Konform avbilding – Wikipedia

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aria-controls="toc-Matematisk_beskrivelse-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Vis/skjul underseksjonen Matematisk beskrivelse</span> </button> <ul id="toc-Matematisk_beskrivelse-sublist" class="vector-toc-list"> <li id="toc-Eksempel:_Sirkelinversjon" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eksempel:_Sirkelinversjon"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Eksempel: Sirkelinversjon</span> </div> </a> <ul id="toc-Eksempel:_Sirkelinversjon-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Komplekse_funksjoner" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Komplekse_funksjoner"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Komplekse funksjoner</span> </div> </a> <button aria-controls="toc-Komplekse_funksjoner-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Vis/skjul underseksjonen Komplekse funksjoner</span> </button> <ul id="toc-Komplekse_funksjoner-sublist" class="vector-toc-list"> <li id="toc-Cauchy-Riemanns_ligninger" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cauchy-Riemanns_ligninger"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Cauchy-Riemanns ligninger</span> </div> </a> <ul id="toc-Cauchy-Riemanns_ligninger-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kompleks_avbildning_er_vinkeltro" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kompleks_avbildning_er_vinkeltro"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Kompleks avbildning er vinkeltro</span> </div> </a> <ul id="toc-Kompleks_avbildning_er_vinkeltro-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Konforme_kartprojeksjoner" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Konforme_kartprojeksjoner"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Konforme kartprojeksjoner</span> </div> </a> <button aria-controls="toc-Konforme_kartprojeksjoner-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Vis/skjul underseksjonen Konforme kartprojeksjoner</span> </button> <ul id="toc-Konforme_kartprojeksjoner-sublist" class="vector-toc-list"> <li id="toc-Stereografisk_projeksjon" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stereografisk_projeksjon"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Stereografisk projeksjon</span> </div> </a> <ul id="toc-Stereografisk_projeksjon-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Andre_anvendelser" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Andre_anvendelser"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Andre anvendelser</span> </div> </a> <button aria-controls="toc-Andre_anvendelser-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Vis/skjul underseksjonen Andre anvendelser</span> </button> <ul id="toc-Andre_anvendelser-sublist" class="vector-toc-list"> <li id="toc-Penrose-diagram" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Penrose-diagram"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Penrose-diagram</span> </div> </a> <ul id="toc-Penrose-diagram-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Referanser" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referanser"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Referanser</span> </div> </a> <ul id="toc-Referanser-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eksterne_lenker" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Eksterne_lenker"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Eksterne lenker</span> </div> </a> <ul id="toc-Eksterne_lenker-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Innhold" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Vis/skjul innholdsfortegnelsen" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Vis/skjul innholdsfortegnelsen</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Konform avbilding</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Gå til en artikkel på et annet språk. Tilgjengelig på 29 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-29" 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">29 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Konform_avbilding" title="Konform avbilding – norsk nynorsk" lang="nn" hreflang="nn" data-title="Konform avbilding" data-language-autonym="Norsk nynorsk" data-language-local-name="norsk nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Konform_avbildning" title="Konform avbildning – svensk" lang="sv" hreflang="sv" data-title="Konform avbildning" data-language-autonym="Svenska" data-language-local-name="svensk" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A5%D8%B3%D9%82%D8%A7%D8%B7_%D8%AA%D8%B4%D9%83%D9%8A%D9%84%D9%8A" title="إسقاط تشكيلي – arabisk" lang="ar" hreflang="ar" data-title="إسقاط تشكيلي" data-language-autonym="العربية" data-language-local-name="arabisk" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Transformaci%C3%B3_conforme" title="Transformació conforme – katalansk" lang="ca" hreflang="ca" data-title="Transformació conforme" data-language-autonym="Català" data-language-local-name="katalansk" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Konformn%C3%AD_zobrazen%C3%AD" title="Konformní zobrazení – tsjekkisk" lang="cs" hreflang="cs" data-title="Konformní zobrazení" data-language-autonym="Čeština" data-language-local-name="tsjekkisk" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Konforme_Abbildung" title="Konforme Abbildung – tysk" lang="de" hreflang="de" data-title="Konforme Abbildung" data-language-autonym="Deutsch" data-language-local-name="tysk" 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/Conformal_map" title="Conformal map – engelsk" lang="en" hreflang="en" data-title="Conformal map" data-language-autonym="English" data-language-local-name="engelsk" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Transformaci%C3%B3n_conforme" title="Transformación conforme – spansk" lang="es" hreflang="es" data-title="Transformación conforme" data-language-autonym="Español" data-language-local-name="spansk" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Konforma_bildigo" title="Konforma bildigo – esperanto" lang="eo" hreflang="eo" data-title="Konforma bildigo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%DA%AF%D8%A7%D8%B4%D8%AA_%D9%87%D9%85%D8%AF%DB%8C%D8%B3" title="نگاشت همدیس – persisk" lang="fa" hreflang="fa" data-title="نگاشت همدیس" data-language-autonym="فارسی" data-language-local-name="persisk" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Transformation_conforme" title="Transformation conforme – fransk" lang="fr" hreflang="fr" data-title="Transformation conforme" data-language-autonym="Français" data-language-local-name="fransk" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%93%B1%EA%B0%81_%EC%82%AC%EC%83%81" title="등각 사상 – koreansk" lang="ko" hreflang="ko" data-title="등각 사상" data-language-autonym="한국어" data-language-local-name="koreansk" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%B8%D5%B6%D6%86%D5%B8%D6%80%D5%B4_%D5%A1%D6%80%D5%BF%D5%A1%D5%BA%D5%A1%D5%BF%D5%AF%D5%A5%D6%80%D5%B8%D6%82%D5%B4" title="Կոնֆորմ արտապատկերում – armensk" lang="hy" hreflang="hy" data-title="Կոնֆորմ արտապատկերում" data-language-autonym="Հայերեն" data-language-local-name="armensk" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%95%E0%A5%8B%E0%A4%A3_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A4%BF%E0%A4%9A%E0%A4%BF%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%A3" title="अनुकोण प्रतिचित्रण – hindi" lang="hi" hreflang="hi" data-title="अनुकोण प्रतिचित्रण" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Mappa_conforme" title="Mappa conforme – italiensk" lang="it" hreflang="it" data-title="Mappa conforme" data-language-autonym="Italiano" data-language-local-name="italiensk" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%A7%D7%95%D7%A0%D7%A4%D7%95%D7%A8%D7%9E%D7%99%D7%AA" title="העתקה קונפורמית – hebraisk" lang="he" hreflang="he" data-title="העתקה קונפורמית" data-language-autonym="עברית" data-language-local-name="hebraisk" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B4%D1%8B_%D0%B1%D0%B5%D0%B9%D0%BD%D0%B5%D0%BB%D0%B5%D1%83" title="Конформды бейнелеу – kasakhisk" lang="kk" hreflang="kk" data-title="Конформды бейнелеу" data-language-autonym="Қазақша" data-language-local-name="kasakhisk" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Conforme_afbeelding" title="Conforme afbeelding – nederlandsk" lang="nl" hreflang="nl" data-title="Conforme