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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Punkt_og_koordinater"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Punkt og koordinater</span> </div> </a> <button aria-controls="toc-Punkt_og_koordinater-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>Vis/skjul underseksjonen Punkt og koordinater</span> </button> <ul id="toc-Punkt_og_koordinater-sublist" class="vector-toc-list"> <li id="toc-Kartesiske_koordinater" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kartesiske_koordinater"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Kartesiske koordinater</span> </div> </a> <ul id="toc-Kartesiske_koordinater-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polarkoordinater_for_planet" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polarkoordinater_for_planet"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Polarkoordinater for planet</span> </div> </a> <ul id="toc-Polarkoordinater_for_planet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sylinderkoordinater" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sylinderkoordinater"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Sylinderkoordinater</span> </div> </a> <ul id="toc-Sylinderkoordinater-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kulekoordinater" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kulekoordinater"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Kulekoordinater</span> </div> </a> <ul id="toc-Kulekoordinater-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vektorer" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vektorer"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Vektorer</span> </div> </a> <ul id="toc-Vektorer-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linjer,_kurver_og_plan" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Linjer,_kurver_og_plan"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Linjer, kurver og plan</span> </div> </a> <ul id="toc-Linjer,_kurver_og_plan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Skifte_av_koordinatsystem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Skifte_av_koordinatsystem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Skifte av koordinatsystem</span> </div> </a> <ul id="toc-Skifte_av_koordinatsystem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referanser" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referanser"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Referanser</span> </div> </a> <ul id="toc-Referanser-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Litteratur" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Litteratur"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Litteratur</span> </div> </a> <ul id="toc-Litteratur-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eksterne_lenker" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Eksterne_lenker"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Eksterne lenker</span> </div> </a> <ul id="toc-Eksterne_lenker-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="Innhold" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Innholdsfortegnelse" > <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="Vis/skjul innholdsfortegnelsen" > <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">Vis/skjul innholdsfortegnelsen</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">Analytisk geometri</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="Gå til en artikkel på et annet språk. Tilgjengelig på 75 språk" > <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-75" 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">75 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Analytisk_geometri" title="Analytisk geometri – norsk nynorsk" lang="nn" hreflang="nn" data-title="Analytisk geometri" data-language-autonym="Norsk nynorsk" data-language-local-name="norsk nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Analytisk_geometri" title="Analytisk geometri – dansk" lang="da" hreflang="da" data-title="Analytisk geometri" data-language-autonym="Dansk" data-language-local-name="dansk" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Analytisk_geometri" title="Analytisk geometri – svensk" lang="sv" hreflang="sv" data-title="Analytisk geometri" data-language-autonym="Svenska" data-language-local-name="svensk" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Analitiese_meetkunde" title="Analitiese meetkunde – afrikaans" lang="af" hreflang="af" data-title="Analitiese meetkunde" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Analytische_Geometrie" title="Analytische Geometrie – sveitsertysk" lang="gsw" hreflang="gsw" data-title="Analytische Geometrie" data-language-autonym="Alemannisch" data-language-local-name="sveitsertysk" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%87%D9%86%D8%AF%D8%B3%D8%A9_%D8%AA%D8%AD%D9%84%D9%8A%D9%84%D9%8A%D8%A9" title="هندسة تحليلية – arabisk" lang="ar" hreflang="ar" data-title="هندسة تحليلية" data-language-autonym="العربية" data-language-local-name="arabisk" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%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%9C%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%AE%E0%A6%BF%E0%A6%A4%E0%A6%BF" title="স্থানাংক জ্যামিতি – assamesisk" lang="as" hreflang="as" data-title="স্থানাংক জ্যামিতি" data-language-autonym="অসমীয়া" data-language-local-name="assamesisk" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Xeometr%C3%ADa_anal%C3%ADtica" title="Xeometría analítica – asturisk" lang="ast" hreflang="ast" data-title="Xeometría analítica" data-language-autonym="Asturianu" data-language-local-name="asturisk" 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/Analitik_h%C9%99nd%C9%99s%C9%99" title="Analitik həndəsə – aserbajdsjansk" lang="az" hreflang="az" data-title="Analitik