Barycentriska koordinater

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vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Barycentriska_koordinater_i_två_dimensioner"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Barycentriska koordinater i två dimensioner</span> </div> </a> <ul id="toc-Barycentriska_koordinater_i_två_dimensioner-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Barycentriska_koordinater_i_tre_eller_flera_dimensioner" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Barycentriska_koordinater_i_tre_eller_flera_dimensioner"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Barycentriska koordinater i tre eller flera dimensioner</span> </div> </a> <ul id="toc-Barycentriska_koordinater_i_tre_eller_flera_dimensioner-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ortsvektorer_och_kartesiska_koordinater" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ortsvektorer_och_kartesiska_koordinater"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Ortsvektorer och kartesiska koordinater</span> </div> </a> <button aria-controls="toc-Ortsvektorer_och_kartesiska_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>Växla underavsnittet Ortsvektorer och kartesiska koordinater</span> </button> <ul id="toc-Ortsvektorer_och_kartesiska_koordinater-sublist" class="vector-toc-list"> <li id="toc-Från_barycentriska_till_kartesiska_koordinater" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Från_barycentriska_till_kartesiska_koordinater"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Från barycentriska till kartesiska koordinater</span> </div> </a> <ul id="toc-Från_barycentriska_till_kartesiska_koordinater-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Från_kartesiska_till_barycentriska_koordinater" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Från_kartesiska_till_barycentriska_koordinater"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Från kartesiska till barycentriska koordinater</span> </div> </a> <ul id="toc-Från_kartesiska_till_barycentriska_koordinater-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Trilinjära_koordinater" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Trilinjära_koordinater"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Trilinjära koordinater</span> </div> </a> <ul id="toc-Trilinjära_koordinater-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Barycentriska_koordinater_för_vissa_punkter" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Barycentriska_koordinater_för_vissa_punkter"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Barycentriska koordinater för vissa punkter</span> </div> </a> <button aria-controls="toc-Barycentriska_koordinater_för_vissa_punkter-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>Växla underavsnittet Barycentriska koordinater för vissa punkter</span> </button> <ul id="toc-Barycentriska_koordinater_för_vissa_punkter-sublist" class="vector-toc-list"> <li id="toc-Den_geometriska_tyngdpunkten" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Den_geometriska_tyngdpunkten"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Den geometriska tyngdpunkten</span> </div> </a> <ul id="toc-Den_geometriska_tyngdpunkten-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Den_inskrivna_cirkelns_medelpunkt" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Den_inskrivna_cirkelns_medelpunkt"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Den inskrivna cirkelns medelpunkt</span> </div> </a> <ul id="toc-Den_inskrivna_cirkelns_medelpunkt-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-De_vidskrivna_cirklarnas_medelpunkter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#De_vidskrivna_cirklarnas_medelpunkter"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>De vidskrivna cirklarnas medelpunkter</span> </div> </a> <ul id="toc-De_vidskrivna_cirklarnas_medelpunkter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Den_omskrivna_cirkelns_medelpunkt" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Den_omskrivna_cirkelns_medelpunkt"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Den omskrivna cirkelns medelpunkt</span> </div> </a> <ul id="toc-Den_omskrivna_cirkelns_medelpunkt-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ortocentrum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ortocentrum"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Ortocentrum</span> </div> </a> <ul id="toc-Ortocentrum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmedianpunkten" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmedianpunkten"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6</span> <span>Symmedianpunkten</span> </div> </a> <ul id="toc-Symmedianpunkten-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Barycentrisk_interpolation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Barycentrisk_interpolation"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Barycentrisk interpolation</span> </div> </a> <ul id="toc-Barycentrisk_interpolation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Se_även" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Se_även"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Se även</span> </div> </a> <ul id="toc-Se_även-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referenser" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referenser"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Referenser</span> </div> </a> <ul id="toc-Referenser-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="Innehåll" 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="Växla innehållsförteckningen" > <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">Växla innehållsförteckningen</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">Barycentriska koordinater</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 till en artikel på ett annat språk. Tillgänglig på 21 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-21" 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">21 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <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%85%D8%B1%D8%AC%D8%AD%D9%8A%D8%A9" title="نظام الإحداثيات المرجحية – arabiska" lang="ar" hreflang="ar" data-title="نظام الإحداثيات المرجحية" data-language-autonym="العربية" data-language-local-name="arabiska" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%D0%B0%D1%80%D0%B8%D1%86%D0%B5%D0%BD%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B8_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B8" title="Барицентрични координати – bulgariska" lang="bg" hreflang="bg" data-title="Барицентрични координати" data-language-autonym="Български" data-language-local-name="bulgariska" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Baryzentrische_Koordinaten" title="Baryzentrische Koordinaten – tyska" lang="de" hreflang="de" data-title="Baryzentrische Koordinaten" data-language-autonym="Deutsch" data-language-local-name="tyska" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Barycentric_coordinate_system" title="Barycentric coordinate system – engelska" lang="en" hreflang="en" data-title="Barycentric coordinate system" data-language-autonym="English" data-language-local-name="engelska" 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/Coordenadas_baric%C3%A9ntricas_(n-simplex)" title="Coordenadas baricéntricas (n-simplex) – spanska" lang="es" hreflang="es" data-title="Coordenadas baricéntricas (n-simplex)" data-language-autonym="Español" data-language-local-name="spanska" class="interlanguage-link-target"><span>Español</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_%DA%AF%D8%B1%D8%A7%D9%86%DB%8C%DA%AF%D8%A7%D9%87%DB%8C" title="دستگاه مختصات گرانیگاهی – persiska" lang="fa" hreflang="fa" data-title="دستگاه مختصات گرانیگاهی" data-language-autonym="فارسی" data-language-local-name="persiska" 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_barycentriques" title="Coordonnées barycentriques – franska" lang="fr" hreflang="fr" data-title="Coordonnées barycentriques" data-language-autonym="Français" data-language-local-name="franska" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Coordenadas_baric%C3%A9ntricas" title="Coordenadas baricéntricas – galiciska" lang="gl" hreflang="gl" data-title="Coordenadas baricéntricas" data-language-autonym="Galego" data-language-local-name="galiciska" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Coordinate_baricentriche" title="Coordinate baricentriche – italienska" lang="it" hreflang="it" data-title="Coordinate baricentriche" data-language-autonym="Italiano" data-language-local-name="italienska" 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%91%D7%A8%D7%99%D7%A6%D7%A0%D7%98%D7%A8%D7%99%D7%95%D7%AA" title="קואורדינטות בריצנטריות – hebreiska" lang="he" hreflang="he" data-title="קואורדינטות בריצנטריות" data-language-autonym="עברית" data-language-local-name="hebreiska" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Barycentrische_co%C3%B6rdinaten" title="Barycentrische coördinaten – nederländska" lang="nl" hreflang="nl" data-title="Barycentrische coördinaten" data-language-autonym="Nederlands" data-language-local-name="nederländska" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Barysentriske_koordinater" title="Barysentriske koordinater – norskt bokmål" lang="nb" hreflang="nb" data-title="Barysentriske koordinater" data-language-autonym="Norsk bokmål" data-language-local-name="norskt bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wsp%C3%B3%C5%82rz%C4%99dne_barycentryczne_(matematyka)" title="Współrzędne barycentryczne (matematyka) – polska" lang="pl" hreflang="pl" data-title="Współrzędne barycentryczne (matematyka)" data-language-autonym="Polski" data-language-local-name="polska" 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_baric%C3%AAntricas" title="Coordenadas baricêntricas – portugisiska" lang="pt" hreflang="pt" data-title="Coordenadas baricêntricas" data-language-autonym="Português" data-language-local-name="portugisiska" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B0%D1%80%D0%B8%D1%86%D0%B5%D0%BD%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D1%8B" title="Барицентрические координаты – ryska" lang="ru" hreflang="ru" data-title="Барицентрические координаты" data-language-autonym="Русский" data-language-local-name="ryska" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Te%C5%BEi%C5%A1%C4%8Dni_koordinatni_sistem" title="Težiščni koordinatni sistem – slovenska" lang="sl" hreflang="sl" data-title="Težiščni koordinatni sistem" data-language-autonym="Slovenščina" data-language-local-name="slovenska" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Barysentrinen_koordinaatti_(geometria)" title="Barysentrinen koordinaatti (geometria) – finska" lang="fi" hreflang="fi" data-title="Barysentrinen koordinaatti (geometria)" data-language-autonym="Suomi" data-language-local-name="finska" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D0%B0%D1%80%D0%B8%D1%86%D0%B5%D0%BD%D1%82%D1%80%D0%B8%D1%87%D0%BD%D1%96_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B8" title="Барицентричні координати – ukrainska" lang="uk" hreflang="uk" data-title="Барицентричні координати" data-language-autonym="Українська" data-language-local-name="ukrainska" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%E1%BB%8Da_%C4%91%E1%BB%99_t%E1%BB%89_c%E1%BB%B1" title="Tọa độ tỉ cự – vietnamesiska" lang="vi" hreflang="vi" data-title="Tọa độ tỉ cự" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesiska" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%87%8D%E5%BF%83%E5%BA%A7%E6%A8%99" title="重心座標 – kantonesiska" lang="yue" hreflang="yue" data-title="重心座標" data-language-autonym="粵語" data-language-local-name="kantonesiska" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%87%8D%E5%BF%83%E5%9D%90%E6%A0%87" title="重心坐标 – kinesiska" lang="zh" hreflang="zh" data-title="重心坐标" data-language-autonym="中文" data-language-local-name="kinesiska" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q738422#sitelinks-wikipedia" title="Redigera interwikilänkar" class="wbc-editpage">Redigera länkar</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namnrymder"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Barycentriska_koordinater" title="Visa innehållssidan [c]" accesskey="c"><span>Artikel</span></a></li><li id="ca-talk" class="new vector-tab-noicon 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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">flytta till sidofältet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">dölj</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">Från Wikipedia</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="sv" dir="ltr"><p>Inom <a href="/wiki/Geometri" title="Geometri">geometri</a> betecknar <b>barycentriska koordinater</b> (från grekiska βαρύς, <i>barys</i>, "tung" och κέντρον, <i>kentron</i>, "centrum") en uppsättning av <i>n+1</i> tal, vilka anger en punkts läge i förhållande till en <i>n</i>-dimensionell <a href="/wiki/Simplex" title="Simplex">simplex</a> (<a href="/wiki/Str%C3%A4cka" title="Sträcka">sträcka</a>, <a href="/wiki/Triangel" title="Triangel">triangel</a>, <a href="/wiki/Tetraeder" title="Tetraeder">tetraeder</a>, etcetera) i det <i>n</i>-dimensionella rummet genom att ange relativa vikter som, om de placeras i hörnen på denna simplex, gör punkten till simplexens geometriska tyngdpunkt. I allmänhet avses läget av en punkt i planet i förhållande till en triangel.