afbeelding" data-language-autonym="Nederlands" data-language-local-name="nederlandsk" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%AD%89%E8%A7%92%E5%86%99%E5%83%8F" title="等角写像 – japansk" lang="ja" hreflang="ja" data-title="等角写像" data-language-autonym="日本語" data-language-local-name="japansk" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Odwzorowanie_r%C3%B3wnok%C4%85tne" title="Odwzorowanie równokątne – polsk" lang="pl" hreflang="pl" data-title="Odwzorowanie równokątne" data-language-autonym="Polski" data-language-local-name="polsk" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Proje%C3%A7%C3%A3o_conforme" title="Projeção conforme – portugisisk" lang="pt" hreflang="pt" data-title="Projeção conforme" data-language-autonym="Português" data-language-local-name="portugisisk" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Transformare_conform%C4%83" title="Transformare conformă – rumensk" lang="ro" hreflang="ro" data-title="Transformare conformă" data-language-autonym="Română" data-language-local-name="rumensk" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%BD%D0%BE%D0%B5_%D0%BE%D1%82%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Конформное отображение – russisk" lang="ru" hreflang="ru" data-title="Конформное отображение" data-language-autonym="Русский" data-language-local-name="russisk" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Konformikuvaus" title="Konformikuvaus – finsk" lang="fi" hreflang="fi" data-title="Konformikuvaus" data-language-autonym="Suomi" data-language-local-name="finsk" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/A%C3%A7%C4%B1korur_g%C3%B6nderim" title="Açıkorur gönderim – tyrkisk" lang="tr" hreflang="tr" data-title="Açıkorur gönderim" data-language-autonym="Türkçe" data-language-local-name="tyrkisk" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%BD%D0%B5_%D0%B2%D1%96%D0%B4%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%BD%D1%8F" title="Конформне відображення – ukrainsk" lang="uk" hreflang="uk" data-title="Конформне відображення" data-language-autonym="Українська" data-language-local-name="ukrainsk" 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/%C3%81nh_x%E1%BA%A1_b%E1%BA%A3o_gi%C3%A1c" title="Ánh xạ bảo giác – vietnamesisk" lang="vi" hreflang="vi" data-title="Ánh xạ bảo giác" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesisk" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BF%9D%E8%A7%92%E6%98%A0%E5%B0%84" title="保角映射 – kantonesisk" lang="yue" hreflang="yue" data-title="保角映射" data-language-autonym="粵語" data-language-local-name="kantonesisk" 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/%E5%85%B1%E5%BD%A2%E6%98%A0%E5%B0%84" title="共形映射 – kinesisk" lang="zh" hreflang="zh" data-title="共形映射" data-language-autonym="中文" data-language-local-name="kinesisk" 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/Q850275#sitelinks-wikipedia" title="Rediger lenker til artikkelen på andre språk" class="wbc-editpage">Rediger lenker</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Fra Wikipedia, den frie encyklopedi</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Omdirigert fra «<a href="/w/index.php?title=Konform_avbildning&redirect=no" class="mw-redirect" title="Konform avbildning">Konform avbildning</a>»)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="nb" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Conformal_map.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Conformal_map.svg/220px-Conformal_map.svg.png" decoding="async" width="220" height="385" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Conformal_map.svg/330px-Conformal_map.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Conformal_map.svg/440px-Conformal_map.svg.png 2x" data-file-width="535" data-file-height="937" /></a><figcaption>En konform avbildning <i>f </i> trans-formerer linjer som skjærer hverandre med 90° til kurver som skjærer hverandre med den samme vinkel.</figcaption></figure> <p>En <b>konform avbilding</b> gir et bilde av en <a href="/wiki/Flate" title="Flate">flate</a> eller <a href="/wiki/Riemanns_differensialgeometri" title="Riemanns differensialgeometri">metrisk rom</a> på en tilsvarende <a href="/wiki/Mangfoldighet" title="Mangfoldighet">mangfoldighet</a> slik at vinkelen mellom to linjer som skjærer hverandre, forblir den samme. Derfor sies avbildningen å være <i>vinkeltro</i>. En tilstrekkelig liten figur, vil få samme form ved avbildningen og har derfor gitt opphav til betegnelsen konform. Derimot vil en endelig <a href="/wiki/Polygon" title="Polygon">polygon</a> avbildes som en ny polygon med de samme hjørnevinklene, men generelt med en ganske annen form. </p><p>I alminnelighet vil alle lengder bli forandret ved denne transformasjonen. Et lite <a href="/wiki/Metrisk_tensor#Riemannske_rom" title="Metrisk tensor">linjeelement</a> <i>dσ</i> vil få lengden <i>ds</i> i bildet. Matematisk er den konforme avbildningen definert ved sammenhengen </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 d\sigma =k(x)ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>σ</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\sigma =k(x)ds}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70d5d9f7eadde5f8b89ed8b22ea4186e982baf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.301ex; height:2.843ex;" alt="{\displaystyle d\sigma =k(x)ds}"></span></dd></dl> <p>hvor skalafaktoren <i>k</i>(<i>x</i>) varierer fra punkt til punkt og er uavhengig av retningen til linjeelementet. Det betyr for eksempel at koordinatlinjene i et <a href="/wiki/Kartesisk_koordinatsystem" title="Kartesisk koordinatsystem">kartesisk koordinatsystem</a> vil bli <a href="/wiki/Krumning" title="Krumning">krumme</a> <a href="/wiki/Kurve" title="Kurve">kurver</a> som står vinkelrett på hverandre i bildet. </p><p>Når skalafaktoren <i>k</i> = 1, sies avbildning å være <i>isometrisk</i>. Translasjoner og rotasjoner i et <a href="/wiki/Euklidsk_rom" title="Euklidsk rom">euklidsk rom</a> er eksempel på slike spesielle transformasjoner. Derimot er <a href="/wiki/Sirkelinversjon" title="Sirkelinversjon">inversjoner</a> konforme transformasjoner med variabel skalafaktor. </p><p>Ved fremstilling av <a href="/wiki/Kart" title="Kart">kart</a> ønsker man ofte at avbildningen skal være konform. Både <a href="/wiki/Mercator-projeksjon" title="Mercator-projeksjon">Mercator-projeksjonen</a> og den <a href="/wiki/Stereografisk_projeksjon" title="Stereografisk projeksjon">stereografiske projeksjonen</a> oppfyller dette kravet i motsetning til den <a href="/wiki/Gnomonisk_projeksjon" title="Gnomonisk projeksjon">gnomoniske projeksjonen</a>. Likedan kan konforme transformasjoner benyttes til å løse problem innen <a href="/wiki/Hydrodynamikk" title="Hydrodynamikk">hydrodynamikk</a>, <a href="/wiki/Elektromagnetisme" title="Elektromagnetisme">elektromagnetisme</a> og andre grener innen <a href="/wiki/Teoretisk_fysikk" title="Teoretisk fysikk">teoretisk fysikk</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Matematisk_beskrivelse">Matematisk beskrivelse</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konform_avbilding&veaction=edit&section=1" title="Rediger avsnitt: Matematisk beskrivelse" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konform_avbilding&action=edit&section=1" title="Rediger kildekoden til seksjonen Matematisk beskrivelse"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For en <i>N</i>-dimensjonal <a href="/wiki/Mangfoldighet" title="Mangfoldighet">mangfoldighet</a> <i>Σ</i> med <a href="/wiki/Metrisk_tensor" title="Metrisk tensor">metrisk tensor</a> <i>g<sub>μν</sub></i> og koordinater <i>u<sup>μ</sup></i> kan man skrive det kvadrerte <a href="/wiki/Metrisk_tensor#Riemannske_rom" title="Metrisk tensor">linjeelementet</a> 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 d\sigma ^{2}=g_{\mu \nu }du^{\mu }du^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\sigma ^{2}=g_{\mu \nu }du^{\mu }du^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23a6c1c8b28f1a9f7a340ef67602f45357eb3069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.321ex; height:3.343ex;" alt="{\displaystyle d\sigma ^{2}=g_{\mu \nu }du^{\mu }du^{\nu }}"></span></dd></dl> <p>Her benyttes <a href="/wiki/Einsteins_summekonvensjon" title="Einsteins summekonvensjon">Einsteins summekonvensjon</a> hvor man summer fra 1 til <i>N</i> over alle par med like indekser. Denne mangfoldigheten skal nå avbildes på en annen <i>Σ' </i> med samme dimensjon.<sup id="cite_ref-TL-2_1-0" class="reference"><a href="#cite_note-TL-2-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> For enkelhets skyld kan man anta at dette er et <a href="/wiki/Euklidsk_rom" title="Euklidsk rom">euklidsk rom</a> med koordinater <i>x<sup>α</sup></i> og metrikk som kan settes lik med <a href="/wiki/Kronecker-delta" title="Kronecker-delta">Kronecker-deltaet</a> <i>δ<sub>αβ</sub></i>. Hvis denne avbildningen er konform, må det da finnes <i>N</i> <a href="/wiki/Derivasjon" title="Derivasjon">deriverbare</a> funksjoner <i>u<sup>μ</sup></i>(<i>x</i>) mellom koordinatene på disse to mangfoldighetene slik at man har </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 d\sigma ^{2}=k^{2}(x)\delta _{\alpha \beta }dx^{\alpha }dx^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\sigma ^{2}=k^{2}(x)\delta _{\alpha \beta }dx^{\alpha }dx^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef59fee4c46b8ae8bcf653ed81f69bbbf01d8ff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.911ex; height:3.343ex;" alt="{\displaystyle d\sigma ^{2}=k^{2}(x)\delta _{\alpha \beta }dx^{\alpha }dx^{\beta }}"></span></dd></dl> <p>der <i>k</i>(<i>x</i>) er en skalafaktor. Her er <i>ds</i><sup> 2</sup> = <i>δ<sub>αβ</sub></i> <i>dx<sup>α</sup> dx<sup>β</sup></i> = (<i>dx</i><sup>1</sup>)<sup>2</sup> + (<i>dx</i><sup> 2</sup>)<sup>2</sup> + ... + (<i>dx</i><sup><i>N</i></sup>)<sup> 2</sup> det kvadrerte linjeelement på den euklidske mangfoldigheten <i>Σ' </i>. </p><p>En <a href="/wiki/Vektor_(matematikk)" title="Vektor (matematikk)">vektor</a> med komponenter <i>A<sup>μ</sup></i> på mangfoldigheten <i>Σ</i> vil avbildes på <i>Σ' </i> med komponenter <i>A'<sup> μ</sup></i>. De er forbundet ved ligningene </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 A^{\mu }={\partial u^{\mu } \over \partial x^{\alpha }}A'^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>A</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\mu }={\partial u^{\mu } \over \partial x^{\alpha }}A'^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/581cc25fe2ab6fc46ee71375f21d42b283cb85fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.313ex; height:5.509ex;" alt="{\displaystyle A^{\mu }={\partial u^{\mu } \over \partial x^{\alpha }}A'^{\alpha }}"></span></dd></dl> <p>som følger fra transformasjonen mellom disse to <a href="/wiki/Koordinatsystem" title="Koordinatsystem">koordinatsystemene</a>. <a href="/wiki/Indreprodukt" title="Indreprodukt">Indreproduktet</a> mellom to slike vektorer blir dermed </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}\mathbf {A} \cdot \mathbf {B} &=g_{\mu \nu }A^{\mu }B^{\nu }=g_{\mu \nu }{\partial u^{\mu } \over \partial x^{\alpha }}{\partial u^{\nu } \over \partial x^{\beta }}A'^{\alpha }B'^{\beta }\\&=k^{2}(x)\delta _{\alpha \beta }A'^{\alpha }B'^{\beta }=k^{2}(x)\mathbf {A'} \cdot \mathbf {B'} \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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>A</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </mrow> </msup> <msup> <mi>B</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <msup> <mi>A</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </mrow> </msup> <msup> <mi>B</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">A</mi> <mo>′</mo> </msup> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">B</mi> <mo>′</mo> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {A} \cdot \mathbf {B} &=g_{\mu \nu }A^{\mu }B^{\nu }=g_{\mu \nu }{\partial u^{\mu } \over \partial x^{\alpha }}{\partial u^{\nu } \over \partial x^{\beta }}A'^{\alpha }B'^{\beta }\\&=k^{2}(x)\delta _{\alpha \beta }A'^{\alpha }B'^{\beta }=k^{2}(x)\mathbf {A'} \cdot \mathbf {B'} \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c95b5a41bddb86859cf98cbde73bacf275e53dc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:41.086ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {A} \cdot \mathbf {B} &=g_{\mu \nu }A^{\mu }B^{\nu }=g_{\mu \nu }{\partial u^{\mu } \over \partial x^{\alpha }}{\partial u^{\nu } \over \partial x^{\beta }}A'^{\alpha }B'^{\beta }\\&=k^{2}(x)\delta _{\alpha \beta }A'^{\alpha }B'^{\beta }=k^{2}(x)\mathbf {A'} \cdot \mathbf {B'} \end{aligned}}}"></span></dd></dl> <p>Indreproduktene mellom vektorene kan uttrykkes ved deres lengder <i>A</i> og <i>B</i> og vinkelen <i>θ</i> mellom dem som <span class="nowrap"><b>A</b>⋅<b>B</b> = <i>AB</i> cos<i>θ</i></span>. Da skalafaktoren <i>k</i>(<i>x</i>) også opptrer mellom lengden <i>A</i> og <i>A' </i> før og etter transformasjonen, må cos<i>θ</i> = cos<i>θ' </i>. Avbildningen forandrer derfor ikke vinkelen mellom de to vektorene slik at den er konform.<sup id="cite_ref-Blair_2-0" class="reference"><a href="#cite_note-Blair-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Det samme gjelder for vinkelen mellom to <a href="/wiki/Kurve" title="Kurve">kurver</a> som skjærer hverandre. Vinkelen mellom dem er da definert som vinkelen mellom deres <a href="/wiki/Tangent_(matematikk)" title="Tangent (matematikk)">tangentvektorer</a> i skjæringspunktet. Denne vil på samme måte forbli uforandret ved en slik konform transformasjon. </p> <div class="mw-heading mw-heading3"><h3 id="Eksempel:_Sirkelinversjon">Eksempel: Sirkelinversjon</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konform_avbilding&veaction=edit&section=2" title="Rediger avsnitt: Eksempel: Sirkelinversjon" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konform_avbilding&action=edit&section=2" title="Rediger kildekoden til seksjonen Eksempel: Sirkelinversjon"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kanskje det eldste og mens kjente eksempel på en konform avbildning er <a href="/wiki/Sirkelinversjon" title="Sirkelinversjon">inversjon</a> i en sirkel.<sup id="cite_ref-CG_3-0" class="reference"><a href="#cite_note-CG-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Den befinner seg i et todimensjonalt plan, og man kan her sette dens radius <span class="nowrap"><i>R</i> = 1</span>. Hvert punkt <b>r</b> med koordinater (<i>x,y</i>) blir transformert til det inverse punktet <span class="nowrap"><b>r</b> → <b>r</b>/<i>r</i><sup> 2</sup></span> med koordinater (<i>u,v</i>) hvor </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}x&\rightarrow u={x \over x^{2}+y^{2}}\\y&\rightarrow v={y \over x^{2}+y^{2}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&\rightarrow u={x \over x^{2}+y^{2}}\\y&\rightarrow v={y \over x^{2}+y^{2}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/028d30896a16c2b0d220d17565cb860928d15866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.716ex; margin-bottom: -0.289ex; width:18.399ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}x&\rightarrow u={x \over x^{2}+y^{2}}\\y&\rightarrow v={y \over x^{2}+y^{2}}\end{aligned}}}"></span></dd></dl> <p>Det transformerte linjeelementet er <i>dσ</i><sup> 2</sup> = <i>du</i><sup> 2</sup> + <i>dv</i><sup> 2</sup> hvor de to <a href="/wiki/Differensial_(matematikk)" title="Differensial (matematikk)">differensialene</a> blir </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}du&={x^{2}-y^{2} \over (x^{2}+y^{2})^{2}}dx-{2xy \over (x^{2}+y^{2})^{2}}dy\\dv&={y^{2}-x^{2} \over (x^{2}+y^{2})^{2}}dy-{2xy \over (x^{2}+y^{2})^{2}}dx\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>d</mi> <mi>u</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mi>y</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>v</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mi>y</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}du&={x^{2}-y^{2} \over (x^{2}+y^{2})^{2}}dx-{2xy \over (x^{2}+y^{2})^{2}}dy\\dv&={y^{2}-x^{2} \over (x^{2}+y^{2})^{2}}dy-{2xy \over (x^{2}+y^{2})^{2}}dx\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f05fcc867966e5879a9891a3d8cf1cc964c2ad2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:36.43ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}du&={x^{2}-y^{2} \over (x^{2}+y^{2})^{2}}dx-{2xy \over (x^{2}+y^{2})^{2}}dy\\dv&={y^{2}-x^{2} \over (x^{2}+y^{2})^{2}}dy-{2xy \over (x^{2}+y^{2})^{2}}dx\end{aligned}}}"></span></dd></dl> <p>Dermed er </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 d\sigma ^{2}={dx^{2}+dy^{2} \over (x^{2}+y^{2})^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\sigma ^{2}={dx^{2}+dy^{2} \over (x^{2}+y^{2})^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd3a2f43fbf473560a582a3415b90a717193a18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.838ex; height:6.509ex;" alt="{\displaystyle d\sigma ^{2}={dx^{2}+dy^{2} \over (x^{2}+y^{2})^{2}}}"></span></dd></dl> <p>slik at transformasjonen er konform. Den er derfor vinkelbevarende som man også kan bevise med rent geometriske metoder.