həndəsə" data-language-autonym="Azərbaycanca" data-language-local-name="aserbajdsjansk" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%DB%8C%D9%84%DB%8C_%D9%87%D9%86%D8%AF%D8%B3%D9%87" title="تحلیلی هندسه – søraserbajdsjansk" lang="azb" hreflang="azb" data-title="تحلیلی هندسه" data-language-autonym="تۆرکجه" data-language-local-name="søraserbajdsjansk" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE%E0%A6%A8%E0%A6%BE%E0%A6%99%E0%A7%8D%E0%A6%95_%E0%A6%9C%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%AE%E0%A6%BF%E0%A6%A4%E0%A6%BF" title="স্থানাঙ্ক জ্যামিতি – bengali" lang="bn" hreflang="bn" data-title="স্থানাঙ্ক জ্যামিতি" data-language-autonym="বাংলা" data-language-local-name="bengali" 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%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D0%BA_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Аналитик геометрия – basjkirsk" lang="ba" hreflang="ba" data-title="Аналитик геометрия" data-language-autonym="Башҡортса" data-language-local-name="basjkirsk" 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%90%D0%BD%D0%B0%D0%BB%D1%96%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%B3%D0%B5%D0%B0%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%8F" title="Аналітычная геаметрыя – belarusisk" lang="be" hreflang="be" data-title="Аналітычная геаметрыя" data-language-autonym="Беларуская" data-language-local-name="belarusisk" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D1%96%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%B3%D0%B5%D0%B0%D0%BC%D1%8D%D1%82%D1%80%D1%8B%D1%8F" title="Аналітычная геамэтрыя – belarusisk (klassisk ortografi)" lang="be-tarask" hreflang="be-tarask" data-title="Аналітычная геамэтрыя" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="belarusisk (klassisk ortografi)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Аналитична геометрия – bulgarsk" lang="bg" hreflang="bg" data-title="Аналитична геометрия" data-language-autonym="Български" data-language-local-name="bulgarsk" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Geometria_anal%C3%ADtica" title="Geometria analítica – katalansk" lang="ca" hreflang="ca" data-title="Geometria analítica" data-language-autonym="Català" data-language-local-name="katalansk" 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%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%C4%83_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8" title="Аналитикăллă геометри – tsjuvasjisk" lang="cv" hreflang="cv" data-title="Аналитикăллă геометри" data-language-autonym="Чӑвашла" data-language-local-name="tsjuvasjisk" 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/Analytick%C3%A1_geometrie" title="Analytická geometrie – tsjekkisk" lang="cs" hreflang="cs" data-title="Analytická geometrie" data-language-autonym="Čeština" data-language-local-name="tsjekkisk" 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/Geometreg_ddadansoddol" title="Geometreg ddadansoddol – walisisk" lang="cy" hreflang="cy" data-title="Geometreg ddadansoddol" data-language-autonym="Cymraeg" data-language-local-name="walisisk" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Analytische_Geometrie" title="Analytische Geometrie – tysk" lang="de" hreflang="de" data-title="Analytische Geometrie" data-language-autonym="Deutsch" data-language-local-name="tysk" 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/Anal%C3%BC%C3%BCtiline_geomeetria" title="Analüütiline geomeetria – estisk" lang="et" hreflang="et" data-title="Analüütiline geomeetria" data-language-autonym="Eesti" data-language-local-name="estisk" 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%91%CE%BD%CE%B1%CE%BB%CF%85%CF%84%CE%B9%CE%BA%CE%AE_%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Αναλυτική γεωμετρία – gresk" lang="el" hreflang="el" data-title="Αναλυτική γεωμετρία" data-language-autonym="Ελληνικά" data-language-local-name="gresk" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Analytic_geometry" title="Analytic geometry – engelsk" lang="en" hreflang="en" data-title="Analytic geometry" data-language-autonym="English" data-language-local-name="engelsk" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Geometr%C3%ADa_anal%C3%ADtica" title="Geometría analítica – spansk" lang="es" hreflang="es" data-title="Geometría analítica" data-language-autonym="Español" data-language-local-name="spansk" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Analitika_geometrio" title="Analitika geometrio – esperanto" lang="eo" hreflang="eo" data-title="Analitika geometrio" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Geometria_analitiko" title="Geometria analitiko – baskisk" lang="eu" hreflang="eu" data-title="Geometria analitiko" data-language-autonym="Euskara" data-language-local-name="baskisk" 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/%D9%87%D9%86%D8%AF%D8%B3%D9%87_%D8%AA%D8%AD%D9%84%DB%8C%D9%84%DB%8C" title="هندسه تحلیلی – persisk" lang="fa" hreflang="fa" data-title="هندسه تحلیلی" data-language-autonym="فارسی" data-language-local-name="persisk" 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/G%C3%A9om%C3%A9trie_analytique" title="Géométrie analytique – fransk" lang="fr" hreflang="fr" data-title="Géométrie analytique" data-language-autonym="Français" data-language-local-name="fransk" 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/Geoim%C3%A9adracht_anail%C3%ADseach" title="Geoiméadracht anailíseach – irsk" lang="ga" hreflang="ga" data-title="Geoiméadracht anailíseach" data-language-autonym="Gaeilge" data-language-local-name="irsk" 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/Xeometr%C3%ADa_anal%C3%ADtica" title="Xeometría analítica – galisisk" lang="gl" hreflang="gl" data-title="Xeometría