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-reference-link-bracket">[</span>1<span class="cite-reference-link-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-reference-link-bracket">[</span>2<span class="cite-reference-link-bracket">]</span></a></sup> De skall inte förväxlas med begreppet "barycentrum" som används inom <a href="/wiki/Astronomi" title="Astronomi">astronomi</a> för att ange den gemensamma tyngdpunkten för en uppsättning himlakroppar (även om begreppet är närbesläktat). Barycentriska koordinater infördes av <a href="/wiki/August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">August Ferdinand Möbius</a> 1827 i <i>Der Barycentrische Calcul</i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-reference-link-bracket">[</span>3<span class="cite-reference-link-bracket">]</span></a></sup> </p><p>Barycentriska koordinater skrivs vanligtvis separerade av kolon (exempelvis <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 \alpha :\beta :\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> <mi>β</mi> <mo>:</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :\beta :\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdeb88b91752ebc42398e577d5455edc63aab124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.956ex; height:2.676ex;" alt="{\displaystyle \alpha :\beta :\gamma }"></span> för en punkt i planet i förhållande till en triangel i samma plan). </p><p>Om alla koordinaterna är större än noll ligger punkten innanför simplexens begränsningar och är en eller flera koordinater noll ligger punkten på begränsningarna. Alla koordinater kan inte vara noll. Är någon koordinat negativ ligger punkten utanför simplexen (det motsvarar att en "negativ vikt", eller en "lyftkraft", måste placeras i hörnet). Någon koordinat måste ha ett positivt värde. </p><p>De barycentriska koordinaterna är relativa, vilket innebär att endast deras inbördes förhållanden spelar roll: <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 \scriptstyle 2:1:0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>2</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 2:1:0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0f7cea1870f05bc63bd8cac38c11960c5d3bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.381ex; height:1.676ex;" alt="{\displaystyle \scriptstyle 2:1:0}"></span> är detsamma 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 \scriptstyle 4:2:0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>4</mn> <mo>:</mo> <mn>2</mn> <mo>:</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 4:2:0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78266cf9ded76c63a412fb05a07231b129e101c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.381ex; height:1.676ex;" alt="{\displaystyle \scriptstyle 4:2:0}"></span> eller <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 \scriptstyle 100:50:0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>100</mn> <mo>:</mo> <mn>50</mn> <mo>:</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 100:50:0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/201f5e24f0a3373bf4cc85f9b68958d0db060a5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.847ex; height:1.676ex;" alt="{\displaystyle \scriptstyle 100:50:0}"></span>. </p><p>Med <b>absoluta barycentriska koordinater</b> menas att koordinaterna normerats så att deras summa blir lika med ett. För att normera koordinaterna delar man dem med deras summa. Exempelvis om koordinaterna <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 \scriptstyle 2:1:0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>2</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 2:1:0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0f7cea1870f05bc63bd8cac38c11960c5d3bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.381ex; height:1.676ex;" alt="{\displaystyle \scriptstyle 2:1:0}"></span> divideras med summan av dem (<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 \scriptstyle 2+1+0=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mn>3</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 2+1+0=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26d870c2de3fc90e2d81f4709b7bd06bcfc63291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.124ex; height:1.676ex;" alt="{\displaystyle \scriptstyle 2+1+0=3}"></span>) får vi de absoluta barycentriska koordinaterna <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 \scriptstyle {\frac {2}{3}}:{\frac {1}{3}}:0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>:</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\frac {2}{3}}:{\frac {1}{3}}:0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9832b4890bbf17c440405e462fe058e62866ed1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.744ex; height:3.176ex;" alt="{\displaystyle \scriptstyle {\frac {2}{3}}:{\frac {1}{3}}:0}"></span>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-reference-link-bracket">[</span>4<span class="cite-reference-link-bracket">]</span></a></sup> </p><p>Inom <a href="/wiki/Astronomi" title="Astronomi">astronomi</a> används termen barycentriskt koordinatsystem för att ange ett koordinatsystem (<a href="/wiki/Sf%C3%A4riska_koordinater" title="Sfäriska koordinater">sfäriskt</a> eller <a href="/wiki/Kartesiska_koordinater" class="mw-redirect" title="Kartesiska koordinater">kartesiskt</a>) med origo i systemets tyngdpunkt (exempelvis solsystemets tyngdpunkt). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Barycentriska_koordinater_i_en_dimension">Barycentriska koordinater i en dimension</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=1" title="Redigera avsnitt: Barycentriska koordinater i en dimension" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=1" title="Redigera avsnitts källkod: Barycentriska koordinater i en dimension"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:BaryzKoord1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/BaryzKoord1.png/250px-BaryzKoord1.png" decoding="async" width="250" height="63" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/BaryzKoord1.png/375px-BaryzKoord1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/BaryzKoord1.png/500px-BaryzKoord1.png 2x" data-file-width="600" data-file-height="150" /></a><figcaption>Figur 1. Endimensionella barycentriska koordinater är väldigt triviala, men mönstret följer med när antalet dimensioner ökar.</figcaption></figure> <p>Endimensionella barycentriska koordinater beskriver läget av en punkt på en linje i förhållande till en given sträcka på linjen. Låt oss kalla sträckans ändpunkter för <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> och <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/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> som i figur 1. De barycentriska koordinaterna anger då hur mycket massa vi skall placera i (eller hur mycket kraft vi skall applicera på) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> respektive <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/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> för att tyngdpunkten skall befinna sig i <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> relativt sett. Placerar vi all massa i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> ligger tyngdpunkten i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> och <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> ligger alltså i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> och har, exempelvis, koordinaterna <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 1:0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>:</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1:0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd598e1b4bbd66a7f2a56c7994c2ee61a10bd02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.262ex; height:2.176ex;" alt="{\displaystyle 1:0}"></span> (och motsvarande för <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/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> såklart). För att ange de barycentriska koordinaterna för <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> skall vi alltså beräkna två krafter applicerade i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> respektive <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/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> (vilket ger oss koordinaterna <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_{A}:F_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{A}:F_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f27132cf9d4540b0ddd43dda6a0ec2afc6e1b51b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.871ex; height:2.509ex;" alt="{\displaystyle F_{A}:F_{B}}"></span>) som ger ett motverkande <a href="/wiki/Vridmoment" title="Vridmoment">vridmoment</a> i förhållande till <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>, det vill säga att <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_{A}\cdot {\vec {AP}}=-F_{B}\cdot {\vec {BP}}=F_{B}\cdot {\vec {PB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>P</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>P</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{A}\cdot {\vec {AP}}=-F_{B}\cdot {\vec {BP}}=F_{B}\cdot {\vec {PB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e579b5202809c8c4bd37ed533089354cd9e34e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.458ex; height:4.009ex;" alt="{\displaystyle F_{A}\cdot {\vec {AP}}=-F_{B}\cdot {\vec {BP}}=F_{B}\cdot {\vec {PB}}}"></span>, vilket ger oss koordinaterna <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_{A}:F_{A}\cdot {\frac {\vec {AP}}{\vec {PB}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>P</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{A}:F_{A}\cdot {\frac {\vec {AP}}{\vec {PB}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1785aacdd8404e6e893974867d5cf76d815e4f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:13.881ex; height:8.343ex;" alt="{\displaystyle F_{A}:F_{A}\cdot {\frac {\vec {AP}}{\vec {PB}}}}"></span>. Eftersom endast koordinaternas relativa värden är av intresse kan vi multiplicera dem 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 {\frac {\vec {PB}}{F_{A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\vec {PB}}{F_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d447126726bdf076b3dfd07fd5d9c08a2949549" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.346ex; height:7.176ex;" alt="{\displaystyle {\frac {\vec {PB}}{F_{A}}}}"></span>, vilket ger oss de likvärdiga koordinaterna <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 {\vec {PB}}:{\vec {AP}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>P</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {PB}}:{\vec {AP}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f137bed8854bb3d646931898d0197907ea6de99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.935ex; height:3.676ex;" alt="{\displaystyle {\vec {PB}}:{\vec {AP}}}"></span>. Dessa kan "normeras" genom att dividera dem 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 {\vec {AP}}+{\vec {PB}}={\vec {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>P</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {AP}}+{\vec {PB}}={\vec {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e40fbe86f307939245650f9d39b200d9455198" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.444ex; height:3.843ex;" alt="{\displaystyle {\vec {AP}}+{\vec {PB}}={\vec {AB}}}"></span> varvid deras summa blir lika med ett. Vi ser att respektive koordinat är proportionell mot det "riktade" avståndet från den andra ändpunkten. I det fall man anger de fakiska avstånden <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 {\vec {PB}}:{\vec {AP}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>P</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {PB}}:{\vec {AP}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f137bed8854bb3d646931898d0197907ea6de99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.935ex; height:3.676ex;" alt="{\displaystyle {\vec {PB}}:{\vec {AP}}}"></span> talar man om de <b>homogena barycentriska koordinaterna</b> med avseende på sträckan <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 {\overline {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f91c39c977790f3cc4e768d5aad89bb1696110" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.622ex; height:3.009ex;" alt="{\displaystyle {\overline {AB}}}"></span>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-reference-link-bracket">[</span>5<span class="cite-reference-link-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Barycentriska_koordinater_i_två_dimensioner"><span id="Barycentriska_koordinater_i_tv.C3.A5_dimensioner"></span>Barycentriska koordinater i två dimensioner</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=2" title="Redigera avsnitt: Barycentriska koordinater i två dimensioner" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=2" title="Redigera avsnitts källkod: Barycentriska koordinater i två dimensioner"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Geovanni_Ceva_1.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Geovanni_Ceva_1.JPG/250px-Geovanni_Ceva_1.JPG" decoding="async" width="250" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/2c/Geovanni_Ceva_1.JPG 1.5x" data-file-width="331" data-file-height="281" /></a><figcaption>Figur 2.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:TriangleBarycentricCoordinates.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/TriangleBarycentricCoordinates.svg/250px-TriangleBarycentricCoordinates.svg.png" decoding="async" width="250" height="374" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/TriangleBarycentricCoordinates.svg/375px-TriangleBarycentricCoordinates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/TriangleBarycentricCoordinates.svg/500px-TriangleBarycentricCoordinates.svg.png 2x" data-file-width="1344" data-file-height="2012" /></a><figcaption>Absoluta barycentriska koordinater för vissa punkter i en liksidig respektive rätvinklig triangel.</figcaption></figure> <p>Tvådimensionella barycentriska koordinater beskriver läget av en 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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> i planet i förhållande till en triangel <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 \triangle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821677f03b63c3c2e448dffc2ae9c8eea31d9d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.339ex; height:2.176ex;" alt="{\displaystyle \triangle ABC}"></span> i samma plan (se figur 2). Genom att placera tre "vikter" i de tre triangelhörnen (eller applicera tre krafter på hörnen) skall vi "balansera" triangeln i <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>. </p><p>Vi börjar med att placera all vikt <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_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e81f21959b4900c0378dfd12ffe739823c6da342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.961ex; height:2.509ex;" alt="{\displaystyle F_{P}}"></span> i <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>. Därefter balanserar vi vikten längs linjen <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 {\overline {AL}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>L</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AL}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d131807a282d91eea1374fe6d573b97d46996dc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.441ex; height:3.009ex;" alt="{\displaystyle {\overline {AL}}}"></span> (i enlighet med resonemanget för endimensionella koordinater) så att vi placerar <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_{A}=F_{P}\cdot {\frac {\vec {PL}}{\vec {AL}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>L</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>L</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{A}=F_{P}\cdot {\frac {\vec {PL}}{\vec {AL}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/876ff91d510867adddfeaab5bd94e0f4d57af8eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:14.862ex; height:8.343ex;" alt="{\displaystyle F_{A}=F_{P}\cdot {\frac {\vec {PL}}{\vec {AL}}}}"></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> och <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_{L}=F_{P}\cdot {\frac {\vec {AP}}{\vec {AL}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>P</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>L</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{L}=F_{P}\cdot {\frac {\vec {AP}}{\vec {AL}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6569bc04e8a5eed4d905ad02c9a481d2f41efa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:14.909ex; height:8.509ex;" alt="{\displaystyle F_{L}=F_{P}\cdot {\frac {\vec {AP}}{\vec {AL}}}}"></span> i <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>, varefter vi balanserar vikten i <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> längs <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 {\overline {BC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/650fedad268814c3e14377171e917b73a0698ace" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.713ex; height:3.009ex;" alt="{\displaystyle {\overline {BC}}}"></span> genom att flytta <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_{B}=F_{L}\cdot {\frac {\vec {LC}}{\vec {BC}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>L</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{B}=F_{L}\cdot {\frac {\vec {LC}}{\vec {BC}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0183112332d40809dc52ad6161fe0147b6fab402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:14.964ex; height:8.509ex;" alt="{\displaystyle F_{B}=F_{L}\cdot {\frac {\vec {LC}}{\vec {BC}}}}"></span> till <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/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> och <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_{C}=F_{L}\cdot {\frac {\vec {BL}}{\vec {BC}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>L</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{C}=F_{L}\cdot {\frac {\vec {BL}}{\vec {BC}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7889a29599e239fbfafab7e6e494faf66eddfadc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:14.966ex; height:8.343ex;" alt="{\displaystyle F_{C}=F_{L}\cdot {\frac {\vec {BL}}{\vec {BC}}}}"></span> till <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. </p><p>Detta innebär att <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> fortfarande är tyngdpunkt på <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 {\overline {BC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/650fedad268814c3e14377171e917b73a0698ace" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.713ex; height:3.009ex;" alt="{\displaystyle {\overline {BC}}}"></span>, vilket innebär att triangeln fortfarande är balanserad i <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> och att all vikt befinner sig fördelad på <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, <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/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> eller <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. </p><p>Vi noterar av det ovanstående att <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 {F_{B}}{F_{C}}}={\frac {\vec {LC}}{\vec {BL}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>L</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>L</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{B}}{F_{C}}}={\frac {\vec {LC}}{\vec {BL}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26aeb9a5cf9c9715410b3f47687271b1e8bd54fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:11.096ex; height:8.343ex;" alt="{\displaystyle {\frac {F_{B}}{F_{C}}}={\frac {\vec {LC}}{\vec {BL}}}}"></span>. På samma sätt kan vi visa att <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 {F_{A}}{F_{B}}}={\frac {\vec {NB}}{\vec {AN}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>N</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>N</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{A}}{F_{B}}}={\frac {\vec {NB}}{\vec {AN}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58bd6043ee1c3ae8f560e9b9598f26fd9a8c27f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:11.573ex; height:8.343ex;" alt="{\displaystyle {\frac {F_{A}}{F_{B}}}={\frac {\vec {NB}}{\vec {AN}}}}"></span> och <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 {F_{A}}{F_{C}}}={\frac {\vec {MC}}{\vec {AM}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>M</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{A}}{F_{C}}}={\frac {\vec {MC}}{\vec {AM}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf2115e17b91d47f2ddf319e7f5a851106dcc37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:11.955ex; height:8.509ex;" alt="{\displaystyle {\frac {F_{A}}{F_{C}}}={\frac {\vec {MC}}{\vec {AM}}}}"></span>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-reference-link-bracket">[</span>6<span class="cite-reference-link-bracket">]</span></a></sup> </p><p>Detta leder till att <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 \triangle BPC:\triangle APC:\triangle APB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>P</mi> <mi>C</mi> <mo>:</mo> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>C</mi> <mo>:</mo> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle BPC:\triangle APC:\triangle APB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dccabfbe9b7a6c7161ee5d83474e619a8d569c0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:25.855ex; height:2.176ex;" alt="{\displaystyle \triangle BPC:\triangle APC:\triangle APB}"></span> (i det fall arean av en deltriangel ligger helt utanför <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 \triangle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821677f03b63c3c2e448dffc2ae9c8eea31d9d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.339ex; height:2.176ex;" alt="{\displaystyle \triangle ABC}"></span> är dess area negativ.) är barycentiska koordinter för <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>, eftersom </p><p><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 {F_{B}}{F_{C}}}={\frac {\vec {LC}}{\vec {BL}}}={\frac {\triangle CLA}{\triangle BLA}}={\frac {\triangle CLP}{\triangle BLP}}={\frac {\triangle CLA-\triangle CLP}{\triangle BLA-\triangle BLP}}={\frac {\triangle APC}{\triangle APB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>L</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>L</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>C</mi> <mi>L</mi> <mi>A</mi> </mrow> <mrow> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>L</mi> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>C</mi> <mi>L</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>L</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>C</mi> <mi>L</mi> <mi>A</mi> <mo>−</mo> <mi mathvariant="normal">△</mi> <mi>C</mi> <mi>L</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>L</mi> <mi>A</mi> <mo>−</mo> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>L</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>C</mi> </mrow> <mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>B</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{B}}{F_{C}}}={\frac {\vec {LC}}{\vec {BL}}}={\frac {\triangle CLA}{\triangle BLA}}={\frac {\triangle CLP}{\triangle BLP}}={\frac {\triangle CLA-\triangle CLP}{\triangle BLA-\triangle BLP}}={\frac {\triangle APC}{\triangle APB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/355aa80042db6c546544845175549b992e543879" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:65.633ex; height:8.343ex;" alt="{\displaystyle {\frac {F_{B}}{F_{C}}}={\frac {\vec {LC}}{\vec {BL}}}={\frac {\triangle CLA}{\triangle BLA}}={\frac {\triangle CLP}{\triangle BLP}}={\frac {\triangle CLA-\triangle CLP}{\triangle BLA-\triangle BLP}}={\frac {\triangle APC}{\triangle APB}}}"></span> </p><p>och på samma sätt är <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 {F_{A}}{F_{B}}}={\frac {\triangle BPC}{\triangle APC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>P</mi> <mi>C</mi> </mrow> <mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>C</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{A}}{F_{B}}}={\frac {\triangle BPC}{\triangle APC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80cbb3ae638f923ea7706fb384e2ed3a19778b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.087ex; height:5.676ex;" alt="{\displaystyle {\frac {F_{A}}{F_{B}}}={\frac {\triangle BPC}{\triangle APC}}}"></span> och <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 {F_{A}}{F_{C}}}={\frac {\triangle BPC}{\triangle APB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>P</mi> <mi>C</mi> </mrow> <mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>B</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{A}}{F_{C}}}={\frac {\triangle BPC}{\triangle APB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57267c0192ed7170eb453c78f9bbeb6c197b88e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.088ex; height:5.843ex;" alt="{\displaystyle {\frac {F_{A}}{F_{C}}}={\frac {\triangle BPC}{\triangle APB}}}"></span>. </p><p>Detta ger oss att </p><p><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_{A}:F_{B}:F_{C}=F_{A}:F_{A}\cdot {\frac {\triangle APC}{\triangle BPC}}:F_{A}\cdot {\frac {\triangle APB}{\triangle BPC}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>C</mi> </mrow> <mrow> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>P</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>B</mi> </mrow> <mrow> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>P</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{A}:F_{B}:F_{C}=F_{A}:F_{A}\cdot {\frac {\triangle APC}{\triangle BPC}}:F_{A}\cdot {\frac {\triangle APB}{\triangle BPC}}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23256cfcf1ed5883d8ff42ea11432deeed1c4842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.802ex; height:5.509ex;" alt="{\displaystyle F_{A}:F_{B}:F_{C}=F_{A}:F_{A}\cdot {\frac {\triangle APC}{\triangle BPC}}:F_{A}\cdot {\frac {\triangle APB}{\triangle BPC}}=}"></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 {\begin{aligned}&={\frac {\triangle BPC}{F_{A}}}\cdot (F_{A}:F_{A}\cdot {\frac {\triangle APC}{\triangle BPC}}:F_{A}\cdot {\frac {\triangle APB}{\triangle BPC}})=\\&=\triangle BPC:\triangle APC:\triangle APB\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 /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>P</mi> <mi>C</mi> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>C</mi> </mrow> <mrow> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>P</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>B</mi> </mrow> <mrow> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>P</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>P</mi> <mi>C</mi> <mo>:</mo> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>C</mi> <mo>:</mo> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>B</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&={\frac {\triangle BPC}{F_{A}}}\cdot (F_{A}:F_{A}\cdot {\frac {\triangle APC}{\triangle BPC}}:F_{A}\cdot {\frac {\triangle APB}{\triangle BPC}})=\\&=\triangle BPC:\triangle APC:\triangle APB\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0a7983ef2f6f22218ba5e4203b3f50ca141ed0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:50.436ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}&={\frac {\triangle BPC}{F_{A}}}\cdot (F_{A}:F_{A}\cdot {\frac {\triangle APC}{\triangle BPC}}:F_{A}\cdot {\frac {\triangle APB}{\triangle BPC}})=\\&=\triangle BPC:\triangle APC:\triangle APB\end{aligned}}}"></span>.</dd></dl> <p>Dessa kallas de <b>homogena barycentriska koordinaterna</b> relativt triangeln <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 \triangle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821677f03b63c3c2e448dffc2ae9c8eea31d9d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.339ex; height:2.176ex;" alt="{\displaystyle \triangle ABC}"></span>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-reference-link-bracket">[</span>7<span class="cite-reference-link-bracket">]</span></a></sup><sup id="cite_ref-ref_1_8-0" class="reference"><a href="#cite_note-ref_1-8"><span class="cite-reference-link-bracket">[</span>8<span class="cite-reference-link-bracket">]</span></a></sup> Vi kan normera dem genom att dividera var och en av dem 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 \triangle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821677f03b63c3c2e448dffc2ae9c8eea31d9d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.339ex; height:2.176ex;" alt="{\displaystyle \triangle ABC}"></span> så att deras summa blir lika med ett, vilket ger oss de <i>absoluta</i> barycentriska koordinaterna (vilka även kallas <i>areal coordinates</i>, "areella koordinater", på engelska<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-reference-link-bracket">[</span>9<span class="cite-reference-link-bracket">]</span></a></sup>): </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 {\frac {\triangle BPC}{\triangle ABC}}:{\frac {\triangle APC}{\triangle ABC}}:{\frac {\triangle APB}{\triangle ABC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>P</mi> <mi>C</mi> </mrow> <mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>C</mi> </mrow> <mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>P</mi> <mi>B</mi> </mrow> <mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\triangle BPC}{\triangle ABC}}:{\frac {\triangle APC}{\triangle ABC}}:{\frac {\triangle APB}{\triangle ABC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f109f5a5a213aa1864f2e9142ff6dfc7b427689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.403ex; height:5.509ex;" alt="{\displaystyle {\frac {\triangle BPC}{\triangle ABC}}:{\frac {\triangle APC}{\triangle ABC}}:{\frac {\triangle APB}{\triangle ABC}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Barycentriska_koordinater_i_tre_eller_flera_dimensioner">Barycentriska koordinater i tre eller flera dimensioner</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=3" title="Redigera avsnitt: Barycentriska koordinater i tre eller flera dimensioner" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=3" title="Redigera avsnitts källkod: Barycentriska koordinater i tre eller flera dimensioner"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Tetraedre_Isobarycentre.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Tetraedre_Isobarycentre.png/250px-Tetraedre_Isobarycentre.png" decoding="async" width="250" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Tetraedre_Isobarycentre.png/375px-Tetraedre_Isobarycentre.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Tetraedre_Isobarycentre.png/500px-Tetraedre_Isobarycentre.png 2x" data-file-width="916" data-file-height="800" /></a><figcaption>Figur 3.