f760433546c11670df316e7656e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.149ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}.}"></span> The result <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> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <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/d8a15d455ed3a65797ebe92e0c6a42e7ce98f007" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.703ex; height:3.009ex;" alt="{\displaystyle (x',y')}"></span> of applying an affine transformation to a point <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> is given by the formula <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}=A{\begin{pmatrix}x\\y\end{pmatrix}}+b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}=A{\begin{pmatrix}x\\y\end{pmatrix}}+b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234b1eba3d12960348d3be9ea98868ccc3525035" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.016ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}=A{\begin{pmatrix}x\\y\end{pmatrix}}+b,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{pmatrix}A_{1,1}&A_{1,2}\\A_{2,1}&A_{2,2}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}A_{1,1}&A_{1,2}\\A_{2,1}&A_{2,2}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b846e076103f02659c165063157cee699648c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.49ex; height:6.509ex;" alt="{\displaystyle A={\begin{pmatrix}A_{1,1}&A_{1,2}\\A_{2,1}&A_{2,2}\end{pmatrix}}}"></span> is a 2×2 <a href="/wiki/Square_matrix" title="Square matrix">matrix</a> 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 b={\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b={\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3073cf890eb6e8c4211c337ce7aaf839ce786cd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.32ex; height:6.176ex;" alt="{\displaystyle b={\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}}"></span> is a column matrix.<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> That is, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x'&=xA_{1,1}+yA_{1,1}+b_{1}\\y'&=xA_{2,1}+yA_{2,2}+b_{2}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>y</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>y</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x'&=xA_{1,1}+yA_{1,1}+b_{1}\\y'&=xA_{2,1}+yA_{2,2}+b_{2}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a1878a255e842f1bfd8e4526acc19d4d551e7f7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.882ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}x'&=xA_{1,1}+yA_{1,1}+b_{1}\\y'&=xA_{2,1}+yA_{2,2}+b_{2}.\end{aligned}}}"></span> </p><p>Among the affine transformations, the <a href="/wiki/Euclidean_transformation" class="mw-redirect" title="Euclidean transformation">Euclidean transformations</a> are characterized by the fact that the matrix <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> is <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a>; that is, its columns are <a href="/wiki/Orthogonal_vectors" class="mw-redirect" title="Orthogonal vectors">orthogonal vectors</a> of <a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a> one, or, explicitly, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1,1}A_{1,2}+A_{2,1}A_{2,2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1,1}A_{1,2}+A_{2,1}A_{2,2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e71755433af57b5200ab0e3b3cde09c51b3be3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.408ex; height:2.843ex;" alt="{\displaystyle A_{1,1}A_{1,2}+A_{2,1}A_{2,2}=0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1,1}^{2}+A_{2,1}^{2}=A_{1,2}^{2}+A_{2,2}^{2}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1,1}^{2}+A_{2,1}^{2}=A_{1,2}^{2}+A_{2,2}^{2}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dae118e2762ce9b91929932b7dcc19d19ae34098" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:29.994ex; height:3.509ex;" alt="{\displaystyle A_{1,1}^{2}+A_{2,1}^{2}=A_{1,2}^{2}+A_{2,2}^{2}=1.}"></span> </p><p>This is equivalent to saying that <span class="texhtml"><i>A</i></span> times its <a href="/wiki/Transpose" title="Transpose">transpose</a> is the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>. If these conditions do not hold, the formula describes a more general <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a>. </p><p>The transformation is a translation <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <span class="texhtml"><i>A</i></span> is the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>. The transformation is a rotation around some point if and only if <span class="texhtml"><i>A</i></span> is a <a href="/wiki/Rotation_matrix" title="Rotation matrix">rotation matrix</a>, meaning that it is orthogonal and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b240b759b71f8ed4fe21d934d4f3326a374d1c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.055ex; height:2.843ex;" alt="{\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=1.}"></span> </p><p>A reflection or glide reflection is obtained when, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627678f49648e2b518f28763b45f37e77003f01b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.863ex; height:2.843ex;" alt="{\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=-1.