analítica" data-language-autonym="Galego" data-language-local-name="galisisk" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%B4%EC%84%9D%EA%B8%B0%ED%95%98%ED%95%99" title="해석기하학 – koreansk" lang="ko" hreflang="ko" data-title="해석기하학" data-language-autonym="한국어" data-language-local-name="koreansk" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%B6%D5%A1%D5%AC%D5%AB%D5%BF%D5%AB%D5%AF_%D5%A5%D6%80%D5%AF%D6%80%D5%A1%D5%B9%D5%A1%D6%83%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Անալիտիկ երկրաչափություն – armensk" lang="hy" hreflang="hy" data-title="Անալիտիկ երկրաչափություն" data-language-autonym="Հայերեն" data-language-local-name="armensk" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A5%88%E0%A4%B6%E0%A5%8D%E2%80%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%BF%E0%A4%95_%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A4%BF" title="वैश्‍लेषिक ज्यामिति – hindi" lang="hi" hreflang="hi" data-title="वैश्‍लेषिक ज्यामिति" 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/Analiti%C4%8Dka_geometrija" title="Analitička geometrija – kroatisk" lang="hr" hreflang="hr" data-title="Analitička geometrija" data-language-autonym="Hrvatski" data-language-local-name="kroatisk" 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/Analitikala_geometrio" title="Analitikala geometrio – ido" lang="io" hreflang="io" data-title="Analitikala geometrio" 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/Geometri_analitis" title="Geometri analitis – indonesisk" lang="id" hreflang="id" data-title="Geometri analitis" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesisk" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Geometria_analitica" title="Geometria analitica – italiensk" lang="it" hreflang="it" data-title="Geometria analitica" data-language-autonym="Italiano" data-language-local-name="italiensk" 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%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94_%D7%90%D7%A0%D7%9C%D7%99%D7%98%D7%99%D7%AA" title="גאומטריה אנליטית – hebraisk" lang="he" hreflang="he" data-title="גאומטריה אנליטית" data-language-autonym="עברית" data-language-local-name="hebraisk" 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%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Аналитикалық геометрия – kasakhisk" lang="kk" hreflang="kk" data-title="Аналитикалық геометрия" data-language-autonym="Қазақша" data-language-local-name="kasakhisk" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Аналитикалык геометрия – kirgisisk" lang="ky" hreflang="ky" data-title="Аналитикалык геометрия" data-language-autonym="Кыргызча" data-language-local-name="kirgisisk" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Geometria_analytica" title="Geometria analytica – latin" lang="la" hreflang="la" data-title="Geometria analytica" 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/Anal%C4%ABtisk%C4%81_%C4%A3eometrija" title="Analītiskā ģeometrija – latvisk" lang="lv" hreflang="lv" data-title="Analītiskā ģeometrija" data-language-autonym="Latviešu" data-language-local-name="latvisk" 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/Analizin%C4%97_geometrija" title="Analizinė geometrija – litauisk" lang="lt" hreflang="lt" data-title="Analizinė geometrija" data-language-autonym="Lietuvių" data-language-local-name="litauisk" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Koordin%C3%A1tageometria" title="Koordinátageometria – ungarsk" lang="hu" hreflang="hu" data-title="Koordinátageometria" data-language-autonym="Magyar" data-language-local-name="ungarsk" 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%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0" title="Аналитичка геометрија – makedonsk" lang="mk" hreflang="mk" data-title="Аналитичка геометрија" data-language-autonym="Македонски" data-language-local-name="makedonsk" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%BF%E0%B4%B6%E0%B5%8D%E0%B4%B2%E0%B5%87%E0%B4%B7%E0%B4%95%E0%B4%9C%E0%B5%8D%E0%B4%AF%E0%B4%BE%E0%B4%AE%E0%B4%BF%E0%B4%A4%E0%B4%BF" title="വിശ്ലേഷകജ്യാമിതി – malayalam" lang="ml" hreflang="ml" data-title="വിശ്ലേഷകജ്യാമിതി" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D0%BA_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80" title="Аналитик геометр – mongolsk" lang="mn" hreflang="mn" data-title="Аналитик геометр" data-language-autonym="Монгол" data-language-local-name="mongolsk" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Analytische_meetkunde" title="Analytische meetkunde – nederlandsk" lang="nl" hreflang="nl" data-title="Analytische meetkunde" data-language-autonym="Nederlands" data-language-local-name="nederlandsk" 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/%E8%A7%A3%E6%9E%90%E5%B9%BE%E4%BD%95%E5%AD%A6" title="解析幾何学 – japansk" lang="ja" hreflang="ja" data-title="解析幾何学" data-language-autonym="日本語" data-language-local-name="japansk" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Analitik_geometriya" title="Analitik geometriya – usbekisk" lang="uz" hreflang="uz" data-title="Analitik geometriya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbekisk" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Geometr%C3%ACa_anal%C3%ACtica" title="Geometrìa analìtica – piemontesisk" lang="pms" hreflang="pms" data-title="Geometrìa analìtica" data-language-autonym="Piemontèis" data-language-local-name="piemontesisk" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Geometria_analityczna" title="Geometria analityczna – polsk" lang="pl" hreflang="pl" data-title="Geometria analityczna" data-language-autonym="Polski" data-language-local-name="polsk" 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/Geometria_anal%C3%ADtica" title="Geometria analítica – portugisisk" lang="pt" hreflang="pt" data-title="Geometria analítica" data-language-autonym="Português" data-language-local-name="portugisisk" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Analitikal%C4%B1q_geometriya" title="Analitikalıq