</figcaption></figure> <p><b>Det tredimensionella fallet:</b> Analogt med det tvådimensionella fallet (i vilket vikterna i triangelhörnen är proportionella mot respektive "motstående deltriangels" area) är vikten i respektive tetraederhörn proportionell mot volymen av dess "motstående deltetraeders" volym. </p> <dl><dd><b>Resonemang</b></dd> <dd>Betrakta tetraedern <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 ABCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412b7d8df4db6ca8093d971320c405598c49c339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.198ex; height:2.176ex;" alt="{\displaystyle ABCD}"></span> och punkten <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> i figur 3. Placera all vikt i <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> och fördela sedan denna vikt på <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> och <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 O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> så att <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> fortfarande är tyngdpunkt. Den tilldelade vikten i <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 O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> fördelas sedan på hörnen i triangeln <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 BCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89e9a67408451aa307e2ef7aa7eb1326750357da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.455ex; height:2.176ex;" alt="{\displaystyle BCD}"></span> (se ovan under <i>två dimensioner</i>) så att denna triangel är balanserad i <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 O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> och varvid tetraedern fortfarande balanserar i <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. För areorna av deltrianglarna <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 BCO}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>C</mi> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BCO}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/988548871a1365a79418298ec983966a7a386f6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.304ex; height:2.176ex;" alt="{\displaystyle BCO}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BDO}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>D</mi> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BDO}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/119b46fb3ef62cc6665dfab358f1baaf098a38da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.462ex; height:2.176ex;" alt="{\displaystyle BDO}"></span> och <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 CDO}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>D</mi> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CDO}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69ac9d88ee3fd3cc422b170b0a1db149733e48b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.464ex; height:2.176ex;" alt="{\displaystyle CDO}"></span> gäller <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 {\frac {BCO}{\mu _{D}}}={\frac {BDO}{\mu _{C}}}={\frac {BCO}{\mu _{B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mi>C</mi> <mi>O</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mi>D</mi> <mi>O</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mi>C</mi> <mi>O</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {BCO}{\mu _{D}}}={\frac {BDO}{\mu _{C}}}={\frac {BCO}{\mu _{B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dedf19cfb85feaf1a5fc65f9d15e0345c1ab7fd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.774ex; height:5.843ex;" alt="{\displaystyle {\frac {BCO}{\mu _{D}}}={\frac {BDO}{\mu _{C}}}={\frac {BCO}{\mu _{B}}}}"></span></dd></dl></dd> <dd>där <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 \mu _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfe6d3f115b8d6cb595119ea9bc7962a11db65a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.034ex; height:2.176ex;" alt="{\displaystyle \mu _{X}}"></span> betecknar vikten i hörnet <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. Dessa deltrianglar utgör baserna för tetraedrar med det fjärde hörnet i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, vilka sålunda har volymer som är proportionella mot bastriangelns yta (och därmed mot vikten i det "motstående triangelhörnet"). På samma sätt är volymerna av de tre tetraedrarna med de tre deltrianglarna som basytor och det fjärde hörnet i <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> proportionella mot vikten i respektive "motstående hörn". Detta ger <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 {\frac {ABCO-GBCO}{\mu _{D}}}={\frac {ABDO-GBDO}{\mu _{C}}}={\frac {ACDO-GCDO}{\mu _{B}}}\Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>O</mi> <mo>−</mo> <mi>G</mi> <mi>B</mi> <mi>C</mi> <mi>O</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>B</mi> <mi>D</mi> <mi>O</mi> <mo>−</mo> <mi>G</mi> <mi>B</mi> <mi>D</mi> <mi>O</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>C</mi> <mi>D</mi> <mi>O</mi> <mo>−</mo> <mi>G</mi> <mi>C</mi> <mi>D</mi> <mi>O</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {ABCO-GBCO}{\mu _{D}}}={\frac {ABDO-GBDO}{\mu _{C}}}={\frac {ACDO-GCDO}{\mu _{B}}}\Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ad1c52b725ce8486136574b2f3388bbf259eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:63.363ex; height:5.843ex;" alt="{\displaystyle {\frac {ABCO-GBCO}{\mu _{D}}}={\frac {ABDO-GBDO}{\mu _{C}}}={\frac {ACDO-GCDO}{\mu _{B}}}\Leftrightarrow }"></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 \Leftrightarrow {\frac {ABCG}{\mu _{D}}}={\frac {ABDG}{\mu _{C}}}={\frac {ACDG}{\mu _{B}}}(={\frac {BCDG}{\mu _{A}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>G</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>B</mi> <mi>D</mi> <mi>G</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>C</mi> <mi>D</mi> <mi>G</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mi>C</mi> <mi>D</mi> <mi>G</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow {\frac {ABCG}{\mu _{D}}}={\frac {ABDG}{\mu _{C}}}={\frac {ACDG}{\mu _{B}}}(={\frac {BCDG}{\mu _{A}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66563bb3a0b09e7f5f90f0af3be38e2fde057ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.673ex; height:5.843ex;" alt="{\displaystyle \Leftrightarrow {\frac {ABCG}{\mu _{D}}}={\frac {ABDG}{\mu _{C}}}={\frac {ACDG}{\mu _{B}}}(={\frac {BCDG}{\mu _{A}}})}"></span>.</dd></dl></dd> <dd>Det vill säga att vikterna i tetraederhörnen är proportionella mot volymerna av de "motstående deltetraedrarna", vilka har den till hörnet motstående tetraedersidan som basyta och det fjärde hörnet i <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> (detta gäller såklart även för <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 \mu _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f50df49aa96673fa1a3853443075a6ac869499" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.867ex; height:2.176ex;" alt="{\displaystyle \mu _{A}}"></span> och <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 BCDG}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>C</mi> <mi>D</mi> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BCDG}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cde274e17f6f0c02be32734047487dba526aa881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.281ex; height:2.176ex;" alt="{\displaystyle BCDG}"></span>, vilket ju visas på samma sätt genom att utgå från ett annat hörn och dess motstående tetraedersida).</dd> <dd>Punkten <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> har alltså de barycentriska koordinaterna <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 \mu _{A}:\mu _{B}:\mu _{C}:\mu _{D}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{A}:\mu _{B}:\mu _{C}:\mu _{D}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afc4fb758adeb8f64a57c03759127feb5c8e364" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.437ex; height:2.176ex;" alt="{\displaystyle \mu _{A}:\mu _{B}:\mu _{C}:\mu _{D}}"></span> i förhållande till tetraedern <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 ABCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412b7d8df4db6ca8093d971320c405598c49c339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.198ex; height:2.176ex;" alt="{\displaystyle ABCD}"></span>. De absoluta barycentriska koordinaterna för <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> är <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 {BCDG}{ABCD}}:{\frac {ACDG}{ABCD}}:{\frac {ABDG}{ABCD}}:{\frac {ABCG}{ABCD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mi>C</mi> <mi>D</mi> <mi>G</mi> </mrow> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>C</mi> <mi>D</mi> <mi>G</mi> </mrow> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>B</mi> <mi>D</mi> <mi>G</mi> </mrow> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>G</mi> </mrow> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {BCDG}{ABCD}}:{\frac {ACDG}{ABCD}}:{\frac {ABDG}{ABCD}}:{\frac {ABCG}{ABCD}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd521f2c8302f8ab4e3a49bd191f89bdbe554f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:38.154ex; height:5.509ex;" alt="{\displaystyle {\frac {BCDG}{ABCD}}:{\frac {ACDG}{ABCD}}:{\frac {ABDG}{ABCD}}:{\frac {ABCG}{ABCD}}}"></span>, eftersom deras summa ju är lika med ett.</dd></dl> <p><b>Fler dimensioner:</b> Ökar vi på antalet dimensioner förfar vi på samma sätt. För fyra dimensioner balanserar vi först mellan ett hörn och skärningspunkten (för linjen genom hörnet och <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>) med den till hörnet motstående tetraedern, varefter vi balanserar denna tetraeders tilldelade vikt på dess fyra hörn enligt ovan. Vi kan fortsätta att öka på med en dimension i taget på samma sätt. Detta innebär att vi för en <i>n</i>-dimensionell simplex <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_{1}A_{2}A_{3}...A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}A_{2}A_{3}...A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51981116defeca46c1613546613639edd09e8dfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.455ex; height:2.509ex;" alt="{\displaystyle A_{1}A_{2}A_{3}...A_{n}}"></span> och en 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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> får </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 {\frac {A_{2}A_{3}...A_{n}G}{\mu _{A_{1}}}}={\frac {A_{1}A_{3}...A_{n}G}{\mu _{A_{2}}}}=...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>G</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>G</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {A_{2}A_{3}...A_{n}G}{\mu _{A_{1}}}}={\frac {A_{1}A_{3}...A_{n}G}{\mu _{A_{2}}}}=...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b94c89f90d4668dc10ea9aa6c78da0a5722bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.908ex; height:6.009ex;" alt="{\displaystyle {\frac {A_{2}A_{3}...A_{n}G}{\mu _{A_{1}}}}={\frac {A_{1}A_{3}...A_{n}G}{\mu _{A_{2}}}}=...}"></span>.</dd> <dd>Punkten <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> har de barycentriska koordinaterna <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 {\mu _{A_{1}}}:{\mu _{A_{2}}}:{\mu _{A_{3}}}:...:{\mu _{A_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> </mrow> <mo>:</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mu _{A_{1}}}:{\mu _{A_{2}}}:{\mu _{A_{3}}}:...:{\mu _{A_{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b7c04cd003ad2baea24fd7a1b42319c963cc81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.486ex; height:2.343ex;" alt="{\displaystyle {\mu _{A_{1}}}:{\mu _{A_{2}}}:{\mu _{A_{3}}}:...:{\mu _{A_{n}}}}"></span> i förhållande till simplexen <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_{1}A_{2}A_{3}...A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}A_{2}A_{3}...A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51981116defeca46c1613546613639edd09e8dfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.455ex; height:2.509ex;" alt="{\displaystyle A_{1}A_{2}A_{3}...A_{n}}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Ortsvektorer_och_kartesiska_koordinater">Ortsvektorer och kartesiska koordinater</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=4" title="Redigera avsnitt: Ortsvektorer och kartesiska koordinater" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=4" title="Redigera avsnitts källkod: Ortsvektorer och kartesiska koordinater"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Barycentricvector.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Barycentricvector.svg/250px-Barycentricvector.svg.png" decoding="async" width="250" height="340" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Barycentricvector.svg/375px-Barycentricvector.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Barycentricvector.svg/500px-Barycentricvector.svg.png 2x" data-file-width="472" data-file-height="641" /></a><figcaption>Figur 4.</figcaption></figure> <p>I figur 4 visas en triangel <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 ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></span> och en 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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> som har de absoluta barycentriska koordinaterna <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 \mu _{A}:\mu _{B}:\mu _{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{A}:\mu _{B}:\mu _{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f119f150f6da4d1a1de19e15ca96bc468785609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.505ex; height:2.176ex;" alt="{\displaystyle \mu _{A}:\mu _{B}:\mu _{C}}"></span>. Vi noterar att, i enlighet med endimensionella koodinater ovan, <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 {\vec {AP_{AB}}}=\mu _{B}\cdot {\vec {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {AP_{AB}}}=\mu _{B}\cdot {\vec {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a58a744637326d03f7d1c6dc7ddd4106a2ce858" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.113ex; height:4.176ex;" alt="{\displaystyle {\vec {AP_{AB}}}=\mu _{B}\cdot {\vec {AB}}}"></span> och <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 {\vec {AP_{AC}}}=\mu _{C}\cdot {\vec {AC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>C</mi> </mrow> </msub> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {AP_{AC}}}=\mu _{C}\cdot {\vec {AC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbd595c226618fb4d9e70b58f49c701d1cdc2ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.119ex; height:4.176ex;" alt="{\displaystyle {\vec {AP_{AC}}}=\mu _{C}\cdot {\vec {AC}}}"></span>. Därför har vi också att <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 {\vec {AP}}={\vec {AP_{AB}}}+{\vec {AP_{AC}}}=\mu _{B}\cdot {\vec {AB}}+\mu _{C}\cdot {\vec {AC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>P</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>C</mi> </mrow> </msub> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {AP}}={\vec {AP_{AB}}}+{\vec {AP_{AC}}}=\mu _{B}\cdot {\vec {AB}}+\mu _{C}\cdot {\vec {AC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46744a14b5410b24134e5395073556c21273869b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.402ex; height:4.176ex;" alt="{\displaystyle {\vec {AP}}={\vec {AP_{AB}}}+{\vec {AP_{AC}}}=\mu _{B}\cdot {\vec {AB}}+\mu _{C}\cdot {\vec {AC}}}"></span>. Om nu triangelhörnen har <a href="/wiki/Ortsvektor" title="Ortsvektor">ortsvektorerna</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {OA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {OA}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8f56dec8d7ea2a7c57e9ff32dae814da42670f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.516ex; height:3.676ex;" alt="{\displaystyle {\vec {OA}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {OB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {OB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c17ebf9f2ea32fac3c1832a9e781cff4334ecab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.537ex; height:3.676ex;" alt="{\displaystyle {\vec {OB}}}"></span> respektive <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 {\vec {OC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {OC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/001a34507625971421c9e70411aa4582e5d6ec26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.54ex; height:3.676ex;" alt="{\displaystyle {\vec {OC}}}"></span> i förhållande till en 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 O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> får vi (i sista ledet utnyttjas att de absoluta koordinaternas summa är ett): </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 {\vec {OP}}={\vec {OA}}+{\vec {AP}}={\vec {OA}}+\mu _{B}\cdot {\vec {AB}}+\mu _{C}\cdot {\vec {AC}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>P</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>P</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {OP}}={\vec {OA}}+{\vec {AP}}={\vec {OA}}+\mu _{B}\cdot {\vec {AB}}+\mu _{C}\cdot {\vec {AC}}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b2a7a82469136acc0d0d3ebdc1f6c085c70029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.35ex; height:4.176ex;" alt="{\displaystyle {\vec {OP}}={\vec {OA}}+{\vec {AP}}={\vec {OA}}+\mu _{B}\cdot {\vec {AB}}+\mu _{C}\cdot {\vec {AC}}=}"></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 {\begin{aligned}&={\vec {OA}}+\mu _{B}\cdot ({\vec {OB}}-{\vec {OA}})+{\vec {OA}}+\mu _{C}\cdot ({\vec {OC}}-{\vec {OA}})=\\&=(1-\mu _{B}-\mu _{C})\cdot {\vec {OA}}+\mu _{B}\cdot {\vec {OB}}+\mu _{C}\cdot {\vec {OC}}=\\&=\mu _{A}\cdot {\vec {OA}}+\mu _{B}\cdot {\vec {OB}}+\mu _{C}\cdot {\vec {OC}}\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 /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&={\vec {OA}}+\mu _{B}\cdot ({\vec {OB}}-{\vec {OA}})+{\vec {OA}}+\mu _{C}\cdot ({\vec {OC}}-{\vec {OA}})=\\&=(1-\mu _{B}-\mu _{C})\cdot {\vec {OA}}+\mu _{B}\cdot {\vec {OB}}+\mu _{C}\cdot {\vec {OC}}=\\&=\mu _{A}\cdot {\vec {OA}}+\mu _{B}\cdot {\vec {OB}}+\mu _{C}\cdot {\vec {OC}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48fb9ec01abedd52ff6f704925d3910805316b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:54.389ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}&={\vec {OA}}+\mu _{B}\cdot ({\vec {OB}}-{\vec {OA}})+{\vec {OA}}+\mu _{C}\cdot ({\vec {OC}}-{\vec {OA}})=\\&=(1-\mu _{B}-\mu _{C})\cdot {\vec {OA}}+\mu _{B}\cdot {\vec {OB}}+\mu _{C}\cdot {\vec {OC}}=\\&=\mu _{A}\cdot {\vec {OA}}+\mu _{B}\cdot {\vec {OB}}+\mu _{C}\cdot {\vec {OC}}\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Från_barycentriska_till_kartesiska_koordinater"><span id="Fr.C3.A5n_barycentriska_till_kartesiska_koordinater"></span>Från barycentriska till kartesiska koordinater</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=5" title="Redigera avsnitt: Från barycentriska till kartesiska koordinater" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=5" title="Redigera avsnitts källkod: Från barycentriska till kartesiska koordinater"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uttrycket för punktens ortsvektor ger direkt att, om triangelhörnen har de <a href="/wiki/Kartesiska_koordinater" class="mw-redirect" title="Kartesiska koordinater">kartesiska koordinaterna</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(x_{A},y_{A})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(x_{A},y_{A})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4d2071ed1b22b9b4fef8a6d7a47c13186bb553f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.083ex; height:2.843ex;" alt="{\displaystyle A=(x_{A},y_{A})}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(x_{B},y_{B})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=(x_{B},y_{B})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d35f5a1ca10c30afbb0286c496a94bb8e5251b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.134ex; height:2.843ex;" alt="{\displaystyle B=(x_{B},y_{B})}"></span> respektive <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 C=(x_{C},y_{C})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=(x_{C},y_{C})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64992045f933f91bd171ab366a0cd16a7dec6f92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.139ex; height:2.843ex;" alt="{\displaystyle C=(x_{C},y_{C})}"></span>, så är de kartesiska koordinaterna för <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></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 P=(x_{P},y_{P})=(\mu _{A}x_{A}+\mu _{B}x_{B}+\mu _{C}x_{C},\ \mu _{A}y_{A}+\mu _{B}y_{B}+\mu _{C}y_{C})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(x_{P},y_{P})=(\mu _{A}x_{A}+\mu _{B}x_{B}+\mu _{C}x_{C},\ \mu _{A}y_{A}+\mu _{B}y_{B}+\mu _{C}y_{C})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb828d014897631a63e348f63c1e46be59d87b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.492ex; height:2.843ex;" alt="{\displaystyle P=(x_{P},y_{P})=(\mu _{A}x_{A}+\mu _{B}x_{B}+\mu _{C}x_{C},\ \mu _{A}y_{A}+\mu _{B}y_{B}+\mu _{C}y_{C})}"></span></dd></dl> <p>Speciellt märker vi att om triangelhörnen har de kartesiska koordinaterna <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 (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b53cc1773694affcc1d4d6c2c778d43156a1206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,0)}"></span> och <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 (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span> så har punkten <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> de absoluta barycentriska koordinaterna <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-x-y:\ x:\ y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>:</mo> <mtext> </mtext> <mi>x</mi> <mo>:</mo> <mtext> </mtext> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-x-y:\ x:\ y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3563996eb6895091c36831d51072d5edb7061614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.849ex; height:2.509ex;" alt="{\displaystyle 1-x-y:\ x:\ y}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Från_kartesiska_till_barycentriska_koordinater"><span id="Fr.C3.A5n_kartesiska_till_barycentriska_koordinater"></span>Från kartesiska till barycentriska koordinater</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=6" title="Redigera avsnitt: Från kartesiska till barycentriska koordinater" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=6" title="Redigera avsnitts källkod: Från kartesiska till barycentriska koordinater"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Om vi skriver om uttrycken för punktens kartesiska koordinater och utnyttjar att <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 \mu _{C}=1-\mu _{A}-\mu _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{C}=1-\mu _{A}-\mu _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b81f1162c9c9c58edbddb4aaa3a116f53696164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.572ex; height:2.676ex;" alt="{\displaystyle \mu _{C}=1-\mu _{A}-\mu _{B}}"></span> får vi två ekvationer med två obekanta: </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 (x_{A}-x_{C})\mu _{A}+(x_{B}-x_{C})\mu _{B}+x_{C}-x_{P}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{A}-x_{C})\mu _{A}+(x_{B}-x_{C})\mu _{B}+x_{C}-x_{P}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f37d43fc58e109c6789bf9476199bfe853139f95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.662ex; height:2.843ex;" alt="{\displaystyle (x_{A}-x_{C})\mu _{A}+(x_{B}-x_{C})\mu _{B}+x_{C}-x_{P}=0}"></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 (y_{A}-y_{C})\mu _{A}+(y_{B}-y_{C})\mu _{B}+y_{C}-y_{P}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (y_{A}-y_{C})\mu _{A}+(y_{B}-y_{C})\mu _{B}+y_{C}-y_{P}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e77c08b7ff4a204d38b56bf665c50ad6f7d348b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.519ex; height:2.843ex;" alt="{\displaystyle (y_{A}-y_{C})\mu _{A}+(y_{B}-y_{C})\mu _{B}+y_{C}-y_{P}=0}"></span></dd></dl> <p>vilka har lösningen </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 \mu _{A}={\frac {(y_{B}-y_{C})(x_{P}-x_{C})+(x_{C}-x_{B})(y_{P}-y_{C})}{(y_{B}-y_{C})(x_{A}-x_{C})+(x_{C}-x_{B})(y_{A}-y_{C})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{A}={\frac {(y_{B}-y_{C})(x_{P}-x_{C})+(x_{C}-x_{B})(y_{P}-y_{C})}{(y_{B}-y_{C})(x_{A}-x_{C})+(x_{C}-x_{B})(y_{A}-y_{C})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/413f68166610a78acae0d6d9ff9822fb81aed40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.933ex; height:6.509ex;" alt="{\displaystyle \mu _{A}={\frac {(y_{B}-y_{C})(x_{P}-x_{C})+(x_{C}-x_{B})(y_{P}-y_{C})}{(y_{B}-y_{C})(x_{A}-x_{C})+(x_{C}-x_{B})(y_{A}-y_{C})}}}"></span> och</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 \mu _{B}={\frac {(y_{C}-y_{A})(x_{P}-x_{C})+(x_{A}-x_{C})(y_{P}-y_{C})}{(y_{B}-y_{C})(x_{A}-x_{C})+(x_{C}-x_{B})(y_{A}-y_{C})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{B}={\frac {(y_{C}-y_{A})(x_{P}-x_{C})+(x_{A}-x_{C})(y_{P}-y_{C})}{(y_{B}-y_{C})(x_{A}-x_{C})+(x_{C}-x_{B})(y_{A}-y_{C})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ed0ab180982cbe5a5dde2d22ada1a2ef4338be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.945ex; height:6.509ex;" alt="{\displaystyle \mu _{B}={\frac {(y_{C}-y_{A})(x_{P}-x_{C})+(x_{A}-x_{C})(y_{P}-y_{C})}{(y_{B}-y_{C})(x_{A}-x_{C})+(x_{C}-x_{B})(y_{A}-y_{C})}}}"></span>.</dd> <dd>(ur vilka vi även får <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 \mu _{C}=1-\mu _{A}-\mu _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{C}=1-\mu _{A}-\mu _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b81f1162c9c9c58edbddb4aaa3a116f53696164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.572ex; height:2.676ex;" alt="{\displaystyle \mu _{C}=1-\mu _{A}-\mu _{B}}"></span>)</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Trilinjära_koordinater"><span id="Trilinj.C3.A4ra_koordinater"></span>Trilinjära koordinater</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=7" title="Redigera avsnitt: Trilinjära koordinater" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=7" title="Redigera avsnitts källkod: Trilinjära koordinater"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Låt <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>, <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> och <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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> beteckna längden av de motstående sidorna till triangelhörnen. En punkt med de <a href="/wiki/Trilinj%C3%A4ra_koordinater" title="Trilinjära koordinater">trilinjära koordinaterna</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x:y:z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>:</mo> <mi>y</mi> <mo>:</mo> <mi>z</mi> </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/4fc1d4f62fa3da71416dec491efd083a8c473361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.448ex; height:2.009ex;" alt="{\displaystyle x:y:z}"></span> har då de barycentriska koordinaterna <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:by:cz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>:</mo> <mi>b</mi> <mi>y</mi> <mo>:</mo> <mi>c</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax:by:cz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0bf3007ae90d3b3ebbe5783f97db9836ff5a6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.682ex; height:2.509ex;" alt="{\displaystyle ax:by:cz}"></span>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-reference-link-bracket">[</span>10<span class="cite-reference-link-bracket">]</span></a></sup> </p><p>Omvänt har därför en punkt med de barycentriska koordinaterna <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 \alpha :\beta :\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> <mi>β</mi> <mo>:</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :\beta :\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdeb88b91752ebc42398e577d5455edc63aab124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.956ex; height:2.676ex;" alt="{\displaystyle \alpha :\beta :\gamma }"></span> de trilinjära koodinaterna <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 {\alpha }{a}}:{\frac {\beta }{b}}:{\frac {\gamma }{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>a</mi> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mi>b</mi> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha }{a}}:{\frac {\beta }{b}}:{\frac {\gamma }{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352355cd7657a92270e54a1305a827fcaf38a81e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.465ex; height:5.509ex;" alt="{\displaystyle {\frac {\alpha }{a}}:{\frac {\beta }{b}}:{\frac {\gamma }{c}}}"></span>. </p><p>De <i>homogena</i> barycentriska koordinaterna <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 \alpha :\beta :\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> <mi>β</mi> <mo>:</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :\beta :\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdeb88b91752ebc42398e577d5455edc63aab124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.956ex; height:2.676ex;" alt="{\displaystyle \alpha :\beta :\gamma }"></span> motsvaras av de <i>exakta</i> trilinjära koordinaterna <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':b':c'={\frac {2\alpha }{a}}:{\frac {2\beta }{b}}:{\frac {2\gamma }{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>:</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>:</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>α</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>β</mi> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>γ</mi> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a':b':c'={\frac {2\alpha }{a}}:{\frac {2\beta }{b}}:{\frac {2\gamma }{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9e777736c7d60c3fa8b736093339f39ee67ac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.213ex; height:5.509ex;" alt="{\displaystyle a':b':c'={\frac {2\alpha }{a}}:{\frac {2\beta }{b}}:{\frac {2\gamma }{c}}}"></span>.