Polar coordinate system

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aria-controls="toc-Conventions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Conventions subsection</span> </button> <ul id="toc-Conventions-sublist" class="vector-toc-list"> <li id="toc-Uniqueness_of_polar_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniqueness_of_polar_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Uniqueness of polar coordinates</span> </div> </a> <ul id="toc-Uniqueness_of_polar_coordinates-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Converting_between_polar_and_Cartesian_coordinates" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Converting_between_polar_and_Cartesian_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Converting between polar and Cartesian coordinates</span> </div> </a> <button aria-controls="toc-Converting_between_polar_and_Cartesian_coordinates-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 Converting between polar and Cartesian coordinates subsection</span> </button> <ul id="toc-Converting_between_polar_and_Cartesian_coordinates-sublist" class="vector-toc-list"> <li id="toc-Complex_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Complex numbers</span> </div> </a> <ul id="toc-Complex_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Polar_equation_of_a_curve" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Polar_equation_of_a_curve"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Polar equation of a curve</span> </div> </a> <button aria-controls="toc-Polar_equation_of_a_curve-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 Polar equation of a curve subsection</span> </button> <ul id="toc-Polar_equation_of_a_curve-sublist" class="vector-toc-list"> <li id="toc-Circle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Circle"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Circle</span> </div> </a> <ul id="toc-Circle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Line" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Line"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Line</span> </div> </a> <ul id="toc-Line-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polar_rose" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polar_rose"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Polar rose</span> </div> </a> <ul id="toc-Polar_rose-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Archimedean_spiral" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Archimedean_spiral"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Archimedean spiral</span> </div> </a> <ul id="toc-Archimedean_spiral-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conic_sections" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conic_sections"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Conic sections</span> </div> </a> <ul id="toc-Conic_sections-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quadratrix" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quadratrix"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Quadratrix</span> </div> </a> <ul id="toc-Quadratrix-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Intersection_of_two_polar_curves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Intersection_of_two_polar_curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Intersection of two polar curves</span> </div> </a> <ul id="toc-Intersection_of_two_polar_curves-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Calculus" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Calculus</span> </div> </a> <button aria-controls="toc-Calculus-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 Calculus subsection</span> </button> <ul id="toc-Calculus-sublist" class="vector-toc-list"> <li id="toc-Differential_calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differential_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Differential calculus</span> </div> </a> <ul id="toc-Differential_calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integral_calculus_(arc_length)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_calculus_(arc_length)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Integral calculus (arc length)</span> </div> </a> <ul id="toc-Integral_calculus_(arc_length)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integral_calculus_(area)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_calculus_(area)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Integral calculus (area)</span> </div> </a> <ul id="toc-Integral_calculus_(area)-sublist" class="vector-toc-list"> <li id="toc-Generalization" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Generalization"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.1</span> <span>Generalization</span> </div> </a> <ul id="toc-Generalization-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vector_calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Vector calculus</span> </div> </a> <ul id="toc-Vector_calculus-sublist" class="vector-toc-list"> <li id="toc-Centrifugal_and_Coriolis_terms" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Centrifugal_and_Coriolis_terms"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4.1</span> <span>Centrifugal and Coriolis terms</span> </div> </a> <ul id="toc-Centrifugal_and_Coriolis_terms-sublist" class="vector-toc-list"> <li id="toc-Co-rotating_frame" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Co-rotating_frame"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4.1.1</span> <span>Co-rotating frame</span> </div> </a> <ul id="toc-Co-rotating_frame-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Differential_geometry" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Differential_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Differential geometry</span> </div> </a> <ul id="toc-Differential_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extensions_in_3D" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Extensions_in_3D"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Extensions in 3D</span> </div> </a> <ul id="toc-Extensions_in_3D-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-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 Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Position_and_navigation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Position_and_navigation"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Position and navigation</span> </div> </a> <ul id="toc-Position_and_navigation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modeling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modeling"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Modeling</span> </div> </a> <ul id="toc-Modeling-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <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-General_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>General references</span> </div> </a> <ul id="toc-General_references-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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">Polar 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 68 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-68" 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">68 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 badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://af.wikipedia.org/wiki/Poolko%C3%B6rdinatestelsel" title="Poolkoördinatestelsel – Afrikaans" lang="af" hreflang="af" data-title="Poolkoördinatestelsel" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%A7%D9%85_%D8%A7%D9%84%D8%A5%D8%AD%D8%AF%D8%A7%D8%AB%D9%8A%D8%A7%D8%AA_%D8%A7%D9%84%D9%82%D8%B7%D8%A8%D9%8A%D8%A9" title="نظام الإحداثيات القطبية – Arabic" lang="ar" hreflang="ar" data-title="نظام الإحداثيات القطبية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Coordenaes_polares" title="Coordenaes polares – Asturian" lang="ast" hreflang="ast" data-title="Coordenaes polares" 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/Q%C3%BCtb_koordinat_sistemi" title="Qütb koordinat sistemi – Azerbaijani" lang="az" hreflang="az" data-title="Qütb 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%AA%E0%A7%8B%E0%A6%B2%E0%A6%BE%E0%A6%B0_%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE%E0%A6%A8%E0%A6%BE%E0%A6%82%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="পোলার স্থানাংক ব্যবস্থা – Bangla" lang="bn" hreflang="bn" data-title="পোলার স্থানাংক ব্যবস্থা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%8F%D1%80_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B0%D0%BB%D0%B0%D1%80_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0%D2%BB%D1%8B" title="Поляр координаталар системаһы – Bashkir" lang="ba" hreflang="ba" data-title="Поляр 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%9F%D0%B0%D0%BB%D1%8F%D1%80%D0%BD%D0%B0%D1%8F_%D1%81%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="Палярная сістэма каардынат – Belarusian" lang="be" hreflang="be" data-title="Палярная 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-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%8F%D1%80%D0%BD%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Полярна координатна система – Bulgarian" lang="bg" hreflang="bg" data-title="Полярна 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/Polarni_koordinatni_sistem" title="Polarni koordinatni sistem – Bosnian" lang="bs" hreflang="bs" data-title="Polarni 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 badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ca.wikipedia.org/wiki/Coordenades_polars" title="Coordenades polars – Catalan" lang="ca" hreflang="ca" data-title="Coordenades polars" 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_%D0%BF%D0%BE%D0%BB%D1%8F%D1%80%D0%BB%D0%B0_%D1%82%D1%8B%D1%82%C4%83%D0%BC%C4%95" title="Координатсен полярла тытăмĕ – Chuvash" lang="cv" hreflang="cv" data-title="Координатсен полярла 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/Pol%C3%A1rn%C3%AD_soustava_sou%C5%99adnic" title="Polární soustava souřadnic – Czech" lang="cs" hreflang="cs" data-title="Polární soustava souřadnic" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/System_cyfesurynnau_polar" title="System cyfesurynnau polar – Welsh" lang="cy" hreflang="cy" data-title="System cyfesurynnau polar" 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/Pol%C3%A6rt_koordinatsystem" title="Polært koordinatsystem – Danish" lang="da" hreflang="da" data-title="Polært 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/Polarkoordinaten" title="Polarkoordinaten – German" lang="de" hreflang="de" data-title="Polarkoordinaten" 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/Polaarkoordinaadid" title="Polaarkoordinaadid – Estonian" lang="et" hreflang="et" data-title="Polaarkoordinaadid" 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%A0%CE%BF%CE%BB%CE%B9%CE%BA%CF%8C_%CF%83%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1_%CF%83%CF%85%CE%BD%CF%84%CE%B5%CF%84%CE%B1%CE%B3%CE%BC%CE%AD%CE%BD%CF%89%CE%BD" title="Πολικό σύστημα συντεταγμένων – Greek" lang="el" hreflang="el" data-title="Πολικό σύστημα συντεταγμένων" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://es.wikipedia.org/wiki/Coordenadas_polares" title="Coordenadas polares – Spanish" lang="es" hreflang="es" data-title="Coordenadas polares" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://eo.wikipedia.org/wiki/Polusa_koordinatsistemo" title="Polusa koordinatsistemo – Esperanto" lang="eo" hreflang="eo" data-title="Polusa 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_polar" title="Koordenatu polar – Basque" lang="eu" hreflang="eu" data-title="Koordenatu polar" 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_%D9%82%D8%B7%D8%A8%DB%8C" title="دستگاه مختصات قطبی – Persian" lang="fa" hreflang="fa" data-title="دستگاه مختصات قطبی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Coordonn%C3%A9es_polaires" title="Coordonnées polaires – French" lang="fr" hreflang="fr" data-title="Coordonnées polaires" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Comhordan%C3%A1id%C3%AD_polacha" title="Comhordanáidí polacha – Irish" lang="ga" hreflang="ga" data-title="Comhordanáidí polacha" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Coordenadas_polares" title="Coordenadas polares – Galician" lang="gl" hreflang="gl" data-title="Coordenadas polares" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ko.wikipedia.org/wiki/%EA%B7%B9%EC%A2%8C%ED%91%9C%EA%B3%84" title="극좌표계 – Korean" lang="ko" hreflang="ko" data-title="극좌표계" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A7%E0%A5%8D%E0%A4%B0%E0%A5%81%E0%A4%B5%E0%A5%80%E0%A4%AF_%E0%A4%A8%E0%A4%BF%E0%A4%B0%E0%A5%8D%E0%A4%A6%E0%A5%87%E0%A4%B6%E0%A4%BE%E0%A4%82%E0%A4%95_%E0%A4%AA%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%A4%E0%A4%BF" title="ध्रुवीय निर्देशांक पद्धति – Hindi" lang="hi" hreflang="hi" data-title="ध्रुवीय 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/Polarni_koordinatni_sustav" title="Polarni koordinatni sustav – Croatian" lang="hr" hreflang="hr" data-title="Polarni koordinatni sustav" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Polala_koordinati" title="Polala koordinati – Ido" lang="io" hreflang="io" data-title="Polala koordinati" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sistem_koordinat_polar" title="Sistem koordinat polar – Indonesian" lang="id" hreflang="id" data-title="Sistem koordinat polar" 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/Skauthnitakerfi" title="Skauthnitakerfi – Icelandic" lang="is" hreflang="is" data-title="Skauthnitakerfi" 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_polari" title="Sistema di coordinate polari – Italian" lang="it" hreflang="it" data-title="Sistema di coordinate polari" 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_%D7%A7%D7%95%D7%98%D7%91%D7%99%D7%95%D7%AA" title="קואורדינטות קוטביות – Hebrew" lang="he" hreflang="he" data-title="קואורדינטות קוטביות" 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%9F%D0%BE%D0%BB%D1%8F%D1%80%D0%BB%D1%8B%D2%9B_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B0%D0%BB%D0%B0%D1%80" title="Полярлық координаталар – Kazakh" lang="kk" hreflang="kk" data-title="Полярлық координаталар" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Systema_polare_coordinatarum" title="Systema polare coordinatarum – Latin" lang="la" hreflang="la" data-title="Systema polare 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/Pol%C4%81r%C4%81_koordin%C4%81tu_sist%C4%93ma" title="Polārā koordinātu sistēma – Latvian" lang="lv" hreflang="lv" data-title="Polārā koordinātu sistēma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Polin%C4%97_koordina%C4%8Di%C5%B3_sistema" title="Polinė koordinačių sistema – Lithuanian" lang="lt" hreflang="lt" data-title="Polinė 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/Pol%C3%A1rkoordin%C3%A1ta-rendszer" title="Polárkoordináta-rendszer – Hungarian" lang="hu" hreflang="hu" data-title="Polárkoordináta-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%9F%D0%BE%D0%BB%D0%B0%D1%80%D0%B5%D0%BD_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Поларен координатен систем – Macedonian" lang="mk" hreflang="mk" data-title="Поларен 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-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sistem_koordinat_berkutub" title="Sistem koordinat berkutub – Malay" lang="ms" hreflang="ms" data-title="Sistem koordinat berkutub" 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%9D%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%B8%E1%80%9C%E1%80%BE%E1%80%8A%E1%80%B7%E1%80%BA_%E1%80%A1%E1%80%99%E1%80%BE%E1%80%90%E1%80%BA%E1%80%81%E1%80%BB%E1%80%A1%E1%80%AD%E1%80%99%E1%80%BA" title="ဝိုင်းလှည့် အမှတ်ချအိမ် – Burmese" lang="my" hreflang="my" data-title="ဝိုင်းလှည့် အမှတ်ချအိမ်" 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/Poolco%C3%B6rdinaten" title="Poolcoördinaten – Dutch" lang="nl" hreflang="nl" data-title="Poolcoördinaten" 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/%E6%A5%B5%E5%BA%A7%E6%A8%99%E7%B3%BB" title="極座標系 – Japanese" lang="ja" hreflang="ja" data-title="極座標系" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Polaarkoordinaaten" title="Polaarkoordinaaten – Northern Frisian" lang="frr" hreflang="frr" data-title="Polaarkoordinaaten" 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/Polarkoordinatsystem" title="Polarkoordinatsystem – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Polarkoordinatsystem" 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/Polarkoordinatsystem" title="Polarkoordinatsystem – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Polarkoordinatsystem" 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-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A9%8B%E0%A8%B2%E0%A8%B0_%E0%A8%A8%E0%A8%BF%E0%A8%B0%E0%A8%A6%E0%A9%87%E0%A8%B8%E0%A8%BC%E0%A8%BE%E0%A8%82%E0%A8%95" title="ਪੋਲਰ ਨਿਰਦੇਸ਼ਾਂਕ – Punjabi" lang="pa" hreflang="pa" data-title="ਪੋਲਰ ਨਿਰਦੇਸ਼ਾਂਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</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_biegunowych" title="Układ współrzędnych biegunowych – Polish" lang="pl" hreflang="pl" data-title="Układ współrzędnych biegunowych" 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/Coordenadas_polares" title="Coordenadas polares – Portuguese" lang="pt" hreflang="pt" data-title="Coordenadas polares" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ro.wikipedia.org/wiki/Coordonate_polare" title="Coordonate polare – Romanian" lang="ro" hreflang="ro" data-title="Coordonate polare" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%8F%D1%80%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="Полярная система координат – Russian" lang="ru" hreflang="ru" data-title="Полярная 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-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Polar_coordinate_seestem" title="Polar coordinate seestem – Scots" lang="sco" hreflang="sco" data-title="Polar coordinate seestem" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sistemi_koordinativ_polar" title="Sistemi koordinativ polar – Albanian" lang="sq" hreflang="sq" data-title="Sistemi koordinativ polar" 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/Polar_coordinate_system" title="Polar coordinate system – Simple English" lang="en-simple" hreflang="en-simple" data-title="Polar coordinate system" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Pol%C3%A1rna_s%C3%BAstava_s%C3%BAradn%C3%ADc" title="Polárna sústava súradníc – Slovak" lang="sk" hreflang="sk" data-title="Polárna 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/Polarni_koordinatni_sistem" title="Polarni koordinatni sistem – Slovenian" lang="sl" hreflang="sl" data-title="Polarni 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-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%8C_%D8%AC%DB%95%D9%85%D8%B3%DB%95%D8%B1%DB%8C" title="سیستمی پۆتانی جەمسەری – Central Kurdish" lang="ckb" hreflang="ckb" data-title="سیستمی پۆتانی جەمسەری" 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%9F%D0%BE%D0%BB%D0%B0%D1%80%D0%BD%D0%B8_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%BD%D0%B8_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Поларни координатни систем – Serbian" lang="sr" hreflang="sr" data-title="Поларни координатни систем" 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/Polarne_koordinate" title="Polarne koordinate – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Polarne 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/Napakoordinaatisto" title="Napakoordinaatisto – Finnish" lang="fi" hreflang="fi" data-title="Napakoordinaatisto" 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/Pol%C3%A4ra_koordinater" title="Polära koordinater – Swedish" lang="sv" hreflang="sv" data-title="Polära koordinater" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%BE%E0%AE%B3%E0%AF%8D%E0%AE%AE%E0%AF%81%E0%AE%A9%E0%AF%88_%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="வாள்முனை ஆள்கூற்று முறைமை – Tamil" lang="ta" hreflang="ta" data-title="வாள்முனை 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%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%82%E0%B8%B1%E0%B9%89%E0%B8%A7" title="ระบบพิกัดเชิงขั้ว – Thai" lang="th" hreflang="th" data-title="ระบบพิกัดเชิงขั้ว" 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/Kutupsal_koordinat_sistemi" title="Kutupsal koordinat sistemi – Turkish" lang="tr" hreflang="tr" data-title="Kutupsal 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 badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a 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href="/w/index.php?title=Polar_coordinates&redirect=no" class="mw-redirect" title="Polar coordinates">Polar coordinates</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Coordinates comprising a distance and an angle</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Examples_of_Polar_Coordinates.