geometriya – karakalpakisk" lang="kaa" hreflang="kaa" data-title="Analitikalıq geometriya" data-language-autonym="Qaraqalpaqsha" data-language-local-name="karakalpakisk" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Geometrie_analitic%C4%83" title="Geometrie analitică – rumensk" lang="ro" hreflang="ro" data-title="Geometrie analitică" data-language-autonym="Română" data-language-local-name="rumensk" 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%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Аналитическая геометрия – russisk" lang="ru" hreflang="ru" data-title="Аналитическая геометрия" data-language-autonym="Русский" data-language-local-name="russisk" 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/Analytic_geometry" title="Analytic geometry – skotsk" lang="sco" hreflang="sco" data-title="Analytic geometry" data-language-autonym="Scots" data-language-local-name="skotsk" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Analytic_geometry" title="Analytic geometry – enkel engelsk" lang="en-simple" hreflang="en-simple" data-title="Analytic geometry" data-language-autonym="Simple English" data-language-local-name="enkel engelsk" 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/Analytick%C3%A1_geometria" title="Analytická geometria – slovakisk" lang="sk" hreflang="sk" data-title="Analytická geometria" data-language-autonym="Slovenčina" data-language-local-name="slovakisk" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A6%DB%95%D9%86%D8%AF%D8%A7%D8%B2%DB%95%DB%8C_%D8%B4%DB%8C%DA%A9%D8%A7%D8%B1%D8%A7%D9%86%DB%95" title="ئەندازەی شیکارانە – sentralkurdisk" lang="ckb" hreflang="ckb" data-title="ئەندازەی شیکارانە" data-language-autonym="کوردی" data-language-local-name="sentralkurdisk" 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%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0" title="Аналитичка геометрија – serbisk" lang="sr" hreflang="sr" data-title="Аналитичка геометрија" data-language-autonym="Српски / srpski" data-language-local-name="serbisk" 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/Analiti%C4%8Dka_geometrija" title="Analitička geometrija – serbokroatisk" lang="sh" hreflang="sh" data-title="Analitička geometrija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroatisk" 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/Analyyttinen_geometria" title="Analyyttinen geometria – finsk" lang="fi" hreflang="fi" data-title="Analyyttinen geometria" data-language-autonym="Suomi" data-language-local-name="finsk" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Heometriyang_pasuri" title="Heometriyang pasuri – tagalog" lang="tl" hreflang="tl" data-title="Heometriyang pasuri" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%95%E0%AF%81%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%88_%E0%AE%B5%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D" title="பகுமுறை வடிவவியல் – tamil" lang="ta" hreflang="ta" data-title="பகுமுறை வடிவவியல்" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A3%E0%B8%82%E0%B8%B2%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A7%E0%B8%B4%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B2%E0%B8%B0%E0%B8%AB%E0%B9%8C" title="เรขาคณิตวิเคราะห์ – thai" lang="th" hreflang="th" data-title="เรขาคณิตวิเคราะห์" data-language-autonym="ไทย" data-language-local-name="thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D2%B2%D0%B0%D0%BD%D0%B4%D0%B0%D1%81%D0%B0%D0%B8_%D1%82%D0%B0%D2%B3%D0%BB%D0%B8%D0%BB%D3%A3" title="Ҳандасаи таҳлилӣ – tadsjikisk" lang="tg" hreflang="tg" data-title="Ҳандасаи таҳлилӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="tadsjikisk" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Analitik_geometri" title="Analitik geometri – tyrkisk" lang="tr" hreflang="tr" data-title="Analitik geometri" data-language-autonym="Türkçe" data-language-local-name="tyrkisk" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D1%96%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F" title="Аналітична геометрія – ukrainsk" lang="uk" hreflang="uk" data-title="Аналітична геометрія" data-language-autonym="Українська" data-language-local-name="ukrainsk" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%DB%8C%D9%84%DB%8C_%DB%81%D9%86%D8%AF%D8%B3%DB%81" title="تحلیلی ہندسہ – urdu" lang="ur" hreflang="ur" data-title="تحلیلی ہندسہ" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%ACnh_h%E1%BB%8Dc_gi%E1%BA%A3i_t%C3%ADch" title="Hình học giải tích – vietnamesisk" lang="vi" hreflang="vi" data-title="Hình học giải tích" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesisk" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a 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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Utseende</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">flytt til sidefeltet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skjul</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Fra Wikipedia, den frie encyklopedi</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="nb" dir="ltr"><p><b>Analytisk geometri</b> eller <b>koordinatgeometri</b> er en gren av <a href="/wiki/Geometri" title="Geometri">geometri</a> der geometriske figurer og objekt blir beskrevet ved hjelp av <a href="/wiki/Koordinatsystem" title="Koordinatsystem">koordinater</a> og der metoder fra <a href="/wiki/Algebra" title="Algebra">algebra</a> og <a href="/wiki/Matematisk_analyse" title="Matematisk analyse">matematisk analyse</a> anvendes for å løse problemer.