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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Common_coordinate_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Common_coordinate_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Common coordinate systems</span> </div> </a> <button aria-controls="toc-Common_coordinate_systems-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 Common coordinate systems subsection</span> </button> <ul id="toc-Common_coordinate_systems-sublist" class="vector-toc-list"> <li id="toc-Number_line" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_line"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Number line</span> </div> </a> <ul id="toc-Number_line-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cartesian_coordinate_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cartesian_coordinate_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Cartesian coordinate system</span> </div> </a> <ul id="toc-Cartesian_coordinate_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polar_coordinate_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polar_coordinate_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Polar coordinate system</span> </div> </a> <ul id="toc-Polar_coordinate_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cylindrical_and_spherical_coordinate_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cylindrical_and_spherical_coordinate_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Cylindrical and spherical coordinate systems</span> </div> </a> <ul id="toc-Cylindrical_and_spherical_coordinate_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Homogeneous_coordinate_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Homogeneous_coordinate_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Homogeneous coordinate system</span> </div> </a> <ul id="toc-Homogeneous_coordinate_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_commonly_used_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_commonly_used_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Other commonly used systems</span> </div> </a> <ul id="toc-Other_commonly_used_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Coordinates_of_geometric_objects" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Coordinates_of_geometric_objects"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Coordinates of geometric objects</span> </div> </a> <ul id="toc-Coordinates_of_geometric_objects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transformations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Transformations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Transformations</span> </div> </a> <ul id="toc-Transformations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coordinate_lines/curves" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Coordinate_lines/curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Coordinate lines/curves</span> </div> </a> <ul id="toc-Coordinate_lines/curves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coordinate_planes/surfaces" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Coordinate_planes/surfaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Coordinate planes/surfaces</span> </div> </a> <ul id="toc-Coordinate_planes/surfaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coordinate_maps" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Coordinate_maps"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Coordinate maps</span> </div> </a> <ul id="toc-Coordinate_maps-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orientation-based_coordinates" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Orientation-based_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Orientation-based coordinates</span> </div> </a> <ul id="toc-Orientation-based_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geographic_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geographic_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Geographic systems</span> </div> </a> <ul id="toc-Geographic_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <button aria-controls="toc-See_also-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 See also subsection</span> </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-Relativistic_coordinate_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativistic_coordinate_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Relativistic coordinate systems</span> </div> </a> <ul id="toc-Relativistic_coordinate_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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 References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">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" > <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">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 94 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-94" 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">94 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/Ko%C3%B6rdinatestelsel" title="Koördinatestelsel – Afrikaans" lang="af" hreflang="af" data-title="Koördinatestelsel" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Koordinatensystem" title="Koordinatensystem – Alemannic" lang="gsw" hreflang="gsw" data-title="Koordinatensystem" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</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" title="نظام إحداثيات – Arabic" lang="ar" hreflang="ar" data-title="نظام إحداثيات" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Sistema_de_coordenadas" title="Sistema de coordenadas – Aragonese" lang="an" hreflang="an" data-title="Sistema de coordenadas" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Sistema_de_coordenaes" title="Sistema de coordenaes – Asturian" lang="ast" hreflang="ast" data-title="Sistema de coordenaes" 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/Koordinat_sistemi" title="Koordinat sistemi – Azerbaijani" lang="az" hreflang="az" data-title="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%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="স্থানাঙ্ক ব্যবস্থা – Bangla" lang="bn" hreflang="bn" data-title="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-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ch%C5%8D-piau-h%C4%93" title="Chō-piau-hē – Minnan" lang="nan" hreflang="nan" data-title="Chō-piau-hē" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%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="Координаталар системаһы – Bashkir" lang="ba" hreflang="ba" data-title="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-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%96%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%BA%D0%B0%D0%B0%D1%80%D0%B4%D1%8B%D0%BD%D0%B0%D1%82" title="Сістэма каардынат – Belarusian" lang="be" hreflang="be" data-title="Сістэма каардынат" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A1%D1%8B%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%BA%D0%B0%D0%B0%D1%80%D0%B4%D1%8B%D0%BD%D0%B0%D1%82" title="Сыстэма каардынат – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Сыстэма каардынат" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%95%E0%A5%8B%E0%A4%86%E0%A4%B0%E0%A5%8D%E0%A4%A1%E0%A4%BF%E0%A4%A8%E0%A5%87%E0%A4%9F_%E0%A4%B8%E0%A4%BF%E0%A4%B8%E0%A5%8D%E0%A4%9F%E0%A4%AE" title="कोआर्डिनेट सिस्टम – Bhojpuri" lang="bh" hreflang="bh" data-title="कोआर्डिनेट सिस्टम" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%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="Координатна система – Bulgarian" lang="bg" hreflang="bg" data-title="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/Koordinatni_sistem" title="Koordinatni sistem – Bosnian" lang="bs" hreflang="bs" data-title="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" title="Sistema de coordenades – Catalan" lang="ca" hreflang="ca" data-title="Sistema de coordenades" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D1%81%D0%B5%D0%BD_%D1%82%D1%8B%D1%82%C4%83%D0%BC%C4%95" title="Координатсен тытăмĕ – Chuvash" lang="cv" hreflang="cv" data-title="Координатсен 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/Soustava_sou%C5%99adnic" title="Soustava