<sup id="cite_ref-ref_1_8-1" class="reference"><a href="#cite_note-ref_1-8"><span class="cite-reference-link-bracket">[</span>8<span class="cite-reference-link-bracket">]</span></a></sup> </p> <dl><dd><b>Bevis</b></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 \alpha :\beta :\gamma =|\triangle BCP|:|\triangle ACP|:|\triangle ABP|={\frac {a'a}{2}}:{\frac {b'b}{2}}:{\frac {c'c}{2}}=a'a:b'b:c'c=ax:by:cy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> <mi>β</mi> <mo>:</mo> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>C</mi> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>C</mi> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>a</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mo>′</mo> </msup> <mi>b</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>c</mi> <mo>′</mo> </msup> <mi>c</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>a</mi> <mo>:</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mi>b</mi> <mo>:</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mo>:</mo> <mi>b</mi> <mi>y</mi> <mo>:</mo> <mi>c</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :\beta :\gamma =|\triangle BCP|:|\triangle ACP|:|\triangle ABP|={\frac {a'a}{2}}:{\frac {b'b}{2}}:{\frac {c'c}{2}}=a'a:b'b:c'c=ax:by:cy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b24acb0d8e7497ceab56849dddc9d6079880ff10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:88.138ex; height:5.509ex;" alt="{\displaystyle \alpha :\beta :\gamma =|\triangle BCP|:|\triangle ACP|:|\triangle ABP|={\frac {a'a}{2}}:{\frac {b'b}{2}}:{\frac {c'c}{2}}=a'a:b'b:c'c=ax:by:cy}"></span></dd> <dd>Om <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 \alpha :\beta :\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> <mi>β</mi> <mo>:</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :\beta :\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdeb88b91752ebc42398e577d5455edc63aab124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.956ex; height:2.676ex;" alt="{\displaystyle \alpha :\beta :\gamma }"></span> är <i>homogena</i> är de alltså lika 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 {\frac {a'a}{2}}:{\frac {b'b}{2}}:{\frac {c'c}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>a</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mo>′</mo> </msup> <mi>b</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>c</mi> <mo>′</mo> </msup> <mi>c</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a'a}{2}}:{\frac {b'b}{2}}:{\frac {c'c}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a639e5eb8ffc83a0b7715f7334c13d8bdc2a7e7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.905ex; height:5.509ex;" alt="{\displaystyle {\frac {a'a}{2}}:{\frac {b'b}{2}}:{\frac {c'c}{2}}}"></span>, och omvänt är de <i>exakta</i> trilinjära koordinaterna <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':b':c'={\frac {2\alpha }{a}}:{\frac {2\beta }{b}}:{\frac {2\gamma }{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>:</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>:</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>α</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>β</mi> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>γ</mi> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a':b':c'={\frac {2\alpha }{a}}:{\frac {2\beta }{b}}:{\frac {2\gamma }{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9e777736c7d60c3fa8b736093339f39ee67ac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.213ex; height:5.509ex;" alt="{\displaystyle a':b':c'={\frac {2\alpha }{a}}:{\frac {2\beta }{b}}:{\frac {2\gamma }{c}}}"></span>.</dd></dl> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Barycentriska_koordinater_för_vissa_punkter"><span id="Barycentriska_koordinater_f.C3.B6r_vissa_punkter"></span>Barycentriska koordinater för vissa punkter</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=8" title="Redigera avsnitt: Barycentriska koordinater för vissa punkter" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=8" title="Redigera avsnitts källkod: Barycentriska koordinater för vissa punkter"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Den_geometriska_tyngdpunkten">Den geometriska tyngdpunkten</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=9" title="Redigera avsnitt: Den geometriska tyngdpunkten" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=9" title="Redigera avsnitts källkod: Den geometriska tyngdpunkten"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Triangelns geometriska tyngdpunkt har barycentriska koordinater </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 1:1:1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1:1:1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f807f34ce5a0799a9c736a216f231aab754a2b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.362ex; height:2.176ex;" alt="{\displaystyle 1:1:1}"></span></dd></dl> <dl><dd><b>Bevis</b></dd> <dd>Beviset följer direkt ur att den geometriska tyngdpunkten är <a href="/wiki/Median_(geometri)" title="Median (geometri)">medianernas</a> skärningspunkt och att <a href="/wiki/Median_(geometri)#Sex_likstora_trianglar" title="Median (geometri)">medianerna delar triangeln i sex likstora trianglar</a>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Den_inskrivna_cirkelns_medelpunkt">Den inskrivna cirkelns medelpunkt</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=10" title="Redigera avsnitt: Den inskrivna cirkelns medelpunkt" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=10" title="Redigera avsnitts källkod: Den inskrivna cirkelns medelpunkt"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Den <a href="/wiki/Inskriven_cirkel" title="Inskriven cirkel">inskrivna cirkelns</a> medelpunkt har barycentriska koordinater som kan skrivas som </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 {\overline {BC}}\ :\ {\overline {AC}}\ :\ {\overline {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}\ :\ {\overline {AC}}\ :\ {\overline {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b8213d6f18c5e2eeaf96e2078222a849275ff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.225ex; height:3.009ex;" alt="{\displaystyle {\overline {BC}}\ :\ {\overline {AC}}\ :\ {\overline {AB}}}"></span></dd></dl> <dl><dd><b>Bevis</b></dd> <dd>Den inskrivna cirkelns medelpunkt <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> har samma avstånd till triangelns sidor, dess radie <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>. Radien är lika med deltrianglarnas höjd och de barycentriska koordinaterna är därför <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 \triangle BCP:\triangle ACP:\triangle ABP={\frac {r\cdot {\overline {BC}}}{2}}:{\frac {r\cdot {\overline {AC}}}{2}}:{\frac {r\cdot {\overline {AB}}}{2}}={\overline {BC}}\ :\ {\overline {AC}}\ :\ {\overline {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>C</mi> <mi>P</mi> <mo>:</mo> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>C</mi> <mi>P</mi> <mo>:</mo> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle BCP:\triangle ACP:\triangle ABP={\frac {r\cdot {\overline {BC}}}{2}}:{\frac {r\cdot {\overline {AC}}}{2}}:{\frac {r\cdot {\overline {AB}}}{2}}={\overline {BC}}\ :\ {\overline {AC}}\ :\ {\overline {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9a35c392b0c1a8a2741285ad3713b3b09bf8cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:74.871ex; height:6.176ex;" alt="{\displaystyle \triangle BCP:\triangle ACP:\triangle ABP={\frac {r\cdot {\overline {BC}}}{2}}:{\frac {r\cdot {\overline {AC}}}{2}}:{\frac {r\cdot {\overline {AB}}}{2}}={\overline {BC}}\ :\ {\overline {AC}}\ :\ {\overline {AB}}}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="De_vidskrivna_cirklarnas_medelpunkter">De vidskrivna cirklarnas medelpunkter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=11" title="Redigera avsnitt: De vidskrivna cirklarnas medelpunkter" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=11" title="Redigera avsnitts källkod: De vidskrivna cirklarnas medelpunkter"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Medelpunkterna för de <a href="/wiki/Vidskriven_cirkel" title="Vidskriven cirkel">vidskrivna cirklarna</a> till <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 {\overline {BC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/650fedad268814c3e14377171e917b73a0698ace" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.713ex; height:3.009ex;" alt="{\displaystyle {\overline {BC}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea69bf8ccc972402f794c261e00b222f396bcce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.692ex; height:3.009ex;" alt="{\displaystyle {\overline {AC}}}"></span> och <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 {\overline {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f91c39c977790f3cc4e768d5aad89bb1696110" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.622ex; height:3.009ex;" alt="{\displaystyle {\overline {AB}}}"></span> har de barycentriska koordinaterna </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 -|{\overline {BC}}|:|{\overline {AC}}|:|{\overline {AB}}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -|{\overline {BC}}|:|{\overline {AC}}|:|{\overline {AB}}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c07bcd000dcd70dc44278a1f754a51fc4d34a63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.591ex; height:3.509ex;" alt="{\displaystyle -|{\overline {BC}}|:|{\overline {AC}}|:|{\overline {AB}}|}"></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 |{\overline {BC}}|:-|{\overline {AC}}|:|{\overline {AB}}|}"> <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"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\overline {BC}}|:-|{\overline {AC}}|:|{\overline {AB}}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5e41b896f52f3603232f22cb361fd2bac5244a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.591ex; height:3.509ex;" alt="{\displaystyle |{\overline {BC}}|:-|{\overline {AC}}|:|{\overline {AB}}|}"></span> respektive</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 |{\overline {BC}}|:|{\overline {AC}}|:-|{\overline {AB}}|}"> <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"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\overline {BC}}|:|{\overline {AC}}|:-|{\overline {AB}}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94ef63f24e04a9138ea7e82e48f58d25e944e11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.591ex; height:3.509ex;" alt="{\displaystyle |{\overline {BC}}|:|{\overline {AC}}|:-|{\overline {AB}}|}"></span>.</dd></dl> <dl><dd><b>Bevis</b></dd> <dd>Eftersom medelpunkten för den vidskrivna cirkeln, med radien <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>, till <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 {\overline {BC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/650fedad268814c3e14377171e917b73a0698ace" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.713ex; height:3.009ex;" alt="{\displaystyle {\overline {BC}}}"></span> har de <a href="/wiki/Trilinj%C3%A4ra_koordinater" title="Trilinjära koordinater">trilinjära koordinaterna</a> <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:r:r=-1:1:1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>r</mi> <mo>:</mo> <mi>r</mi> <mo>:</mo> <mi>r</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -r:r:r=-1:1:1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9e8e4507a807b3991738393c19ce0fd73dbe3b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.097ex; height:2.343ex;" alt="{\displaystyle -r:r:r=-1:1:1}"></span></dd></dl></dd> <dd>har den de barycentriska koordinaterna <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 -|{\overline {BC}}|:|{\overline {AC}}|:|{\overline {AB}}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -|{\overline {BC}}|:|{\overline {AC}}|:|{\overline {AB}}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c07bcd000dcd70dc44278a1f754a51fc4d34a63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.591ex; height:3.509ex;" alt="{\displaystyle -|{\overline {BC}}|:|{\overline {AC}}|:|{\overline {AB}}|}"></span></dd></dl></dd> <dd>i enlighet med <a class="mw-selflink-fragment" href="#Trilinjära_koordinater">avsnittet om trilinjära koordinater</a> ovan. Motsvarande gäller de båda övriga vidskrivna cirklarnas medelpunkter.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Den_omskrivna_cirkelns_medelpunkt">Den omskrivna cirkelns medelpunkt</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=12" title="Redigera avsnitt: Den omskrivna cirkelns medelpunkt" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=12" title="Redigera avsnitts källkod: Den omskrivna cirkelns medelpunkt"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Circumcentre_barycentric_proof.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Circumcentre_barycentric_proof.svg/250px-Circumcentre_barycentric_proof.svg.png" decoding="async" width="250" height="261" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Circumcentre_barycentric_proof.svg/375px-Circumcentre_barycentric_proof.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Circumcentre_barycentric_proof.svg/500px-Circumcentre_barycentric_proof.svg.png 2x" data-file-width="387" data-file-height="404" /></a><figcaption>Figur 5.