<sup id="cite_ref-COLLINS1_1-0" class="reference"><a href="#cite_note-COLLINS1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Analytisk geometri danner grunnlaget for moderne geometri, inkludert retninger som <a href="/wiki/Algebraisk_geometri" title="Algebraisk geometri">algebraisk geometri</a> og <a href="/wiki/Differensialgeometri" title="Differensialgeometri">differensialgeometri</a>. Det er også mye brukt som verktøy i andre naturvitenskaper, som fysikk og astronomi. Metoder fra analytisk geometri har mange industrielle anvendelser, for eksempel i <a href="/wiki/Datagrafikk" title="Datagrafikk">datagrafikk</a> og alle former for <a href="/wiki/Geometrisk_modellering" title="Geometrisk modellering">geometrisk modellering</a>. </p><p>Vanligvis studeres i analytisk geometri det todimensjonale planet eller det tredimensjonale rommet, i en <a href="/wiki/Euklidsk_geometri" title="Euklidsk geometri">euklidsk geometri</a>. Hvert punkt er definert med et ordnet sett av tall, kalt koordinater. Geometriske objekt som <a href="/wiki/Linje" title="Linje">linjer</a>, <a href="/wiki/Kurve" title="Kurve">kurver</a> og <a href="/wiki/Plan" class="mw-disambig" title="Plan">plan</a> blir definert ved ligninger som gir relasjoner mellom koordinatene. Til å beskrive en orientert lengde brukes en <a href="/wiki/Vektor_(matematikk)" title="Vektor (matematikk)">vektor</a>. </p><p>I motsetning til analytisk geometri brukes i <a href="/wiki/Syntetisk_geometri" title="Syntetisk geometri">syntetisk geometri</a> ikke koordinater, men problemstillinger løses ved logiske resonnement og konstruksjon med passer og linjal. </p><p>Som grunnlegger av den analytiske geometri regnes <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> med <i>Discours de la méthode</i> (1637).<sup id="cite_ref-BOYER_2-0" class="reference"><a href="#cite_note-BOYER-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Også <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> ga tidlig viktige bidrag til fagområdet. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Punkt_og_koordinater">Punkt og koordinater</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytisk_geometri&veaction=edit&section=1" title="Rediger avsnitt: Punkt og koordinater" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analytisk_geometri&action=edit&section=1" title="Rediger kildekoden til seksjonen Punkt og koordinater"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I analytisk geometri er hvert <a href="/wiki/Punkt" title="Punkt">punkt</a> i planet <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} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feb97520ae70482ae41b49980ec140d871cb8243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.057ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{2}}" /></span> definert ved to koordinater, mens et punkt i rommet er gitt ved tre koordinater. Verdiene til koordinatene vil avhenge av valg av koordinatsystem og definisjon av <a href="/wiki/Origo" title="Origo">origo</a> og koordinatakser. Flere alternative koordinatsystem er i bruk, og avsnittet omtaler de vanligste systemene.<sup id="cite_ref-THOMAS1_3-0" class="reference"><a href="#cite_note-THOMAS1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Kartesiske_koordinater">Kartesiske koordinater</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytisk_geometri&veaction=edit&section=2" title="Rediger avsnitt: Kartesiske koordinater" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analytisk_geometri&action=edit&section=2" title="Rediger kildekoden til seksjonen Kartesiske koordinater"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="utdypende-artikkel" style="padding-left:2em; font-style: italic;"> <p>Utdypende artikkel: <a href="/wiki/Kartesisk_koordinatsystem" title="Kartesisk koordinatsystem">Kartesisk koordinatsystem</a> </p> </div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fil:Cartesian-coordinate-system.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/250px-Cartesian-coordinate-system.svg.png" decoding="async" width="250" height="247" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/375px-Cartesian-coordinate-system.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/500px-Cartesian-coordinate-system.svg.png 2x" data-file-width="661" data-file-height="654" /></a><figcaption>Definisjon av punkt i et kartesisk koordinatsystem i planet</figcaption></figure> <p>Det vanligste koordinatsystemet er et kartesisk system definert med to eller tre akser som står vinkelrett på hverandre. Koordinatene til et punkt blir bestemt ved <a href="/wiki/Projeksjon" class="mw-disambig" title="Projeksjon">projeksjoner</a> ned på koordinataksene. Et vilkårlig punkt i planet betegnes ofte <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" /></span> og et punkt i rommet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}" /></span>. </p><p><a href="/wiki/Basis_(matematikk)" title="Basis (matematikk)">Standardbasisen</a> i det kartesiske koordinatsystemet for <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} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ee047387e551a89e8481e1a9e974dcc5fd5acc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.057ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{3}}" /></span> er definert ved </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 {\begin{alignedat}{2}\mathbf {i} &=(1,0,0)\\\mathbf {j} &=(0,1,0)\\\mathbf {k} &=(0,0,1).\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}\mathbf {i} &=(1,0,0)\\\mathbf {j} &=(0,1,0)\\\mathbf {k} &=(0,0,1).\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74f3be05f99c55608b3b2d115bfa04aaf171752b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.272ex; height:9.176ex;" alt="{\displaystyle {\begin{alignedat}{2}\mathbf {i} &=(1,0,0)\\\mathbf {j} &=(0,1,0)\\\mathbf {k} &=(0,0,1).\end{alignedat}}}" /></span></dd></dl> <p>På vektorform kan ligningen for et vilkårlig punkt da skrives </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 \mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7936e3b57a8bd57b8e249bdc00bb841dfcb7ae2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.071ex; height:2.509ex;" alt="{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k} .