souřadnic – Czech" lang="cs" hreflang="cs" data-title="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-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Kupanga_Mazano" title="Kupanga Mazano – Shona" lang="sn" hreflang="sn" data-title="Kupanga Mazano" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/System_gyfesurynnau" title="System gyfesurynnau – Welsh" lang="cy" hreflang="cy" data-title="System gyfesurynnau" 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/Koordinatsystem" title="Koordinatsystem – Danish" lang="da" hreflang="da" data-title="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/Koordinatensystem" title="Koordinatensystem – German" lang="de" hreflang="de" data-title="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/Koordinaadis%C3%BCsteem" title="Koordinaadisüsteem – Estonian" lang="et" hreflang="et" data-title="Koordinaadisüsteem" 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%A3%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1_%CE%B1%CE%BD%CE%B1%CF%86%CE%BF%CF%81%CE%AC%CF%82" title="Σύστημα αναφοράς – Greek" lang="el" hreflang="el" data-title="Σύστημα αναφοράς" 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/Sistema_de_coordenadas" title="Sistema de coordenadas – Spanish" lang="es" hreflang="es" data-title="Sistema de coordenadas" 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/Koordinatsistemo" title="Koordinatsistemo – Esperanto" lang="eo" hreflang="eo" data-title="Koordinatsistemo" 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_sistema" title="Koordenatu sistema – Basque" lang="eu" hreflang="eu" data-title="Koordenatu sistema" 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" title="دستگاه مختصات – Persian" lang="fa" hreflang="fa" data-title="دستگاه مختصات" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Coordinate_system" title="Coordinate system – Fiji Hindi" lang="hif" hreflang="hif" data-title="Coordinate system" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_de_coordonn%C3%A9es" title="Système de coordonnées – French" lang="fr" hreflang="fr" data-title="Système de coordonnées" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Siostaman_cho-chomharran" title="Siostaman cho-chomharran – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Siostaman cho-chomharran" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Sistema_de_coordenadas" title="Sistema de coordenadas – Galician" lang="gl" hreflang="gl" data-title="Sistema de coordenadas" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%BA%A7%E6%A8%99" title="座標 – Gan" lang="gan" hreflang="gan" data-title="座標" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A2%8C%ED%91%9C%EA%B3%84" title="좌표계 – Korean" lang="ko" hreflang="ko" data-title="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%BF%D5%B8%D5%B8%D6%80%D5%A4%D5%AB%D5%B6%D5%A1%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B0%D5%A1%D5%B4%D5%A1%D5%AF%D5%A1%D6%80%D5%A3" title="Կոորդինատային համակարգ – Armenian" lang="hy" hreflang="hy" data-title="Կոորդինատային 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%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="निर्देशांक पद्धति – Hindi" lang="hi" hreflang="hi" data-title="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/Koordinatni_sustav" title="Koordinatni sustav – Croatian" lang="hr" hreflang="hr" data-title="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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sistem_koordinat" title="Sistem koordinat – Indonesian" lang="id" hreflang="id" data-title="Sistem koordinat" 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/Hnit_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Hnit (stærðfræði) – Icelandic" lang="is" hreflang="is" data-title="Hnit (stærðfræði)" 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_coordinate" title="Sistema di coordinate – Italian" lang="it" hreflang="it" data-title="Sistema di coordinate" 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%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA" title="קואורדינטות – Hebrew" lang="he" hreflang="he" data-title="קואורדינטות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%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="Координаттар жүйесі – Kazakh" lang="kk" hreflang="kk" data-title="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-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Majiranukta" title="Majiranukta – Swahili" lang="sw" hreflang="sw" data-title="Majiranukta" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Sist%C3%A8m_kouw%C3%B2done" title="Sistèm kouwòdone – Haitian Creole" lang="ht" hreflang="ht" data-title="Sistèm kouwòdone" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Sist%C3%A8m_di_ko%C3%B2rdon%C3%A9" title="Sistèm di koòrdoné – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Sistèm di koòrdoné" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B0%D0%BB%D0%B0%D1%80_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%B4%D0%B0)" title="Координаталар (математикада) – Kyrgyz" lang="ky" hreflang="ky" data-title="Координаталар (математикада)" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Systema_coordinatarum" title="Systema coordinatarum – Latin" lang="la" hreflang="la" data-title="Systema 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/Koordin%C4%81tu_sist%C4%93ma" title="Koordinātu sistēma – Latvian" lang="lv" hreflang="lv" data-title="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-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Koordinatesystem" title="Koordinatesystem – Luxembourgish" lang="lb" hreflang="lb" data-title="Koordinatesystem" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Koordina%C4%8Di%C5%B3_sistema" title="Koordinačių sistema – Lithuanian" lang="lt" hreflang="lt" data-title="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/Koordin%C3%A1ta-rendszer" title="Koordináta-rendszer – Hungarian" lang="hu" hreflang="hu" data-title="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%9A%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="Координатен систем – Macedonian" lang="mk" hreflang="mk" data-title="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%A8%E0%B4%BF%E0%B5%BC%E0%B4%A6%E0%B5%87%E0%B4%B6%E0%B4%BE%E0%B4%99%E0%B5%8D%E0%B4%95%E0%B4%B5%E0%B5%8D%E0%B4%AF%E0%B4%B5%E0%B4%B8%E0%B5%8D%E0%B4%A5" title="നിർദേശാങ്കവ്യവസ്ഥ – Malayalam" lang="ml" hreflang="ml" data-title="നിർദേശാങ്കവ്യവസ്ഥ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sistem_koordinat" title="Sistem koordinat – Malay" lang="ms" hreflang="ms" data-title="Sistem koordinat" 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%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="ကိုဩဒိနိတ်စနစ် – Burmese" lang="my" hreflang="my" data-title="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/Co%C3%B6rdinatenstelsel" title="Coördinatenstelsel – Dutch" lang="nl" hreflang="nl" data-title="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/%E5%BA%A7%E6%A8%99" title="座標 – Japanese" lang="ja" hreflang="ja" data-title="座標" 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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Koordinaatens%C3%BCsteem" title="Koordinaatensüsteem – Northern Frisian" lang="frr" hreflang="frr" data-title="Koordinaatensüsteem" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Koordinatsystem" title="Koordinatsystem – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="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/Koordinatsystem" title="Koordinatsystem – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Sist%C3%A8ma_de_coordenadas" title="Sistèma de coordenadas – Occitan" lang="oc" hreflang="oc" data-title="Sistèma de coordenadas" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Sirna_Diriiroo_Ko%E2%80%99oordineetotaa" title="Sirna Diriiroo Ko’oordineetotaa – Oromo" lang="om" hreflang="om" data-title="Sirna Diriiroo Ko’oordineetotaa" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Koordinatalar" title="Koordinatalar – Uzbek" lang="uz" hreflang="uz" data-title="Koordinatalar" 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%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="ਗੁਣਕ ਪ੍ਰਬੰਧ – Punjabi" lang="pa" hreflang="pa" data-title="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-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%A9%D9%88%D8%A2%D8%B1%DA%88%DB%8C%D9%86%DB%8C%D9%B9_%D9%BE%D8%B1%D8%A8%D9%86%D8%AF%DA%BE" title="کوآرڈینیٹ پربندھ – Western Punjabi" lang="pnb" hreflang="pnb" data-title="کوآرڈینیٹ پربندھ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Kwaadinet_(matimatix)" title="Kwaadinet (matimatix) – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Kwaadinet (matimatix)" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Coordin%C3%A0" title="Coordinà – Piedmontese" lang="pms" hreflang="pms" data-title="Coordinà" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Koordinatensysteem" title="Koordinatensysteem – Low German" lang="nds" hreflang="nds" data-title="Koordinatensysteem" 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" title="Układ współrzędnych – Polish" lang="pl" hreflang="pl" data-title="Układ współrzędnych" 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" title="Sistema de coordenadas – Portuguese" lang="pt" hreflang="pt" data-title="Sistema de coordenadas" 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/Sistem_de_coordonate" title="Sistem de coordonate – Romanian" lang="ro" hreflang="ro" data-title="Sistem de coordonate" 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%A1%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="Система координат – Russian" lang="ru" hreflang="ru" data-title="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" title="Sistemi koordinativ – Albanian" lang="sq" hreflang="sq" data-title="Sistemi koordinativ" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Coordinate_system" title="Coordinate system – Simple English" lang="en-simple" hreflang="en-simple" data-title="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-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%DA%AA%D9%88%D8%A2%D8%B1%DA%8A%D9%8A%D9%86%D9%8A%D9%BD_%D8%B3%D8%B3%D9%BD%D9%85" title="ڪوآرڊينيٽ سسٽم – Sindhi" lang="sd" hreflang="sd" data-title="ڪوآرڊينيٽ سسٽم" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/S%C3%BAstava_s%C3%BAradn%C3%ADc" title="Sústava súradníc – Slovak" lang="sk" hreflang="sk" data-title="Sústava súradníc" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Koordinatni_sistem" title="Koordinatni sistem – Slovenian" lang="sl" hreflang="sl" data-title="Koordinatni sistem" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Uk%C5%82od_wsp%C5%AF%C5%82rzyndnych" title="Ukłod wspůłrzyndnych – Silesian" lang="szl" hreflang="szl" data-title="Ukłod wspůłrzyndnych" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B3%DB%8C%D8%B3%D8%AA%D9%85%DB%8C_%D9%BE%DB%86%D8%AA%D8%A7%D9%86%DB%95%DA%A9%D8%A7%D9%86" title="سیستمی پۆتانەکان – Central Kurdish" lang="ckb" hreflang="ckb" data-title="سیستمی پۆتانەکان" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B5" title="Координате – Serbian" lang="sr" hreflang="sr" data-title="Координате" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Koordinate" title="Koordinate – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Koordinate" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Koordinaatisto" title="Koordinaatisto – Finnish" lang="fi" hreflang="fi" data-title="Koordinaatisto" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Koordinatsystem" title="Koordinatsystem – Swedish" lang="sv" hreflang="sv" data-title="Koordinatsystem" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Sistema_ng_koordinado" title="Sistema ng koordinado – Tagalog" lang="tl" hreflang="tl" data-title="Sistema ng koordinado" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%86%E0%AE%B3%E0%AF%8D%E0%AE%95%E0%AF%82%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81_%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%88%E0%AE%AE%E0%AF%88" title="ஆள்கூற்று முறைமை – Tamil" lang="ta" hreflang="ta" data-title="ஆள்கூற்று முறைமை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%9A%E0%B8%9A%E0%B8%9E%E0%B8%B4%E0%B8%81%E0%B8%B1%E0%B8%94" title="ระบบพิกัด – Thai" lang="th" hreflang="th" data-title="ระบบพิกัด" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Koordinat_sistemi" title="Koordinat sistemi – Turkish" lang="tr" hreflang="tr" data-title="Koordinat sistemi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%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="Система координат – Ukrainian" lang="uk" hreflang="uk" data-title="Система координат" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%AA%D9%86%D8%A7%D8%B3%D9%82_%D9%86%D8%B8%D8%A7%D9%85" title="متناسق نظام – Urdu" lang="ur" hreflang="ur" data-title="متناسق نظام" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_t%E1%BB%8Da_%C4%91%E1%BB%99" title="Hệ tọa độ – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ tọa độ" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Sistema_coordinata" title="Sistema coordinata – Waray" lang="war" hreflang="war" data-title="Sistema coordinata" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%9D%90%E6%A0%87%E7%B3%BB" title="坐标系 – Wu" lang="wuu" hreflang="wuu" data-title="坐标系" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%90%D7%90%D7%A8%D7%93%D7%99%D7%A0%D7%90%D7%98%D7%9F_%D7%A1%D7%99%D7%A1%D7%98%D7%A2%D7%9D" title="קאארדינאטן סיסטעם – Yiddish" lang="yi" hreflang="yi" data-title="קאארדינאטן סיסטעם" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%9D%90%E6%A8%99" 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</div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" 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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">"Coordinate" redirects here. For coordinates on the Earth, see <a href="/wiki/Spatial_reference_system" title="Spatial reference system">Spatial reference system</a>. For other uses, see <a href="/wiki/Coordinate_(disambiguation)" class="mw-disambig" title="Coordinate (disambiguation)">Coordinate (disambiguation)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:3D_Spherical.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/3D_Spherical.svg/220px-3D_Spherical.svg.png" decoding="async" width="220" height="191" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/3D_Spherical.svg/330px-3D_Spherical.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/3D_Spherical.svg/440px-3D_Spherical.svg.png 2x" data-file-width="600" data-file-height="520" /></a><figcaption>The <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinate system</a> is commonly used in <i>physics</i>. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance <i>r</i>, polar angle <i>θ</i> (<a href="/wiki/Theta" title="Theta">theta</a>), and azimuthal angle <i>φ</i> (<a href="/wiki/Phi" title="Phi">phi</a>). The symbol <i>ρ</i> (<a href="/wiki/Rho" title="Rho">rho</a>) is often used instead of <i>r</i>.</figcaption></figure> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>coordinate system</b> is a system that uses one or more <a href="/wiki/Number" title="Number">numbers</a>, or <b>coordinates</b>, to uniquely determine the <a href="/wiki/Position_(geometry)" title="Position (geometry)">position</a> of the <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a> or other geometric elements on a <a href="/wiki/Manifold" title="Manifold">manifold</a> such as <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>.<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><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> The order of the coordinates is significant, and they are sometimes identified by their position in an ordered <a href="/wiki/Tuple" title="Tuple">tuple</a> and sometimes by a letter, as in "the <i>x</i>-coordinate". The coordinates are taken to be <a href="/wiki/Real_number" title="Real number">real numbers</a> in <a href="/wiki/Elementary_mathematics" title="Elementary mathematics">elementary mathematics</a>, but may be <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> or elements of a more abstract system such as a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a>. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and <i>vice versa</i>; this is the basis of <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Common_coordinate_systems">Common coordinate systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=1" title="Edit section: Common coordinate systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Number_line">Number line</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=2" title="Edit section: Number line"><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/Number_line" title="Number line">Number line</a></div> <p><span class="anchor" id="Number_line"></span> The simplest example of a coordinate system is the identification of points on a <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a> with real numbers using the <i><a href="/wiki/Number_line" title="Number line">number line</a></i>. In this system, an arbitrary point <i>O</i> (the <i>origin</i>) is chosen on a given line. The coordinate of a point <i>P</i> is defined as the signed distance from <i>O</i> to <i>P</i>, where the signed distance is the distance taken as positive or negative depending on which side of the line <i>P</i> lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point.<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> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:Number-line-right-arrow.svg" class="mw-file-description" title="The number line"><img alt="The number line" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Number-line-right-arrow.svg/512px-Number-line-right-arrow.svg.png" decoding="async" width="512" height="34" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Number-line-right-arrow.svg/768px-Number-line-right-arrow.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Number-line-right-arrow.svg/1024px-Number-line-right-arrow.svg.png 2x" data-file-width="512" data-file-height="34" /></a><figcaption>The number line</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Cartesian_coordinate_system">Cartesian coordinate system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=3" title="Edit section: Cartesian coordinate system"><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/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:442px;max-width:442px"><div class="trow"><div class="tsingle" style="width:219px;max-width:219px"><div class="thumbimage" style="height:214px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Cartesian-coordinate-system.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/217px-Cartesian-coordinate-system.svg.png" decoding="async" width="217" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/326px-Cartesian-coordinate-system.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/434px-Cartesian-coordinate-system.svg.png 2x" data-file-width="661" data-file-height="654" /></a></span></div><div class="thumbcaption">The <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a> in the plane</div></div><div class="tsingle" style="width:219px;max-width:219px"><div class="thumbimage" style="height:214px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Rectangular_coordinates.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Rectangular_coordinates.svg/217px-Rectangular_coordinates.svg.png" decoding="async" width="217" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Rectangular_coordinates.svg/326px-Rectangular_coordinates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Rectangular_coordinates.svg/434px-Rectangular_coordinates.svg.png 2x" data-file-width="429" data-file-height="425" /></a></span></div><div class="thumbcaption">The Cartesian coordinate system in three-dimensional space</div></div></div></div></div> <p>The prototypical example of a coordinate system is the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>. In the <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>, two <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> lines are chosen and the coordinates of a point are taken to be the signed distances to the lines.<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> In three dimensions, three mutually <a href="/wiki/Orthogonality" title="Orthogonality">orthogonal</a> planes are chosen and the three coordinates of a point are the signed distances to each of the planes.<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> This can be generalized to create <i>n</i> coordinates for any point in <i>n</i>-dimensional Euclidean space. </p><p>Depending on the direction and order of the <a href="#Coordinate_axis">coordinate axes</a>, the three-dimensional system may be a <a href="/wiki/Right-hand_rule" title="Right-hand rule">right-handed</a> or a left-handed system. </p> <div class="mw-heading mw-heading3"><h3 id="Polar_coordinate_system">Polar coordinate system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=4" title="Edit section: Polar coordinate system"><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/Polar_coordinate_system" title="Polar coordinate system">Polar coordinate system</a></div> <p>Another common coordinate system for the plane is the <i>polar coordinate system</i>.<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> A point is chosen as the <i>pole</i> and a ray from this point is taken as the <i>polar axis</i>. For a given angle <i>θ</i>, there is a single line through the pole whose angle with the polar axis is <i>θ</i> (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is <i>r</i> for given number <i>r</i>. For a given pair of coordinates (<i>r</i>, <i>θ</i>) there is a single point, but any point is represented by many pairs of coordinates. For example, (<i>r</i>, <i>θ</i>), (<i>r</i>, <i>θ</i>+2<i>π</i>) and (−<i>r</i>, <i>θ</i>+<i>π</i>) are all polar coordinates for the same point. The pole is represented by (0, <i>θ</i>) for any value of <i>θ</i>. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Cylindrical_and_spherical_coordinate_systems">Cylindrical and spherical coordinate systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=5" title="Edit section: Cylindrical and spherical coordinate systems"><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/Cylindrical_coordinate_system" title="Cylindrical coordinate system">Cylindrical coordinate system</a> and <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">Spherical coordinate system</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cylindrical_Coordinates.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Cylindrical_Coordinates.svg/220px-Cylindrical_Coordinates.svg.png" decoding="async" width="220" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Cylindrical_Coordinates.svg/330px-Cylindrical_Coordinates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Cylindrical_Coordinates.svg/440px-Cylindrical_Coordinates.svg.png 2x" data-file-width="748" data-file-height="745" /></a><figcaption>Cylindrical coordinate system</figcaption></figure> <p>There are two common methods for extending the polar coordinate system to three dimensions. In the <b>cylindrical coordinate system</b>, a <i>z</i>-coordinate with the same meaning as in Cartesian coordinates is added to the <i>r</i> and <i>θ</i> polar coordinates giving a triple (<i>r</i>, <i>θ</i>, <i>z</i>).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (<i>r</i>, <i>z</i>) to polar coordinates (<i>ρ</i>, <i>φ</i>) giving a triple (<i>ρ</i>, <i>θ</i>, <i>φ</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> </p> <div class="mw-heading mw-heading3"><h3 id="Homogeneous_coordinate_system">Homogeneous coordinate system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=6" title="Edit section: Homogeneous coordinate system"><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/Homogeneous_coordinates" title="Homogeneous coordinates">Homogeneous coordinates</a></div> <p>A point in the plane may be represented in <i>homogeneous coordinates</i> by a triple (<i>x</i>, <i>y</i>, <i>z</i>) where <i>x</i>/<i>z</i> and <i>y</i>/<i>z</i> are the Cartesian coordinates of the point.