<sup id="cite_ref-CG_3-1" class="reference"><a href="#cite_note-CG-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Inversjon kan vises på samme måte å være konform når den foretas mellom to euklidske rom med dimensjon <i>N</i> > 2. Et geometrisk bevis er ikke enkelt i dette generelle tilfellet.<sup id="cite_ref-Blair_2-1" class="reference"><a href="#cite_note-Blair-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Komplekse_funksjoner">Komplekse funksjoner</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konform_avbilding&veaction=edit&section=3" title="Rediger avsnitt: Komplekse funksjoner" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konform_avbilding&action=edit&section=3" title="Rediger kildekoden til seksjonen Komplekse funksjoner"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fil:Biholomorphism_illustration.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Biholomorphism_illustration.svg/240px-Biholomorphism_illustration.svg.png" decoding="async" width="240" height="436" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Biholomorphism_illustration.svg/360px-Biholomorphism_illustration.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Biholomorphism_illustration.svg/480px-Biholomorphism_illustration.svg.png 2x" data-file-width="4709" data-file-height="8557" /></a><figcaption>Under en typisk, kompleks avbildning som <i>z</i> → <i>e<sup>z</sup></i> bevares vinkler mellom linjer.</figcaption></figure> <p>Konforme transformasjoner i <i>N</i> = 2 dimensjoner er av spesiell stor betydning. Det kommer klart frem ved å betrakte en slik transformasjon <span class="nowrap"><i>x<sup>μ</sup></i> → <i>u<sup>μ</sup></i>(<i>x</i>)</span> som involverer koordinater <span class="nowrap"><i>x<sup>μ</sup></i> = (<i>x,y</i>)</span> og <span class="nowrap"><i>u<sup>μ</sup></i> = (<i>u,v</i>)</span> i to euklidske plan.<sup id="cite_ref-TL-2_1-1" class="reference"><a href="#cite_note-TL-2-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Da er </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}d\sigma ^{2}&=du^{2}+dv^{2}\\&=\left[{\Big (}{\partial u \over \partial x}{\Big )}^{2}+{\Big (}{\partial v \over \partial x}{\Big )}^{2}\right]dx^{2}+\left[{\Big (}{\partial u \over \partial y}{\Big )}^{2}+{\Big (}{\partial v \over \partial y}{\Big )}^{2}\right]dy^{2}\\&+2\left({\partial u \over \partial x}{\partial u \over \partial y}+{\partial v \over \partial x}{\partial v \over \partial y}\right)dxdy\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>d</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}d\sigma ^{2}&=du^{2}+dv^{2}\\&=\left[{\Big (}{\partial u \over \partial x}{\Big )}^{2}+{\Big (}{\partial v \over \partial x}{\Big )}^{2}\right]dx^{2}+\left[{\Big (}{\partial u \over \partial y}{\Big )}^{2}+{\Big (}{\partial v \over \partial y}{\Big )}^{2}\right]dy^{2}\\&+2\left({\partial u \over \partial x}{\partial u \over \partial y}+{\partial v \over \partial x}{\partial v \over \partial y}\right)dxdy\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999207222c48f64ca22f5822a09dd20c0bd52b5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; margin-top: -0.225ex; width:57.767ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}d\sigma ^{2}&=du^{2}+dv^{2}\\&=\left[{\Big (}{\partial u \over \partial x}{\Big )}^{2}+{\Big (}{\partial v \over \partial x}{\Big )}^{2}\right]dx^{2}+\left[{\Big (}{\partial u \over \partial y}{\Big )}^{2}+{\Big (}{\partial v \over \partial y}{\Big )}^{2}\right]dy^{2}\\&+2\left({\partial u \over \partial x}{\partial u \over \partial y}+{\partial v \over \partial x}{\partial v \over \partial y}\right)dxdy\end{aligned}}}"></span></dd></dl> <p>For at dette skal være proporsjonalt med <i>ds</i><sup> 2</sup> = <i>dx</i><sup> 2</sup> + <i>dy</i><sup> 2</sup>, må derfor de to funksjonene <span class="nowrap"><i>u</i> = <i>u</i>(<i>x,y</i>)</span> og <span class="nowrap"><i>v</i> = <i>v</i>(<i>x,y</i>)</span> oppfylle betingelsene </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 {\Big (}{\partial u \over \partial x}{\Big )}^{2}+{\Big (}{\partial v \over \partial x}{\Big )}^{2}={\Big (}{\partial u \over \partial y}{\Big )}^{2}+{\Big (}{\partial v \over \partial y}{\Big )}^{2}=k^{2}(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Big (}{\partial u \over \partial x}{\Big )}^{2}+{\Big (}{\partial v \over \partial x}{\Big )}^{2}={\Big (}{\partial u \over \partial y}{\Big )}^{2}+{\Big (}{\partial v \over \partial y}{\Big )}^{2}=k^{2}(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b8f5a8b4bace1b106e6ba0fa97cdb5545c3e98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.552ex; height:6.009ex;" alt="{\displaystyle {\Big (}{\partial u \over \partial x}{\Big )}^{2}+{\Big (}{\partial v \over \partial x}{\Big )}^{2}={\Big (}{\partial u \over \partial y}{\Big )}^{2}+{\Big (}{\partial v \over \partial y}{\Big )}^{2}=k^{2}(x,y)}"></span></dd></dl> <p>sammen med </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 {\partial u \over \partial x}{\partial u \over \partial y}+{\partial v \over \partial x}{\partial v \over \partial y}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial u \over \partial x}{\partial u \over \partial y}+{\partial v \over \partial x}{\partial v \over \partial y}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235895d6539402d3a798f598b2de35c3f3f1cdee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.863ex; height:6.009ex;" alt="{\displaystyle {\partial u \over \partial x}{\partial u \over \partial y}+{\partial v \over \partial x}{\partial v \over \partial y}=0}"></span></dd></dl> <p>I denne siste ligningen kan man sette <span class="nowrap"><i>∂u</i> /<i>∂x</i> = <i>ε</i> <i>∂v</i> /<i>∂y</i></span> hvor <i>ε</i> er en ukjent størrelse. Den gir da at <span class="nowrap"><i>∂v</i> /<i>∂x</i> = -<i>ε</i> <i>∂u</i> /<i>∂y</i></span>. Kombineres dette med de to første betingelsene, ser man at <span class="nowrap"><i>ε</i><sup> 2</sup> = 1</span>. Det finnes derfor uendelig mange konforme transformasjoner i to dimensjoner kun gitt ved disse to kravene til deres deriverte.<sup id="cite_ref-TL-2_1-2" class="reference"><a href="#cite_note-TL-2-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Cauchy-Riemanns_ligninger">Cauchy-Riemanns ligninger</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konform_avbilding&veaction=edit&section=4" title="Rediger avsnitt: Cauchy-Riemanns ligninger" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konform_avbilding&action=edit&section=4" title="Rediger kildekoden til seksjonen Cauchy-Riemanns ligninger"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Når <i>ε</i> = 1, er kravet for en konform avbildning at </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 {\partial u \over \partial x}={\partial v \over \partial y},\;\;\;{\partial u \over \partial y}=-{\partial v \over \partial x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial u \over \partial x}={\partial v \over \partial y},\;\;\;{\partial u \over \partial y}=-{\partial v \over \partial x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/527d32ba48f86265cf89382d855d6f1d7948983d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.736ex; height:6.009ex;" alt="{\displaystyle {\partial u \over \partial x}={\partial v \over \partial y},\;\;\;{\partial u \over \partial y}=-{\partial v \over \partial x}}"></span></dd></dl> <p>Dette er <a href="/wiki/Cauchy%E2%80%93Riemanns_ligninger" title="Cauchy–Riemanns ligninger">Cauchy–Riemanns ligninger</a> for en <a href="/wiki/Kompleks_analyse" title="Kompleks analyse">kompleks funksjon</a> <i>w</i>(<i>z</i>) = <i>u</i>(<i>x,y</i>) + <i>iv</i>(<i>x,y</i>) hvor <i>z</i> = <i>x</i> + <i>iy</i> er den <a href="/wiki/Komplekst_tall" title="Komplekst tall">komplekse</a> variable. Enhver slik analytisk funksjon <span class="nowrap"><i>w</i> = <i>f</i>(<i>z</i>)</span> gir derfor opphav til en konform avbildning i det komplekse planet. En spesielt viktig rolle har <a href="/w/index.php?title=M%C3%B6bius-transformasjon&action=edit&redlink=1" class="new" title="Möbius-transformasjon (ikke skrevet ennå)">Möbius-transformasjoner</a> som overfører linjer og sirkler på linjer og sirkler. </p><p>I det motsatte tilfelle