</figcaption></figure> <p>Den <a href="/wiki/Omskriven_cirkel" title="Omskriven cirkel">omskrivna cirkelns</a> medelpunkt har barycentriska koordinater som kan skrivas som </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 {\overline {BC}}\cdot \cos \angle BAC\ :\ {\overline {AC}}\cdot \cos \angle CBA\ :\ {\overline {AB}}\cdot \cos \angle BCA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>A</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>C</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}\cdot \cos \angle BAC\ :\ {\overline {AC}}\cdot \cos \angle CBA\ :\ {\overline {AB}}\cdot \cos \angle BCA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2506838adc7ccf39bff934d57ac3d37590e6ef9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:53.611ex; height:3.009ex;" alt="{\displaystyle {\overline {BC}}\cdot \cos \angle BAC\ :\ {\overline {AC}}\cdot \cos \angle CBA\ :\ {\overline {AB}}\cdot \cos \angle BCA}"></span></dd></dl> <dl><dd><b>Bevis</b></dd> <dd>Den omskrivna cirkelns medelpunkt <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> har samma avstånd till triangelhörnen, cirkelns radie <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>. Låt <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> vara <a href="/wiki/Fotpunkt" title="Fotpunkt">fotpunkt</a> åt <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> på <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 {\overline {BC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/650fedad268814c3e14377171e917b73a0698ace" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.713ex; height:3.009ex;" alt="{\displaystyle {\overline {BC}}}"></span> (figur 5). Vi har då</dd> <dd> <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 \triangle BCP=\triangle BFP+\triangle CFP=2\cdot \triangle BFP={\overline {BF}}\cdot {\overline {FP}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>C</mi> <mi>P</mi> <mo>=</mo> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>F</mi> <mi>P</mi> <mo>+</mo> <mi mathvariant="normal">△</mi> <mi>C</mi> <mi>F</mi> <mi>P</mi> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>F</mi> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>F</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>F</mi> <mi>P</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle BCP=\triangle BFP+\triangle CFP=2\cdot \triangle BFP={\overline {BF}}\cdot {\overline {FP}}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd9ecac0fbc39843e2cdab50a9da4f850d7877eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:55.948ex; height:3.176ex;" alt="{\displaystyle \triangle BCP=\triangle BFP+\triangle CFP=2\cdot \triangle BFP={\overline {BF}}\cdot {\overline {FP}}=}"></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 \ ={\overline {BF}}\cdot r\cos \angle BPF={\overline {BF}}\cdot r\cos {\frac {\angle BPC}{2}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>F</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>P</mi> <mi>F</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>F</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>P</mi> <mi>C</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ ={\overline {BF}}\cdot r\cos \angle BPF={\overline {BF}}\cdot r\cos {\frac {\angle BPC}{2}}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f645265273fb77b047fb4bcce18c91d1f51674" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.735ex; height:5.343ex;" alt="{\displaystyle \ ={\overline {BF}}\cdot r\cos \angle BPF={\overline {BF}}\cdot r\cos {\frac {\angle BPC}{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 \ ={\overline {BF}}\cdot r\cos \angle BAC={\frac {\overline {BC}}{2}}\cdot r\cos \angle BAC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>F</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ ={\overline {BF}}\cdot r\cos \angle BAC={\frac {\overline {BC}}{2}}\cdot r\cos \angle BAC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e331369311ba3c9f522cbc9f0331e0c1b2c0d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.236ex; height:6.009ex;" alt="{\displaystyle \ ={\overline {BF}}\cdot r\cos \angle BAC={\frac {\overline {BC}}{2}}\cdot r\cos \angle BAC}"></span></dd></dl></dd> <dd>I näst sista ledet utnyttjas <a href="/wiki/Randvinkelsatsen" title="Randvinkelsatsen">randvinkelsatsen</a>. (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74e0f24a49061dcd63874f7d81f395b5f38800f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.176ex;" alt="{\displaystyle BC}"></span> är ju en <a href="/wiki/Korda" title="Korda">korda</a> i den omskrivna cirkeln och vinkeln i medelpunkten är enligt denna sats dubbelt så stor som vinkeln i en punkt på omkretsen.)</dd> <dd></dd> <dd>Samma resonemang för de båda andra triangelsidorna ger oss barycentriska koordinater</dd> <dd> <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 {\frac {\overline {BC}}{2}}\cdot r\cos \angle BAC\ :\ {\frac {\overline {AC}}{2}}\cdot r\cos \angle CBA\ :\ {\frac {\overline {AB}}{2}}\cdot r\cos \angle BCA=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>A</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>C</mi> <mi>A</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\overline {BC}}{2}}\cdot r\cos \angle BAC\ :\ {\frac {\overline {AC}}{2}}\cdot r\cos \angle CBA\ :\ {\frac {\overline {AB}}{2}}\cdot r\cos \angle BCA=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6b8da91fb98389d2688a2136552be2367a1ed15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:62.88ex; height:6.176ex;" alt="{\displaystyle {\frac {\overline {BC}}{2}}\cdot r\cos \angle BAC\ :\ {\frac {\overline {AC}}{2}}\cdot r\cos \angle CBA\ :\ {\frac {\overline {AB}}{2}}\cdot r\cos \angle BCA=}"></span></dd></dl></dd></dl> <dl><dd><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 \ ={\overline {BC}}\cdot \cos \angle BAC\ :\ {\overline {AC}}\cdot \cos \angle CBA\ :\ {\overline {AB}}\cdot \cos \angle BCA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>A</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>C</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ ={\overline {BC}}\cdot \cos \angle BAC\ :\ {\overline {AC}}\cdot \cos \angle CBA\ :\ {\overline {AB}}\cdot \cos \angle BCA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7062154dea5faace902b81ce66df4d01f2cde7ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:57.29ex; height:3.009ex;" alt="{\displaystyle \ ={\overline {BC}}\cdot \cos \angle BAC\ :\ {\overline {AC}}\cdot \cos \angle CBA\ :\ {\overline {AB}}\cdot \cos \angle BCA}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Ortocentrum">Ortocentrum</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=13" title="Redigera avsnitt: Ortocentrum" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=13" title="Redigera avsnitts källkod: Ortocentrum"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Ortocentrum" title="Ortocentrum">Ortocentrum</a> har barycentriska koordinater som kan skrivas som </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 {\frac {\overline {BC}}{\cos \angle BAC}}:{\frac {\overline {AC}}{\cos \angle CBA}}:{\frac {\overline {AB}}{\cos \angle ACB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\overline {BC}}{\cos \angle BAC}}:{\frac {\overline {AC}}{\cos \angle CBA}}:{\frac {\overline {AB}}{\cos \angle ACB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991f5e7b1b79de710c90e6e165fa4a9fc63f9cc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.732ex; height:6.343ex;" alt="{\displaystyle {\frac {\overline {BC}}{\cos \angle BAC}}:{\frac {\overline {AC}}{\cos \angle CBA}}:{\frac {\overline {AB}}{\cos \angle ACB}}}"></span></dd></dl> <p>eller </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 {\overline {BC}}\cdot \cos \angle CBA\cdot \cos \angle ACB:{\overline {AC}}\cdot \cos \angle ACB\cdot \cos \angle BAC:{\overline {BC}}\cdot \cos \angle BAC\cdot \cos \angle CBA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>A</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}\cdot \cos \angle CBA\cdot \cos \angle ACB:{\overline {AC}}\cdot \cos \angle ACB\cdot \cos \angle BAC:{\overline {BC}}\cdot \cos \angle BAC\cdot \cos \angle CBA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11647206b560c4c2e310fa555cecd77a4ee17b50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:87.766ex; height:3.009ex;" alt="{\displaystyle {\overline {BC}}\cdot \cos \angle CBA\cdot \cos \angle ACB:{\overline {AC}}\cdot \cos \angle ACB\cdot \cos \angle BAC:{\overline {BC}}\cdot \cos \angle BAC\cdot \cos \angle CBA}"></span></dd></dl> <p>Det andra uttrycket erhålls genom multiplikation 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 \cos \angle BAC\cos \angle CBA\cos \angle ACB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \angle BAC\cos \angle CBA\cos \angle ACB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8f845896ed9cf5956f15621d45a2fa417199dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:32.123ex; height:2.176ex;" alt="{\displaystyle \cos \angle BAC\cos \angle CBA\cos \angle ACB}"></span> </p> <dl><dd><b>Bevis</b></dd> <dd>Vi utnyttjar förhållandet mellan barycentriska koordinater och trilinjära koordinater (se <a class="mw-selflink-fragment" href="#Trilinjära_koordinater">avsnittet ovan</a>).</dd> <dd>Den omskrivna cirkelns medelpunkt har barycentriska koordinater (se ovan) <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 \ {\overline {BC}}\cdot \cos \angle BAC\ :\ {\overline {AC}}\cdot \cos \angle CBA\ :\ {\overline {AB}}\cdot \cos \angle BCA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>A</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>C</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\overline {BC}}\cdot \cos \angle BAC\ :\ {\overline {AC}}\cdot \cos \angle CBA\ :\ {\overline {AB}}\cdot \cos \angle BCA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed024ca4b48bec9cb4ed5067fffae27ad491994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:54.192ex; height:3.009ex;" alt="{\displaystyle \ {\overline {BC}}\cdot \cos \angle BAC\ :\ {\overline {AC}}\cdot \cos \angle CBA\ :\ {\overline {AB}}\cdot \cos \angle BCA}"></span></dd></dl></dd> <dd>och därför trilinjära koordinater <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 \angle BAC:\cos \angle CBA:\cos \angle ACB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> <mo>:</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>A</mi> <mo>:</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \angle BAC:\cos \angle CBA:\cos \angle ACB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67c885a1697b95eab6946a75fb04e44211fd399a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:35.223ex; height:2.176ex;" alt="{\displaystyle \cos \angle BAC:\cos \angle CBA:\cos \angle ACB}"></span></dd></dl></dd> <dd>Ortocentrum är <a href="/wiki/Isogonalkonjugat" title="Isogonalkonjugat">isogonalkonjugat</a> till den omskrivna cirkelns medelpunkt och dess trilinjära koordinater är därför <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 {\frac {1}{\cos \angle BAC}}:{\frac {1}{\cos \angle CBA}}:{\frac {1}{\cos \angle ACB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\cos \angle BAC}}:{\frac {1}{\cos \angle CBA}}:{\frac {1}{\cos \angle ACB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d389f24225b7d70bf9154e857d9005b33bad4dfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.732ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{\cos \angle BAC}}:{\frac {1}{\cos \angle CBA}}:{\frac {1}{\cos \angle ACB}}}"></span></dd></dl></dd> <dd>vilka motsvarar de barycentriska koordinaterna <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 {\frac {\overline {BC}}{\cos \angle BAC}}:{\frac {\overline {AC}}{\cos \angle CBA}}:{\frac {\overline {AB}}{\cos \angle ACB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\overline {BC}}{\cos \angle BAC}}:{\frac {\overline {AC}}{\cos \angle CBA}}:{\frac {\overline {AB}}{\cos \angle ACB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991f5e7b1b79de710c90e6e165fa4a9fc63f9cc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.732ex; height:6.343ex;" alt="{\displaystyle {\frac {\overline {BC}}{\cos \angle BAC}}:{\frac {\overline {AC}}{\cos \angle CBA}}:{\frac {\overline {AB}}{\cos \angle ACB}}}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Symmedianpunkten">Symmedianpunkten</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=14" title="Redigera avsnitt: Symmedianpunkten" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=14" title="Redigera avsnitts källkod: Symmedianpunkten"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Symmedian" title="Symmedian">Symmedianpunkten</a> har de barycentriska koordinaterna </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 {\overline {BC}}^{2}:{\overline {AC}}^{2}:{\overline {AB}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}^{2}:{\overline {AC}}^{2}:{\overline {AB}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb05cc5726c51be4cc2934526b84959c3b4ad59a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.065ex; height:3.509ex;" alt="{\displaystyle {\overline {BC}}^{2}:{\overline {AC}}^{2}:{\overline {AB}}^{2}}"></span></dd></dl> <dl><dd><b>Bevis</b></dd> <dd>För beviset utnyttjar vi <a class="mw-selflink-fragment" href="#Trilinjära_koordinater">sambandet</a> mellan barycentriska och <a href="/wiki/Trilinj%C3%A4ra_koordinater" title="Trilinjära koordinater">trilinjära koordinater</a> och att symmedianpunkten är <a href="/wiki/Isogonalkonjugat" title="Isogonalkonjugat">isogonalkonjugat</a> till den geometriska tyngdpunkten.</dd> <dd>Den geometriska tyngdpunkten har barycentriska koordinater <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 1:1:1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1:1:1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f807f34ce5a0799a9c736a216f231aab754a2b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.362ex; height:2.176ex;" alt="{\displaystyle 1:1:1}"></span></dd></dl></dd> <dd>vilket motsvarar de trilinjära koordinaterna <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 {\frac {1}{\overline {BC}}}:{\frac {1}{\overline {AC}}}:{\frac {1}{\overline {AB}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\overline {BC}}}:{\frac {1}{\overline {AC}}}:{\frac {1}{\overline {AB}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50df3905ed7d7aac49d220e3ae3cfb2b6b7a4f95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.41ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\overline {BC}}}:{\frac {1}{\overline {AC}}}:{\frac {1}{\overline {AB}}}}"></span></dd></dl></dd> <dd>Symmedianpunkten har, eftersom den är isogonalkonjugat till den geometriska tyngdpunkten, de trilinjära koordinaterna <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 {\overline {BC}}:{\overline {AC}}:{\overline {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}:{\overline {AC}}:{\overline {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5baf2a502c4e305eda64c2332a2e7b7bbfef577d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.902ex; height:3.009ex;" alt="{\displaystyle {\overline {BC}}:{\overline {AC}}:{\overline {AB}}}"></span></dd></dl></dd> <dd>vilka motsvarar de barycentriska koordinaterna <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 {\overline {BC}}^{2}:{\overline {AC}}^{2}:{\overline {AB}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}^{2}:{\overline {AC}}^{2}:{\overline {AB}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb05cc5726c51be4cc2934526b84959c3b4ad59a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.065ex; height:3.509ex;" alt="{\displaystyle {\overline {BC}}^{2}:{\overline {AC}}^{2}:{\overline {AB}}^{2}}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Barycentrisk_interpolation">Barycentrisk interpolation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=15" title="Redigera avsnitt: Barycentrisk interpolation" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=15" title="Redigera avsnitts källkod: Barycentrisk interpolation"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Barycentric_RGB.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Barycentric_RGB.png/250px-Barycentric_RGB.png" decoding="async" width="250" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Barycentric_RGB.png/375px-Barycentric_RGB.