}"></span> </p><p>Assuming that translations are not used (that 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_{1}=b_{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{1}=b_{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c723fb6d6a3638ef8ab8e83a53d560f1696b97a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.463ex; height:2.509ex;" alt="{\displaystyle b_{1}=b_{2}=0}"></span>) transformations can be <a href="/wiki/Function_composition" title="Function composition">composed</a> by simply multiplying the associated transformation matrices. In the general case, it is useful to use the <a href="/wiki/Augmented_matrix" title="Augmented matrix">augmented matrix</a> of the transformation; that is, to rewrite the transformation formula <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}x'\\y'\\1\end{pmatrix}}=A'{\begin{pmatrix}x\\y\\1\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x'\\y'\\1\end{pmatrix}}=A'{\begin{pmatrix}x\\y\\1\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5538a625293fa065aa9b23a55796bd156fa30662" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:19.154ex; height:9.509ex;" alt="{\displaystyle {\begin{pmatrix}x'\\y'\\1\end{pmatrix}}=A'{\begin{pmatrix}x\\y\\1\end{pmatrix}},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'={\begin{pmatrix}A_{1,1}&A_{1,2}&b_{1}\\A_{2,1}&A_{2,2}&b_{2}\\0&0&1\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'={\begin{pmatrix}A_{1,1}&A_{1,2}&b_{1}\\A_{2,1}&A_{2,2}&b_{2}\\0&0&1\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3b2f2b320591de641d8f6047d60da8405e8f5f1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:25.842ex; height:9.843ex;" alt="{\displaystyle A'={\begin{pmatrix}A_{1,1}&A_{1,2}&b_{1}\\A_{2,1}&A_{2,2}&b_{2}\\0&0&1\end{pmatrix}}.}"></span> With this trick, the composition of affine transformations is obtained by multiplying the augmented matrices. </p> <div class="mw-heading mw-heading3"><h3 id="Affine_transformation">Affine transformation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=18" title="Edit section: Affine transformation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:2D_affine_transformation_matrix.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/2D_affine_transformation_matrix.svg/220px-2D_affine_transformation_matrix.svg.png" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/2D_affine_transformation_matrix.svg/330px-2D_affine_transformation_matrix.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/2D_affine_transformation_matrix.svg/440px-2D_affine_transformation_matrix.svg.png 2x" data-file-width="512" data-file-height="683" /></a><figcaption>Effect of applying various 2D affine transformation matrices on a unit square (reflections are special cases of scaling)</figcaption></figure> <p><a href="/wiki/Affine_transformation" title="Affine transformation">Affine transformations</a> of the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a> are transformations that map lines to lines, but may change distances and angles. As said in the preceding section, they can be represented with augmented matrices: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}A_{1,1}&A_{2,1}&b_{1}\\A_{1,2}&A_{2,2}&b_{2}\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}={\begin{pmatrix}x'\\y'\\1\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <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> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </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> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}A_{1,1}&A_{2,1}&b_{1}\\A_{1,2}&A_{2,2}&b_{2}\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}={\begin{pmatrix}x'\\y'\\1\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41c3e0951139f1cc03919447df9dee7e20d6f5b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:36.395ex; height:9.843ex;" alt="{\displaystyle {\begin{pmatrix}A_{1,1}&A_{2,1}&b_{1}\\A_{1,2}&A_{2,2}&b_{2}\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}={\begin{pmatrix}x'\\y'\\1\end{pmatrix}}.}"></span> </p><p>The Euclidean transformations are the affine transformations such that the 2×2 matrix of the <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_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i,j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f379f42961dd620c7a05dc1c538117ec105877d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.678ex; height:2.843ex;" alt="{\displaystyle A_{i,j}}"></span> is <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a>. </p><p>The augmented matrix that represents the <a href="/wiki/Function_composition" title="Function composition">composition</a> of two affine transformations is obtained by multiplying their augmented matrices. </p><p>Some affine transformations that are not Euclidean transformations have received specific names. </p> <div class="mw-heading mw-heading4"><h4 id="Scaling">Scaling</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=19" title="Edit section: Scaling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An example of an affine transformation which is not Euclidean is given by scaling. To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number <i>m</i>. If <span class="nowrap">(<i>x</i>, <i>y</i>)</span> are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x',y')=(mx,my).