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Examples_of_Polar_Coordinates.svg/250px-Examples_of_Polar_Coordinates.svg.png" decoding="async" width="250" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Examples_of_Polar_Coordinates.svg/375px-Examples_of_Polar_Coordinates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Examples_of_Polar_Coordinates.svg/500px-Examples_of_Polar_Coordinates.svg.png 2x" data-file-width="813" data-file-height="605" /></a><figcaption>Points in the polar coordinate system with pole <i>O</i> and polar axis <i>L</i>. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,<span class="nowrap"> </span>60°). In blue, the point (4,<span class="nowrap"> </span>210°).</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>polar coordinate system</b> is a <a href="/wiki/Two-dimensional" class="mw-redirect" title="Two-dimensional">two-dimensional</a> <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> in which each <a href="/wiki/Point_(mathematics)" class="mw-redirect" title="Point (mathematics)">point</a> on a <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a> is determined by a <a href="/wiki/Distance" title="Distance">distance</a> from a reference point and an <a href="/wiki/Angle" title="Angle">angle</a> from a reference direction. The reference point (analogous to the origin of a <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>) is called the <i>pole</i>, and the <a href="/wiki/Ray_(geometry)" class="mw-redirect" title="Ray (geometry)">ray</a> from the pole in the reference direction is the <i>polar axis</i>. The distance from the pole is called the <i>radial coordinate</i>, <i>radial distance</i> or simply <i>radius</i>, and the angle is called the <i>angular coordinate</i>, <i>polar angle</i>, or <i><a href="/wiki/Azimuth" title="Azimuth">azimuth</a></i>.<sup id="cite_ref-brown_1-0" class="reference"><a href="#cite_note-brown-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Angles in polar notation are generally expressed in either <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a> or <a href="/wiki/Radian" title="Radian">radians</a> (<a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> rad being equal to 180° and 2<a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> rad being equal to 360°). </p><p><a href="/wiki/Gr%C3%A9goire_de_Saint-Vincent" title="Grégoire de Saint-Vincent">Grégoire de Saint-Vincent</a> and <a href="/wiki/Bonaventura_Cavalieri" title="Bonaventura Cavalieri">Bonaventura Cavalieri</a> independently introduced the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to <a href="/wiki/Gregorio_Fontana" title="Gregorio Fontana">Gregorio Fontana</a> in the 18th century. The initial motivation for the introduction of the polar system was the study of <a href="/wiki/Circular_motion" title="Circular motion">circular</a> and <a href="/wiki/Orbital_motion" class="mw-redirect" title="Orbital motion">orbital motion</a>. </p><p>Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as <a href="/wiki/Spiral" title="Spiral">spirals</a>. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates. </p><p>The polar coordinate system is extended to three dimensions in two ways: the <a href="/wiki/Cylindrical_coordinate_system" title="Cylindrical coordinate system">cylindrical</a> and <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical</a> coordinate systems. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_trigonometry" title="History of trigonometry">History of trigonometry</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Head_of_Hipparchus_(cropped).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Head_of_Hipparchus_%28cropped%29.jpg/180px-Head_of_Hipparchus_%28cropped%29.jpg" decoding="async" width="180" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Head_of_Hipparchus_%28cropped%29.jpg/270px-Head_of_Hipparchus_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Head_of_Hipparchus_%28cropped%29.jpg/360px-Head_of_Hipparchus_%28cropped%29.jpg 2x" data-file-width="434" data-file-height="540" /></a><figcaption>Hipparchus</figcaption></figure> <p>The concepts of angle and radius were already used by ancient peoples of the first millennium <a href="/wiki/Before_Christ" class="mw-redirect" title="Before Christ">BC</a>. The <a href="/wiki/Greek_astronomy" class="mw-redirect" title="Greek astronomy">Greek astronomer</a> and <a href="/wiki/Astrologer" class="mw-redirect" title="Astrologer">astrologer</a> <a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a> (190–120 BC) created a table of <a href="/wiki/Chord_(geometry)" title="Chord (geometry)">chord</a> functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.<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> In <i><a href="/wiki/On_Spirals" title="On Spirals">On Spirals</a></i>, <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> describes the <a href="/wiki/Archimedean_spiral" title="Archimedean spiral">Archimedean spiral</a>, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system. </p><p>From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to <a href="/wiki/Mecca" title="Mecca">Mecca</a> (<a href="/wiki/Qibla" title="Qibla">qibla</a>)—and its distance—from any location on the Earth.<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> From the 9th century onward they were using <a href="/wiki/Spherical_trigonometry" title="Spherical trigonometry">spherical trigonometry</a> and <a href="/wiki/Map_projection" title="Map projection">map projection</a> methods to determine these quantities accurately. The calculation is essentially the conversion of the <a href="/wiki/Geodetic_coordinates#Coordinates" title="Geodetic coordinates">equatorial polar coordinates</a> of Mecca (i.e. its <a href="/wiki/Longitude" title="Longitude">longitude</a> and <a href="/wiki/Latitude" title="Latitude">latitude</a>) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the <a href="/wiki/Great_circle" title="Great circle">great circle</a> through the given location and the Earth's poles and whose polar axis is the line through the location and its <a href="/wiki/Antipodal_point" title="Antipodal point">antipodal point</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in <a href="/wiki/Harvard_University" title="Harvard University">Harvard</a> professor <a href="/wiki/Julian_Lowell_Coolidge" class="mw-redirect" title="Julian Lowell Coolidge">Julian Lowell Coolidge</a>'s <i>Origin of Polar Coordinates.</i><sup id="cite_ref-coolidge_5-0" class="reference"><a href="#cite_note-coolidge-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an <a href="/wiki/Archimedean_spiral" title="Archimedean spiral">Archimedean spiral</a>. <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a> subsequently used polar coordinates to calculate the length of <a href="/wiki/Parabola" title="Parabola">parabolic arcs</a>. </p><p>In <i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></i> (written 1671, published 1736), Sir <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems.<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> In the journal <i><a href="/wiki/Acta_Eruditorum" title="Acta Eruditorum">Acta Eruditorum</a></i> (1691), <a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a> used a system with a point on a line, called the <i>pole</i> and <i>polar axis</i> respectively. Coordinates were specified by the distance from the pole and the angle from the <i>polar axis</i>. Bernoulli's work extended to finding the <a href="/wiki/Radius_of_curvature_(mathematics)" class="mw-redirect" title="Radius of curvature (mathematics)">radius of curvature</a> of curves expressed in these coordinates. </p><p>The actual term <i>polar coordinates</i> has been attributed to <a href="/wiki/Gregorio_Fontana" title="Gregorio Fontana">Gregorio Fontana</a> and was used by 18th-century Italian writers. The term appeared in <a href="/wiki/English_language" title="English language">English</a> in <a href="/wiki/George_Peacock_(mathematician)" class="mw-redirect" title="George Peacock (mathematician)">George Peacock</a>'s 1816 translation of <a href="/wiki/Sylvestre_Fran%C3%A7ois_Lacroix" title="Sylvestre François Lacroix">Lacroix</a>'s <i>Differential and Integral Calculus</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><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> <a href="/wiki/Alexis_Clairaut" title="Alexis Clairaut">Alexis Clairaut</a> was the first to think of polar coordinates in three dimensions, and <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> was the first to actually develop them.<sup id="cite_ref-coolidge_5-1" class="reference"><a href="#cite_note-coolidge-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Conventions">Conventions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=2" title="Edit section: Conventions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Polar_graph_paper.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Polar_graph_paper.svg/300px-Polar_graph_paper.svg.png" decoding="async" width="300" height="284" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Polar_graph_paper.svg/450px-Polar_graph_paper.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Polar_graph_paper.svg/600px-Polar_graph_paper.svg.png 2x" data-file-width="495" data-file-height="469" /></a><figcaption>A polar grid with several angles, increasing in counterclockwise orientation and labelled in degrees</figcaption></figure> <p>The radial coordinate is often denoted by <i>r</i> or <a href="/wiki/Rho" title="Rho"><i>ρ</i></a>, and the angular coordinate by <a href="/wiki/Phi" title="Phi"><i>φ</i></a>, <a href="/wiki/Theta" title="Theta"><i>θ</i></a>, or <i>t</i>. The angular coordinate is specified as <i>φ</i> by <a href="/wiki/International_Organization_for_Standardization" title="International Organization for Standardization">ISO</a> standard <a href="/wiki/ISO_31-11" title="ISO 31-11">31-11</a>. However, in mathematical literature the angle is often denoted by θ instead. </p><p>Angles in polar notation are generally expressed in either <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a> or <a href="/wiki/Radian" title="Radian">radians</a> (2<a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> rad being equal to 360°). Degrees are traditionally used in <a href="/wiki/Navigation" title="Navigation">navigation</a>, <a href="/wiki/Surveying" title="Surveying">surveying</a>, and many applied disciplines, while radians are more common in mathematics and mathematical <a href="/wiki/Physics" title="Physics">physics</a>.<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><p>The angle <i>φ</i> is defined to start at 0° from a <i>reference direction</i>, and to increase for rotations in either <a href="/wiki/Clockwise" title="Clockwise">clockwise (cw)</a> or counterclockwise (ccw) orientation. For example, in mathematics, the reference direction is usually drawn as a <a href="/wiki/Ray_(geometry)" class="mw-redirect" title="Ray (geometry)">ray</a> from the pole horizontally to the right, and the polar angle increases to positive angles for ccw rotations, whereas in navigation (<a href="/wiki/Bearing_(navigation)" title="Bearing (navigation)">bearing</a>, <a href="/wiki/Heading_(navigation)" title="Heading (navigation)">heading</a>) the 0°-heading is drawn vertically upwards and the angle increases for cw rotations. The polar angles decrease towards negative values for rotations in the respectively opposite orientations. </p> <div class="mw-heading mw-heading3"><h3 id="Uniqueness_of_polar_coordinates">Uniqueness of polar coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=3" title="Edit section: Uniqueness of polar coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Adding any number of full <a href="/wiki/Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">turns</a> (360°) to the angular coordinate does not change the corresponding direction. Similarly, any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction (adding 180° to the polar angle). Therefore, the same point (<i>r</i>, <i>φ</i>) can be expressed with an infinite number of different polar coordinates <span class="nowrap">(<i>r</i>, <i>φ</i> + <i>n</i> × 360°)</span> and <span class="nowrap">(−<i>r</i>, <i>φ</i> + 180° + <i>n</i> × 360°) = (−<i>r</i>, <i>φ</i> + (2<i>n</i> + 1) × 180°)</span>, where <i>n</i> is an arbitrary <a href="/wiki/Integer" title="Integer">integer</a>.<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> Moreover, the pole itself can be expressed as (0, <i>φ</i>) for any angle <i>φ</i>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Where a unique representation is needed for any point besides the pole, it is usual to limit <i>r</i> to positive numbers (<span class="nowrap"><i>r</i> > 0</span>) and <i>φ</i> to either the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> <span class="texhtml">[0, 360°)</span> or the interval <span class="texhtml">(−180°, 180°]</span>, which in radians are <span class="texhtml">[0, 2π)</span> or <span class="texhtml">(−π, π]</span>.<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> Another convention, in reference to the usual <a href="/wiki/Codomain" title="Codomain">codomain</a> of the <a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">arctan function</a>, is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to <span class="texhtml">(−90°,<span class="nowrap"> </span>90°]</span>. In all cases a unique azimuth for the pole (<i>r</i> = 0) must be chosen, e.g., <i>φ</i> = 0. </p> <div class="mw-heading mw-heading2"><h2 id="Converting_between_polar_and_Cartesian_coordinates">Converting between polar and Cartesian coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=4" title="Edit section: Converting between polar and Cartesian coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Polar_to_cartesian.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Polar_to_cartesian.svg/250px-Polar_to_cartesian.svg.png" decoding="async" width="250" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Polar_to_cartesian.svg/375px-Polar_to_cartesian.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Polar_to_cartesian.svg/500px-Polar_to_cartesian.svg.png 2x" data-file-width="429" data-file-height="425" /></a><figcaption>A diagram illustrating the relationship between polar and Cartesian coordinates.</figcaption></figure><p>The polar coordinates <i>r</i> and <i>φ</i> can be converted to the Cartesian coordinates <i>x</i> and <i>y</i> by using the <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> sine and cosine: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&=r\cos \varphi ,\\y&=r\sin \varphi .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=r\cos \varphi ,\\y&=r\sin \varphi .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c7f6c81b7b59f338ab20da873bdd8e714f347b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.281ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}x&=r\cos \varphi ,\\y&=r\sin \varphi .\end{aligned}}}"></span> </p><p>The Cartesian coordinates <i>x</i> and <i>y</i> can be converted to polar coordinates <i>r</i> and <i>φ</i> with <i>r</i> ≥ 0 and <i>φ</i> in the interval (−<span class="texhtml mvar" style="font-style:italic;">π</span>, <span class="texhtml mvar" style="font-style:italic;">π</span>] by:<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> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}}}=\operatorname {hypot} (x,y)\\\varphi &=\operatorname {atan2} (y,x),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>r</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mi>hypot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>φ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>atan2</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}}}=\operatorname {hypot} (x,y)\\\varphi &=\operatorname {atan2} (y,x),\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3a8927c8a4129125bc15d5b0c62d3f4056aae2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:29.439ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}}}=\operatorname {hypot} (x,y)\\\varphi &=\operatorname {atan2} (y,x),\end{aligned}}}"></span> where hypot is the <a href="/wiki/Pythagorean_addition" title="Pythagorean addition">Pythagorean sum</a> and <a href="/wiki/Atan2" title="Atan2">atan2</a> is a common variation on the <a href="/wiki/Arctangent" class="mw-redirect" title="Arctangent">arctangent</a> function defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\mbox{if }}x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 0\\\arctan \left({\frac {y}{x}}\right)-\pi &{\mbox{if }}x<0{\mbox{ and }}y<0\\{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y>0\\-{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y<0\\{\text{undefined}}&{\mbox{if }}x=0{\mbox{ and }}y=0.