med <span class="nowrap"><i>ε</i> = -1</span> vil man på samme vis ha en konform transformasjonen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\rightarrow f(z^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">→</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\rightarrow f(z^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e14580e09b6a15cebc925f99d5936d6dd202a956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.935ex; height:2.843ex;" alt="{\displaystyle z\rightarrow f(z^{*})}"></span> hvor den kompleks konjugerte variable er <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{*}=x-iy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{*}=x-iy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3321be6fb7bbb29e3a9004308a0e00fa2f01bcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.371ex; height:2.676ex;" alt="{\displaystyle z^{*}=x-iy}"></span>. Dette er en antiholomorf transformasjon som tar en figur med en viss orientering til en tilsvarende figur med motsatt orientering. Dette skjer for eksempel ved <a href="/wiki/Sirkelinversjon#Komplekse_koordinater" title="Sirkelinversjon">sirkelinversjon</a> i planet. </p><p>De to funksjonene <i>u</i> = <i>u</i>(<i>x,y</i>) og <i>v</i> = <i>v</i>(<i>x,y</i>) som gir konforme avbildninger i to dimensjoner, er <a href="/wiki/Harmonisk_funksjon" title="Harmonisk funksjon">harmoniske funksjoner</a>. Det følger direkte fra Cauchy–Riemanns ligninger som gir at </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 {\partial ^{2}u \over \partial x^{2}}={\partial ^{2}v \over \partial y\partial x}=-{\partial ^{2}u \over \partial y^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial ^{2}u \over \partial x^{2}}={\partial ^{2}v \over \partial y\partial x}=-{\partial ^{2}u \over \partial y^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0b4900617bd69e8dd7813543d245c3efec09f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.089ex; height:6.343ex;" alt="{\displaystyle {\partial ^{2}u \over \partial x^{2}}={\partial ^{2}v \over \partial y\partial x}=-{\partial ^{2}u \over \partial y^{2}}}"></span></dd></dl> <p>Derfor oppfyller begge funksjonene <a href="/wiki/Laplace-ligningen" title="Laplace-ligningen">Laplace-ligningen</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 \nabla ^{2}\Phi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\Phi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3dc05852dd419ed76a30cfd3f97dd24878aef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.929ex; height:2.676ex;" alt="{\displaystyle \nabla ^{2}\Phi =0}"></span> i to dimensjoner. Den opptrer også i mange forskjellige anvendelser, for eksempler innen <a href="/wiki/Elektrostatikk" title="Elektrostatikk">elektrostatikken</a> for det <a href="/wiki/Elektrisk_potensial" title="Elektrisk potensial">elektriske potensialet</a>. Beregnes dette i et bestemt område i to dimensjoner, kan man da med en kompleks transformasjon finne det i et annet område med en forskjellig, geometrisk form.<sup id="cite_ref-Churchill_4-0" class="reference"><a href="#cite_note-Churchill-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Kompleks_avbildning_er_vinkeltro">Kompleks avbildning er vinkeltro</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konform_avbilding&veaction=edit&section=5" title="Rediger avsnitt: Kompleks avbildning er vinkeltro" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konform_avbilding&action=edit&section=5" title="Rediger kildekoden til seksjonen Kompleks avbildning er vinkeltro"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En <a href="/wiki/Kurve" title="Kurve">kurve</a> i det komplekse planet kan generelt skrives som <i>z</i>(<i>t</i> ) = <i>x</i>(<i>t</i> ) + <i>iy</i>(<i>t</i> ) der <i>t</i> er dens reelle parameter. Under en kompleks transformasjon <i>z</i> → <i>f</i>(<i>z</i> ) vil den avbildes på en ny kurve <i>w</i>(<i>t</i> ) = <i>f</i>(<i>z</i>(<i>t</i> ). Dens retning er gitt ved <a href="/wiki/Tangent_(matematikk)" title="Tangent (matematikk)">tangentvektoren</a> </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 {dw \over dt}=f'(z){dz \over dt}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>w</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dw \over dt}=f'(z){dz \over dt}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b231ce3fbfce5170964c7849a966fee6ce70e0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.857ex; height:5.509ex;" alt="{\displaystyle {dw \over dt}=f'(z){dz \over dt}}"></span></dd></dl> <p>Hvis man skriver den opprinnelige tangenten 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 dz/dt=|dz/dt|e^{i\theta },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dz/dt=|dz/dt|e^{i\theta },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8103276bb6ca96554f0e7a0c73da6a8b1ca0de9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.737ex; height:3.176ex;" alt="{\displaystyle dz/dt=|dz/dt|e^{i\theta },}"></span> har den en vinkel <i>θ</i> med <i>x</i>-aksen. Kalles den tilsvarende vinkelen for den transformerte tangenten for <i>θ' </i>, vil derfor denne være <span class="nowrap"><i>θ' </i> = <i>θ</i> + <i>ψ</i></span> når man på samme vis skriver <span class="mwe-math-element"><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'(z)=|f'(z)|e^{i\psi }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ψ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(z)=|f'(z)|e^{i\psi }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcbe7ea0ce9678dd82efd659b4c52b6a68bc1652" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.797ex; height:3.176ex;" alt="{\displaystyle f'(z)=|f'(z)|e^{i\psi }.}"></span> Retningen til den opprinnelige tangentvektoren blir dermed dreidd en vinkel <i>ψ</i> som er uavhengig av kurven og gitt ved transformasjonen alene. Skjærer to kurver derfor hverandre i et punkt <i>z</i><sub>0</sub>, vil begge deres tangentvektorer dreies like mye under avbildningen. Dermed forblir vinklene mellom dem uforandret under transformasjonen.<sup id="cite_ref-TL-2_1-3" class="reference"><a href="#cite_note-TL-2-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Konforme_kartprojeksjoner">Konforme kartprojeksjoner</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konform_avbilding&veaction=edit&section=6" title="Rediger avsnitt: Konforme kartprojeksjoner" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konform_avbilding&action=edit&section=6" title="Rediger kildekoden til seksjonen Konforme kartprojeksjoner"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mange <a href="/wiki/Kartografi" title="Kartografi">kartprojeksjoner</a> er konforme. Selv om slike avbildninger er vinkeltro, vil likevel deler av kartet ha mer eller mindre forvrengning da skalafaktoren eller <a href="/wiki/M%C3%A5lestokk_(kart)" title="Målestokk (kart)">målestokken</a> varierer med stedet. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Mercator_projection_SW.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Mercator_projection_SW.jpg/320px-Mercator_projection_SW.jpg" decoding="async" width="320" height="271" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Mercator_projection_SW.jpg/480px-Mercator_projection_SW.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Mercator_projection_SW.jpg/640px-Mercator_projection_SW.jpg 2x" data-file-width="2058" data-file-height="1746" /></a><figcaption> Verdenskart i <a href="/wiki/Mercator-projeksjon" title="Mercator-projeksjon">Mercator-projeksjon</a> som tangerer ekvator. <a href="/wiki/Breddegrad" title="Breddegrad">Breddegrader</a> er horisontale linjer parallell med <i>x</i>-aksen, mens <a href="/wiki/Lengdegrad" title="Lengdegrad">lengdegradene</a> er parallelle med <i>y</i>-aksen.