}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Polarkoordinater_for_planet">Polarkoordinater for planet</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytisk_geometri&veaction=edit&section=3" title="Rediger avsnitt: Polarkoordinater for planet" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analytisk_geometri&action=edit&section=3" title="Rediger kildekoden til seksjonen Polarkoordinater for planet"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="utdypende-artikkel" style="padding-left:2em; font-style: italic;"> <p>Utdypende artikkel: <a href="/wiki/Polarkoordinatsystem" title="Polarkoordinatsystem">Polarkoordinatsystem</a> </p> </div> <p>Polarkoordinatene <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,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8396fdc359fb06c93722137c959e7496e47ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.982ex; height:2.843ex;" alt="{\displaystyle (r,\theta )}" /></span> til et punkt i planet er definert ved avstanden <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> fra origo til punktet, samt vinkelen <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> som forbindelseslinjen mellom punktet og origo danner med en gitt referanselinje gjennom origo. </p><p>Dersom referanselinjen er <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>-aksen i et kartesisk koordinatsystem, så er sammenhengen mellom de to systemene gitt ved </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 {\begin{alignedat}{2}x&=r\cos \theta \\y&=r\sin \theta \\r&={\sqrt {x^{2}+y^{2}}}\\\theta &=\arctan({\frac {y}{x}})\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <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> </mtd> </mtr> <mtr> <mtd> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}x&=r\cos \theta \\y&=r\sin \theta \\r&={\sqrt {x^{2}+y^{2}}}\\\theta &=\arctan({\frac {y}{x}})\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3723938a0246d9b32dffe4163a97226a930e6386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:15.621ex; height:15.843ex;" alt="{\displaystyle {\begin{alignedat}{2}x&=r\cos \theta \\y&=r\sin \theta \\r&={\sqrt {x^{2}+y^{2}}}\\\theta &=\arctan({\frac {y}{x}})\end{alignedat}}}" /></span></dd></dl> <p>Noen plane kurver uttrykkes enklest i polarform, for eksempel vil den følgende ligningen definere en kurve kalt <a href="/w/index.php?title=Arkimedes_spiral&action=edit&redlink=1" class="new" title="Arkimedes spiral (ikke skrevet ennå)">Arkimedes spiral</a> for vilkårlige parametre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /></span> og <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a{\theta }+b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo>+</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a{\theta }+b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cda3bcd0e74b6937e9fac5a9c715ed2e5c55053a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.952ex; height:2.343ex;" alt="{\displaystyle r=a{\theta }+b.}" /></span></dd></dl> <p>Polarkoordinater kan generalisere til rommet i form av sylinderkoordinater eller sfæriske koordinater. </p> <div class="mw-heading mw-heading3"><h3 id="Sylinderkoordinater">Sylinderkoordinater</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytisk_geometri&veaction=edit&section=4" title="Rediger avsnitt: Sylinderkoordinater" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analytisk_geometri&action=edit&section=4" title="Rediger kildekoden til seksjonen Sylinderkoordinater"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="utdypende-artikkel" style="padding-left:2em; font-style: italic;"> <p>Utdypende artikkel: <a href="/w/index.php?title=Sylinderkoordinatsystem&action=edit&redlink=1" class="new" title="Sylinderkoordinatsystem (ikke skrevet ennå)">Sylinderkoordinatsystem</a> </p> </div> <p>Et punkt i rommet kan i sylinderkoordinater defineres som <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,\theta ,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\theta ,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad38045a44383ed963b8a8d7fd0f09a25d62caee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.104ex; height:2.843ex;" alt="{\displaystyle (r,\theta ,z)}" /></span>. Definisjonen av <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> og <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> er tilsvarende som for polarkoordinater, men nå basert på projeksjonen av forbindelseslinjen mellom punktet og origo ned på et referanseplan. Den tredje koordinaten <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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /></span> er høyden fra referanseplanet til punktet. </p><p>Dersom referanseplanet er det kartesiske <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" /></span>-planet, så er sammenhengen mellom de to koordinatsystemene gitt ved </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 {\begin{alignedat}{2}x&=r\cos \theta \\y&=r\sin \theta \\z&=z\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}x&=r\cos \theta \\y&=r\sin \theta \\z&=z\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8fb40238d1f8fde83d7babcef83414c0642fcf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:11.204ex; height:8.843ex;" alt="{\displaystyle {\begin{alignedat}{2}x&=r\cos \theta \\y&=r\sin \theta \\z&=z\end{alignedat}}}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Kulekoordinater">Kulekoordinater</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytisk_geometri&veaction=edit&section=5" title="Rediger avsnitt: Kulekoordinater" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analytisk_geometri&action=edit&section=5" title="Rediger kildekoden til seksjonen