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>atan2</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>x</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>π</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> and </mtext> </mstyle> </mrow> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>π</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> and </mtext> </mstyle> </mrow> <mi>y</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> and </mtext> </mstyle> </mrow> <mi>y</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> and </mtext> </mstyle> </mrow> <mi>y</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>undefined</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> and </mtext> </mstyle> </mrow> <mi>y</mi> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\mbox{if }}x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 0\\\arctan \left({\frac {y}{x}}\right)-\pi &{\mbox{if }}x<0{\mbox{ and }}y<0\\{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y>0\\-{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y<0\\{\text{undefined}}&{\mbox{if }}x=0{\mbox{ and }}y=0.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/020a176bd95008f91e938bb68e78eb0f3b4be9c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.956ex; margin-bottom: -0.215ex; width:52.298ex; height:19.509ex;" alt="{\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\mbox{if }}x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 0\\\arctan \left({\frac {y}{x}}\right)-\pi &{\mbox{if }}x<0{\mbox{ and }}y<0\\{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y>0\\-{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y<0\\{\text{undefined}}&{\mbox{if }}x=0{\mbox{ and }}y=0.\end{cases}}}"></span> </p><p>If <i>r</i> is calculated first as above, then this formula for <i>φ</i> may be stated more simply using the <a href="/wiki/Arccosine" class="mw-redirect" title="Arccosine">arccosine</a> function: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ={\begin{cases}\arccos \left({\frac {x}{r}}\right)&{\mbox{if }}y\geq 0{\mbox{ and }}r\neq 0\\-\arccos \left({\frac {x}{r}}\right)&{\mbox{if }}y<0\\{\text{undefined}}&{\mbox{if }}r=0.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>y</mi> <mo>≥</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> and </mtext> </mstyle> </mrow> <mi>r</mi> <mo>≠</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>y</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>undefined</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>r</mi> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ={\begin{cases}\arccos \left({\frac {x}{r}}\right)&{\mbox{if }}y\geq 0{\mbox{ and }}r\neq 0\\-\arccos \left({\frac {x}{r}}\right)&{\mbox{if }}y<0\\{\text{undefined}}&{\mbox{if }}r=0.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f358ec57db4fdf72027c6003fe741c65ae335a9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:39.652ex; height:9.509ex;" alt="{\displaystyle \varphi ={\begin{cases}\arccos \left({\frac {x}{r}}\right)&{\mbox{if }}y\geq 0{\mbox{ and }}r\neq 0\\-\arccos \left({\frac {x}{r}}\right)&{\mbox{if }}y<0\\{\text{undefined}}&{\mbox{if }}r=0.\end{cases}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Complex_numbers">Complex numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=5" title="Edit section: Complex numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Imaginarynumber2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Imaginarynumber2.svg/265px-Imaginarynumber2.svg.png" decoding="async" width="265" height="261" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Imaginarynumber2.svg/398px-Imaginarynumber2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Imaginarynumber2.svg/530px-Imaginarynumber2.svg.png 2x" data-file-width="881" data-file-height="869" /></a><figcaption>An illustration of a complex number <i>z</i> plotted on the complex plane</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Euler%27s_formula.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/265px-Euler%27s_formula.svg.png" decoding="async" width="265" height="273" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/398px-Euler%27s_formula.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/530px-Euler%27s_formula.svg.png 2x" data-file-width="760" data-file-height="782" /></a><figcaption>An illustration of a complex number plotted on the complex plane using <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a></figcaption></figure> <p>Every <a href="/wiki/Complex_number" title="Complex number">complex number</a> can be represented as a point in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form). </p><p>In polar form, the distance and angle coordinates are often referred to as the number's <b>magnitude</b> and <b>argument</b> respectively. Two complex numbers can be multiplied by adding their arguments and multiplying their magnitudes. </p><p>The complex number <i>z</i> can be represented in rectangular form as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+iy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+iy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e90bb6b36fef59c6113eed2a08f10d77240741" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle z=x+iy}"></span> where <i>i</i> is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, or can alternatively be written in polar form as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=r(\cos \varphi +i\sin \varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=r(\cos \varphi +i\sin \varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe097f200e7ea38fe974bf69e6af9a50711f431" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.856ex; height:2.843ex;" alt="{\displaystyle z=r(\cos \varphi +i\sin \varphi )}"></span> and from there, by <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>,<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> as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=re^{i\varphi }=r\exp i\varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> <mo>=</mo> <mi>r</mi> <mi>exp</mi> <mo>⁡</mo> <mi>i</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=re^{i\varphi }=r\exp i\varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa4851a3817d44e19d4432ddd1d920a750cf299d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.636ex; height:3.176ex;" alt="{\displaystyle z=re^{i\varphi }=r\exp i\varphi .}"></span> where <i>e</i> is <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">Euler's number</a>, and <i>φ</i>, expressed in radians, is the <a href="/wiki/Principal_value" title="Principal value">principal value</a> of the complex number function <a href="/wiki/Argument_(complex_analysis)" title="Argument (complex analysis)">arg</a> applied to <i>x</i> + <i>iy</i>. To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used. Equivalent are the <span class="texhtml"><a href="/wiki/Cis_(mathematics)" title="Cis (mathematics)">cis</a></span> and <a href="/wiki/Angle_notation" class="mw-redirect" title="Angle notation">angle notations</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=r\operatorname {\mathrm {cis} } \varphi =r\angle \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>r</mi> <mi mathvariant="normal">∠</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=r\operatorname {\mathrm {cis} } \varphi =r\angle \varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac4c954a9291a076a06678cb802ce1f2c091ee0a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.117ex; height:2.676ex;" alt="{\displaystyle z=r\operatorname {\mathrm {cis} } \varphi =r\angle \varphi .}"></span> </p><p>For the operations of <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>, <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>, and <a href="/wiki/Root_extraction" class="mw-redirect" title="Root extraction">root extraction</a> of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form. From the laws of exponentiation: </p> <dl><dt>Multiplication</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}e^{i\varphi _{0}}\,r_{1}e^{i\varphi _{1}}=r_{0}r_{1}e^{i\left(\varphi _{0}+\varphi _{1}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{0}e^{i\varphi _{0}}\,r_{1}e^{i\varphi _{1}}=r_{0}r_{1}e^{i\left(\varphi _{0}+\varphi _{1}\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0ab9451e932e31b25551bb0fac473da6f91ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.73ex; height:3.176ex;" alt="{\displaystyle r_{0}e^{i\varphi _{0}}\,r_{1}e^{i\varphi _{1}}=r_{0}r_{1}e^{i\left(\varphi _{0}+\varphi _{1}\right)}}"></span></dd> <dt>Division</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {r_{0}e^{i\varphi _{0}}}{r_{1}e^{i\varphi _{1}}}}={\frac {r_{0}}{r_{1}}}e^{i(\varphi _{0}-\varphi _{1})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {r_{0}e^{i\varphi _{0}}}{r_{1}e^{i\varphi _{1}}}}={\frac {r_{0}}{r_{1}}}e^{i(\varphi _{0}-\varphi _{1})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6feef865ff906bab2170b9d6a0bc02a4ef12cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.02ex; height:6.176ex;" alt="{\displaystyle {\frac {r_{0}e^{i\varphi _{0}}}{r_{1}e^{i\varphi _{1}}}}={\frac {r_{0}}{r_{1}}}e^{i(\varphi _{0}-\varphi _{1})}}"></span></dd> <dt>Exponentiation (<a href="/wiki/De_Moivre%27s_formula" title="De Moivre's formula">De Moivre's formula</a>)</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(re^{i\varphi }\right)^{n}=r^{n}e^{in\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>φ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(re^{i\varphi }\right)^{n}=r^{n}e^{in\varphi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf55ee93e12eb2ac00715c6ea258ea537c19ac0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.665ex; height:3.343ex;" alt="{\displaystyle \left(re^{i\varphi }\right)^{n}=r^{n}e^{in\varphi }}"></span></dd> <dt>Root Extraction (Principal root)</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{re^{i\varphi }}}={\sqrt[{n}]{r}}e^{i\varphi \over n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>φ</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{re^{i\varphi }}}={\sqrt[{n}]{r}}e^{i\varphi \over n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9702297c60884f24c748553ecc7246eccf95a448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.899ex; height:4.176ex;" alt="{\displaystyle {\sqrt[{n}]{re^{i\varphi }}}={\sqrt[{n}]{r}}e^{i\varphi \over n}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Polar_equation_of_a_curve">Polar equation of a curve</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=6" title="Edit section: Polar equation of a curve"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Cartesian_to_polar.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Cartesian_to_polar.gif/251px-Cartesian_to_polar.gif" decoding="async" width="251" height="251" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Cartesian_to_polar.gif/377px-Cartesian_to_polar.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/5/53/Cartesian_to_polar.gif 2x" data-file-width="500" data-file-height="500" /></a><figcaption>A curve on the Cartesian plane can be mapped into polar coordinates. In this animation, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\sin(6\!\cdot \!x)+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mspace width="negativethinmathspace" /> <mo>⋅</mo> <mspace width="negativethinmathspace" /> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\sin(6\!\cdot \!x)+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf591d693bd935388272e00709852d7bf7546464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.319ex; height:2.843ex;" alt="{\displaystyle y=\sin(6\!\cdot \!x)+2}"></span> is mapped onto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=\sin(6\!\cdot \!\theta )+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mspace width="negativethinmathspace" /> <mo>⋅</mo> <mspace width="negativethinmathspace" /> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\sin(6\!\cdot \!\theta )+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce36e6b7aff8172fdcfd13cba463edd2841036e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.973ex; height:2.843ex;" alt="{\displaystyle r=\sin(6\!\cdot \!\theta )+2}"></span>. Click on image for details.</figcaption></figure><p>The equation defining a <a href="/wiki/Plane_curve" title="Plane curve">plane curve</a> expressed in polar coordinates is known as a <i>polar equation</i>. In many cases, such an equation can simply be specified by defining <i>r</i> as a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of <i>φ</i>. The resulting curve then consists of points of the form (<i>r</i>(<i>φ</i>), <i>φ</i>) and can be regarded as the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of the polar function <i>r</i>. Note that, in contrast to Cartesian coordinates, the independent variable <i>φ</i> is the <i>second</i> entry in the ordered pair. </p><p>Different forms of <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> can be deduced from the equation of a polar function <i>r</i>: </p> <ul><li>If <span class="texhtml"><i>r</i>(−<i>φ</i>) = <i>r</i>(<i>φ</i>)</span> the curve will be symmetrical about the horizontal (0°/180°) ray;</li> <li>If <span class="texhtml"><i>r</i>(<i>π</i> − <i>φ</i>) = <i>r</i>(<i>φ</i>)</span> it will be symmetric about the vertical (90°/270°) ray:</li> <li>If <span class="texhtml"><i>r</i>(<i>φ</i> − α) = <i>r</i>(<i>φ</i>)</span> it will be <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotationally symmetric</a> by α clockwise and counterclockwise about the pole.</li></ul> <p>Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the <a href="/wiki/Rose_(mathematics)" title="Rose (mathematics)">polar rose</a>, <a href="/wiki/Archimedean_spiral" title="Archimedean spiral">Archimedean spiral</a>, <a href="/wiki/Lemniscate_of_Bernoulli" title="Lemniscate of Bernoulli">lemniscate</a>, <a href="/wiki/Lima%C3%A7on" title="Limaçon">limaçon</a>, and <a href="/wiki/Cardioid" title="Cardioid">cardioid</a>. </p><p>For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve. </p> <div class="mw-heading mw-heading3"><h3 id="Circle">Circle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=7" title="Edit section: Circle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_r%3D1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Circle_r%3D1.svg/220px-Circle_r%3D1.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Circle_r%3D1.svg/330px-Circle_r%3D1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Circle_r%3D1.svg/440px-Circle_r%3D1.svg.png 2x" data-file-width="386" data-file-height="386" /></a><figcaption>A circle with equation <span class="texhtml"><i>r</i>(<i>φ</i>) = 1</span></figcaption></figure> <p>The general equation for a circle with a center at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r_{0},\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{0},\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ba4978ce2510f55703b6912c4eac28c7b259f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.208ex; height:2.843ex;" alt="{\displaystyle (r_{0},\gamma )}"></span> and radius <i>a</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{2}-2rr_{0}\cos(\varphi -\gamma )+r_{0}^{2}=a^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{2}-2rr_{0}\cos(\varphi -\gamma )+r_{0}^{2}=a^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b2133629adf698f1cbbfb515b96bbf4b98854a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.16ex; height:3.343ex;" alt="{\displaystyle r^{2}-2rr_{0}\cos(\varphi -\gamma )+r_{0}^{2}=a^{2}.}"></span> </p><p>This can be simplified in various ways, to conform to more specific cases, such as the equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(\varphi )=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(\varphi )=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1bd9d6b55e284e3e39e44b031bab3545e094b8d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.706ex; height:2.843ex;" alt="{\displaystyle r(\varphi )=a}"></span> for a circle with a center at the pole and radius <i>a</i>.<sup id="cite_ref-ping_15-0" class="reference"><a href="#cite_note-ping-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>When <span class="texhtml"><i>r</i><sub>0</sub> = <i>a</i></span> or the origin lies on the circle, the equation becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=2a\cos(\varphi -\gamma ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>2</mn> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=2a\cos(\varphi -\gamma ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afe766975258871bd446ee93b07c2e95cc7b4543" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.117ex; height:2.843ex;" alt="{\displaystyle r=2a\cos(\varphi -\gamma ).