<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> This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a> without the use of <a href="/wiki/Infinity" title="Infinity">infinity</a>. In general, a homogeneous coordinate system is one where only the ratios of the coordinates are significant and not the actual values. </p> <div class="mw-heading mw-heading3"><h3 id="Other_commonly_used_systems">Other commonly used systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=7" title="Edit section: Other commonly used systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some other common coordinate systems are the following: </p> <ul><li><a href="/wiki/Curvilinear_coordinates" title="Curvilinear coordinates">Curvilinear coordinates</a> are a generalization of coordinate systems generally; the system is based on the intersection of curves. <ul><li><a href="/wiki/Orthogonal_coordinates" title="Orthogonal coordinates">Orthogonal coordinates</a>: <a href="/wiki/Coordinate_surface" class="mw-redirect" title="Coordinate surface">coordinate surfaces</a> meet at right angles</li> <li><a href="/wiki/Skew_coordinates" title="Skew coordinates">Skew coordinates</a>: <a href="/wiki/Coordinate_surface" class="mw-redirect" title="Coordinate surface">coordinate surfaces</a> are not orthogonal</li></ul></li> <li>The <a href="/wiki/Log-polar_coordinates" title="Log-polar coordinates">log-polar coordinate system</a> represents a point in the plane by the logarithm of the distance from the origin and an angle measured from a reference line intersecting the origin.</li> <li><a href="/wiki/Pl%C3%BCcker_coordinates" title="Plücker coordinates">Plücker coordinates</a> are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as <a href="/wiki/Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a>.</li> <li><a href="/wiki/Generalized_coordinates" title="Generalized coordinates">Generalized coordinates</a> are used in the <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a> treatment of mechanics.</li> <li><a href="/wiki/Canonical_coordinates" title="Canonical coordinates">Canonical coordinates</a> are used in the <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a> treatment of mechanics.</li> <li><a href="/wiki/Barycentric_coordinate_system" title="Barycentric coordinate system">Barycentric coordinate system</a> as used for <a href="/wiki/Ternary_plot" title="Ternary plot">ternary plots</a> and more generally in the analysis of <a href="/wiki/Triangle" title="Triangle">triangles</a>.</li> <li><a href="/wiki/Trilinear_coordinates" title="Trilinear coordinates">Trilinear coordinates</a> are used in the context of triangles.</li></ul> <p>There are ways of describing curves without coordinates, using <a href="/wiki/Intrinsic_equation" title="Intrinsic equation">intrinsic equations</a> that use invariant quantities such as <a href="/wiki/Curvature" title="Curvature">curvature</a> and <a href="/wiki/Arc_length" title="Arc length">arc length</a>. These include: </p> <ul><li>The <a href="/wiki/Whewell_equation" title="Whewell equation">Whewell equation</a> relates arc length and the <a href="/wiki/Tangential_angle" title="Tangential angle">tangential angle</a>.</li> <li>The <a href="/wiki/Ces%C3%A0ro_equation" title="Cesàro equation">Cesàro equation</a> relates arc length and curvature.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Coordinates_of_geometric_objects">Coordinates of geometric objects</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=8" title="Edit section: Coordinates of geometric objects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Coordinates systems are often used to specify the position of a point, but they may also be used to specify the position of more complex figures such as lines, planes, <a href="/wiki/Circle" title="Circle">circles</a> or <a href="/wiki/Sphere" title="Sphere">spheres</a>. For example, <a href="/wiki/Pl%C3%BCcker_coordinates" title="Plücker coordinates">Plücker coordinates</a> are used to determine the position of a line in space.<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> When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the term <i><a href="/wiki/Line_coordinates" title="Line coordinates">line coordinates</a></i> is used for any coordinate system that specifies the position of a line. </p><p>It may occur that systems of coordinates for two different sets of geometric figures are equivalent in terms of their analysis. An example of this is the systems of homogeneous coordinates for points and lines in the projective plane. The two systems in a case like this are said to be <i>dualistic</i>. Dualistic systems have the property that results from one system can be carried over to the other since these results are only different interpretations of the same analytical result; this is known as the <i>principle of <a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">duality</a></i>.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Transformations">Transformations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=9" title="Edit section: Transformations"><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">For broader coverage of this topic, see <a href="/wiki/Geometric_transformation" title="Geometric transformation">Geometric transformation</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For a more comprehensive list, see <a href="/wiki/List_of_common_coordinate_transformations" title="List of common coordinate transformations">List of common coordinate transformations</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/Active_and_passive_transformation" title="Active and passive transformation">Active and passive transformation</a></div> <p>There are often many different possible coordinate systems for describing geometrical figures. The relationship between different systems is described by <i>coordinate transformations</i>, which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (<i>x</i>, <i>y</i>) and polar coordinates (<i>r</i>, <i>θ</i>) have the same origin, and the polar axis is the positive <i>x</i> axis, then the coordinate transformation from polar to Cartesian coordinates is given by <i>x</i> = <i>r</i> cos<i>θ</i> and <i>y</i> = <i>r</i> sin<i>θ</i>. </p><p>With every <a href="/wiki/Bijection" title="Bijection">bijection</a> from the space to itself two coordinate transformations can be associated: </p> <ul><li>Such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)</li> <li>Such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation)</li></ul> <p>For example, in <a href="/wiki/Dimension" title="Dimension">1D</a>, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to −3, so that the coordinate of each point becomes 3 more. </p> <div class="mw-heading mw-heading2"><h2 id="Coordinate_lines/curves"><span id="Coordinate_lines.2Fcurves"></span>Coordinate lines/curves</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=10" title="Edit section: Coordinate lines/curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Coordinate_axis"></span><span class="anchor" id="Coordinate_line"></span><span class="anchor" id="Coordinate_curve"></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Coordinate line" redirects here. Not to be confused with <a href="/wiki/Line_coordinates" title="Line coordinates">Line coordinates</a>.</div> <p>Given a coordinate system, if one of the coordinates of a point varies while the other coordinates are held constant, then the resulting curve is called a <b>coordinate curve</b>. If a coordinate curve is a <a href="/wiki/Straight_line" class="mw-redirect" title="Straight line">straight line</a>, it is called a <b>coordinate line</b>. A coordinate system for which some coordinate curves are not lines is called a <i><a href="/wiki/Curvilinear_coordinate_system" class="mw-redirect" title="Curvilinear coordinate system">curvilinear coordinate system</a></i>.