</figcaption></figure> <p>Et godt eksempel er den mye brukte <a href="/wiki/Mercator-projeksjon" title="Mercator-projeksjon">Mercator-projeksjonen</a> som avbilder en <a href="/wiki/Sf%C3%A6re" class="mw-redirect" title="Sfære">kuleflate</a> på en <a href="/wiki/Sylinder" title="Sylinder">sylinder</a> som tangerer den. Settes dens radius <i>R</i> = 1, kan hvert punkt på sfæren angis med <a href="/wiki/Kulekoordinater" title="Kulekoordinater">kulekoordinater</a> (<i>θ</i>,<i>φ</i>) slik at den er beskrevet med metrikken </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 d\sigma ^{2}=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mi>d</mi> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\sigma ^{2}=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52bdb21c26e86f1a2acac68b23557c672ee618d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.943ex; height:3.009ex;" alt="{\displaystyle d\sigma ^{2}=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}}"></span></dd></dl> <p>Denne kan omskrives 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 d\sigma ^{2}=\sin ^{2}\theta (dx^{2}+dy^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\sigma ^{2}=\sin ^{2}\theta (dx^{2}+dy^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a47af4d4caf39974a166e380b7498e9b585d85a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.767ex; height:3.176ex;" alt="{\displaystyle d\sigma ^{2}=\sin ^{2}\theta (dx^{2}+dy^{2})}"></span></dd></dl> <p>etter å ha innført nye koordinater (<i>x,y</i>) der <i>dx</i> = <i>dφ</i> og <span class="nowrap"><i>dy</i> = -<i>dθ</i>/sin<i>θ</i></span>. De kan betraktes som <a href="/wiki/Kartesisk_koordinatsystem" title="Kartesisk koordinatsystem">kartesiske koordinater</a> på sylinderen med <span class="nowrap"><i>x</i> = <i>φ</i></span>. Fortegnet til <i>y</i>-koordinaten er valgt slik at den avtar med vinkelen <i>θ</i>. Ved å skrive <span class="nowrap">sin<i>θ</i> = 2sin(<i>θ</i> /2)cos(<i>θ</i> /2)</span>, finner man da ved direkte integrasjon at <i>y</i> = ln cot(<i>θ </i>/2) etter å ha bestemt integrasjonskonstanten slik at <span class="nowrap"><i>y</i> = 0</span> for <span class="nowrap"><i>θ</i> = 90°</span>. Dette er nå en konform avbildning av kuleflaten med en skalafaktor <i>k</i> = sin<i>θ</i> som bare varierer i <i>y</i>-retning.<sup id="cite_ref-Kreyszig_5-0" class="reference"><a href="#cite_note-Kreyszig-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Når man i et verdenskart vil ha minst forvrengning i områder på begge sider av <a href="/wiki/Ekvator" title="Ekvator">ekvator</a>, kan man la sylinderen tangere kuleflaten langs denne <a href="/wiki/Storsirkel" title="Storsirkel">storsirkelen</a> <span class="nowrap"><i>θ</i> = 90°</span>. Hvis man benytter vanlige <a href="/wiki/Lengdegrad" title="Lengdegrad">lengdegrader</a> <span class="nowrap"><i>λ</i> = <i>φ</i></span> og <a href="/wiki/Breddegrad" title="Breddegrad">breddegrader</a> <span class="nowrap"><i>β</i> = 90° - <i>θ</i></span>, vil da de kartesiske koordinatene på kartet være </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}x&=\lambda ,\\y&=\ln \tan {\Big (}{\beta \over 2}+{\pi \over 4}{\Big )}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>λ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=\lambda ,\\y&=\ln \tan {\Big (}{\beta \over 2}+{\pi \over 4}{\Big )}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d6c8a13c9263a023af02711c91667b392d1de5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:21.852ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}x&=\lambda ,\\y&=\ln \tan {\Big (}{\beta \over 2}+{\pi \over 4}{\Big )}.\end{aligned}}}"></span></dd></dl> <p>Lengdegradene og breddegradene fremstilles på kartet som rette linjer som står vinkelrett på hverandre. På grunn av skalafaktoren øker forvrengningen mot polene slik at både <a href="/wiki/Arktis" title="Arktis">Arktis</a> og <a href="/wiki/Antarktis" title="Antarktis">Antarktis</a> synes å ha veldig stor utstrekning. </p> <div class="mw-heading mw-heading3"><h3 id="Stereografisk_projeksjon">Stereografisk projeksjon</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konform_avbilding&veaction=edit&section=7" title="Rediger avsnitt: Stereografisk projeksjon" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konform_avbilding&action=edit&section=7" title="Rediger kildekoden til seksjonen Stereografisk projeksjon"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Stereoprojnegone.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Stereoprojnegone.svg/240px-Stereoprojnegone.svg.png" decoding="async" width="240" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Stereoprojnegone.svg/360px-Stereoprojnegone.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Stereoprojnegone.svg/480px-Stereoprojnegone.svg.png 2x" data-file-width="232" data-file-height="232" /></a><figcaption>Stereografisk projeksjon <i>P</i> → <i>P' </i> av kuleflate med nordpol <i>N</i> som projeksjons-sentrum.</figcaption></figure> <p>Den første kartprojeksjon ble ikke benyttet til å gi en avbildning av Jorden, men av <a href="/wiki/Himmelhvelving" title="Himmelhvelving">himmelhvelvingen</a>. Fremgangsmåten ble beskrevet allerede av <a href="/wiki/Klaudius_Ptolemaios" class="mw-redirect" title="Klaudius Ptolemaios">Klaudius Ptolemaios</a> i hans verk <i>Planisphaerium</i>. Navnet benyttes fremdeles i dag for en <a href="/wiki/Planisf%C3%A6re" title="Planisfære">planisfære</a> som er et kart over stjernehimmelen med koordinater. Den er en <a href="/wiki/Stereografisk_projeksjon" title="Stereografisk projeksjon">stereografisk projeksjon</a> av en himmelhvelvingen på et plan med den spesielle egenskap at den er både konform og avbilder alle sirkler på sfæren som sirkler i planet.<sup id="cite_ref-Brummelen-1_6-0" class="reference"><a href="#cite_note-Brummelen-1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Både den <a href="/wiki/Gnomonisk_projeksjon" title="Gnomonisk projeksjon">gnomiske</a> og den stereografiske projeksjonen av en kuleflate er <a href="/wiki/Sentralperspektiv" title="Sentralperspektiv">sentralprojeksjoner</a> fra punkt på en diameter i kulen på et plan som står <a href="/wiki/Vinkelrett" title="Vinkelrett">vinkelrett</a> på diameteren. Mens den første benytter kulens sentrum som projeksjonspunkt, foretas den stereografiske projeksjonen fra en av <a href="/wiki/Geografisk_pol" title="Geografisk pol">polene</a> der diameteren møter kuleflaten. Projeksjonsplanet kan legges gjennom kulens sentrum eller som tangentplan i den motsatte polen. </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Fil:Stereographic_projection_SW.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Stereographic_projection_SW.JPG/240px-Stereographic_projection_SW.JPG" decoding="async" width="240" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Stereographic_projection_SW.JPG/360px-Stereographic_projection_SW.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Stereographic_projection_SW.JPG/480px-Stereographic_projection_SW.JPG 2x" data-file-width="2060" data-file-height="2060" /></a><figcaption>Stereografisk kart av verden nord for 30°S.</figcaption></figure> <p>Hvis man for eksempel vil avbilde områdene på den sydlige halvkule med minst forvrengning, er det naturlig å benytte <a href="/wiki/Nordpolen" title="Nordpolen">Nordpolen</a> som projeksjonspunkt og et projeksjonsplan som tangerer <a href="/wiki/Sydpolen" title="Sydpolen">Sydpolen</a>. Ved bruk av <a href="/wiki/Kulekoordinater" title="Kulekoordinater">kulekoordinater</a> (<i>θ</i>,<i>φ</i>) for et punkt <i>P</i> på overflaten, vil dette bli avbildet på et punkt <i>P' </i> med koordinater (<i>x,y</i>) i kartplanet hvor <span class="nowrap"><i>x</i> = <i>r</i> cos<i>φ</i></span> og <span class="nowrap"><i>y</i> = <i>r</i> sin<i>φ</i></span>. Her er nå <span class="nowrap"><i>r</i> = 2 tan(<i>θ </i>/2)</span> hvis man setter kulens radius <span class="nowrap"><i>R</i> = 1</span> og benytter loven om <a href="/wiki/Periferivinkel" title="Periferivinkel">periferivinkler</a>. Derfor er <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\theta =\cos ^{2}{\theta \over 2}dr}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>θ</mi> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mi>d</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\theta =\cos ^{2}{\theta \over 2}dr}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23ea4e33b7b5f491dabd0503ba13221d46c19cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.22ex; height:5.343ex;" alt="{\displaystyle d\theta =\cos ^{2}{\theta \over 2}dr}"></span> hvor </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}{\theta \over 2}={1 \over 1+r^{2}/4},\;\;\;\sin \theta ={r \over 1+r^{2}/4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}{\theta \over 2}={1 \over 1+r^{2}/4},\;\;\;\sin \theta ={r \over 1+r^{2}/4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821638679ece1177da53fbac7c3b2f84179e551f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.584ex; height:6.176ex;" alt="{\displaystyle \cos ^{2}{\theta \over 2}={1 \over 1+r^{2}/4},\;\;\;\sin \theta ={r \over 