}"></span> </p><p>In the general case, the equation can be solved for <span class="texhtml"><i>r</i></span>, giving <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r_{0}\cos(\varphi -\gamma )+{\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\varphi -\gamma )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r_{0}\cos(\varphi -\gamma )+{\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\varphi -\gamma )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d994793bd0b5f0b27d5f44c0ee5c358eea0dcf0e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:41.301ex; height:4.676ex;" alt="{\displaystyle r=r_{0}\cos(\varphi -\gamma )+{\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\varphi -\gamma )}}}"></span> The solution with a minus sign in front of the square root gives the same curve. </p> <div class="mw-heading mw-heading3"><h3 id="Line">Line</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=8" title="Edit section: Line"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Radial</i> lines (those running through the pole) are represented by the equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\gamma ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>γ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\gamma ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69719ee225311b045858df8195025e200e1cd201" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.528ex; height:2.176ex;" alt="{\displaystyle \varphi =\gamma ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is the angle of elevation of the line; that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\arctan m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\arctan m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e619df44e8373642f59fbbd6624b14b2f8a2b92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.512ex; height:2.509ex;" alt="{\displaystyle \varphi =\arctan m}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the <a href="/wiki/Slope" title="Slope">slope</a> of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92fa30c8d0384ef454ffd19bdd96deb59227d03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.881ex; height:2.176ex;" alt="{\displaystyle \varphi =\gamma }"></span> <a href="/wiki/Perpendicular" title="Perpendicular">perpendicularly</a> at the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r_{0},\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{0},\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ba4978ce2510f55703b6912c4eac28c7b259f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.208ex; height:2.843ex;" alt="{\displaystyle (r_{0},\gamma )}"></span> has the equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(\varphi )=r_{0}\sec(\varphi -\gamma ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(\varphi )=r_{0}\sec(\varphi -\gamma ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef6da7aade8923fd88ff2e33b0770bacd37e37fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.027ex; height:2.843ex;" alt="{\displaystyle r(\varphi )=r_{0}\sec(\varphi -\gamma ).}"></span> </p><p>Otherwise stated <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r_{0},\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{0},\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ba4978ce2510f55703b6912c4eac28c7b259f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.208ex; height:2.843ex;" alt="{\displaystyle (r_{0},\gamma )}"></span> is the point in which the tangent intersects the imaginary circle of radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb12fcfddb65e3d1e6a044215f6e833f0cd4337b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{0}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Polar_rose">Polar rose</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=9" title="Edit section: Polar rose"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rose_2sin(4theta).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Rose_2sin%284theta%29.svg/220px-Rose_2sin%284theta%29.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Rose_2sin%284theta%29.svg/330px-Rose_2sin%284theta%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Rose_2sin%284theta%29.svg/440px-Rose_2sin%284theta%29.svg.png 2x" data-file-width="355" data-file-height="355" /></a><figcaption>A polar rose with equation <span class="texhtml"><i>r</i>(<i>φ</i>) = 2 sin 4<i>φ</i></span></figcaption></figure> <p>A <a href="/wiki/Rose_(mathematics)" title="Rose (mathematics)">polar rose</a> is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(\varphi )=a\cos \left(k\varphi +\gamma _{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mi>φ</mi> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(\varphi )=a\cos \left(k\varphi +\gamma _{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1253e5cbb9692700eebdb3f01dca8bd42d35744" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.844ex; height:2.843ex;" alt="{\displaystyle r(\varphi )=a\cos \left(k\varphi +\gamma _{0}\right)}"></span> </p><p>for any constant γ<sub>0</sub> (including 0). If <i>k</i> is an integer, these equations will produce a <i>k</i>-petaled rose if <i>k</i> is <a href="/wiki/Even_and_odd_numbers" class="mw-redirect" title="Even and odd numbers">odd</a>, or a 2<i>k</i>-petaled rose if <i>k</i> is even. If <i>k</i> is rational, but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The <a href="/wiki/Variable_(math)" class="mw-redirect" title="Variable (math)">variable</a> <i>a</i> directly represents the length or amplitude of the petals of the rose, while <i>k</i> relates to their spatial frequency. The constant γ<sub>0</sub> can be regarded as a phase angle. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Archimedean_spiral">Archimedean spiral</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=10" title="Edit section: Archimedean spiral"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Spiral_of_Archimedes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Spiral_of_Archimedes.svg/220px-Spiral_of_Archimedes.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Spiral_of_Archimedes.svg/330px-Spiral_of_Archimedes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Spiral_of_Archimedes.svg/440px-Spiral_of_Archimedes.svg.png 2x" data-file-width="386" data-file-height="386" /></a><figcaption>One arm of an Archimedean spiral with equation <span class="texhtml"><i>r</i>(<i>φ</i>) = <i>φ</i> / 2<i>π</i></span> for <span class="texhtml">0 < <i>φ</i> < 6<i>π</i></span></figcaption></figure> <p>The <a href="/wiki/Archimedean_spiral" title="Archimedean spiral">Archimedean spiral</a> is a spiral discovered by <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> which can also be expressed as a simple polar equation. It is represented by the equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(\varphi )=a+b\varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(\varphi )=a+b\varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6bc07b3412232dc537a723cc05e6cb670c6bb08" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.711ex; height:2.843ex;" alt="{\displaystyle r(\varphi )=a+b\varphi .}"></span> Changing the parameter <i>a</i> will turn the spiral, while <i>b</i> controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one for <span class="texhtml"><i>φ</i> > 0</span> and one for <span class="texhtml"><i>φ</i> < 0</span>. The two arms are smoothly connected at the pole. If <span class="texhtml"><i>a</i> = 0</span>, taking the mirror image of one arm across the 90°/270° line will yield the other arm. This curve is notable as one of the first curves, after the <a href="/wiki/Conic_section" title="Conic section">conic sections</a>, to be described in a mathematical treatise, and as a prime example of a curve best defined by a polar equation. </p> <div style="clear:both;" class=""></div><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Elps-slr.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Elps-slr.svg/250px-Elps-slr.svg.png" decoding="async" width="250" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Elps-slr.svg/375px-Elps-slr.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Elps-slr.svg/500px-Elps-slr.svg.png 2x" data-file-width="512" data-file-height="285" /></a><figcaption>Ellipse, showing semi-latus rectum</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Conic_sections">Conic sections</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=11" title="Edit section: Conic sections"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Conic_section" title="Conic section">conic section</a> with one focus on the pole and the other somewhere on the 0° ray (so that the conic's <a href="/wiki/Semi-major_axis" class="mw-redirect" title="Semi-major axis">major axis</a> lies along the polar axis) is given by: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\ell \over {1-e\cos \varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\ell \over {1-e\cos \varphi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e9ca91efb44ae1c78c1a9344ec9d9c209cdc98a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.475ex; height:5.843ex;" alt="{\displaystyle r={\ell \over {1-e\cos \varphi }}}"></span> where <i>e</i> is the <a href="/wiki/Eccentricity_(mathematics)" title="Eccentricity (mathematics)">eccentricity</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span> is the <a href="/wiki/Semi-latus_rectum" class="mw-redirect" title="Semi-latus rectum">semi-latus rectum</a> (the perpendicular distance at a focus from the major axis to the curve). If <span class="nowrap"><i>e</i> > 1</span>, this equation defines a <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a>; if <span class="texhtml"><i>e</i> = 1</span>, it defines a <a href="/wiki/Parabola" title="Parabola">parabola</a>; and if <span class="texhtml"><i>e</i> < 1</span>, it defines an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>. The special case <span class="texhtml"><i>e</i> = 0</span> of the latter results in a circle of the radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span>. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Quadratrix">Quadratrix</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=12" title="Edit section: Quadratrix"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Quadratrix_animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Quadratrix_animation.gif/200px-Quadratrix_animation.gif" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Quadratrix_animation.gif/300px-Quadratrix_animation.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Quadratrix_animation.gif/400px-Quadratrix_animation.gif 2x" data-file-width="500" data-file-height="500" /></a><figcaption></figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quadratrix_of_Hippias" title="Quadratrix of Hippias">Quadratrix of Hippias</a></div> <p>A quadratrix in the first quadrant (<i>x, y</i>) is a curve with <i>y</i> = ρ sin θ equal to the fraction of the quarter circle with radius <i>r</i> determined by the radius through the curve point. Since this fraction is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2r\theta }{\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>r</mi> <mi>θ</mi> </mrow> <mi>π</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2r\theta }{\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd79159404d755ca42a659f176826407d1a2962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.138ex; height:5.343ex;" alt="{\displaystyle {\frac {2r\theta }{\pi }}}"></span>, the curve is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\theta )={\frac {2r\theta }{\pi \sin \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>r</mi> <mi>θ</mi> </mrow> <mrow> <mi>π</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\theta )={\frac {2r\theta }{\pi \sin \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b44eee99d11b7f826a87ceec4d6a00d4683a3e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.089ex; height:5.509ex;" alt="{\displaystyle \rho (\theta )={\frac {2r\theta }{\pi \sin \theta }}}"></span>.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Intersection_of_two_polar_curves">Intersection of two polar curves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=13" title="Edit section: Intersection of two polar curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The graphs of two polar functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=f(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=f(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74b1c1243470e52a5b3e0dded9ce187554c998f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.325ex; height:2.843ex;" alt="{\displaystyle r=f(\theta )}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=g(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=g(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b91bd02a323fda2f673914202c9e392ba3de6f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.163ex; height:2.843ex;" alt="{\displaystyle r=g(\theta )}"></span> have possible intersections of three types: </p> <ol><li>In the origin, if the equations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\theta )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\theta )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd8517172980db67ff4d64f1861d97dea3038be9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.439ex; height:2.843ex;" alt="{\displaystyle f(\theta )=0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(\theta )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(\theta )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fee61e9df244268896c1f8197e0fb8e578a51ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.277ex; height:2.843ex;" alt="{\displaystyle g(\theta )=0}"></span> have at least one solution each.</li> <li>All the points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [g(\theta _{i}),\theta _{i}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [g(\theta _{i}),\theta _{i}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c3942e962cea54fd104a33ebac17dde8bffa819" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.033ex; height:2.843ex;" alt="{\displaystyle [g(\theta _{i}),\theta _{i}]}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/302b19204ed378e99ff4575341a67eebdbe5a555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.509ex;" alt="{\displaystyle \theta _{i}}"></span> are solutions to the equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\theta +2k\pi )=g(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\theta +2k\pi )=g(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a2f15bfe8b3ae192a02890befba28345dc3051a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.839ex; height:2.843ex;" alt="{\displaystyle f(\theta +2k\pi )=g(\theta )}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is an integer.</li> <li>All the points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [g(\theta _{i}),\theta _{i}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [g(\theta _{i}),\theta _{i}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c3942e962cea54fd104a33ebac17dde8bffa819" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.033ex; height:2.843ex;" alt="{\displaystyle [g(\theta _{i}),\theta _{i}]}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/302b19204ed378e99ff4575341a67eebdbe5a555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.509ex;" alt="{\displaystyle \theta _{i}}"></span> are solutions to the equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\theta +(2k+1)\pi )=-g(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\theta +(2k+1)\pi )=-g(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4421b2abe8af4eea8f0e49401d48228f35651887" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.459ex; height:2.843ex;" alt="{\displaystyle f(\theta +(2k+1)\pi )=-g(\theta )}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is an integer.</li></ol> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Calculus">Calculus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=14" title="Edit section: Calculus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Calculus" title="Calculus">Calculus</a> can be applied to equations expressed in polar coordinates.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>The angular coordinate <i>φ</i> is expressed in radians throughout this section, which is the conventional choice when doing calculus. </p> <div class="mw-heading mw-heading3"><h3 id="Differential_calculus">Differential calculus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=15" title="Edit section: Differential calculus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using <span class="texhtml"><i>x</i> = <i>r</i> cos <i>φ</i></span> and <span class="texhtml"><i>y</i> = <i>r</i> sin <i>φ</i></span>, one can derive a relationship between derivatives in Cartesian and polar coordinates. For a given function, <i>u</i>(<i>x</i>,<i>y</i>), it follows that (by computing its <a href="/wiki/Total_derivative" title="Total derivative">total derivatives</a>) or <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}r{\frac {du}{dr}}&=r{\frac {\partial u}{\partial x}}\cos \varphi +r{\frac {\partial u}{\partial y}}\sin \varphi =x{\frac {\partial u}{\partial x}}+y{\frac {\partial u}{\partial y}},\\[2pt]{\frac {du}{d\varphi }}&=-{\frac {\partial u}{\partial x}}r\sin \varphi +{\frac {\partial u}{\partial y}}r\cos \varphi =-y{\frac {\partial u}{\partial x}}+x{\frac {\partial u}{\partial y}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}r{\frac {du}{dr}}&=r{\frac {\partial u}{\partial x}}\cos \varphi +r{\frac {\partial u}{\partial y}}\sin \varphi =x{\frac {\partial u}{\partial x}}+y{\frac {\partial u}{\partial y}},\\[2pt]{\frac {du}{d\varphi }}&=-{\frac {\partial u}{\partial x}}r\sin \varphi +{\frac {\partial u}{\partial y}}r\cos \varphi =-y{\frac {\partial u}{\partial x}}+x{\frac {\partial u}{\partial y}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d876b55918333fa5b3792a72f5254b82b86c63" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:50.396ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}r{\frac {du}{dr}}&=r{\frac {\partial u}{\partial x}}\cos \varphi +r{\frac {\partial u}{\partial y}}\sin \varphi =x{\frac {\partial u}{\partial x}}+y{\frac {\partial u}{\partial y}},\\[2pt]{\frac {du}{d\varphi }}&=-{\frac {\partial u}{\partial x}}r\sin \varphi +{\frac {\partial u}{\partial y}}r\cos \varphi =-y{\frac {\partial u}{\partial x}}+x{\frac {\partial u}{\partial y}}.\end{aligned}}}"></span> </p><p>Hence, we have the following formula: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}r{\frac {d}{dr}}&=x{\frac {\partial }{\partial x}}+y{\frac {\partial }{\partial y}}\\[2pt]{\frac {d}{d\varphi }}&=-y{\frac {\partial }{\partial x}}+x{\frac {\partial }{\partial y}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}r{\frac {d}{dr}}&=x{\frac {\partial }{\partial x}}+y{\frac {\partial }{\partial y}}\\[2pt]{\frac {d}{d\varphi }}&=-y{\frac {\partial }{\partial x}}+x{\frac {\partial }{\partial y}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f6d6d1a9e05c093275ecaf66033a4298e41fe1d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:22.573ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}r{\frac {d}{dr}}&=x{\frac {\partial }{\partial x}}+y{\frac {\partial }{\partial y}}\\[2pt]{\frac {d}{d\varphi }}&=-y{\frac {\partial }{\partial x}}+x{\frac {\partial }{\partial y}}.