<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> <i><a href="/wiki/Orthogonal_coordinates" title="Orthogonal coordinates">Orthogonal coordinates</a></i> are a special but extremely common case of curvilinear coordinates. </p><p>A coordinate line with all other constant coordinates equal to zero is called a <b>coordinate axis</b>, an <a href="/wiki/Oriented_line" class="mw-redirect" title="Oriented line">oriented line</a> used for assigning coordinates. In a <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>, all coordinates curves are lines, and, therefore, there are as many coordinate axes as coordinates. Moreover, the coordinate axes are pairwise <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a>. </p><p>A polar coordinate system is a curvilinear system where coordinate curves are lines or <a href="/wiki/Circle" title="Circle">circles</a>. However, one of the coordinate curves is reduced to a single point, the origin, which is often viewed as a circle of radius zero. Similarly, spherical and cylindrical coordinate systems have coordinate curves that are lines, circles or circles of radius zero. </p><p>Many curves can occur as coordinate curves. For example, the coordinate curves of <a href="/wiki/Parabolic_coordinates" title="Parabolic coordinates">parabolic coordinates</a> are <a href="/wiki/Parabola" title="Parabola">parabolas</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Coordinate_planes/surfaces"><span id="Coordinate_planes.2Fsurfaces"></span>Coordinate planes/surfaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=11" title="Edit section: Coordinate planes/surfaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Coordinate_plane"></span><span class="anchor" id="Coordinate_surface"></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Coordinate plane" redirects here. Not to be confused with <a href="/wiki/Plane_coordinates" class="mw-redirect" title="Plane coordinates">Plane coordinates</a>.</div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Parabolic_coordinates_3D.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Parabolic_coordinates_3D.png/220px-Parabolic_coordinates_3D.png" decoding="async" width="220" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Parabolic_coordinates_3D.png/330px-Parabolic_coordinates_3D.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Parabolic_coordinates_3D.png/440px-Parabolic_coordinates_3D.png 2x" data-file-width="560" data-file-height="598" /></a><figcaption>Coordinate surfaces of the three-dimensional paraboloidal coordinates.</figcaption></figure> <p>In three-dimensional space, if one coordinate is held constant and the other two are allowed to vary, then the resulting surface is called a <b>coordinate surface</b>. For example, the coordinate surfaces obtained by holding <i>ρ</i> constant in the <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinate system</a> are the spheres with center at the origin. In three-dimensional space the intersection of two coordinate surfaces is a coordinate curve. In the Cartesian coordinate system we may speak of <b>coordinate planes</b>. Similarly, <b>coordinate hypersurfaces</b> are the <span class="nowrap">(<i>n</i> − 1)</span>-dimensional spaces resulting from fixing a single coordinate of an <i>n</i>-dimensional coordinate system.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Coordinate_maps">Coordinate maps</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=12" title="Edit section: Coordinate maps"><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/Coordinate_map" class="mw-redirect" title="Coordinate map">Coordinate map</a></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/Manifold" title="Manifold">Manifold</a></div> <p>The concept of a <i>coordinate map</i>, or <i>coordinate chart</i> is central to the theory of manifolds. A coordinate map is essentially a coordinate system for a subset of a given space with the property that each point has exactly one set of coordinates. More precisely, a coordinate map is a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> from an open subset of a space <i>X</i> to an open subset of <b>R</b><sup><i>n</i></sup>.<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> It is often not possible to provide one consistent coordinate system for an entire space. In this case, a collection of coordinate maps are put together to form an <a href="/wiki/Atlas_(topology)" title="Atlas (topology)">atlas</a> covering the space. A space equipped with such an atlas is called a <i>manifold</i> and additional structure can be defined on a manifold if the structure is consistent where the coordinate maps overlap. For example, a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function. </p> <div class="mw-heading mw-heading2"><h2 id="Orientation-based_coordinates">Orientation-based coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=13" title="Edit section: Orientation-based coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a> and <a href="/wiki/Kinematics" title="Kinematics">kinematics</a>, coordinate systems are used to describe the (linear) position of points and the <a href="/wiki/Orientation_(geometry)" title="Orientation (geometry)">angular position</a> of axes, planes, and <a href="/wiki/Rigid_body" title="Rigid body">rigid bodies</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> In the latter case, the orientation of a second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientation <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>, which includes, in its three columns, the <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> of three points. These points are used to define the orientation of the axes of the local system; they are the tips of three <a href="/wiki/Unit_vector" title="Unit vector">unit vectors</a> aligned with those axes. </p> <div class="mw-heading mw-heading2"><h2 id="Geographic_systems">Geographic systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=14" title="Edit section: Geographic systems"><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/Spatial_reference_system" title="Spatial reference system">Spatial reference system</a></div> <p>The Earth as a whole is one of the most common geometric spaces requiring the precise measurement of location, and thus coordinate systems. Starting with the Greeks of the <a href="/wiki/Hellenistic_period" title="Hellenistic period">Hellenistic period</a>, a variety of coordinate systems have been developed based on the types above, including: </p> <ul><li><a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">Geographic coordinate system</a>, the <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinates</a> of <a href="/wiki/Latitude" title="Latitude">latitude</a> and <a href="/wiki/Longitude" title="Longitude">longitude</a></li> <li><a href="/wiki/Grid_reference_system" class="mw-redirect" title="Grid reference system">Projected coordinate systems</a>, including thousands of <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">cartesian coordinate systems</a>, each based on a <a href="/wiki/Map_projection" title="Map projection">map projection</a> to create a planar surface of the world or a region.</li> <li><a href="/wiki/Geocentric_coordinate_system" class="mw-redirect" title="Geocentric coordinate system">Geocentric coordinate system</a>, a three-dimensional <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">cartesian coordinate system</a> that models the earth as an object, and are most commonly used for modeling the orbits of <a href="/wiki/Satellite" title="Satellite">satellites</a>, including the <a href="/wiki/Global_Positioning_System" title="Global Positioning System">Global Positioning System</a> and other <a href="/wiki/Satellite_navigation" title="Satellite navigation">satellite navigation</a> systems.