Kulekoordinater"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="utdypende-artikkel" style="padding-left:2em; font-style: italic;"> <p>Utdypende artikkel: <a href="/wiki/Kulekoordinater" title="Kulekoordinater">Kulekoordinater</a> </p> </div> <p>I kulekoordinater, også kalt sfæriske koordinater, kan et punkt defineres ved hjelp av tre koordinater <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,\theta ,\psi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\theta ,\psi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81359092ea47cd29fc7e0a372def3cc779159b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.529ex; height:2.843ex;" alt="{\displaystyle (r,\theta ,\psi )}" /></span>. Nå er <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> avstanden fra punktet til origo. Vinkelen <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> er vinkelen mellom projeksjonen av forbindelseslinjen og den horisontale aksen. Forbindelseslinjen danner vinkelen <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 \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" /></span> med <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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /></span>-aksen. </p> <div class="mw-heading mw-heading2"><h2 id="Vektorer">Vektorer</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytisk_geometri&veaction=edit&section=6" title="Rediger avsnitt: Vektorer" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analytisk_geometri&action=edit&section=6" title="Rediger kildekoden til seksjonen Vektorer"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="utdypende-artikkel" style="padding-left:2em; font-style: italic;"> <p>Utdypende artikkel: <a href="/wiki/Vektor_(matematikk)" title="Vektor (matematikk)">Vektor (matematikk)</a> </p> </div> <p>Et orientert linjestykke kan i analytisk geometri defineres som en <a href="/wiki/Vektor" class="mw-disambig" title="Vektor">vektor</a>, basert på den valgte basisen i koordinatsystemet. Vektorer kan adderes ved å legge sammen koordinatene: </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 \mathbf {v} +\mathbf {u} =(v_{1}+u_{1},v_{2}+u_{2},v_{3}+u_{3})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} +\mathbf {u} =(v_{1}+u_{1},v_{2}+u_{2},v_{3}+u_{3})\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89fe3417065673bfe40fbec13eb72e27cc7fbcb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.318ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} +\mathbf {u} =(v_{1}+u_{1},v_{2}+u_{2},v_{3}+u_{3})\,}" /></span></dd></dl> <p>Lengden av en vektor skrives ofte som <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {v} |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {v} |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a779e65de92152d395f5576ce1001c8e56d7f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.705ex; height:2.843ex;" alt="{\displaystyle |\mathbf {v} |}" /></span> og er definert ved </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 |\mathbf {v} |={\sqrt {v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {v} |={\sqrt {v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8221395f8e071402afed99bd83d2a5603f496d1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.74ex; height:4.843ex;" alt="{\displaystyle |\mathbf {v} |={\sqrt {v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}}\,}" /></span></dd></dl> <p>To vektorer står <a href="/wiki/Normal_(geometri)" title="Normal (geometri)">normalt</a> på hverandre dersom <a href="/wiki/Indreprodukt" title="Indreprodukt">indreproduktet</a> mellom de to vektorene er lik null. </p> <div class="mw-heading mw-heading2"><h2 id="Linjer,_kurver_og_plan"><span id="Linjer.2C_kurver_og_plan"></span>Linjer, kurver og plan</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytisk_geometri&veaction=edit&section=7" title="Rediger avsnitt: Linjer, kurver og plan" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analytisk_geometri&action=edit&section=7" title="Rediger kildekoden til seksjonen Linjer, kurver og plan"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="utdypende-artikkel" style="padding-left:2em; font-style: italic;"> <p>Utdypende artikler: <a href="/wiki/Linje" title="Linje">Linje</a>, <a href="/wiki/Kurve" title="Kurve">Kurve</a> og <a href="/wiki/Plan_(matematikk)" title="Plan (matematikk)">Plan (matematikk)</a> </p> </div> <p>I analytisk geometri vil ligninger som involverer koordinatene kunne definere <a href="/wiki/Delmengde" title="Delmengde">delmengder</a> av punkt i planet eller rommet. Ligninger kan dermed brukes til å definere geometriske objekt som linjer, kurver og plan. </p><p>Ligningen for en rett linje kan i kartesiske koordinater uttrykkes på flere ulike måter, for eksempel i vektorform: </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 \mathbf {r} (t)=\mathbf {r_{0}} +t\mathbf {v} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </msub> </mrow> <mo>+</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t)=\mathbf {r_{0}} +t\mathbf {v} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/748d155ccbf7d94079531c35b797f15ff1094978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.867ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} (t)=\mathbf {r_{0}} +t\mathbf {v} .