\end{aligned}}}"></span> </p><p>Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function <i>u</i>(<i>r</i>,<i>φ</i>), it follows that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {du}{dx}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial x}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial x}},\\[2pt]{\frac {du}{dy}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial y}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial y}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {du}{dx}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial x}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial x}},\\[2pt]{\frac {du}{dy}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial y}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial y}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f446e4f0df07ead889d7039cec3301472fc1f9ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:25.035ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {du}{dx}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial x}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial x}},\\[2pt]{\frac {du}{dy}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial y}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial y}},\end{aligned}}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {du}{dx}}&={\frac {\partial u}{\partial r}}{\frac {x}{\sqrt {x^{2}+y^{2}}}}-{\frac {\partial u}{\partial \varphi }}{\frac {y}{x^{2}+y^{2}}}\\[2pt]&=\cos \varphi {\frac {\partial u}{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial u}{\partial \varphi }},\\[2pt]{\frac {du}{dy}}&={\frac {\partial u}{\partial r}}{\frac {y}{\sqrt {x^{2}+y^{2}}}}+{\frac {\partial u}{\partial \varphi }}{\frac {x}{x^{2}+y^{2}}}\\[2pt]&=\sin \varphi {\frac {\partial u}{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial u}{\partial \varphi }}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.5em 0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {du}{dx}}&={\frac {\partial u}{\partial r}}{\frac {x}{\sqrt {x^{2}+y^{2}}}}-{\frac {\partial u}{\partial \varphi }}{\frac {y}{x^{2}+y^{2}}}\\[2pt]&=\cos \varphi {\frac {\partial u}{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial u}{\partial \varphi }},\\[2pt]{\frac {du}{dy}}&={\frac {\partial u}{\partial r}}{\frac {y}{\sqrt {x^{2}+y^{2}}}}+{\frac {\partial u}{\partial \varphi }}{\frac {x}{x^{2}+y^{2}}}\\[2pt]&=\sin \varphi {\frac {\partial u}{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial u}{\partial \varphi }}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dcc66ee6a6250cc206f488bb56f976a9a8bd2e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:36.104ex; height:27.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {du}{dx}}&={\frac {\partial u}{\partial r}}{\frac {x}{\sqrt {x^{2}+y^{2}}}}-{\frac {\partial u}{\partial \varphi }}{\frac {y}{x^{2}+y^{2}}}\\[2pt]&=\cos \varphi {\frac {\partial u}{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial u}{\partial \varphi }},\\[2pt]{\frac {du}{dy}}&={\frac {\partial u}{\partial r}}{\frac {y}{\sqrt {x^{2}+y^{2}}}}+{\frac {\partial u}{\partial \varphi }}{\frac {x}{x^{2}+y^{2}}}\\[2pt]&=\sin \varphi {\frac {\partial u}{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial u}{\partial \varphi }}.\end{aligned}}}"></span> </p><p>Hence, we have the following formulae: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d}{dx}}&=\cos \varphi {\frac {\partial }{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial }{\partial \varphi }}\\[2pt]{\frac {d}{dy}}&=\sin \varphi {\frac {\partial }{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial }{\partial \varphi }}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dx}}&=\cos \varphi {\frac {\partial }{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial }{\partial \varphi }}\\[2pt]{\frac {d}{dy}}&=\sin \varphi {\frac {\partial }{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial }{\partial \varphi }}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a08bfe016a4d8e15a933032d077bcdf8c49d41e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:29.763ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dx}}&=\cos \varphi {\frac {\partial }{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial }{\partial \varphi }}\\[2pt]{\frac {d}{dy}}&=\sin \varphi {\frac {\partial }{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial }{\partial \varphi }}.\end{aligned}}}"></span> </p><p>To find the Cartesian slope of the tangent line to a polar curve <i>r</i>(<i>φ</i>) at any given point, the curve is first expressed as a system of <a href="/wiki/Parametric_equations" class="mw-redirect" title="Parametric equations">parametric equations</a>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&=r(\varphi )\cos \varphi \\y&=r(\varphi )\sin \varphi \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=r(\varphi )\cos \varphi \\y&=r(\varphi )\sin \varphi \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d57e14dd98cc1b390d69a4f11c9b4086542b45f7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.963ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}x&=r(\varphi )\cos \varphi \\y&=r(\varphi )\sin \varphi \end{aligned}}}"></span> </p><p><a href="/wiki/Derivative" title="Derivative">Differentiating</a> both equations with respect to <i>φ</i> yields <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {dx}{d\varphi }}&=r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi \\[2pt]{\frac {dy}{d\varphi }}&=r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {dx}{d\varphi }}&=r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi \\[2pt]{\frac {dy}{d\varphi }}&=r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/341713e0b0d484e169fb6cc867ab6a6e31b7ca65" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.41ex; margin-bottom: -0.261ex; width:30.906ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {dx}{d\varphi }}&=r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi \\[2pt]{\frac {dy}{d\varphi }}&=r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi .\end{aligned}}}"></span> </p><p>Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point <span class="nowrap">(<i>r</i>(<i>φ</i>), <i>φ</i>)</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dy}{dx}}={\frac {r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi }{r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mrow> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}={\frac {r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi }{r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff730497ec3671286d63415d676817f8f14299e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.8ex; height:6.509ex;" alt="{\displaystyle {\frac {dy}{dx}}={\frac {r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi }{r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi }}.}"></span> </p><p>For other useful formulas including divergence, gradient, and Laplacian in polar coordinates, see <a href="/wiki/Curvilinear_coordinates" title="Curvilinear coordinates">curvilinear coordinates</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Integral_calculus_(arc_length)"><span id="Integral_calculus_.28arc_length.29"></span>Integral calculus (arc length)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=16" title="Edit section: Integral calculus (arc length)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The arc length (length of a line segment) defined by a polar function is found by the integration over the curve <i>r</i>(<i>φ</i>). Let <i>L</i> denote this length along the curve starting from points <i>A</i> through to point <i>B</i>, where these points correspond to <i>φ</i> = <i>a</i> and <i>φ</i> = <i>b</i> such that <span class="texhtml">0 < <i>b</i> − <i>a</i> < 2<i>π</i></span>. The length of <i>L</i> is given by the following integral <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\int _{a}^{b}{\sqrt {\left[r(\varphi )\right]^{2}+\left[{\tfrac {dr(\varphi )}{d\varphi }}\right]^{2}}}d\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>[</mo> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>φ</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi>d</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\int _{a}^{b}{\sqrt {\left[r(\varphi )\right]^{2}+\left[{\tfrac {dr(\varphi )}{d\varphi }}\right]^{2}}}d\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34f62c06ac2868b9ac160abe1e75a075cad9e261" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.137ex; height:6.343ex;" alt="{\displaystyle L=\int _{a}^{b}{\sqrt {\left[r(\varphi )\right]^{2}+\left[{\tfrac {dr(\varphi )}{d\varphi }}\right]^{2}}}d\varphi }"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Integral_calculus_(area)"><span id="Integral_calculus_.28area.29"></span>Integral calculus (area)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=17" title="Edit section: Integral calculus (area)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Polar_coordinates_integration_region.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Polar_coordinates_integration_region.svg/220px-Polar_coordinates_integration_region.svg.png" decoding="async" width="220" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Polar_coordinates_integration_region.svg/330px-Polar_coordinates_integration_region.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Polar_coordinates_integration_region.svg/440px-Polar_coordinates_integration_region.svg.png 2x" data-file-width="850" data-file-height="552" /></a><figcaption>The integration region <i>R</i> is bounded by the curve <i>r</i>(<i>φ</i>) and the rays <i>φ</i> = <i>a</i> and <i>φ</i> = <i>b</i>.</figcaption></figure> <p>Let <i>R</i> denote the region enclosed by a curve <i>r</i>(<i>φ</i>) and the rays <i>φ</i> = <i>a</i> and <i>φ</i> = <i>b</i>, where <span class="nowrap">0 < <i>b</i> − <i>a</i> ≤ 2<span class="texhtml mvar" style="font-style:italic;">π</span></span>. Then, the area of <i>R</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\int _{a}^{b}\left[r(\varphi )\right]^{2}\,d\varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mrow> <mo>[</mo> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\int _{a}^{b}\left[r(\varphi )\right]^{2}\,d\varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc7958a00360514f36fabcac7cab0361c2ae068" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.671ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{2}}\int _{a}^{b}\left[r(\varphi )\right]^{2}\,d\varphi .}"></span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Polar_coordinates_integration_Riemann_sum.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Polar_coordinates_integration_Riemann_sum.svg/220px-Polar_coordinates_integration_Riemann_sum.svg.png" decoding="async" width="220" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Polar_coordinates_integration_Riemann_sum.svg/330px-Polar_coordinates_integration_Riemann_sum.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Polar_coordinates_integration_Riemann_sum.svg/440px-Polar_coordinates_integration_Riemann_sum.svg.png 2x" data-file-width="797" data-file-height="537" /></a><figcaption>The region <i>R</i> is approximated by <i>n</i> sectors (here, <i>n</i> = 5).</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Planimeter.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Planimeter.jpg/220px-Planimeter.jpg" decoding="async" width="220" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Planimeter.jpg/330px-Planimeter.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Planimeter.jpg/440px-Planimeter.jpg 2x" data-file-width="1682" data-file-height="1282" /></a><figcaption>A <a href="/wiki/Planimeter" title="Planimeter">planimeter</a>, which mechanically computes polar integrals</figcaption></figure> <p>This result can be found as follows. First, the interval <span class="nowrap">[<i>a</i>, <i>b</i>]</span> is divided into <i>n</i> subintervals, where <i>n</i> is some positive integer. Thus Δ<i>φ</i>, the angle measure of each subinterval, is equal to <span class="texhtml"><i>b</i> − <i>a</i></span> (the total angle measure of the interval), divided by <i>n</i>, the number of subintervals. For each subinterval <i>i</i> = 1, 2, ..., <i>n</i>, let <i>φ</i><sub><i>i</i></sub> be the midpoint of the subinterval, and construct a <a href="/wiki/Circular_sector" title="Circular sector">sector</a> with the center at the pole, radius <i>r</i>(<i>φ</i><sub><i>i</i></sub>), central angle Δ<i>φ</i> and arc length <i>r</i>(<i>φ</i><sub><i>i</i></sub>)Δ<i>φ</i>. The area of each constructed sector is therefore equal to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[r(\varphi _{i})\right]^{2}\pi \cdot {\frac {\Delta \varphi }{2\pi }}={\frac {1}{2}}\left[r(\varphi _{i})\right]^{2}\Delta \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>[</mo> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>φ</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow> <mo>[</mo> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[r(\varphi _{i})\right]^{2}\pi \cdot {\frac {\Delta \varphi }{2\pi }}={\frac {1}{2}}\left[r(\varphi _{i})\right]^{2}\Delta \varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec0cb32735f6a73076e1db62726c0eace3f000f6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.555ex; height:5.509ex;" alt="{\displaystyle \left[r(\varphi _{i})\right]^{2}\pi \cdot {\frac {\Delta \varphi }{2\pi }}={\frac {1}{2}}\left[r(\varphi _{i})\right]^{2}\Delta \varphi .}"></span> Hence, the total area of all of the sectors is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}{\tfrac {1}{2}}r(\varphi _{i})^{2}\,\Delta \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}{\tfrac {1}{2}}r(\varphi _{i})^{2}\,\Delta \varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3102375792a188288a60e6138e3eb3497d54eed4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.122ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}{\tfrac {1}{2}}r(\varphi _{i})^{2}\,\Delta \varphi .}"></span> </p><p>As the number of subintervals <i>n</i> is increased, the approximation of the area improves. Taking <span class="nowrap"><i>n</i> → ∞</span>, the sum becomes the <a href="/wiki/Riemann_sum" title="Riemann sum">Riemann sum</a> for the above integral. </p><p>A mechanical device that computes area integrals is the <a href="/wiki/Planimeter" title="Planimeter">planimeter</a>, which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element <a href="/wiki/Linkage_(mechanical)" title="Linkage (mechanical)">linkage</a> effects <a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's theorem</a>, converting the quadratic polar integral to a linear integral. </p> <div class="mw-heading mw-heading4"><h4 id="Generalization">Generalization</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=18" title="Edit section: Generalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>, an infinitesimal area element can be calculated as <i>dA</i> = <i>dx</i> <i>dy</i>. The <a href="/wiki/Integration_by_substitution#Substitution_for_multiple_variables" title="Integration by substitution">substitution rule for multiple integrals</a> states that, when using other coordinates, the <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> determinant of the coordinate conversion formula has to be considered: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=\det {\frac {\partial (x,y)}{\partial (r,\varphi )}}={\begin{vmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \varphi }}\\[2pt]{\frac {\partial y}{\partial r}}&{\frac {\partial y}{\partial \varphi }}\end{vmatrix}}={\begin{vmatrix}\cos \varphi &-r\sin \varphi \\\sin \varphi &r\cos \varphi \end{vmatrix}}=r\cos ^{2}\varphi +r\sin ^{2}\varphi =r.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="0.6em 0.4em" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mo>−</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>r</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>r</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=\det {\frac {\partial (x,y)}{\partial (r,\varphi )}}={\begin{vmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \varphi }}\\[2pt]{\frac {\partial y}{\partial r}}&{\frac {\partial y}{\partial \varphi }}\end{vmatrix}}={\begin{vmatrix}\cos \varphi &-r\sin \varphi \\\sin \varphi &r\cos \varphi \end{vmatrix}}=r\cos ^{2}\varphi +r\sin ^{2}\varphi =r.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52a1293fcb0e90b77017c0d9176c62aa06615b9b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:74.756ex; height:9.509ex;" alt="{\displaystyle J=\det {\frac {\partial (x,y)}{\partial (r,\varphi )}}={\begin{vmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \varphi }}\\[2pt]{\frac {\partial y}{\partial r}}&{\frac {\partial y}{\partial \varphi }}\end{vmatrix}}={\begin{vmatrix}\cos \varphi &-r\sin \varphi \\\sin \varphi &r\cos \varphi \end{vmatrix}}=r\cos ^{2}\varphi +r\sin ^{2}\varphi =r.}"></span> </p><p>Hence, an area element in polar coordinates can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dA=dx\,dy\ =J\,dr\,d\varphi =r\,dr\,d\varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mtext> </mtext> <mo>=</mo> <mi>J</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>φ</mi> <mo>=</mo> <mi>r</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dA=dx\,dy\ =J\,dr\,d\varphi =r\,dr\,d\varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/204ed02c6ad48093830a86a8f76829a384e16bac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.855ex; height:2.676ex;" alt="{\displaystyle dA=dx\,dy\ =J\,dr\,d\varphi =r\,dr\,d\varphi .}"></span> </p><p>Now, a function, that is given in polar coordinates, can be integrated as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iint _{R}f(x,y)\,dA=\int _{a}^{b}\int _{0}^{r(\varphi )}f(r,\varphi )\,r\,dr\,d\varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>A</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>r</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iint _{R}f(x,y)\,dA=\int _{a}^{b}\int _{0}^{r(\varphi )}f(r,\varphi )\,r\,dr\,d\varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91b80d79c6bc8a4a265145bedfd8cdbc864eba37" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.953ex; height:6.343ex;" alt="{\displaystyle \iint _{R}f(x,y)\,dA=\int _{a}^{b}\int _{0}^{r(\varphi )}f(r,\varphi )\,r\,dr\,d\varphi .}"></span> </p><p>Here, <i>R</i> is the same region as above, namely, the region enclosed by a curve <i>r</i>(<i>φ</i>) and the rays <i>φ</i> = <i>a</i> and <i>φ</i> = <i>b</i>. The formula for the area of <i>R</i> is retrieved by taking <i>f</i> identically equal to 1. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:E%5E(-x%5E2).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/E%5E%28-x%5E2%29.svg/220px-E%5E%28-x%5E2%29.