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Barycentric_RGB.png/500px-Barycentric_RGB.png 2x" data-file-width="2048" data-file-height="1774" /></a><figcaption>En <a href="/w/index.php?title=Gouraud-skuggning&action=edit&redlink=1" class="new" title="Gouraud-skuggning [inte skriven än]">Gouraud-skuggad</a> triangel <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 \scriptstyle \triangle RGB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">△</mi> <mi>R</mi> <mi>G</mi> <mi>B</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \triangle RGB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/315fbd48f5fdeeb352aba89f170e32d4db63849c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.247ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \triangle RGB}"></span> i vilken punkternas färg (<a href="/wiki/RGB" title="RGB">Röd, Grön, Blå</a>) är <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 \scriptstyle f_{P}=\mu _{R}\cdot f_{R}+\mu _{G}\cdot f_{G}+\mu _{B}\cdot f_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle f_{P}=\mu _{R}\cdot f_{R}+\mu _{G}\cdot f_{G}+\mu _{B}\cdot f_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c52622dcb498d105e2344a993423d12bec7b256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.703ex; height:2.009ex;" alt="{\displaystyle \scriptstyle f_{P}=\mu _{R}\cdot f_{R}+\mu _{G}\cdot f_{G}+\mu _{B}\cdot f_{B}}"></span> (där <i><b>f<sub>R</sub></b></i> = 100% röd, 0% grön, 0% blå, etcetera).</figcaption></figure> <p>För en funktion av två variabler, <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,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29473ed0c4e838ac9dbe074535e507166c0e9101" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.843ex;" alt="{\displaystyle f(x,y)}"></span>, med de kända värdena <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_{A}=f(x_{A},y_{A})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{A}=f(x_{A},y_{A})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51dcbed0ac67f1d20842325731bad98fbaaa1b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.223ex; height:2.843ex;" alt="{\displaystyle f_{A}=f(x_{A},y_{A})}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{B}=f(x_{B},y_{B})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{B}=f(x_{B},y_{B})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bb5d7cf98dfa223c3ebda313863afdedd54814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.267ex; height:2.843ex;" alt="{\displaystyle f_{B}=f(x_{B},y_{B})}"></span> och <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_{C}=f(x_{C},y_{C})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{C}=f(x_{C},y_{C})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a650126b5981bc76d556c78badc2538c07994598" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.272ex; height:2.843ex;" alt="{\displaystyle f_{C}=f(x_{C},y_{C})}"></span> för hörnen i triangeln <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 \triangle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821677f03b63c3c2e448dffc2ae9c8eea31d9d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.339ex; height:2.176ex;" alt="{\displaystyle \triangle ABC}"></span> kan en linjär <a href="/wiki/Interpolation" title="Interpolation">interpolation</a> av värdet i en 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 P=(x_{P},y_{P})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(x_{P},y_{P})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9495647400fbd7b2198cf430d72fa11bd0ee8e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.089ex; height:2.843ex;" alt="{\displaystyle P=(x_{P},y_{P})}"></span> med de <i>absoluta</i> barycentriska koordinaterna <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 \mu _{A}:\mu _{B}:\mu _{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{A}:\mu _{B}:\mu _{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f119f150f6da4d1a1de19e15ca96bc468785609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.505ex; height:2.176ex;" alt="{\displaystyle \mu _{A}:\mu _{B}:\mu _{C}}"></span> göras enligt <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_{P}=f(x_{P},y_{P})\approx \mu _{A}f_{A}+\mu _{B}f_{B}+\mu _{C}f_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≈</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{P}=f(x_{P},y_{P})\approx \mu _{A}f_{A}+\mu _{B}f_{B}+\mu _{C}f_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb5d1ec8d5b31a0dda01cad435a2634b0aac2f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.481ex; height:2.843ex;" alt="{\displaystyle f_{P}=f(x_{P},y_{P})\approx \mu _{A}f_{A}+\mu _{B}f_{B}+\mu _{C}f_{C}}"></span>. Barycentrisk interpolation kan enkelt utsträckas till fler dimenensioner.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-reference-link-bracket">[</span>11<span class="cite-reference-link-bracket">]</span></a></sup> Genom att skapa ett nät av trianglar (ett så kallat "mesh"), eller simplexar av högre dimension, kan beräkningar göras för större områden (ett exempel är beräkningar av <a href="/wiki/Isolinje" class="mw-redirect" title="Isolinje">isolinjer</a> eller värden för olika platser i ett nät av väderstationer). Mer förfinade interpolationer kan göras med polynomapproximationer i stället för linjära sådana.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-reference-link-bracket">[</span>12<span class="cite-reference-link-bracket">]</span></a></sup> </p><p>Barycentrisk interpolation och generaliseringar av denna till godtyckliga polygoner och polyedrar används inom flera områden, exempelvis <a href="/wiki/Finita_elementmetoden" title="Finita elementmetoden">finita elementmetoden</a> (FEM), och speciellt noterbart är applikationer inom datorgrafik för exempelvis skuggning och animation.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-reference-link-bracket">[</span>13<span class="cite-reference-link-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Se_även"><span id="Se_.C3.A4ven"></span>Se även</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=16" title="Redigera avsnitt: Se även" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=16" title="Redigera avsnitts källkod: Se även"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Homogena_koordinater" title="Homogena koordinater">Homogena koordinater</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referenser">Referenser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Barycentriska_koordinater&veaction=edit&section=17" title="Redigera avsnitt: Referenser" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Barycentriska_koordinater&action=edit&section=17" title="Redigera avsnitts källkod: Referenser"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <span class="reference-text">Clifford A. Pickover, 2015, <i><a rel="nofollow" class="external text" href="https://books.google.se/books?id=q80-CwAAQBAJ&pg=PA222">250 milstolpar i matematikens historia från Pythagoras till 57:e dimensionen</a></i>, sid. 222. <a href="/wiki/Special:Bokk%C3%A4llor/9789176172629" title="Special:Bokkällor/9789176172629">ISBN 9789176172629</a></span> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Barycentric_Coordinates"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, "<a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/BarycentricCoordinates.html">Barycentric Coordinates</a>", <i><a href="/wiki/Mathworld" title="Mathworld">MathWorld</a></i>. <span style="font-size:.95em;font-weight:bold;color:var(--color-subtle, #54595d);">(engelska)</span></span></span> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <span class="reference-text">August Ferdinand Möbius, 1827, <i><a rel="nofollow" class="external text" href="https://archive.org/stream/derbarycentrisc00mbgoog#page/n2/mode/2up">Der Barycentrische Calcul</a></i>, Verlag von Johann Ambosius Barth, Leipzig.</span> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <span class="reference-text">Marie-Nicole Gras, 2014, <i><a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/261795525_Distances_between_the_circumcenter_of_the_extouch_triangle_and_the_classical_centers">Distances between the circumcenter of the extouch triangle and the classical centers</a></i>, Forum Geometricorum, 14, sid. 52.</span> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <span class="reference-text">Paul Yiu, 2013, <i><a rel="nofollow" class="external text" href="http://math.fau.edu/Yiu/YIUIntroductionToTriangleGeometry130411.pdf">Introduction to the Geometry of the Triangle</a></i>, sid. 1. Department of Mathematics, Florida Atlantic University.</span> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <span class="reference-text">Med dessa tre uttryck bevisar man dessutom enkelt <a href="/wiki/Cevas_sats" title="Cevas sats">Cevas sats</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\frac {\overline {LC}}{\overline {BL}}}\cdot {\frac {\overline {NB}}{\overline {AN}}}\cdot {\frac {\overline {AM}}{\overline {MC}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>L</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> <mover> <mrow> <mi>B</mi> <mi>L</mi> </mrow> <mo accent="false">¯</mo> </mover> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>N</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> <mover> <mrow> <mi>A</mi> <mi>N</mi> </mrow> <mo accent="false">¯</mo> </mover> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>A</mi> <mi>M</mi> </mrow> <mo accent="false">¯</mo> </mover> <mover> <mrow> <mi>M</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\frac {\overline {LC}}{\overline {BL}}}\cdot {\frac {\overline {NB}}{\overline {AN}}}\cdot {\frac {\overline {AM}}{\overline {MC}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b242479d36a3bd3c52f29b8ddb1e20fce9054e87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:12.497ex; height:4.176ex;" alt="{\displaystyle \scriptstyle {\frac {\overline {LC}}{\overline {BL}}}\cdot {\frac {\overline {NB}}{\overline {AN}}}\cdot {\frac {\overline {AM}}{\overline {MC}}}=1}"></span>.</span> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <span class="reference-text">Paul Yiu, 2000, <i><a rel="nofollow" class="external text" href="https://pdfs.semanticscholar.org/224d/fdc07536ae3dac6b2ff5a5d02f478c08d7e2.pdf">The uses of homogeneous barycentric coordinates in plane euclidean geometry</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180612210917/https://pdfs.semanticscholar.org/224d/fdc07536ae3dac6b2ff5a5d02f478c08d7e2.pdf">Arkiverad</a> 12 juni 2018 hämtat från the <a href="/wiki/Internet_Archive#Wayback_Machine" title="Internet Archive">Wayback Machine</a>.</i> i "International Journal of Mathematical Education in Science and Technology" 31, sid. 570</span> </li> <li id="cite_note-ref_1-8">^ [<a href="#cite_ref-ref_1_8-0"><small>a</small></a> <a href="#cite_ref-ref_1_8-1"><small>b</small></a>] <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Homogeneous_Barycentric_Coordinates"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, "<a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/HomogeneousBarycentricCoordinates.html">Homogeneous Barycentric Coordinates</a>", <i><a href="/wiki/Mathworld" title="Mathworld">MathWorld</a></i>. <span style="font-size:.95em;font-weight:bold;color:var(--color-subtle, #54595d);">(engelska)</span></span></span> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Areal_Coordinates"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, "<a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ArealCoordinates.html">Areal Coordinates</a>", <i><a href="/wiki/Mathworld" title="Mathworld">MathWorld</a></i>. <span style="font-size:.95em;font-weight:bold;color:var(--color-subtle, #54595d);">(engelska)</span></span></span> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <span class="reference-text">Matthew Harvey, 2015, <i><a rel="nofollow" class="external text" href="https://books.google.se/books?id=PPafCgAAQBAJ&pg=PA349">Geometry Illuminated: An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry</a></i>, sid. 349. <a href="/wiki/Special:Bokk%C3%A4llor/9781939512116" title="Special:Bokkällor/9781939512116">ISBN 9781939512116</a></span> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <span class="reference-text">Remi Munos, Andrew Moore, 1998, <i><a rel="nofollow" class="external text" href="https://papers.nips.cc/paper/1565-barycentric-interpolators-for-continuous-space-and-time-reinforcement-learning.pdf">Barycentric Interpolators for Continuous Space & Time Reinforcement Learning</a></i> i "<a rel="nofollow" class="external text" href="https://dl.acm.org/citation.cfm?id=340534&picked=prox">Proceedings of the 1998 conference on Advances in neural information processing systems II</a>", sid. 1024– 1030, Cambridge, MA, USA, 1999. MIT Press.</span> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <span class="reference-text">Tatiana V. Voitovich, Stefan Vandewalle, <i><a rel="nofollow" class="external text" href="https://www.researchgate.net/profile/Stefan_Vandewalle/publication/253438911_Barycentric_Interpolation_and_Exact_Integration_Formulas_for_the_Finite_Volume_Element_Method/links/55b9dd3e08ae9289a0901c02/Barycentric-Interpolation-and-Exact-Integration-Formulas-for-the-Finite-Volume-Element-Method.pdf">Barycentric Interpolation and Exact Integration Formulas for the Finite Volume Element Method</a></i> i "Numerical analysis and applied mathematcs, AIP Conference Proceedings, 1048", (Simos, T., Psihoyios, G., Tsitouras, C. (Eds.)). International Conference on Numerical Analysis and Applied Mathematics. Kos, Grekland, 16 - 20 september 2008", sid. 575-579.</span> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <span class="reference-text">Michael S Floater, Jiři Kosinka, 2010, <i><a rel="nofollow" class="external text" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.180.3050&rep=rep1&type=pdf">Barycentric interpolation and mappings on smooth convex domains</a></i> i "<a rel="nofollow" class="external text" href="https://dl.acm.org/citation.cfm?id=1839778&picked=prox">Proceedings of the 14th ACM Symposium on Solid and Physical Modeling - SPM '10</a>".</span> </li> </ol></div> <style data-mw-deduplicate="TemplateStyles:r56287950">.mw-parser-output table.navbox{border:#aaa 1px solid;width:100%;margin:auto;margin-top:1em;clear:both;font-size:88%;text-align:center;padding:1px}.mw-parser-output link+table.navbox{margin-top:-1px}.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow,.mw-parser-output table.navbox th{text-align:center;padding-left:1em;padding-right:1em}.mw-parser-output .navbox-thlinkcolor .navbox-title button,.mw-parser-output .navbox-thlinkcolor .navbox-title .mw-collapsible-text,.mw-parser-output .navbox-thlinkcolor .navbox-title a{color:inherit}.mw-parser-output .nowraplinks a,.mw-parser-output .nowraplinks .selflink{white-space:nowrap}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right;font-weight:bold;padding-left:1em;padding-right:1em}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background:#fdfdfd}.mw-parser-output .navbox-list{border-color:#fdfdfd}.mw-parser-output .navbox-title,.mw-parser-output table.navbox th{background:#b0c4de}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background:#d0e0f5}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background:#deeafa}.mw-parser-output .navbox-even{background:#f7f7f7}.mw-parser-output .navbox-odd{background:transparent}</style><table class="navbox" style="border-spacing:0; 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Barycentriska koordinater – Wikipedia