1+r^{2}/4}}"></span></dd></dl> <p>Fra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{2}=x^{2}+y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{2}=x^{2}+y^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb2a5f9d5d6102af609d3730471b017b2ac7ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.64ex; height:3.009ex;" alt="{\displaystyle r^{2}=x^{2}+y^{2}}"></span> følger <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rdr=xdx+ydy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>d</mi> <mi>r</mi> <mo>=</mo> <mi>x</mi> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>d</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rdr=xdx+ydy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f58715302bf3d750b984ccc3c8a54129b58e3e34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.654ex; height:2.509ex;" alt="{\displaystyle rdr=xdx+ydy}"></span>. Videre betyr <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \phi =y/x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \phi =y/x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04b4b51f7b981ec89d4d8cbc94b72a922b9ee1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.878ex; height:2.843ex;" alt="{\displaystyle \tan \phi =y/x}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{2}d\phi =xdy-ydx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>ϕ</mi> <mo>=</mo> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>−</mo> <mi>y</mi> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{2}d\phi =xdy-ydx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b37395ac89ab875efd977ca9f91711e041b6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.045ex; height:3.009ex;" alt="{\displaystyle r^{2}d\phi =xdy-ydx}"></span> . Linjeelementet på kuleflaten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\sigma ^{2}=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mi>d</mi> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\sigma ^{2}=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52bdb21c26e86f1a2acac68b23557c672ee618d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.943ex; height:3.009ex;" alt="{\displaystyle d\sigma ^{2}=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}}"></span> transformeres dermed til </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 d\sigma ^{2}={dr^{2} \over (1+r^{2}/4)^{2}}+{r^{2}d\phi ^{2} \over (1+r^{2}/4)^{2}}={dx^{2}+dy^{2} \over {\big (}1+\textstyle {\frac {1}{4}}(x^{2}+y^{2}){\big )}^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\sigma ^{2}={dr^{2} \over (1+r^{2}/4)^{2}}+{r^{2}d\phi ^{2} \over (1+r^{2}/4)^{2}}={dx^{2}+dy^{2} \over {\big (}1+\textstyle {\frac {1}{4}}(x^{2}+y^{2}){\big )}^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d1c33b19f8b95495257b4a37e4b95712b55c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:55.828ex; height:7.509ex;" alt="{\displaystyle d\sigma ^{2}={dr^{2} \over (1+r^{2}/4)^{2}}+{r^{2}d\phi ^{2} \over (1+r^{2}/4)^{2}}={dx^{2}+dy^{2} \over {\big (}1+\textstyle {\frac {1}{4}}(x^{2}+y^{2}){\big )}^{2}}}"></span></dd></dl> <p>Denne delen av kuleflaten er derfor konformt ekvivalent med et euklidsk plan. <a href="/wiki/Lengdegrad" title="Lengdegrad">Lengdegradene</a> er radielle linjer ut fra polen, mens <a href="/wiki/Breddegrad" title="Breddegrad">breddegradene</a> er konsentriske sirkler om dette punktet. Hadde kartplanet istedet gått gjennom ekvator, ville det tilsvare forandringene <i>x</i> → 2<i>x</i> og <i>y</i> → 2<i>y</i> i metrikken. Hvis radius <i>R</i> til kulen hadde blitt tatt med, ville faktoren 1/4 i nevneren i stedet blitt 1/4<i>R</i><sup> 2</sup> hvor <span class="nowrap"><i>K</i> = 1/<i>R</i><sup> 2</sup></span> er den <a href="/wiki/Differensiell_flategeometri#Hovedkrumninger" title="Differensiell flategeometri">gaussiske krumningen</a> til kuleflaten. </p> <div class="mw-heading mw-heading2"><h2 id="Andre_anvendelser">Andre anvendelser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konform_avbilding&veaction=edit&section=8" title="Rediger avsnitt: Andre anvendelser" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konform_avbilding&action=edit&section=8" title="Rediger kildekoden til seksjonen Andre anvendelser"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Droites_disquePoincare.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Droites_disquePoincare.svg/240px-Droites_disquePoincare.svg.png" decoding="async" width="240" height="236" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Droites_disquePoincare.svg/360px-Droites_disquePoincare.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Droites_disquePoincare.svg/480px-Droites_disquePoincare.svg.png 2x" data-file-width="594" data-file-height="583" /></a><figcaption><a href="/wiki/Geodetisk_kurve" title="Geodetisk kurve">Geodetiske linjer</a> i det <a href="/wiki/Hyperbolsk_geometri" title="Hyperbolsk geometri">hyperbolske planet</a> konformt avbildet på innsiden av en sirkel.</figcaption></figure> <p>Konforme avbildninger kan gjøres av <a href="/wiki/Mangfoldighet" title="Mangfoldighet">mangfoldigheter</a> med dimensjoner <i>N</i> > 2. For eksempel kan en <i>N</i>-dimensjonal kuleflate eller <a href="/wiki/Sf%C3%A6risk_geometri" title="Sfærisk geometri">sfærisk rom</a> <b>S</b><sup><i>N</i></sup> avbildes på et <a href="/wiki/Euklidsk_rom" title="Euklidsk rom">euklidsk rom</a> <b>E</b><sup><i>N</i></sup> ved en stereografisk projeksjon. Hvis dette skjer gjennom et hyperplan gjennom kulens sentrum, vil den sfæriske metrikken ta formen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\sigma ^{2}={4d\mathbf {x} \cdot d\mathbf {x} \over (1+\mathbf {x} \cdot \mathbf {x} )^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\sigma ^{2}={4d\mathbf {x} \cdot d\mathbf {x} \over (1+\mathbf {x} \cdot \mathbf {x} )^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe89937ad1483404dd1177ad1261f1323df8803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.903ex; height:6.176ex;" alt="{\displaystyle d\sigma ^{2}={4d\mathbf {x} \cdot d\mathbf {x} \over (1+\mathbf {x} \cdot \mathbf {x} )^{2}}}"></span></dd></dl> <p>på samme måte som i <i>N</i> = 2 dimensjoner. Her er <b>x</b> = (<i>x</i><sup>1</sup>, <i>x</i><sup>2</sup>, ... , <i>x</i><sup><i>N</i></sup>) kartesiske koordinater i det euklidske rommet. </p><p>Mens det sfæriske rommet <b>S</b><sup><i>N</i></sup> har konstant, positiv <a href="/wiki/Krumning" title="Krumning">krumning</a>, har det <a href="/wiki/Hyperbolsk_geometri" title="Hyperbolsk geometri">hyperbolske rommet</a> <b>H</b><sup><i>N</i></sup> konstant, negativ krumning. Det kan formelt beskrives som en kuleflate med <a href="/wiki/Imagin%C3%A6rt_tall" title="Imaginært tall">imaginær</a> radius. Metrikken for dette rommet kan dermed oppnås fra den sfæriske ved substitusjonen <i>R</i><sup> 2</sup> → - <i>R</i><sup> 2</sup>. På den måten finner man det kvadrerte linjeelementet </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 d\sigma ^{2}={4d\mathbf {x} \cdot d\mathbf {x} \over (1-\mathbf {x} \cdot \mathbf {x} )^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\sigma ^{2}={4d\mathbf {x} \cdot d\mathbf {x} \over (1-\mathbf {x} \cdot \mathbf {x} )^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d1d1b9d6c63434135b2bae4fb171ccc6f79a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.903ex; height:6.176ex;" alt="{\displaystyle d\sigma ^{2}={4d\mathbf {x} \cdot d\mathbf {x} \over (1-\mathbf {x} \cdot \mathbf {x} )^{2}}}"></span></dd></dl> <p>Det hyperbolske rommet er derfor også konformt ekvivalent med det euklidske rommet <b>E</b><sup><i>N</i></sup> og blir avbildet på innsiden av en <i>N</i>-dimensjonal kule. </p><p>Denne hyperbolske metrikken ble først etablert av <a href="/w/index.php?title=Eugenio_Beltrami&action=edit&redlink=1" class="new" title="Eugenio Beltrami (ikke skrevet ennå)">Eugenio Beltrami</a> som gjorde bruk av den nye <a href="/wiki/Riemanns_differensialgeometri" title="Riemanns differensialgeometri">differensialgeometrien</a> til <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a>. I <span class="nowrap"><i>N</i> = 2</span> dimensjoner inneholder den viktige symmetrier som ble avdekket av <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a>. Derfor omtales også geometrien ofte som <a href="/w/index.php?title=Poincar%C3%A9s_diskmodell&action=edit&redlink=1" class="new" title="Poincarés diskmodell (ikke skrevet ennå)">Poincarés diskmodell</a> for det hyperbolske planet. </p> <div class="mw-heading mw-heading3"><h3 id="Penrose-diagram">Penrose-diagram</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konform_avbilding&veaction=edit&section=9" title="Rediger avsnitt: Penrose-diagram" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konform_avbilding&action=edit&section=9" title="Rediger kildekoden til seksjonen Penrose-diagram"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Penrose.