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mi>x</mi> <mo>,</mo> <mi>m</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x',y')=(mx,my).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7aa538d48f3d97fca0dc6e6d4ad0c87230fa096" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.857ex; height:3.009ex;" alt="{\displaystyle (x',y')=(mx,my).}"></span> </p><p>If <i>m</i> is greater than 1, the figure becomes larger; if <i>m</i> is between 0 and 1, it becomes smaller. </p> <div class="mw-heading mw-heading4"><h4 id="Shearing">Shearing</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=20" title="Edit section: Shearing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Shear_mapping" title="Shear mapping">shearing transformation</a> will push the top of a square sideways to form a parallelogram. Horizontal shearing is defined by: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x',y')=(x+ys,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>s</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x',y')=(x+ys,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9871cabfbe5f97c991406190280a5ebdea9967f5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.216ex; height:3.009ex;" alt="{\displaystyle (x',y')=(x+ys,y)}"></span> </p><p>Shearing can also be applied vertically: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x',y')=(x,xs+y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mi>s</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x',y')=(x,xs+y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a738a9e3d84c6850e505debbfbe964b9b8c64e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.39ex; height:3.009ex;" alt="{\displaystyle (x',y')=(x,xs+y)}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Orientation_and_handedness">Orientation and handedness</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=21" title="Edit section: Orientation and handedness"><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/Orientability" title="Orientability">Orientability</a></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/Right-hand_rule" title="Right-hand rule">Right-hand rule</a> and <a href="/wiki/Axes_conventions" title="Axes conventions">Axes conventions</a></div> <div class="mw-heading mw-heading3"><h3 id="In_two_dimensions">In two dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=22" title="Edit section: In two dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Rechte-hand-regel.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Rechte-hand-regel.jpg/220px-Rechte-hand-regel.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Rechte-hand-regel.jpg/330px-Rechte-hand-regel.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Rechte-hand-regel.jpg/440px-Rechte-hand-regel.jpg 2x" data-file-width="800" data-file-height="600" /></a><figcaption>The <a href="/wiki/Right-hand_rule" title="Right-hand rule">right-hand rule</a> </figcaption></figure> <p>Fixing or choosing the <i>x</i>-axis determines the <i>y</i>-axis up to direction. Namely, the <i>y</i>-axis is necessarily the <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to the <i>x</i>-axis through the point marked 0 on the <i>x</i>-axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation (also called <i>handedness</i>) of the Cartesian plane. </p><p>The usual way of orienting the plane, with the positive <i>x</i>-axis pointing right and the positive <i>y</i>-axis pointing up (and the <i>x</i>-axis being the "first" and the <i>y</i>-axis the "second" axis), is considered the <i>positive</i> or <i>standard</i> orientation, also called the <i>right-handed</i> orientation. </p><p>A commonly used mnemonic for defining the positive orientation is the <i><a href="/wiki/Right-hand_rule" title="Right-hand rule">right-hand rule</a></i>. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the <i>x</i>-axis to the <i>y</i>-axis, in a positively oriented coordinate system. </p><p>The other way of orienting the plane is following the <i>left-hand rule</i>, placing the left hand on the plane with the thumb pointing up. </p><p>When pointing the thumb away from the origin along an axis towards positive, the curvature of the fingers indicates a positive rotation along that axis. </p><p>Regardless of the rule used to orient the plane, rotating the coordinate system will preserve the orientation. Switching any one axis will reverse the orientation, but switching both will leave the orientation unchanged. </p> <div class="mw-heading mw-heading3"><h3 id="In_three_dimensions">In three dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=23" title="Edit section: In three dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Cartesian_coordinate_system_handedness.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Cartesian_coordinate_system_handedness.svg/220px-Cartesian_coordinate_system_handedness.svg.png" decoding="async" width="220" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Cartesian_coordinate_system_handedness.svg/330px-Cartesian_coordinate_system_handedness.