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/E%5E%28-x%5E2%29.svg/330px-E%5E%28-x%5E2%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/E%5E%28-x%5E2%29.svg/440px-E%5E%28-x%5E2%29.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption>A graph of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=e^{-x^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=e^{-x^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bed0b77b34cab03996deb42d464becab2f05636" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.882ex; height:3.509ex;" alt="{\displaystyle f(x)=e^{-x^{2}}}"></span> and the area between the function and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-axis, which is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae18ec124928c74818b516e6350ca9610966c6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.268ex; height:3.009ex;" alt="{\displaystyle {\sqrt {\pi }}}"></span>.</figcaption></figure> <p>A more surprising application of this result yields the <a href="/wiki/Gaussian_integral" title="Gaussian integral">Gaussian integral</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b06d446e3c625f48f318811eabdfe5902b11508a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.145ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Vector_calculus">Vector calculus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=19" title="Edit section: Vector calculus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a> can also be applied to polar coordinates. For a planar motion, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span> be the position vector <span class="texhtml">(<i>r</i> cos(<i>φ</i>), <i>r</i> sin(<i>φ</i>))</span>, with <i>r</i> and <span class="texhtml"><i>φ</i></span> depending on time <i>t</i>. </p><p><span class="anchor" id="Radial,_transverse,_normal"></span>We define an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> with three unit vectors: <i>radial, transverse, and normal directions</i>. The <i>radial direction</i> is defined by normalizing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {r} }}=(\cos(\varphi ),\sin(\varphi ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {r} }}=(\cos(\varphi ),\sin(\varphi ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea4252337486f32086a8eca580118da61998da8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.73ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathbf {r} }}=(\cos(\varphi ),\sin(\varphi ))}"></span> Radial and velocity directions span the <i>plane of the motion</i>, whose normal direction is denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {k} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {k} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45c643fb60ea71542145705fe801c7ab8c769507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathbf {k} }}}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {k} }}={\hat {\mathbf {v} }}\times {\hat {\mathbf {r} }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {k} }}={\hat {\mathbf {v} }}\times {\hat {\mathbf {r} }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0b0fd52a60edf991ec517a0079d1404c159f84" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.57ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathbf {k} }}={\hat {\mathbf {v} }}\times {\hat {\mathbf {r} }}.}"></span> The <i>transverse direction</i> is perpendicular to both radial and normal directions: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\varphi }}}=(-\sin(\varphi ),\cos(\varphi ))={\hat {\mathbf {k} }}\times {\hat {\mathbf {r} }}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">φ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\varphi }}}=(-\sin(\varphi ),\cos(\varphi ))={\hat {\mathbf {k} }}\times {\hat {\mathbf {r} }}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38836592838cbd2616f64f824d7f20f76c6a1e28" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.238ex; height:3.343ex;" alt="{\displaystyle {\hat {\boldsymbol {\varphi }}}=(-\sin(\varphi ),\cos(\varphi ))={\hat {\mathbf {k} }}\times {\hat {\mathbf {r} }}\ ,}"></span> </p><p>Then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {r} &=(x,\ y)=r(\cos \varphi ,\ \sin \varphi )=r{\hat {\mathbf {r} }}\ ,\\[1.5ex]{\dot {\mathbf {r} }}&=\left({\dot {x}},\ {\dot {y}}\right)={\dot {r}}(\cos \varphi ,\ \sin \varphi )+r{\dot {\varphi }}(-\sin \varphi ,\ \cos \varphi )={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\varphi }}{\hat {\boldsymbol {\varphi }}}\ ,\\[1.5ex]{\ddot {\mathbf {r} }}&=\left({\ddot {x}},\ {\ddot {y}}\right)\\[1ex]&={\ddot {r}}(\cos \varphi ,\ \sin \varphi )+2{\dot {r}}{\dot {\varphi }}(-\sin \varphi ,\ \cos \varphi )+r{\ddot {\varphi }}(-\sin \varphi ,\ \cos \varphi )-r{\dot {\varphi }}^{2}(\cos \varphi ,\ \sin \varphi )\\[1ex]&=\left({\ddot {r}}-r{\dot {\varphi }}^{2}\right){\hat {\mathbf {r} }}+\left(r{\ddot {\varphi }}+2{\dot {r}}{\dot {\varphi }}\right){\hat {\boldsymbol {\varphi }}}\\[1ex]&=\left({\ddot {r}}-r{\dot {\varphi }}^{2}\right){\hat {\mathbf {r} }}+{\frac {1}{r}}\;{\frac {d}{dt}}\left(r^{2}{\dot {\varphi }}\right){\hat {\boldsymbol {\varphi }}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.945em 0.945em 0.73em 0.73em 0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">φ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">φ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">φ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {r} &=(x,\ y)=r(\cos \varphi ,\ \sin \varphi )=r{\hat {\mathbf {r} }}\ ,\\[1.5ex]{\dot {\mathbf {r} }}&=\left({\dot {x}},\ {\dot {y}}\right)={\dot {r}}(\cos \varphi ,\ \sin \varphi )+r{\dot {\varphi }}(-\sin \varphi ,\ \cos \varphi )={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\varphi }}{\hat {\boldsymbol {\varphi }}}\ ,\\[1.5ex]{\ddot {\mathbf {r} }}&=\left({\ddot {x}},\ {\ddot {y}}\right)\\[1ex]&={\ddot {r}}(\cos \varphi ,\ \sin \varphi )+2{\dot {r}}{\dot {\varphi }}(-\sin \varphi ,\ \cos \varphi )+r{\ddot {\varphi }}(-\sin \varphi ,\ \cos \varphi )-r{\dot {\varphi }}^{2}(\cos \varphi ,\ \sin \varphi )\\[1ex]&=\left({\ddot {r}}-r{\dot {\varphi }}^{2}\right){\hat {\mathbf {r} }}+\left(r{\ddot {\varphi }}+2{\dot {r}}{\dot {\varphi }}\right){\hat {\boldsymbol {\varphi }}}\\[1ex]&=\left({\ddot {r}}-r{\dot {\varphi }}^{2}\right){\hat {\mathbf {r} }}+{\frac {1}{r}}\;{\frac {d}{dt}}\left(r^{2}{\dot {\varphi }}\right){\hat {\boldsymbol {\varphi }}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/272fcf3c718a928afd6ddced5527376aa6b5e58b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.008ex; margin-bottom: -0.33ex; width:83.794ex; height:27.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {r} &=(x,\ y)=r(\cos \varphi ,\ \sin \varphi )=r{\hat {\mathbf {r} }}\ ,\\[1.5ex]{\dot {\mathbf {r} }}&=\left({\dot {x}},\ {\dot {y}}\right)={\dot {r}}(\cos \varphi ,\ \sin \varphi )+r{\dot {\varphi }}(-\sin \varphi ,\ \cos \varphi )={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\varphi }}{\hat {\boldsymbol {\varphi }}}\ ,\\[1.5ex]{\ddot {\mathbf {r} }}&=\left({\ddot {x}},\ {\ddot {y}}\right)\\[1ex]&={\ddot {r}}(\cos \varphi ,\ \sin \varphi )+2{\dot {r}}{\dot {\varphi }}(-\sin \varphi ,\ \cos \varphi )+r{\ddot {\varphi }}(-\sin \varphi ,\ \cos \varphi )-r{\dot {\varphi }}^{2}(\cos \varphi ,\ \sin \varphi )\\[1ex]&=\left({\ddot {r}}-r{\dot {\varphi }}^{2}\right){\hat {\mathbf {r} }}+\left(r{\ddot {\varphi }}+2{\dot {r}}{\dot {\varphi }}\right){\hat {\boldsymbol {\varphi }}}\\[1ex]&=\left({\ddot {r}}-r{\dot {\varphi }}^{2}\right){\hat {\mathbf {r} }}+{\frac {1}{r}}\;{\frac {d}{dt}}\left(r^{2}{\dot {\varphi }}\right){\hat {\boldsymbol {\varphi }}}.\end{aligned}}}"></span> </p><p>This equation can be obtain by taking derivative of the function and derivatives of the unit basis vectors. </p><p>For a curve in 2D where the parameter is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> the previous equations simplify to: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {r} &=r(\theta ){\hat {\mathbf {e} }}_{r}\\[1ex]{\frac {d\mathbf {r} }{d\theta }}&={\frac {dr}{d\theta }}{\hat {\mathbf {e} }}_{r}+r{\hat {\mathbf {e} }}_{\theta }\\[1ex]{\frac {d^{2}\mathbf {r} }{d\theta ^{2}}}&=\left({\frac {d^{2}r}{d\theta ^{2}}}-r\right){\hat {\mathbf {e} }}_{r}+{\frac {dr}{d\theta }}{\hat {\mathbf {e} }}_{\theta }\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.73em 0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mi>r</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {r} &=r(\theta ){\hat {\mathbf {e} }}_{r}\\[1ex]{\frac {d\mathbf {r} }{d\theta }}&={\frac {dr}{d\theta }}{\hat {\mathbf {e} }}_{r}+r{\hat {\mathbf {e} }}_{\theta }\\[1ex]{\frac {d^{2}\mathbf {r} }{d\theta ^{2}}}&=\left({\frac {d^{2}r}{d\theta ^{2}}}-r\right){\hat {\mathbf {e} }}_{r}+{\frac {dr}{d\theta }}{\hat {\mathbf {e} }}_{\theta }\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e05ed720cc1c7b0eeb261f3941aade1fcec687a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:30.365ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {r} &=r(\theta ){\hat {\mathbf {e} }}_{r}\\[1ex]{\frac {d\mathbf {r} }{d\theta }}&={\frac {dr}{d\theta }}{\hat {\mathbf {e} }}_{r}+r{\hat {\mathbf {e} }}_{\theta }\\[1ex]{\frac {d^{2}\mathbf {r} }{d\theta ^{2}}}&=\left({\frac {d^{2}r}{d\theta ^{2}}}-r\right){\hat {\mathbf {e} }}_{r}+{\frac {dr}{d\theta }}{\hat {\mathbf {e} }}_{\theta }\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Centrifugal_and_Coriolis_terms">Centrifugal and Coriolis terms</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=20" title="Edit section: Centrifugal and Coriolis terms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/w/index.php?title=Mechanics_of_planar_particle_motion&action=edit&redlink=1" class="new" title="Mechanics of planar particle motion (page does not exist)">Mechanics of planar particle motion</a> and <a href="/wiki/Centrifugal_force" title="Centrifugal force">Centrifugal force</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:462px;max-width:462px"><div class="trow"><div class="tsingle" style="width:102px;max-width:102px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Position_vector_plane_polar_coords.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Position_vector_plane_polar_coords.svg/100px-Position_vector_plane_polar_coords.svg.png" decoding="async" width="100" height="137" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Position_vector_plane_polar_coords.svg/150px-Position_vector_plane_polar_coords.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Position_vector_plane_polar_coords.svg/200px-Position_vector_plane_polar_coords.svg.png 2x" data-file-width="155" data-file-height="212" /></a></span></div><div class="thumbcaption">Position vector <b>r</b>, always points radially from the origin.</div></div><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Velocity_vector_plane_polar_coords.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Velocity_vector_plane_polar_coords.svg/150px-Velocity_vector_plane_polar_coords.svg.png" decoding="async" width="150" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Velocity_vector_plane_polar_coords.svg/225px-Velocity_vector_plane_polar_coords.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Velocity_vector_plane_polar_coords.svg/300px-Velocity_vector_plane_polar_coords.svg.png 2x" data-file-width="255" data-file-height="212" /></a></span></div><div class="thumbcaption">Velocity vector <b>v</b>, always tangent to the path of motion.</div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Acceleration_vector_plane_polar_coords.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Acceleration_vector_plane_polar_coords.svg/200px-Acceleration_vector_plane_polar_coords.svg.png" decoding="async" width="200" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Acceleration_vector_plane_polar_coords.svg/300px-Acceleration_vector_plane_polar_coords.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Acceleration_vector_plane_polar_coords.svg/400px-Acceleration_vector_plane_polar_coords.svg.png 2x" data-file-width="326" data-file-height="201" /></a></span></div><div class="thumbcaption">Acceleration vector <b>a</b>, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.</div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2d space, but a plane in any higher dimension.</div></div></div></div> <p>The term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r{\dot {\varphi }}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r{\dot {\varphi }}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a43d54e4ab6b4e75edf3915099451d1d87ce4cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.638ex; height:3.176ex;" alt="{\displaystyle r{\dot {\varphi }}^{2}}"></span> is sometimes referred to as the <i>centripetal acceleration</i>, and the term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\dot {r}}{\dot {\varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\dot {r}}{\dot {\varphi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31062c81fca84b3560ed160d47ae3b0850001845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.989ex; height:2.676ex;" alt="{\displaystyle 2{\dot {r}}{\dot {\varphi }}}"></span> as the <i>Coriolis acceleration</i>. For example, see Shankar.<sup id="cite_ref-Shankar_19-0" class="reference"><a href="#cite_note-Shankar-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Note: these terms, that appear when acceleration is expressed in polar coordinates, are a mathematical consequence of differentiation; they appear whenever polar coordinates are used. In planar particle dynamics these accelerations appear when setting up Newton's <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">second law of motion</a> in a rotating frame of reference. Here these extra terms are often called <a href="/wiki/Fictitious_force" title="Fictitious force">fictitious forces</a>; fictitious because they are simply a result of a change in coordinate frame. That does not mean they do not exist, rather they exist only in the rotating frame. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Co-rotating_frame_vector.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Co-rotating_frame_vector.svg/220px-Co-rotating_frame_vector.svg.png" decoding="async" width="220" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Co-rotating_frame_vector.svg/330px-Co-rotating_frame_vector.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Co-rotating_frame_vector.svg/440px-Co-rotating_frame_vector.svg.png 2x" data-file-width="1225" data-file-height="876" /></a><figcaption>Inertial frame of reference <i>S</i> and instantaneous non-inertial co-rotating frame of reference <i>S′</i>. The co-rotating frame rotates at angular rate Ω equal to the rate of rotation of the particle about the origin of <i>S′</i> at the particular moment <i>t</i>. Particle is located at vector position <i>r</i>(<i>t</i>) and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angle <i>ϕ</i> normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distance <i>r</i> need not be related to the radius of curvature of the path.</figcaption></figure> <div class="mw-heading mw-heading5"><h5 id="Co-rotating_frame">Co-rotating frame</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=21" title="Edit section: Co-rotating frame"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneous <i>co-rotating frame of reference</i>.<sup id="cite_ref-Taylor_20-0" class="reference"><a href="#cite_note-Taylor-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> To define a co-rotating frame, first an origin is selected from which the distance <i>r</i>(<i>t</i>) to the particle is defined. An axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin. Then, at the selected moment <i>t</i>, the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, <i>dφ</i>/<i>dt</i>. Next, the terms in the acceleration in the inertial frame are related to those in the co-rotating frame. Let the location of the particle in the inertial frame be (<i>r</i>(<i>t</i>), <i>φ</i>(<i>t</i>)), and in the co-rotating frame be (<i>r</i>′(t), <i>φ</i>′(t)<i>). Because the co-rotating frame rotates at the same rate as the particle, </i>dφ<i>′/</i>dt<i> = 0. The fictitious centrifugal force in the co-rotating frame is </i>mr<i>Ω<sup>2</sup>, radially outward. The velocity of the particle in the co-rotating frame also is radially outward, because </i>dφ<i>′/</i>dt<i> = 0. The </i>fictitious Coriolis force<i> therefore has a value −2</i>m<i>(</i>dr<i>/</i>dt<i>)Ω, pointed in the direction of increasing </i>φ<i> only. Thus, using these forces in Newton's second law we find:</i> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} +\mathbf {F} _{\text{cf}}+\mathbf {F} _{\text{Cor}}=m{\ddot {\mathbf {r} }}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>cf</mtext> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Cor</mtext> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} +\mathbf {F} _{\text{cf}}+\mathbf {F} _{\text{Cor}}=m{\ddot {\mathbf {r} }}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05b00a73ed51fd8991ceb02d9ef1711a581355f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.415ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} +\mathbf {F} _{\text{cf}}+\mathbf {F} _{\text{Cor}}=m{\ddot {\mathbf {r} }}\,,}"></span> where over dots represent derivatives with respect to time, and <b>F</b> is the net real force (as opposed to the fictitious forces). In terms of components, this vector equation becomes: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}F_{r}+mr\Omega ^{2}&=m{\ddot {r}}\\F_{\varphi }-2m{\dot {r}}\Omega &=mr{\ddot {\varphi }}\ ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mi>m</mi> <mi>r</mi> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">Ω</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F_{r}+mr\Omega ^{2}&=m{\ddot {r}}\\F_{\varphi }-2m{\dot {r}}\Omega &=mr{\ddot {\varphi }}\ ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3a86cde193c5bce5cdbac8827bf824f44a6a188" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.516ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}F_{r}+mr\Omega ^{2}&=m{\ddot {r}}\\F_{\varphi }-2m{\dot {r}}\Omega &=mr{\ddot {\varphi }}\ ,\end{aligned}}}"></span> which can be compared to the equations for the inertial frame: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}F_{r}&=m{\ddot {r}}-mr{\dot {\varphi }}^{2}\\F_{\varphi }&=mr{\ddot {\varphi }}+2m{\dot {r}}{\dot {\varphi }}\ .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>m</mi> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>+</mo> <mn>2</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F_{r}&=m{\ddot {r}}-mr{\dot {\varphi }}^{2}\\F_{\varphi }&=mr{\ddot {\varphi }}+2m{\dot {r}}{\dot {\varphi }}\ .