}" /></span></dd></dl> <p>En viktig familie av plane kurver er <a href="/wiki/Kjeglesnitt" title="Kjeglesnitt">kjeglesnitt</a>, som inkluderer <a href="/wiki/Ellipse" title="Ellipse">ellipser</a>, <a href="/wiki/Hyperbel" title="Hyperbel">hyperbler</a>, <a href="/wiki/Parabel" title="Parabel">parabler</a> og <a href="/wiki/Sirkel" title="Sirkel">sirkelen</a>. Alle kurvene kan beskrives i et kartesisk koordinatsystem av en ligningen på formen </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 Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>B</mi> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>C</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>D</mi> <mi>x</mi> <mo>+</mo> <mi>E</mi> <mi>y</mi> <mo>+</mo> <mi>F</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a694f3a0805f2f11868a2a69413866d64170409b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.746ex; height:3.009ex;" alt="{\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0}" /></span></dd></dl> <p>Et plan gjennom et punkt <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_{0}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r_{0}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e354fab59b0195a0ac38a2834970572067e7e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.009ex;" alt="{\displaystyle \mathbf {r_{0}} }" /></span> og normalt på vektoren <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 {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }" /></span> kan defineres ved ligningen </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 (\mathbf {r} -\mathbf {r_{0}} )\cdot \mathbf {n} =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {r} -\mathbf {r_{0}} )\cdot \mathbf {n} =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da2fac8df37fb3d986a6e16bb37df13b88df5b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.103ex; height:2.843ex;" alt="{\displaystyle (\mathbf {r} -\mathbf {r_{0}} )\cdot \mathbf {n} =0.}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Skifte_av_koordinatsystem">Skifte av koordinatsystem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytisk_geometri&veaction=edit&section=8" title="Rediger avsnitt: Skifte av koordinatsystem" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analytisk_geometri&action=edit&section=8" title="Rediger kildekoden til seksjonen Skifte av koordinatsystem"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ved skifte mellom alternative koordinatsystemer spiller <a href="/wiki/Line%C3%A6r_algebra" title="Lineær algebra">lineær algebra</a> en viktig rolle. </p> <div class="mw-heading mw-heading2"><h2 id="Referanser">Referanser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytisk_geometri&veaction=edit&section=9" title="Rediger avsnitt: Referanser" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analytisk_geometri&action=edit&section=9" title="Rediger kildekoden til seksjonen Referanser"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-COLLINS1-1"><b><a href="#cite_ref-COLLINS1_1-0">^</a></b> <span class="reference-text"><cite class="citation book">E.J.Borowski, J.M.Borwein (1989). <i>Dictionary of mathematics</i>. Glasgow: Collins. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Spesial:Bokkilder/0-00-434347-6" title="Spesial:Bokkilder/0-00-434347-6">0-00-434347-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fno.wikipedia.org%3AAnalytisk+geometri&rft.au=E.J.Borowski%2C+J.M.Borwein&rft.btitle=Dictionary+of+mathematics&rft.date=1989&rft.genre=book&rft.isbn=0-00-434347-6&rft.place=Glasgow&rft.pub=Collins&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span> </span> </li> <li id="cite_note-BOYER-2"><b><a href="#cite_ref-BOYER_2-0">^</a></b> <span class="reference-text"><cite id="CBB" class="citation book">C.B.Boyer (1968). <i>A history of mathematics</i>. Princeton, USA: John Wiley & Sons, Inc. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Spesial:Bokkilder/0-691-02391-3" title="Spesial:Bokkilder/0-691-02391-3">0-691-02391-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fno.wikipedia.org%3AAnalytisk+geometri&rft.au=C.B.Boyer&rft.btitle=A+history+of+mathematics&rft.date=1968&rft.genre=book&rft.isbn=0-691-02391-3&rft.place=Princeton%2C+USA&rft.pub=John+Wiley+%26+Sons%2C+Inc&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-THOMAS1-3"><b><a href="#cite_ref-THOMAS1_3-0">^</a></b> <span class="reference-text"><a href="#THOMAS">G.B. Thomas, R.L. Finney: <i>Calculus and analytical geometry</i></a>, s.647ff</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Litteratur">Litteratur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytisk_geometri&veaction=edit&section=10" title="Rediger avsnitt: Litteratur" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analytisk_geometri&action=edit&section=10" title="Rediger kildekoden til seksjonen Litteratur"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation book">Fr. Fabricius-Bjerre (1977). <i>Lærebog i geometri. Analytisk geometri og lineær algebra</i>. Lyngby, Danmark: Polyteknisk forlag. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Spesial:Bokkilder/87-502-0440-8" title="Spesial:Bokkilder/87-502-0440-8">87-502-0440-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fno.wikipedia.org%3AAnalytisk+geometri&rft.au=Fr.+Fabricius-Bjerre&rft.btitle=L%C3%A6rebog+i+geometri.++Analytisk+geometri+og+line%C3%A6r+algebra&rft.date=1977&rft.genre=book&rft.isbn=87-502-0440-8&rft.place=Lyngby%2C+Danmark&rft.pub=Polyteknisk+forlag&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <ul><li><cite id="THOMAS" class="citation book">G.B. Thomas, R.L. Finney (1979). <i>Calculus and analytical geometry</i>. Reading, USA: Addison-Wesley Publishing Company. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Spesial:Bokkilder/0-201-07523-7" title="Spesial:Bokkilder/0-201-07523-7">0-201-07523-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fno.wikipedia.org%3AAnalytisk+geometri&rft.au=G.B.+Thomas%2C+R.L.+Finney&rft.btitle=Calculus+and+analytical+geometry&rft.date=1979&rft.genre=book&rft.isbn=0-201-07523-7&rft.place=Reading%2C+USA&rft.pub=Addison-Wesley+Publishing+Company&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Eksterne_lenker">Eksterne lenker</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytisk_geometri&veaction=edit&section=11" title="Rediger avsnitt: Eksterne lenker" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a 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