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Cartesian_coordinate_system_handedness.svg/440px-Cartesian_coordinate_system_handedness.svg.png 2x" data-file-width="400" data-file-height="245" /></a><figcaption>Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Right_hand_cartesian.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Right_hand_cartesian.svg/220px-Right_hand_cartesian.svg.png" decoding="async" width="220" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Right_hand_cartesian.svg/330px-Right_hand_cartesian.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Right_hand_cartesian.svg/440px-Right_hand_cartesian.svg.png 2x" data-file-width="269" data-file-height="207" /></a><figcaption>Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes</figcaption></figure> <p>Once the <i>x</i>- and <i>y</i>-axes are specified, they determine the <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a> along which the <i>z</i>-axis should lie, but there are two possible orientations for this line. The two possible coordinate systems, which result are called 'right-handed' and 'left-handed'.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The standard orientation, where the <i>xy</i>-plane is horizontal and the <i>z</i>-axis points up (and the <i>x</i>- and the <i>y</i>-axis form a positively oriented two-dimensional coordinate system in the <i>xy</i>-plane if observed from <i>above</i> the <i>xy</i>-plane) is called <b>right-handed</b> or <b>positive</b>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:3D_Cartesian_Coodinate_Handedness.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/3D_Cartesian_Coodinate_Handedness.jpg/220px-3D_Cartesian_Coodinate_Handedness.jpg" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/3D_Cartesian_Coodinate_Handedness.jpg/330px-3D_Cartesian_Coodinate_Handedness.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/3D_Cartesian_Coodinate_Handedness.jpg/440px-3D_Cartesian_Coodinate_Handedness.jpg 2x" data-file-width="800" data-file-height="450" /></a><figcaption>3D Cartesian coordinate handedness</figcaption></figure> <p>The name derives from the <a href="/wiki/Right-hand_rule" title="Right-hand rule">right-hand rule</a>. If the <a href="/wiki/Index_finger" title="Index finger">index finger</a> of the right hand is pointed forward, the <a href="/wiki/Middle_finger" title="Middle finger">middle finger</a> bent inward at a right angle to it, and the <a href="/wiki/Thumb" title="Thumb">thumb</a> placed at a right angle to both, the three fingers indicate the relative orientation of the <i>x</i>-, <i>y</i>-, and <i>z</i>-axes in a <i>right-handed</i> system. The thumb indicates the <i>x</i>-axis, the index finger the <i>y</i>-axis and the middle finger the <i>z</i>-axis. Conversely, if the same is done with the left hand, a left-handed system results. </p><p>Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point <i>towards</i> the observer, whereas the "middle"-axis is meant to point <i>away</i> from the observer. The red circle is <i>parallel</i> to the horizontal <i>xy</i>-plane and indicates rotation from the <i>x</i>-axis to the <i>y</i>-axis (in both cases). Hence the red arrow passes <i>in front of</i> the <i>z</i>-axis. </p><p>Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a <a href="https://en.wiktionary.org/wiki/convex" class="extiw" title="wikt:convex">convex</a> cube and a <a href="https://en.wiktionary.org/wiki/concave" class="extiw" title="wikt:concave">concave</a> "corner". This corresponds to the two possible orientations of the space. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the <i>x</i>-axis as pointing <i>towards</i> the observer and thus seeing a concave corner. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Representing_a_vector_in_the_standard_basis">Representing a vector in the standard basis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=24" title="Edit section: Representing a vector in the standard basis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A point in space in a Cartesian coordinate system may also be represented by a position <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a>, which can be thought of as an arrow pointing from the origin of the coordinate system to the point.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> If the coordinates represent spatial positions (displacements), it is common to represent the vector from the origin to the point of interest 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 {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span>. In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b314439fb208f26860eacddbb878e7c83c6a1d78" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.731ex; height:2.509ex;" alt="{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} ,}"></span> </p><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 \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c981894d231e081378f98f474b5b368068ace133" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.176ex; height:6.176ex;" alt="{\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}}"></span> 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 {j} ={\begin{pmatrix}0\\1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753de7491cee3b69ed6401d1f9c56cd0a9f09212" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.164ex; width:9.414ex; height:6.176ex;" alt="{\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}}"></span> are <a href="/wiki/Unit_vectors" class="mw-redirect" title="Unit vectors">unit vectors</a> in the direction of the <i>x</i>-axis and <i>y</i>-axis respectively, generally referred to as the <i><a href="/wiki/Standard_basis" title="Standard basis">standard basis</a></i> (in some application areas these may also be referred to as <a href="/wiki/Versor" title="Versor">versors</a>). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates <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> can be written as:<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/557378b34f29a80fdb8ccf95997c9106ca0fb27e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.071ex; height:2.509ex;" alt="{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k} ,}"></span> </p><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 \mathbf {i} ={\begin{pmatrix}1\\0\\0\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\\0\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cd3d23c199969d4acf470cf1584de38c135b459" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:10.468ex; height:9.176ex;" alt="{\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\\0\end{pmatrix}},}"></span> <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 {j} ={\begin{pmatrix}0\\1\\0\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\\0\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12f338adc9bbe187c14992944691d461f6502e5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.164ex; width:10.706ex; height:9.176ex;" alt="{\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\\0\end{pmatrix}},}"></span> 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 {k} ={\begin{pmatrix}0\\0\\1\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {k} ={\begin{pmatrix}0\\0\\1\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0d046ac2dae27878e4aff6cc08f5dab5bedbe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.137ex; height:9.176ex;" alt="{\displaystyle \mathbf {k} ={\begin{pmatrix}0\\0\\1\end{pmatrix}}.}"></span> </p><p>There is no <i>natural</i> interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> to provide such a multiplication. In a two-dimensional cartesian plane, identify the point with coordinates <span class="nowrap">(<i>x</i>, <i>y</i>)</span> with the complex number <span class="nowrap"><i>z</i> = <i>x</i> + <i>iy</i></span>. Here, <i>i</i> is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a> and is identified with the point with coordinates <span class="nowrap">(0, 1)</span>, so it is <i>not</i> the unit vector in the direction of the <i>x</i>-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three-dimensional cartesian space a similar identification can be made with a subset of the <a href="/wiki/Quaternion" title="Quaternion">quaternions</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=Cartesian_coordinate_system&action=edit&section=25" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cartesian_coordinate_robot" title="Cartesian coordinate robot">Cartesian coordinate robot</a></li> <li><a href="/wiki/Horizontal_and_vertical" class="mw-redirect" title="Horizontal and vertical">Horizontal and vertical</a></li> <li><a href="/wiki/Jones_diagram" title="Jones diagram">Jones diagram</a>, which plots four variables rather than two</li> <li><a href="/wiki/Orthogonal_coordinates" title="Orthogonal coordinates">Orthogonal coordinates</a></li> <li><a href="/wiki/Polar_coordinate_system" title="Polar coordinate system">Polar coordinate system</a></li> <li><a href="/wiki/Regular_grid" title="Regular grid">Regular grid</a></li> <li><a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">Spherical coordinate system</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=26" title="Edit section: Citations"><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 mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFBixD'Souza" class="citation web cs1">Bix, Robert A.; D'Souza, Harry J. <a rel="nofollow" class="external text" href="https://www.britannica.com/topic/analytic-geometry">"Analytic geometry"</a>. <i>Encyclopædia Britannica</i><span class="reference-accessdate">. Retrieved <span class="nowrap">6 August</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclop%C3%A6dia+Britannica&rft.atitle=Analytic+geometry&rft.aulast=Bix&rft.aufirst=Robert+A.&rft.au=D%27Souza%2C+Harry+J.&rft_id=https%3A%2F%2Fwww.britannica.com%2Ftopic%2Fanalytic-geometry&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" 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"><a href="#CITEREFKentVujakovic2017">Kent & Vujakovic 2017</a>, See <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EVRSDwAAQBAJ&q=Nicole+Oresme+coordinate&pg=PT307">here</a></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="CITEREFKatz2009" class="citation book cs1">Katz, Victor J. (2009). <a rel="nofollow" class="external text" href="https://www.worldcat.org/title/71006826"><i>A history of mathematics: an introduction</i></a> (3rd ed.). Boston: Addison-Wesley. p. 484. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-38700-4" title="Special:BookSources/978-0-321-38700-4"><bdi>978-0-321-38700-4</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/71006826">71006826</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+mathematics%3A+an+introduction&rft.place=Boston&rft.pages=484&rft.edition=3rd&rft.pub=Addison-Wesley&rft.date=2009&rft_id=info%3Aoclcnum%2F71006826&rft.isbn=978-0-321-38700-4&rft.aulast=Katz&rft.aufirst=Victor+J.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Ftitle%2F71006826&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" 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"><a href="#CITEREFBurton2011">Burton 2011</a>, p. 374.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFBerlinski2011">Berlinski 2011</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"><a href="#CITEREFAxler2015">Axler 2015</a>, p. 1</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">Consider the two <a href="/wiki/Ray_(geometry)" class="mw-redirect" title="Ray (geometry)">rays</a> or half-lines resulting from splitting the line at the origin. One of the half-lines can be assigned to positive numbers, and the other half-line to negative numbers.</span> </li> <li id="cite_note-:0-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_8-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_8-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_8-3"><sup><i><b>d</b></i></sup></a></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.encyclopediaofmath.org/index.php/Cartesian_orthogonal_coordinate_system">"Cartesian orthogonal coordinate system"</a>. <i>Encyclopedia of Mathematics</i><span class="reference-accessdate">. Retrieved <span class="nowrap">6 August</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Cartesian+orthogonal+coordinate+system&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%2FCartesian_orthogonal_coordinate_system&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://planetmath.org/cartesiancoordinates">"Cartesian coordinates"</a>. <i>planetmath.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">25 August</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=planetmath.org&rft.atitle=Cartesian+coordinates&rft_id=https%3A%2F%2Fplanetmath.org%2Fcartesiancoordinates&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></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"><a href="#CITEREFHughes-HallettMcCallumGleason2013">Hughes-Hallett, McCallum & Gleason 2013</a></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#CITEREFSmart1998">Smart 1998</a>, Chap. 2</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"><a href="#CITEREFBrannanEsplenGray1998">Brannan, Esplen & Gray 1998</a>, pg. 49</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFAntonBivensDavis2021">Anton, Bivens & Davis 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=001EEAAAQBAJ&pg=PA657">657</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFBrannanEsplenGray1998">Brannan, Esplen & Gray 1998</a>, Appendix 2, pp. 377–382</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="#CITEREFGriffiths1999">Griffiths 1999</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="General_and_cited_references">General and cited references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&section=27" title="Edit section: General and cited references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAxler2015" class="citation book cs1">Axler, Sheldon (2015). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220527195708/https://zenodo.org/record/4461746"><i>Linear Algebra Done Right</i></a>. Undergraduate Texts in Mathematics. Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-11080-6">10.1007/978-3-319-11080-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-11079-0" title="Special:BookSources/978-3-319-11079-0"><bdi>978-3-319-11079-0</bdi></a>. Archived from <a rel="nofollow" class="external text" href="https://zenodo.org/record/4461746">the original</a> on 27 May 2022<span class="reference-accessdate">. Retrieved <span class="nowrap">17 April</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Done+Right&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2015&rft_id=info%3Adoi%2F10.1007%2F978-3-319-11080-6&rft.isbn=978-3-319-11079-0&rft.aulast=Axler&rft.aufirst=Sheldon&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F4461746&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerlinski2011" class="citation book cs1">Berlinski, David (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Com9OzFJgRcC"><i>A Tour of the Calculus</i></a>. Knopf Doubleday Publishing Group. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780307789730" title="Special:BookSources/9780307789730"><bdi>9780307789730</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+Tour+of+the+Calculus&rft.pub=Knopf+Doubleday+Publishing+Group&rft.date=2011&rft.isbn=9780307789730&rft.aulast=Berlinski&rft.aufirst=David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCom9OzFJgRcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrannanEsplenGray1998" class="citation book cs1">Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998). <i>Geometry</i>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-59787-6" title="Special:BookSources/978-0-521-59787-6"><bdi>978-0-521-59787-6</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&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1998&rft.isbn=978-0-521-59787-6&rft.aulast=Brannan&rft.aufirst=David+A.