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a698bd84bbff8bdf676c2a723b63abad1a7f78" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.373ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}F_{r}&=m{\ddot {r}}-mr{\dot {\varphi }}^{2}\\F_{\varphi }&=mr{\ddot {\varphi }}+2m{\dot {r}}{\dot {\varphi }}\ .\end{aligned}}}"></span> </p><p>This comparison, plus the recognition that by the definition of the co-rotating frame at time <i>t</i> it has a rate of rotation Ω = <i>dφ</i>/<i>dt</i>, shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame. </p><p>For general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous <a href="/wiki/Osculating_circle" title="Osculating circle">osculating circle</a> of its motion, not to a fixed center of polar coordinates. For more detail, see <a href="/wiki/Centripetal_force#Local_coordinates" title="Centripetal force">centripetal force</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Differential_geometry">Differential geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=22" title="Edit section: Differential geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the modern terminology of <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, polar coordinates provide <a href="/wiki/Coordinate_charts" class="mw-redirect" title="Coordinate charts">coordinate charts</a> for the <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> <span class="texhtml"><b>R</b><sup>2</sup> \ {(0,0)}</span>, the plane minus the origin. In these coordinates, the Euclidean <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> is given by<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e4367248c8cd078f74f838c99b8b2e0766e4bc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.728ex; height:2.843ex;" alt="{\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2}.}"></span>This can be seen via the change of variables formula for the metric tensor, or by computing the <a href="/wiki/Differential_form" title="Differential form">differential forms</a> <i>dx</i>, <i>dy</i> via the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> of the 0-forms <span class="texhtml"><i>x</i> = <i>r</i> cos(<i>θ</i>)</span>, <span class="texhtml"><i>y</i> = <i>r</i> sin(<i>θ</i>)</span> and substituting them in the Euclidean metric tensor <span class="texhtml"><i>ds</i><sup>2</sup> = <i>dx</i><sup>2</sup> + <i>dy</i><sup>2</sup></span>. </p> <style data-mw-deduplicate="TemplateStyles:r1256386598">.mw-parser-output .cot-header-mainspace{background:#F0F2F5;color:inherit}.mw-parser-output .cot-header-other{background:#CCFFCC;color:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .cot-header-mainspace{background:#14181F;color:inherit}html.skin-theme-clientpref-night .mw-parser-output .cot-header-other{background:#003500;color:inherit}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cot-header-mainspace{background:#14181F;color:inherit}html.skin-theme-clientpref-os .mw-parser-output .cot-header-other{background:#003500;color:inherit}}</style> <div style="margin-left:0"> <table class="mw-collapsible mw-archivedtalk mw-collapsed" style="color:inherit; background: transparent; text-align: left; border: 1px solid Silver; margin: 0.2em auto auto; width:100%; clear: both; padding: 1px;"> <tbody><tr> <th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">An elementary proof of the formula</div> </th></tr> <tr> <td style="color:inherit; border: solid 1px Silver; padding: 0.6em; background: var(--background-color-base, #fff);"> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}=(x_{1},y_{1})=(r_{1},\theta _{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}=(x_{1},y_{1})=(r_{1},\theta _{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24a89bb206b44f2fedf322ef8adb6bed3f36587d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:23.021ex; height:2.843ex;" alt="{\displaystyle p_{1}=(x_{1},y_{1})=(r_{1},\theta _{1})}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{2}=(x_{2},y_{2})=(r_{2},\theta _{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{2}=(x_{2},y_{2})=(r_{2},\theta _{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bea2a177c765b80efe065428bd5eec3f1144983" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:23.021ex; height:2.843ex;" alt="{\displaystyle p_{2}=(x_{2},y_{2})=(r_{2},\theta _{2})}"></span> be two points in the plane given by their cartesian and polar coordinates. Then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=dx^{2}+dy^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=dx^{2}+dy^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e269a03041ad37930bf2a7d4d9c3002868812e61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.478ex; height:3.176ex;" alt="{\displaystyle ds^{2}=dx^{2}+dy^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}.}"></span></dd></dl> <p>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx^{2}=(r_{2}\cos \theta _{2}-r_{1}\cos \theta _{1})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx^{2}=(r_{2}\cos \theta _{2}-r_{1}\cos \theta _{1})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46d4f7c2c0934aa842ce3473cfb3603ba037c381" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.668ex; height:3.176ex;" alt="{\displaystyle dx^{2}=(r_{2}\cos \theta _{2}-r_{1}\cos \theta _{1})^{2}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dy^{2}=(r_{2}\sin \theta _{2}-r_{1}\sin \theta _{1})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dy^{2}=(r_{2}\sin \theta _{2}-r_{1}\sin \theta _{1})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af09247c26fde2acecc1cbd248f89bc27dce4216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.988ex; height:3.176ex;" alt="{\displaystyle dy^{2}=(r_{2}\sin \theta _{2}-r_{1}\sin \theta _{1})^{2}}"></span>, we get that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=r_{2}^{2}\cos ^{2}\theta _{2}-2r_{1}r_{2}\cos \theta _{1}\cos \theta _{2}+r_{1}^{2}\cos ^{2}\theta _{1}+r_{2}^{2}\sin ^{2}\theta _{2}-2r_{1}r_{2}\sin \theta _{1}\sin \theta _{2}+r_{1}^{2}\sin ^{2}\theta _{1}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=r_{2}^{2}\cos ^{2}\theta _{2}-2r_{1}r_{2}\cos \theta _{1}\cos \theta _{2}+r_{1}^{2}\cos ^{2}\theta _{1}+r_{2}^{2}\sin ^{2}\theta _{2}-2r_{1}r_{2}\sin \theta _{1}\sin \theta _{2}+r_{1}^{2}\sin ^{2}\theta _{1}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec2e9e5df0fa75543fc012e6e5f5fcf580420f89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:93.697ex; height:3.343ex;" alt="{\displaystyle ds^{2}=r_{2}^{2}\cos ^{2}\theta _{2}-2r_{1}r_{2}\cos \theta _{1}\cos \theta _{2}+r_{1}^{2}\cos ^{2}\theta _{1}+r_{2}^{2}\sin ^{2}\theta _{2}-2r_{1}r_{2}\sin \theta _{1}\sin \theta _{2}+r_{1}^{2}\sin ^{2}\theta _{1}=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}^{2}(\cos ^{2}\theta _{2}+\sin ^{2}\theta _{2})+r_{1}^{2}(\cos ^{2}\theta _{1}+\sin ^{2}\theta _{1})-2r_{1}r_{2}(\cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2})=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}^{2}(\cos ^{2}\theta _{2}+\sin ^{2}\theta _{2})+r_{1}^{2}(\cos ^{2}\theta _{1}+\sin ^{2}\theta _{1})-2r_{1}r_{2}(\cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2})=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb5c48631364b2ca69314016015e23c740b7488" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:80.77ex; height:3.343ex;" alt="{\displaystyle r_{2}^{2}(\cos ^{2}\theta _{2}+\sin ^{2}\theta _{2})+r_{1}^{2}(\cos ^{2}\theta _{1}+\sin ^{2}\theta _{1})-2r_{1}r_{2}(\cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2})=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}(1-1+\cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2})=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}(1-1+\cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2})=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1eb646f79e055ea55e2a85b79860587bf03654f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.198ex; height:3.176ex;" alt="{\displaystyle r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}(1-1+\cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2})=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r_{2}-r_{1})^{2}+2r_{1}r_{2}(1-\cos \theta _{1}\cos \theta _{2}-\sin \theta _{1}\sin \theta _{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{2}-r_{1})^{2}+2r_{1}r_{2}(1-\cos \theta _{1}\cos \theta _{2}-\sin \theta _{1}\sin \theta _{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b696d30f4211f7466c2a68885dbe3bb265fe8fae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.253ex; height:3.176ex;" alt="{\displaystyle (r_{2}-r_{1})^{2}+2r_{1}r_{2}(1-\cos \theta _{1}\cos \theta _{2}-\sin \theta _{1}\sin \theta _{2}).}"></span></dd></dl> <p>Now we use the trigonometric identity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\theta _{2}-\theta _{1})=\cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\theta _{2}-\theta _{1})=\cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75fd64c78f2fbc630ce57e4d0fa622314d1705ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.824ex; height:2.843ex;" alt="{\displaystyle \cos(\theta _{2}-\theta _{1})=\cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}}"></span> to proceed: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=dr^{2}+2r_{1}r_{2}(1-\cos d\theta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>d</mi> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=dr^{2}+2r_{1}r_{2}(1-\cos d\theta ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed38cc678a1fcd4997dd251f8020d56e674c165" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.25ex; height:3.176ex;" alt="{\displaystyle ds^{2}=dr^{2}+2r_{1}r_{2}(1-\cos d\theta ).}"></span></dd></dl> <p>If the radial and angular quantities are near to each other, and therefore near to a common quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>, we have that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}r_{2}\approx r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≈</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}r_{2}\approx r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3bc8304486a0d1e230104dde407e2a499885344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.407ex; height:3.009ex;" alt="{\displaystyle r_{1}r_{2}\approx r^{2}}"></span>. Moreover, the cosine of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75ae6ca1248d081ef3fcfdd3e17ba0e3f6c02ee9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.306ex; height:2.176ex;" alt="{\displaystyle d\theta }"></span> can be approximated with the Taylor series of the cosine up to linear terms: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos d\theta \approx 1-{\frac {d\theta ^{2}}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>d</mi> <mi>θ</mi> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos d\theta \approx 1-{\frac {d\theta ^{2}}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea3229e6fb55e20caea0b9ed2ba2821a77dc01f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.749ex; height:5.676ex;" alt="{\displaystyle \cos d\theta \approx 1-{\frac {d\theta ^{2}}{2}},}"></span></dd></dl> <p>so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\cos d\theta \approx {\frac {d\theta ^{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>d</mi> <mi>θ</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\cos d\theta \approx {\frac {d\theta ^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/084a928582c1b0231ac8429d76176edacc13e4d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.103ex; height:5.676ex;" alt="{\displaystyle 1-\cos d\theta \approx {\frac {d\theta ^{2}}{2}}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2r_{1}r_{2}(1-\cos d\theta )\approx 2r^{2}{\frac {d\theta ^{2}}{2}}=r^{2}d\theta ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>d</mi> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>2</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2r_{1}r_{2}(1-\cos d\theta )\approx 2r^{2}{\frac {d\theta ^{2}}{2}}=r^{2}d\theta ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4525d31de3a3237144c515f309d3684a25f9925" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.107ex; height:5.676ex;" alt="{\displaystyle 2r_{1}r_{2}(1-\cos d\theta )\approx 2r^{2}{\frac {d\theta ^{2}}{2}}=r^{2}d\theta ^{2}}"></span>. Therefore, around an infinitesimally small domain of any point, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94033f289cb986e49f994aacfec12bd61fec03b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.728ex; height:3.009ex;" alt="{\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2},}"></span></dd></dl> <p>as stated. </p> </td></tr></tbody></table></div> <p>An <a href="/wiki/Orthonormality" title="Orthonormality">orthonormal</a> <a href="/wiki/Moving_frame" title="Moving frame">frame</a> with respect to this metric is given by<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{r}={\frac {\partial }{\partial r}},\quad e_{\theta }={\frac {1}{r}}{\frac {\partial }{\partial \theta }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{r}={\frac {\partial }{\partial r}},\quad e_{\theta }={\frac {1}{r}}{\frac {\partial }{\partial \theta }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6758e0f65d688a8afc9ece6d3c3d516f42e6e9e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.79ex; height:5.509ex;" alt="{\displaystyle e_{r}={\frac {\partial }{\partial r}},\quad e_{\theta }={\frac {1}{r}}{\frac {\partial }{\partial \theta }},}"></span>with <a href="/wiki/Moving_frame#Coframes" title="Moving frame">dual coframe</a><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{r}=dr,\quad e^{\theta }=rd\theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>=</mo> <mi>d</mi> <mi>r</mi> <mo>,</mo> <mspace width="1em" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <mi>r</mi> <mi>d</mi> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{r}=dr,\quad e^{\theta }=rd\theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1c06d78c84de73182642ed63b3a4f89015c669" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.964ex; height:3.009ex;" alt="{\displaystyle e^{r}=dr,\quad e^{\theta }=rd\theta .}"></span>The <a href="/wiki/Connection_form" title="Connection form">connection form</a> relative to this frame and the <a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a> is given by the skew-symmetric matrix of 1-forms<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\omega ^{i}}_{j}={\begin{pmatrix}0&-d\theta \\d\theta &0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>d</mi> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>θ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\omega ^{i}}_{j}={\begin{pmatrix}0&-d\theta \\d\theta &0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7ae10ad77fd9535b2536bcb66e9bcc013a0fb1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.17ex; height:6.176ex;" alt="{\displaystyle {\omega ^{i}}_{j}={\begin{pmatrix}0&-d\theta \\d\theta &0\end{pmatrix}}}"></span>and hence the <a href="/wiki/Curvature_form" title="Curvature form">curvature form</a> <span class="texhtml">Ω = <i>dω</i> + <i>ω</i>∧<i>ω</i></span> vanishes. Therefore, as expected, the punctured plane is a <a href="/wiki/Flat_manifold" title="Flat manifold">flat manifold</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Extensions_in_3D">Extensions in 3D</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=23" title="Edit section: Extensions in 3D"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The polar coordinate system is extended into three dimensions with two different coordinate systems, the <a href="/wiki/Cylindrical_coordinate_system" title="Cylindrical coordinate system">cylindrical</a> and <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinate system</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=24" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. They are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. For instance, the examples above show how elementary polar equations suffice to define curves—such as the Archimedean spiral—whose equation in the Cartesian coordinate system would be much more intricate. Moreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates. The initial motivation for the introduction of the polar system was the study of <a href="/wiki/Circular_motion" title="Circular motion">circular</a> and <a href="/wiki/Orbital_motion" class="mw-redirect" title="Orbital motion">orbital motion</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Position_and_navigation">Position and navigation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=25" title="Edit section: Position and navigation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polar coordinates are used often in <a href="/wiki/Navigation" title="Navigation">navigation</a> as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance, <a href="/wiki/Aircraft" title="Aircraft">aircraft</a> use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a clockwise direction, rather than counterclockwise, as in the mathematical system. Heading 360 corresponds to <a href="/wiki/Magnetic_north" class="mw-redirect" title="Magnetic north">magnetic north</a>, while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read <a href="/wiki/ICAO_spelling_alphabet" class="mw-redirect" title="ICAO spelling alphabet">zero-niner-zero</a> by <a href="/wiki/Air_traffic_control" title="Air traffic control">air traffic control</a>).<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Modeling">Modeling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=26" title="Edit section: Modeling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Systems displaying <a href="/wiki/Radial_symmetry" class="mw-redirect" title="Radial symmetry">radial symmetry</a> provide natural settings for the polar coordinate system, with the central point acting as the pole. A prime example of this usage is the <a href="/wiki/Groundwater_flow_equation" title="Groundwater flow equation">groundwater flow equation</a> when applied to radially symmetric wells. Systems with a <a href="/wiki/Central_force" title="Central force">radial force</a> are also good candidates for the use of the polar coordinate system. These systems include <a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitational fields</a>, which obey the <a href="/wiki/Inverse-square_law" title="Inverse-square law">inverse-square law</a>, as well as systems with <a href="/wiki/Point_source" title="Point source">point sources</a>, such as <a href="/wiki/Antenna_(radio)" title="Antenna (radio)">radio antennas</a>. </p><p>Radially asymmetric systems may also be modeled with polar coordinates. For example, a <a href="/wiki/Microphone" title="Microphone">microphone</a>'s <a href="/wiki/Microphone_pick_up_patterns" class="mw-redirect" title="Microphone pick up patterns">pickup pattern</a> illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves. The curve for a standard cardioid microphone, the most common unidirectional microphone, can be represented as <span class="nowrap"><i>r</i> = 0.5 + 0.5sin(<i>ϕ</i>)</span> at its target design frequency.