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Penrose.PNG/240px-Penrose.PNG" decoding="async" width="240" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Penrose.PNG/360px-Penrose.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Penrose.PNG/480px-Penrose.PNG 2x" data-file-width="651" data-file-height="588" /></a><figcaption>Penrose-diagram for det todimen-sjonale <a href="/wiki/Spesiell_relativitet#Minkowski-rom" class="mw-redirect" title="Spesiell relativitet">Minkowski-rommet</a> med konform tid <i>T</i> i vertikal retning og konform avstand <i>R</i> langs den horisontale aksen.</figcaption></figure> <p><a href="/wiki/Spesiell_relativitetsteori" class="mw-redirect" title="Spesiell relativitetsteori">Spesiell relativitetsteori</a> kan beskrives i et 4-dimensjonalt <a href="/wiki/Spesiell_relativitet#Minkowski-rom" class="mw-redirect" title="Spesiell relativitet">Minkowski-rom</a>. Når lyshastigheten settes lik med <span class="nowrap"><i>c</i> = 1</span>, kan det beskrives ved koordinater (<i>t</i>,<i>x,y,z</i>) eller tilsvarende <a href="/wiki/Kulekoordinater" title="Kulekoordinater">kulekoordinater</a> (<i>t</i>;<i>r,θ,φ</i>). Mange prosesser i dette rommet er uavhengige av den radielle retningen gitt ved vinklene (<i>θ,φ</i>) slik at linjeelementet effektivt er <span class="nowrap"><i>dσ</i><sup> 2</sup> = <i>dt</i><sup> 2</sup> - <i>dr</i><sup> 2</sup>.</span> Lysstråler følger nå baner <i>r</i> = ± <i>t</i> som er rette linjer som danner 45° med aksene i det 2-dimensjonale Minkowski-rommet. </p><p>Dette uendelig store <a href="/wiki/Tidrom" title="Tidrom">tidrom</a> kan konformt avbildes på et endelig tidrom med koordinater (<i>p,q</i>) hvor </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 t+r=\tan p,\;\;\;t-r=\tan q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>+</mo> <mi>r</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>p</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t+r=\tan p,\;\;\;t-r=\tan q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e304e21fee3e8de773d90d3140fcecdc40f0637" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.356ex; height:2.343ex;" alt="{\displaystyle t+r=\tan p,\;\;\;t-r=\tan q}"></span></dd></dl> <p>og som tar verdier i intervallet fra -<i>π</i> /2 til <i>π</i> /2. Den reduserte Minkowski-metrikken tar dermed formen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\sigma ^{2}={dpdq \over \cos ^{2}p\cos ^{2}p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> <mi>d</mi> <mi>q</mi> </mrow> <mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>p</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>p</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\sigma ^{2}={dpdq \over \cos ^{2}p\cos ^{2}p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63d8734d323122060640720755b75fc024953a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.366ex; height:6.009ex;" alt="{\displaystyle d\sigma ^{2}={dpdq \over \cos ^{2}p\cos ^{2}p}}"></span></dd></dl> <p>Ved å innføre konform tid <i>T</i> = <i>p</i> + <i>q</i> og radiell avstand <span class="nowrap"><i>R</i> = <i>p</i> - <i>q</i></span>, er denne proporsjonal med <span class="nowrap"><i>ds</i><sup> 2</sup> = <i>dT</i><sup> 2</sup> - <i>dR</i><sup> 2</sup></span> slik at hele Minkowski-rommet befinner seg innen et endelig kvadrat. En lysstråle som fulgte en bane med <span class="nowrap"><i>dσ</i> = 0</span>, vil etter transformasjonen følge <span class="nowrap"><i>ds</i> = 0</span> som betyr at den fortsatt danner 45° med disse nye koordinataksene. Det resulterende bildet av tidrommet blir vanligvis omtalt som et <i>Penrose-diagram</i> etter den britiske fysiker <a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a>.<sup id="cite_ref-MTW_7-0" class="reference"><a href="#cite_note-MTW-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Tidrommet rundt et sfærisk symmetrisk, <a href="/wiki/Sort_hull" title="Sort hull">sort hull</a> kan også fremstilles i et slikt Penrose-diagram. Man benytter da <a href="/wiki/Schwarzschilds_l%C3%B8sning" title="Schwarzschilds løsning">Schwarzschild-løsningen</a> av <a href="/wiki/Generell_relativitet" class="mw-redirect" title="Generell relativitet">Einsteins ligninger</a> ved bruk av <a href="/wiki/Schwarzschilds_l%C3%B8sning#Singulariteter_og_analytisk_forlengelse" title="Schwarzschilds løsning">Kruskal-Szekeres-koordinater</a>. Fordelen med denne konforme fremstillingen er at den gir en bedre forståelse av geometrien og fysiske prosesser innenfor horisonten gitt ved <a href="/wiki/Schwarzschild-radius" title="Schwarzschild-radius">Schwarzschild-radien</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Referanser">Referanser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konform_avbilding&veaction=edit&section=10" title="Rediger avsnitt: Referanser" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konform_avbilding&action=edit&section=10" title="Rediger kildekoden til seksjonen Referanser"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-TL-2-1"><b>^</b> <a href="#cite_ref-TL-2_1-0"><sup>a</sup></a> <a href="#cite_ref-TL-2_1-1"><sup>b</sup></a> <a href="#cite_ref-TL-2_1-2"><sup>c</sup></a> <a href="#cite_ref-TL-2_1-3"><sup>d</sup></a> <span class="reference-text">R. Tambs Lyche, <i>Matematisk Analyse</i> II, Gyldendal Norsk Forlag, Oslo (1961).</span> </li> <li id="cite_note-Blair-2"><b>^</b> <a href="#cite_ref-Blair_2-0"><sup>a</sup></a> <a href="#cite_ref-Blair_2-1"><sup>b</sup></a> <span class="reference-text"> D.E. Blair, <i>Inversion Theory and Conformal Mapping</i>, Student Mathematical Library. No. 9, AMS (2000).</span> </li> <li id="cite_note-CG-3"><b>^</b> <a href="#cite_ref-CG_3-0"><sup>a</sup></a> <a href="#cite_ref-CG_3-1"><sup>b</sup></a> <span class="reference-text"> H.S.M. Coxeter and S.L. Greitzer, <i>Geometry Revisited</i>, Mathematical Association of America, Washington, DC (1967). <a href="/wiki/Spesial:Bokkilder/0883856190" class="internal mw-magiclink-isbn">ISBN 0-8838-5619-0</a>.</span> </li> <li id="cite_note-Churchill-4"><b><a href="#cite_ref-Churchill_4-0">^</a></b> <span class="reference-text"> R.V. Churchill, <i>Complex Variables and Applications</i>, McGraw–Hill, New York (1974). <a href="/wiki/Spesial:Bokkilder/9780070108554" class="internal mw-magiclink-isbn">ISBN 978-0-07-010855-4</a>.</span> </li> <li id="cite_note-Kreyszig-5"><b><a href="#cite_ref-Kreyszig_5-0">^</a></b> <span class="reference-text"> E. Kreyszig, <i>Differential Geometry</i>, Dover Publications, New York (1991). <a href="/wiki/Spesial:Bokkilder/0486667219" class="internal mw-magiclink-isbn">ISBN 0-486-66721-9</a>.</span> </li> <li id="cite_note-Brummelen-1-6"><b><a href="#cite_ref-Brummelen-1_6-0">^</a></b> <span class="reference-text"> G. van Brummelen, <i>Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry</i>, Princeton University Press, New Jersey (2013). <a href="/wiki/Spesial:Bokkilder/9780691148922" class="internal mw-magiclink-isbn">ISBN 978-0-691-14892-2</a>.</span> </li> <li id="cite_note-MTW-7"><b><a href="#cite_ref-MTW_7-0">^</a></b> <span class="reference-text">C.W. Misner, K.S. Thorne and J.A. Wheeler, <i>Gravitation</i>, W. H. Freeman, San Francisco (1973). <a href="/wiki/Spesial:Bokkilder/0716703440" class="internal mw-magiclink-isbn">ISBN 0-7167-0344-0</a>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Eksterne_lenker">Eksterne lenker</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konform_avbilding&veaction=edit&section=11" title="Rediger avsnitt: Eksterne lenker" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konform_avbilding&action=edit&section=11" title="Rediger kildekoden til seksjonen Eksterne lenker"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>E. Weisstein, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ConformalMapping.html"><i>Conformal Mapping</i></a>, Wolfram MathWorld</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r23230704">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": 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Konform avbilding – Wikipedia