</li></ul> <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=Coordinate_system&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 21em;"> <ul><li><a href="/wiki/Absolute_angular_momentum" title="Absolute angular momentum">Absolute angular momentum</a></li> <li><a href="/wiki/Alphanumeric_grid" title="Alphanumeric grid">Alphanumeric grid</a></li> <li><a href="/wiki/Axes_conventions" title="Axes conventions">Axes conventions</a> in engineering</li> <li><a href="/wiki/Celestial_coordinate_system" class="mw-redirect" title="Celestial coordinate system">Celestial coordinate system</a></li> <li><a href="/wiki/Coordinate-free" title="Coordinate-free">Coordinate-free</a></li> <li><a href="/wiki/Fractional_coordinates" title="Fractional coordinates">Fractional coordinates</a></li> <li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a></li> <li><a href="/wiki/Grid_reference" class="mw-redirect" title="Grid reference">Grid reference</a></li> <li><a href="/wiki/Nomogram" title="Nomogram">Nomogram</a>, graphical representations of different coordinate systems</li> <li><a href="/wiki/Reference_system" class="mw-redirect" title="Reference system">Reference system</a></li> <li><a href="/wiki/Rotation_of_axes" class="mw-redirect" title="Rotation of axes">Rotation of axes</a></li> <li><a href="/wiki/Translation_of_axes" title="Translation of axes">Translation of axes</a></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Relativistic_coordinate_systems">Relativistic coordinate systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=16" title="Edit section: Relativistic coordinate systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col"> <ul><li><a href="/wiki/Eddington%E2%80%93Finkelstein_coordinates" title="Eddington–Finkelstein coordinates">Eddington–Finkelstein coordinates</a></li> <li><a href="/wiki/Gaussian_polar_coordinates" title="Gaussian polar coordinates">Gaussian polar coordinates</a></li> <li><a href="/wiki/Gullstrand%E2%80%93Painlev%C3%A9_coordinates" title="Gullstrand–Painlevé coordinates">Gullstrand–Painlevé coordinates</a></li> <li><a href="/wiki/Isotropic_coordinates" title="Isotropic coordinates">Isotropic coordinates</a></li> <li><a href="/wiki/Kruskal%E2%80%93Szekeres_coordinates" title="Kruskal–Szekeres coordinates">Kruskal–Szekeres coordinates</a></li> <li><a href="/wiki/Schwarzschild_coordinates" title="Schwarzschild coordinates">Schwarzschild coordinates</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=18" 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">Woods p. 1</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"><span class="citation mathworld" id="Reference-Mathworld-Coordinate_System"><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="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/CoordinateSystem.html">"Coordinate System"</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=Coordinate+System&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCoordinateSystem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+system" class="Z3988"></span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-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/Coordinates.html">"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=Coordinates&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCoordinates.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+system" class="Z3988"></span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewartRedlinWatson2008" class="citation book cs1"><a href="/wiki/James_Stewart_(mathematician)" title="James Stewart (mathematician)">Stewart, James B.</a>; Redlin, Lothar; Watson, Saleem (2008). <i>College Algebra</i> (5th ed.). <a href="/wiki/Brooks_Cole" class="mw-redirect" title="Brooks Cole">Brooks Cole</a>. pp. 13–19. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-56521-5" title="Special:BookSources/978-0-495-56521-5"><bdi>978-0-495-56521-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=College+Algebra&rft.pages=13-19&rft.edition=5th&rft.pub=Brooks+Cole&rft.date=2008&rft.isbn=978-0-495-56521-5&rft.aulast=Stewart&rft.aufirst=James+B.&rft.au=Redlin%2C+Lothar&rft.au=Watson%2C+Saleem&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite 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&pg=PA657"><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%26pg%3DPA657&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite 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%3ACoordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFinneyGeorge_ThomasFranklin_DemanaBert_Waits1994" class="citation book cs1">Finney, Ross; George Thomas; Franklin Demana; Bert Waits (June 1994). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/calculusgraphica00ross"><i>Calculus: Graphical, Numerical, Algebraic</i></a></span> (Single Variable Version ed.). Addison-Wesley Publishing Co. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-55478-X" title="Special:BookSources/0-201-55478-X"><bdi>0-201-55478-X</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+Graphical%2C+Numerical%2C+Algebraic&rft.edition=Single+Variable+Version&rft.pub=Addison-Wesley+Publishing+Co.&rft.date=1994-06&rft.isbn=0-201-55478-X&rft.aulast=Finney&rft.aufirst=Ross&rft.au=George+Thomas&rft.au=Franklin+Demana&rft.au=Bert+Waits&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcalculusgraphica00ross&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><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, Henry</a>; Murphy, George M. (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 City: D. van Nostrand. p. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsofphy0002marg/page/178">178</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88275-423-9" title="Special:BookSources/978-0-88275-423-9"><bdi>978-0-88275-423-9</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/55010911">55010911</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/3017486">3017486</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+City&rft.pages=178&rft.pub=D.+van+Nostrand&rft.date=1956&rft_id=info%3Aoclcnum%2F3017486&rft_id=info%3Alccn%2F55010911&rft.isbn=978-0-88275-423-9&rft.aulast=Margenau&rft.aufirst=Henry&rft.au=Murphy%2C+George+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicsofphy0002marg&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+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 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. p. 658. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-043316-X" title="Special:BookSources/0-07-043316-X"><bdi>0-07-043316-X</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/52011515">52011515</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.pages=658&rft.pub=McGraw-Hill&rft.date=1953&rft_id=info%3Alccn%2F52011515&rft.isbn=0-07-043316-X&rft.aulast=Morse&rft.aufirst=PM&rft.au=Feshbach%2C+H&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJones1912" class="citation book cs1">Jones, Alfred Clement (1912). <i>An Introduction to Algebraical Geometry</i>. Clarendon.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Algebraical+Geometry&rft.pub=Clarendon&rft.date=1912&rft.aulast=Jones&rft.aufirst=Alfred+Clement&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+system" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodgeD._Pedoe1994" class="citation book cs1"><a href="/wiki/W._V._D._Hodge" title="W. V. D. Hodge">Hodge, W.V.D.</a>; <a href="/wiki/D._Pedoe" class="mw-redirect" title="D. Pedoe">D. Pedoe</a> (1994) [1947]. <i>Methods of Algebraic Geometry, Volume I (Book II)</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-46900-5" title="Special:BookSources/978-0-521-46900-5"><bdi>978-0-521-46900-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Algebraic+Geometry%2C+Volume+I+%28Book+II%29&rft.pub=Cambridge+University+Press&rft.date=1994&rft.isbn=978-0-521-46900-5&rft.aulast=Hodge&rft.aufirst=W.V.D.&rft.au=D.+Pedoe&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Woods p. 2</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTang2006" class="citation book cs1">Tang, K. T. (2006). <i>Mathematical Methods for Engineers and Scientists</i>. Vol. 2. Springer. p. 13. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-30268-9" title="Special:BookSources/3-540-30268-9"><bdi>3-540-30268-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+for+Engineers+and+Scientists&rft.pages=13&rft.pub=Springer&rft.date=2006&rft.isbn=3-540-30268-9&rft.aulast=Tang&rft.aufirst=K.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+system" class="Z3988"></span></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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiseikin2007" class="citation book cs1">Liseikin, Vladimir D. (2007). <i>A Computational Differential Geometry Approach to Grid Generation</i>. Springer. p. 38. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-34235-9" title="Special:BookSources/978-3-540-34235-9"><bdi>978-3-540-34235-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Computational+Differential+Geometry+Approach+to+Grid+Generation&rft.pages=38&rft.pub=Springer&rft.date=2007&rft.isbn=978-3-540-34235-9&rft.aulast=Liseikin&rft.aufirst=Vladimir+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+system" class="Z3988"></span></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">Munkres, James R. (2000) <i>Topology</i>. Prentice Hall. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-181629-2" title="Special:BookSources/0-13-181629-2">0-13-181629-2</a>.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHanspeter_SchaubJohn_L._Junkins2003" class="citation book cs1"><a href="/wiki/Hanspeter_Schaub" title="Hanspeter Schaub">Hanspeter Schaub</a>; <a href="/wiki/John_Junkins" title="John Junkins">John L. Junkins</a> (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qXvESNWrfpUC&pg=PA71">"Rigid body kinematics"</a>. <i>Analytical Mechanics of Space Systems</i>. American Institute of Aeronautics and Astronautics. p. 71. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56347-563-4" title="Special:BookSources/1-56347-563-4"><bdi>1-56347-563-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Rigid+body+kinematics&rft.btitle=Analytical+Mechanics+of+Space+Systems&rft.pages=71&rft.pub=American+Institute+of+Aeronautics+and+Astronautics&rft.date=2003&rft.isbn=1-56347-563-4&rft.au=Hanspeter+Schaub&rft.au=John+L.+Junkins&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqXvESNWrfpUC%26pg%3DPA71&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+system" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Coordinate_system&action=edit&section=19" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVoitsekhovskiiIvanov2001" class="citation cs2">Voitsekhovskii, M.I.; Ivanov, A.B. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Coordinates">"Coordinates"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Coordinates&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Voitsekhovskii&rft.aufirst=M.I.&rft.au=Ivanov%2C+A.B.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DCoordinates&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWoods1922" class="citation book cs1">Woods, Frederick S. (1922). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3ZULAAAAYAAJ&pg=PA1"><i>Higher Geometry</i></a>. Ginn and Co. pp. 1ff.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Higher+Geometry&rft.pages=1ff&rft.pub=Ginn+and+Co.&rft.date=1922&rft.aulast=Woods&rft.aufirst=Frederick+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3ZULAAAAYAAJ%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShigeyuki_MoritaTeruko_NagaseKatsumi_Nomizu2001" class="citation book cs1">Shigeyuki Morita; Teruko Nagase; <a href="/wiki/Katsumi_Nomizu" title="Katsumi Nomizu">Katsumi Nomizu</a> (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5N33Of2RzjsC&pg=PA12"><i>Geometry of Differential Forms</i></a>. AMS Bookstore. p. 12. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-1045-6" title="Special:BookSources/0-8218-1045-6"><bdi>0-8218-1045-6</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+of+Differential+Forms&rft.pages=12&rft.pub=AMS+Bookstore&rft.date=2001&rft.isbn=0-8218-1045-6&rft.au=Shigeyuki+Morita&rft.au=Teruko+Nagase&rft.au=Katsumi+Nomizu&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5N33Of2RzjsC%26pg%3DPA12&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACoordinate+system" class="Z3988"></span></li></ul> </div> <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=Coordinate_system&action=edit&section=20" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/coordinate_system" class="extiw" title="wiktionary:coordinate system">coordinate system</a></b></i> or <i><b><a href="https://en.wiktionary.org/wiki/coordinate" class="extiw" title="wiktionary:coordinate">coordinate</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Coordinate_systems" class="extiw" title="commons:Category:Coordinate systems">Coordinate systems</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/AV0405/MARTIN/Hex.pdf">Hexagonal 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tensor theory</a></i></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Scope</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Coordinate system</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></li> <li><a href="/wiki/Dyadics" title="Dyadics">Dyadic algebra</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li> <li><a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior calculus</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li> <li><a href="/wiki/Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="/wiki/Physics" title="Physics">Physics</a></li><li><a href="/wiki/Engineering" title="Engineering">Engineering</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Transport_phenomena" title="Transport phenomena">Transport phenomena</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_index_notation" title="Abstract index notation">Abstract index notation</a></li> <li><a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a></li> <li><a href="/wiki/Index_notation" title="Index notation">Index notation</a></li> <li><a href="/wiki/Multi-index_notation" title="Multi-index notation">Multi-index notation</a></li> <li><a href="/wiki/Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li> <li><a href="/wiki/Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li> <li><a href="/wiki/Tetrad_(index_notation)" class="mw-redirect" title="Tetrad (index notation)">Tetrad (index notation)</a></li> <li><a href="/wiki/Van_der_Waerden_notation" title="Van der Waerden notation">Van der Waerden notation</a></li> <li><a href="/wiki/Voigt_notation" title="Voigt notation">Voigt notation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tensor<br />definitions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a></li> <li><a href="/wiki/Tensor_density" title="Tensor density">Tensor density</a></li> <li><a href="/wiki/Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li> <li><a href="/wiki/Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li> <li><a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li> <li><a href="/wiki/Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li> <li><a href="/wiki/Tensor_operator" title="Tensor operator">Tensor operator</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li> <li><a href="/wiki/Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li> <li><a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">Raising and lowering indices</a></li> <li><a href="/wiki/Symmetrization" title="Symmetrization">Symmetrization</a></li> <li><a href="/wiki/Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />abstractions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine connection</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Connection form</a></li> <li><a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Dimension" title="Dimension">Dimension</a></li> <li><a href="/wiki/Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Pseudotensor" title="Pseudotensor">Pseudotensor</a></li> <li><a href="/wiki/Spinor" title="Spinor">Spinor</a></li> <li><a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li> <li><a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li> <li><a href="/wiki/Metric_tensor" title="Metric tensor">Metric tensor</a></li> <li><a href="/wiki/Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li> <li><a href="/wiki/Weyl_tensor" title="Weyl tensor">Weyl tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li> <li><a href="/wiki/Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li> <li><a href="/wiki/Spin_tensor" title="Spin tensor">Spin tensor</a></li> <li><a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li> <li><a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li> <li><a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li> <li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li> <li><a href="/wiki/Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li> <li><a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematician" title="Mathematician">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li> <li><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li> <li><a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li> <li><a href="/wiki/Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li> <li><a href="/wiki/Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Jan Arnoldus Schouten</a></li> <li><a href="/wiki/Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" 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