&rft.au=Esplen%2C+Matthew+F.&rft.au=Gray%2C+Jeremy+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurton2011" class="citation book cs1">Burton, David M. (2011). <i>The History of Mathematics/An Introduction</i> (7th ed.). New York: McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-338315-6" title="Special:BookSources/978-0-07-338315-6"><bdi>978-0-07-338315-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+History+of+Mathematics%2FAn+Introduction&rft.place=New+York&rft.edition=7th&rft.pub=McGraw-Hill&rft.date=2011&rft.isbn=978-0-07-338315-6&rft.aulast=Burton&rft.aufirst=David+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths1999" class="citation book cs1">Griffiths, David J. (1999). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoel00grif_0"><i>Introduction to Electrodynamics</i></a></span>. Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-805326-0" title="Special:BookSources/978-0-13-805326-0"><bdi>978-0-13-805326-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Electrodynamics&rft.pub=Prentice+Hall&rft.date=1999&rft.isbn=978-0-13-805326-0&rft.aulast=Griffiths&rft.aufirst=David+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoel00grif_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHughes-HallettMcCallumGleason2013" class="citation book cs1">Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013). <i>Calculus: Single and Multivariable</i> (6th ed.). <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0470-88861-2" title="Special:BookSources/978-0470-88861-2"><bdi>978-0470-88861-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Single+and+Multivariable&rft.edition=6th&rft.pub=John+Wiley+%26+Sons&rft.date=2013&rft.isbn=978-0470-88861-2&rft.aulast=Hughes-Hallett&rft.aufirst=Deborah&rft.au=McCallum%2C+William+G.&rft.au=Gleason%2C+Andrew+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKentVujakovic2017" class="citation book cs1">Kent, Alexander J.; Vujakovic, Peter (4 October 2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EVRSDwAAQBAJ"><i>The Routledge Handbook of Mapping and Cartography</i></a>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781317568216" title="Special:BookSources/9781317568216"><bdi>9781317568216</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Routledge+Handbook+of+Mapping+and+Cartography&rft.pub=Routledge&rft.date=2017-10-04&rft.isbn=9781317568216&rft.aulast=Kent&rft.aufirst=Alexander+J.&rft.au=Vujakovic%2C+Peter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEVRSDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmart1998" class="citation cs2">Smart, James R. (1998), <i>Modern Geometries</i> (5th ed.), Pacific Grove: Brooks/Cole, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-35188-5" title="Special:BookSources/978-0-534-35188-5"><bdi>978-0-534-35188-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Geometries&rft.place=Pacific+Grove&rft.edition=5th&rft.pub=Brooks%2FCole&rft.date=1998&rft.isbn=978-0-534-35188-5&rft.aulast=Smart&rft.aufirst=James+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAntonBivensDavis2021" class="citation book cs1">Anton, Howard; Bivens, Irl C.; Davis, Stephen (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=001EEAAAQBAJ"><i>Calculus: Multivariable</i></a>. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. p. 657. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-77798-4" title="Special:BookSources/978-1-119-77798-4"><bdi>978-1-119-77798-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Multivariable&rft.pages=657&rft.pub=John+Wiley+%26+Sons&rft.date=2021&rft.isbn=978-1-119-77798-4&rft.aulast=Anton&rft.aufirst=Howard&rft.au=Bivens%2C+Irl+C.&rft.au=Davis%2C+Stephen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D001EEAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li></ul> <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=Cartesian_coordinate_system&action=edit&section=28" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDescartes,_René2001" class="citation book cs1"><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes, René</a> (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XKVvclclrnwC"><i>Discourse on Method, Optics, Geometry, and Meteorology</i></a>. Translated by Paul J. Oscamp (Revised ed.). Indianapolis, IN: Hackett Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-87220-567-3" title="Special:BookSources/978-0-87220-567-3"><bdi>978-0-87220-567-3</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/488633510">488633510</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discourse+on+Method%2C+Optics%2C+Geometry%2C+and+Meteorology&rft.place=Indianapolis%2C+IN&rft.edition=Revised&rft.pub=Hackett+Publishing&rft.date=2001&rft_id=info%3Aoclcnum%2F488633510&rft.isbn=978-0-87220-567-3&rft.au=Descartes%2C+Ren%C3%A9&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXKVvclclrnwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKornKorn1961" class="citation book cs1">Korn GA, <a href="/wiki/Theresa_M._Korn" title="Theresa M. Korn">Korn TM</a> (1961). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalhand0000korn"><i>Mathematical Handbook for Scientists and Engineers</i></a></span> (1st ed.). New York: McGraw-Hill. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalhand0000korn/page/55">55–79</a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/59-14456">59-14456</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/19959906">19959906</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Handbook+for+Scientists+and+Engineers&rft.place=New+York&rft.pages=55-79&rft.edition=1st&rft.pub=McGraw-Hill&rft.date=1961&rft_id=info%3Aoclcnum%2F19959906&rft_id=info%3Alccn%2F59-14456&rft.aulast=Korn&rft.aufirst=GA&rft.au=Korn%2C+TM&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalhand0000korn&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMargenauMurphy1956" class="citation book cs1"><a href="/wiki/Henry_Margenau" title="Henry Margenau">Margenau H</a>, Murphy GM (1956). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsofphy0002marg"><i>The Mathematics of Physics and Chemistry</i></a></span>. New York: D. van Nostrand. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/55-10911">55-10911</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematics+of+Physics+and+Chemistry&rft.place=New+York&rft.pub=D.+van+Nostrand&rft.date=1956&rft_id=info%3Alccn%2F55-10911&rft.aulast=Margenau&rft.aufirst=H&rft.au=Murphy%2C+GM&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicsofphy0002marg&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoonSpencer1988" class="citation book cs1">Moon P, Spencer DE (1988). "Rectangular Coordinates (x, y, z)". <i>Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions</i> (corrected 2nd, 3rd print ed.). New York: Springer-Verlag. pp. 9–11 (Table 1.01). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-18430-2" title="Special:BookSources/978-0-387-18430-2"><bdi>978-0-387-18430-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Rectangular+Coordinates+%28x%2C+y%2C+z%29&rft.btitle=Field+Theory+Handbook%2C+Including+Coordinate+Systems%2C+Differential+Equations%2C+and+Their+Solutions&rft.place=New+York&rft.pages=9-11+%28Table+1.01%29&rft.edition=corrected+2nd%2C+3rd+print&rft.pub=Springer-Verlag&rft.date=1988&rft.isbn=978-0-387-18430-2&rft.aulast=Moon&rft.aufirst=P&rft.au=Spencer%2C+DE&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorseFeshbach1953" class="citation book cs1"><a href="/wiki/Philip_M._Morse" title="Philip M. Morse">Morse PM</a>, <a href="/wiki/Herman_Feshbach" title="Herman Feshbach">Feshbach H</a> (1953). <i>Methods of Theoretical Physics, Part I</i>. New York: McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-043316-8" title="Special:BookSources/978-0-07-043316-8"><bdi>978-0-07-043316-8</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/52-11515">52-11515</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Theoretical+Physics%2C+Part+I&rft.place=New+York&rft.pub=McGraw-Hill&rft.date=1953&rft_id=info%3Alccn%2F52-11515&rft.isbn=978-0-07-043316-8&rft.aulast=Morse&rft.aufirst=PM&rft.au=Feshbach%2C+H&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSauerSzabó1967" class="citation book cs1">Sauer R, Szabó I (1967). <i>Mathematische Hilfsmittel des Ingenieurs</i>. New York: Springer Verlag. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/67-25285">67-25285</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematische+Hilfsmittel+des+Ingenieurs&rft.place=New+York&rft.pub=Springer+Verlag&rft.date=1967&rft_id=info%3Alccn%2F67-25285&rft.aulast=Sauer&rft.aufirst=R&rft.au=Szab%C3%B3%2C+I&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></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=Cartesian_coordinate_system&action=edit&section=29" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.cut-the-knot.org/Curriculum/Calculus/Coordinates.shtml">Cartesian Coordinate System</a></li> <li><span class="citation mathworld" id="Reference-Mathworld-Cartesian_Coordinates"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CartesianCoordinates.html">"Cartesian Coordinates"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Cartesian+Coordinates&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCartesianCoordinates.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACartesian+coordinate+system" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="https://www.random-science-tools.com/maths/coordinate-converter.htm">Coordinate Converter – converts between polar, Cartesian and spherical coordinates</a></li> <li><a rel="nofollow" class="external text" href="https://www.mathopenref.com/coordpoint.html">Coordinates of a point</a> – interactive tool to explore coordinates of a point</li> <li><a rel="nofollow" class="external text" href="https://github.com/DanIsraelMalta/CoordSysJS">open source JavaScript class for 2D/3D Cartesian coordinate system manipulation</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist 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Cartesian coordinate system - Wikipedia