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> The pattern shifts toward omnidirectionality at lower frequencies. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=27" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <ul><li><a href="/wiki/Curvilinear_coordinates" title="Curvilinear coordinates">Curvilinear coordinates</a></li> <li><a href="/wiki/List_of_common_coordinate_transformations" title="List of common coordinate transformations">List of common coordinate transformations</a></li> <li><a href="/wiki/Log-polar_coordinates" title="Log-polar coordinates">Log-polar coordinates</a></li> <li><a href="/wiki/Polar_decomposition" title="Polar decomposition">Polar decomposition</a></li> <li><a href="/wiki/Unit_circle" title="Unit circle">Unit circle</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=28" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-brown-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-brown_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBrown1997" class="citation book cs1">Brown, Richard G. (1997). Andrew M. Gleason (ed.). <a rel="nofollow" class="external text" href="https://archive.org/details/advancedmathemat00rich_0"><i>Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis</i></a>. Evanston, Illinois: McDougal Littell. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-395-77114-5" title="Special:BookSources/0-395-77114-5"><bdi>0-395-77114-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Mathematics%3A+Precalculus+with+Discrete+Mathematics+and+Data+Analysis&rft.place=Evanston%2C+Illinois&rft.pub=McDougal+Littell&rft.date=1997&rft.isbn=0-395-77114-5&rft.aulast=Brown&rft.aufirst=Richard+G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fadvancedmathemat00rich_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFriendly2009" class="citation web cs1">Friendly, Michael (August 24, 2009). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180926124138/http://www.math.yorku.ca/SCS/Gallery/milestone/milestone.pdf">"Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://www.math.yorku.ca/SCS/Gallery/milestone/milestone.pdf">the original</a> <span class="cs1-format">(PDF)</span> on September 26, 2018<span class="reference-accessdate">. Retrieved <span class="nowrap">July 23,</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Milestones+in+the+History+of+Thematic+Cartography%2C+Statistical+Graphics%2C+and+Data+Visualization&rft.date=2009-08-24&rft.aulast=Friendly&rft.aufirst=Michael&rft_id=http%3A%2F%2Fwww.math.yorku.ca%2FSCS%2FGallery%2Fmilestone%2Fmilestone.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKing2005" class="citation book cs1">King, David A. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AMOQZfrZq-EC&pg=PA161">"The Sacred Geography of Islam"</a>. In Koetsier, Teun; Luc, Bergmans (eds.). <i>Mathematics and the Divine: A Historical Study</i>. Amsterdam: Elsevier. pp. 162–78. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-50328-5" title="Special:BookSources/0-444-50328-5"><bdi>0-444-50328-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Sacred+Geography+of+Islam&rft.btitle=Mathematics+and+the+Divine%3A+A+Historical+Study&rft.place=Amsterdam&rft.pages=162-78&rft.pub=Elsevier&rft.date=2005&rft.isbn=0-444-50328-5&rft.aulast=King&rft.aufirst=David+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAMOQZfrZq-EC%26pg%3DPA161&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">King (<a href="#CITEREFKing2005">2005</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AMOQZfrZq-EC&pg=PA169">p. 169</a>). The calculations were as accurate as could be achieved under the limitations imposed by their assumption that the Earth was a perfect sphere.</span> </li> <li id="cite_note-coolidge-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-coolidge_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-coolidge_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoolidge1952" class="citation journal cs1"><a href="/wiki/Julian_Lowell_Coolidge" class="mw-redirect" title="Julian Lowell Coolidge">Coolidge, Julian</a> (1952). <a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/Extras/Coolidge_Polars.html">"The Origin of Polar Coordinates"</a>. <i>American Mathematical Monthly</i>. <b>59</b> (2). Mathematical Association of America: 78–85. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2307104">10.2307/2307104</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2307104">2307104</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=The+Origin+of+Polar+Coordinates&rft.volume=59&rft.issue=2&rft.pages=78-85&rft.date=1952&rft_id=info%3Adoi%2F10.2307%2F2307104&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2307104%23id-name%3DJSTOR&rft.aulast=Coolidge&rft.aufirst=Julian&rft_id=http%3A%2F%2Fwww-history.mcs.st-and.ac.uk%2FExtras%2FCoolidge_Polars.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+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="CITEREFBoyer1949" class="citation journal cs1">Boyer, C. B. (1949). "Newton as an Originator of Polar Coordinates". <i>American Mathematical Monthly</i>. <b>56</b> (2). 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Retrieved <span class="nowrap">2006-09-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Earliest+Known+Uses+of+Some+of+the+Words+of+Mathematics&rft.aulast=Miller&rft.aufirst=Jeff&rft_id=http%3A%2F%2Fjeff560.tripod.com%2Fp.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+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="CITEREFSmith1925" class="citation book cs1">Smith, David Eugene (1925). <i>History of Mathematics, Vol II</i>. 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Retrieved <span class="nowrap">2006-09-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Polar+Coordinates+and+Graphing&rft.date=2006-04-13&rft_id=http%3A%2F%2Fcampuses.fortbendisd.com%2Fcampuses%2Fdocuments%2FTeacher%2F2012%255Cteacher_20120507_1147.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+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="CITEREFLeeDavid_CohenDavid_Sklar2005" class="citation book cs1">Lee, Theodore; David Cohen; David Sklar (2005). <i>Precalculus: With Unit-Circle Trigonometry</i> (Fourth ed.). Thomson Brooks/Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-40230-5" title="Special:BookSources/0-534-40230-5"><bdi>0-534-40230-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Precalculus%3A+With+Unit-Circle+Trigonometry&rft.edition=Fourth&rft.pub=Thomson+Brooks%2FCole&rft.date=2005&rft.isbn=0-534-40230-5&rft.aulast=Lee&rft.aufirst=Theodore&rft.au=David+Cohen&rft.au=David+Sklar&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewartDavid_Tall1983" class="citation book cs1">Stewart, Ian; David Tall (1983). <i>Complex Analysis (the Hitchhiker's Guide to the Plane)</i>. 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(2003). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060915004724/http://ccrma-www.stanford.edu/~jos/mdft/Euler_s_Identity.html">"Euler's Identity"</a>. <i>Mathematics of the Discrete Fourier Transform (DFT)</i>. W3K Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-9745607-0-7" title="Special:BookSources/0-9745607-0-7"><bdi>0-9745607-0-7</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://ccrma-www.stanford.edu/~jos/mdft/Euler_s_Identity.html">the original</a> on 2006-09-15<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-09-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Euler%27s+Identity&rft.btitle=Mathematics+of+the+Discrete+Fourier+Transform+%28DFT%29&rft.pub=W3K+Publishing&rft.date=2003&rft.isbn=0-9745607-0-7&rft.aulast=Smith&rft.aufirst=Julius+O.&rft_id=http%3A%2F%2Fccrma-www.stanford.edu%2F~jos%2Fmdft%2FEuler_s_Identity.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-ping-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-ping_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClaeys" class="citation web cs1">Claeys, Johan. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060427230725/http://www.ping.be/~ping1339/polar.htm">"Polar coordinates"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.ping.be/~ping1339/polar.htm">the original</a> on 2006-04-27<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-05-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Polar+coordinates&rft.aulast=Claeys&rft.aufirst=Johan&rft_id=http%3A%2F%2Fwww.ping.be%2F~ping1339%2Fpolar.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></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">N.H. Lucas, P.J. Bunt & J.D Bedient (1976) <i>Historical Roots of Elementary Mathematics</i>, page 113</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHusch,_Lawrence_S." class="citation web cs1">Husch, Lawrence S. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20000301151724/http://archives.math.utk.edu/visual.calculus/5/polar.1/index.html">"Areas Bounded by Polar Curves"</a>. Archived from <a rel="nofollow" class="external text" href="http://archives.math.utk.edu/visual.calculus/5/polar.1/index.html">the original</a> on 2000-03-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-11-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Areas+Bounded+by+Polar+Curves&rft.au=Husch%2C+Lawrence+S.&rft_id=http%3A%2F%2Farchives.math.utk.edu%2Fvisual.calculus%2F5%2Fpolar.1%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLawrence_S._Husch" class="citation web cs1">Lawrence S. Husch. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191121222301/http://archives.math.utk.edu/visual.calculus/3/polar.1/index.html">"Tangent Lines to Polar Graphs"</a>. Archived from <a rel="nofollow" class="external text" href="http://archives.math.utk.edu/visual.calculus/3/polar.1/index.html">the original</a> on 2019-11-21<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-11-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Tangent+Lines+to+Polar+Graphs&rft.au=Lawrence+S.+Husch&rft_id=http%3A%2F%2Farchives.math.utk.edu%2Fvisual.calculus%2F3%2Fpolar.1%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-Shankar-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-Shankar_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamamurti_Shankar1994" class="citation book cs1">Ramamurti Shankar (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2zypV5EbKuIC&q=Coriolis+%22polar+coordinates%22&pg=PA81"><i>Principles of Quantum Mechanics</i></a> (2nd ed.). Springer. p. 81. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-306-44790-8" title="Special:BookSources/0-306-44790-8"><bdi>0-306-44790-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Quantum+Mechanics&rft.pages=81&rft.edition=2nd&rft.pub=Springer&rft.date=1994&rft.isbn=0-306-44790-8&rft.au=Ramamurti+Shankar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2zypV5EbKuIC%26q%3DCoriolis%2B%2522polar%2Bcoordinates%2522%26pg%3DPA81&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-Taylor-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-Taylor_20-0">^</a></b></span> <span class="reference-text">For the following discussion, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_R_Taylor2005" class="citation book cs1">John R Taylor (2005). <i>Classical Mechanics</i>. University Science Books. p. §9.10, pp. 358–359. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-891389-22-X" title="Special:BookSources/1-891389-22-X"><bdi>1-891389-22-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=Classical+Mechanics&rft.pages=%C2%A79.10%2C+pp.+358-359&rft.pub=University+Science+Books&rft.date=2005&rft.isbn=1-891389-22-X&rft.au=John+R+Taylor&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSanthi" class="citation web cs1">Santhi, Sumrit. <a rel="nofollow" class="external text" href="http://www.thaitechnics.com/nav/adf.html">"Aircraft Navigation System"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2006-11-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Aircraft+Navigation+System&rft.aulast=Santhi&rft.aufirst=Sumrit&rft_id=http%3A%2F%2Fwww.thaitechnics.com%2Fnav%2Fadf.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130603111635/http://www.faa.gov/regulations_policies/handbooks_manuals/aircraft/airplane_handbook/media/faa-h-8083-3a-7of7.pdf">"Emergency Procedures"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="https://www.faa.gov/regulations_policies/handbooks_manuals/aircraft/airplane_handbook/media/faa-h-8083-3a-7of7.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2013-06-03<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-01-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Emergency+Procedures&rft_id=https%3A%2F%2Fwww.faa.gov%2Fregulations_policies%2Fhandbooks_manuals%2Faircraft%2Fairplane_handbook%2Fmedia%2Ffaa-h-8083-3a-7of7.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEargle2005" class="citation book cs1"><a href="/wiki/John_M._Eargle" title="John M. Eargle">Eargle, John</a> (2005). <i>Handbook of Recording Engineering</i> (Fourth ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-28470-2" title="Special:BookSources/0-387-28470-2"><bdi>0-387-28470-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Recording+Engineering&rft.edition=Fourth&rft.pub=Springer&rft.date=2005&rft.isbn=0-387-28470-2&rft.aulast=Eargle&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="General_references">General references</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=29" title="Edit section: General references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdamsChristopher_Essex2013" class="citation book cs1">Adams, Robert; Christopher Essex (2013). <i>Calculus: a complete course</i> (Eighth ed.). Pearson Canada Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-78107-9" title="Special:BookSources/978-0-321-78107-9"><bdi>978-0-321-78107-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+a+complete+course&rft.edition=Eighth&rft.pub=Pearson+Canada+Inc.&rft.date=2013&rft.isbn=978-0-321-78107-9&rft.aulast=Adams&rft.aufirst=Robert&rft.au=Christopher+Essex&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAntonIrl_BivensStephen_Davis2002" class="citation book cs1">Anton, Howard; Irl Bivens; Stephen Davis (2002). <i>Calculus</i> (Seventh ed.). Anton Textbooks, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-38157-8" title="Special:BookSources/0-471-38157-8"><bdi>0-471-38157-8</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&rft.edition=Seventh&rft.pub=Anton+Textbooks%2C+Inc.&rft.date=2002&rft.isbn=0-471-38157-8&rft.aulast=Anton&rft.aufirst=Howard&rft.au=Irl+Bivens&rft.au=Stephen+Davis&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></li> <li><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). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusgraphica00ross"><i>Calculus: Graphical, Numerical, Algebraic</i></a> (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%3APolar+coordinate+system" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polar_coordinate_system&action=edit&section=30" 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 .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></span></span></div> <div class="side-box-text plainlist">The Wikibook <i><a href="https://en.wikibooks.org/wiki/Calculus" class="extiw" title="wikibooks:Calculus">Calculus</a></i> has a page on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Calculus/Polar_Integration" class="extiw" title="wikibooks:Calculus/Polar Integration">Polar Integration</a></b></i></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Polar_coordinates">"Polar 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>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Polar+coordinates&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DPolar_coordinates&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolar+coordinate+system" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.random-science-tools.com/maths/coordinate-converter.htm">Coordinate Converter — converts between polar, Cartesian and spherical coordinates</a></li> <li><a rel="nofollow" class="external text" href="http://scratch.mit.edu/projects/nevit/691690">Polar Coordinate System Dynamic Demo</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist 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style="font-size:114%;margin:0 4em"><a href="/wiki/Orthogonal_coordinate_system" class="mw-redirect" title="Orthogonal coordinate system">Orthogonal coordinate systems</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Two dimensional</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian</a></li> <li><a class="mw-selflink selflink">Polar</a> (<a href="/wiki/Log-polar_coordinates" title="Log-polar coordinates">Log-polar</a>)</li> <li><a href="/wiki/Parabolic_coordinates" title="Parabolic coordinates">Parabolic</a></li> <li><a href="/wiki/Bipolar_coordinate_system" class="mw-redirect" title="Bipolar coordinate system">Bipolar</a></li> <li><a href="/wiki/Elliptic_coordinate_system" title="Elliptic coordinate system">Elliptic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" 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coordinates">Ellipsoidal</a></li> <li><a href="/wiki/Elliptic_cylindrical_coordinates" title="Elliptic cylindrical coordinates">Elliptic cylindrical</a></li> <li><a href="/wiki/Toroidal_coordinates" title="Toroidal coordinates">Toroidal</a></li> <li><a href="/wiki/Bispherical_coordinates" title="Bispherical coordinates">Bispherical</a></li> <li><a href="/wiki/Bipolar_cylindrical_coordinates" title="Bipolar cylindrical coordinates">Bipolar cylindrical</a></li> <li><a href="/wiki/Conical_coordinates" title="Conical coordinates">Conical</a></li> <li><a href="/wiki/6-sphere_coordinates" title="6-sphere coordinates">6-sphere</a></li> <li class="mw-empty-elt"></li> <li class="mw-empty-elt"></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" 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Polar coordinate system - Wikipedia