Ellipsoid – Wikipedia

<!DOCTYPE html> <html class="client-nojs" lang="de" dir="ltr"> <head> <meta charset="UTF-8"> <title>Ellipsoid – Wikipedia</title> <script>(function(){var className="client-js";var cookie=document.cookie.match(/(?:^|; )dewikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t.",".\t,"],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","Januar","Februar","März","April","Mai","Juni","Juli","August","September","Oktober","November","Dezember"],"wgRequestId":"28f7e4ee-a045-43fb-a6f4-57d8582f9594","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Ellipsoid","wgTitle":"Ellipsoid","wgCurRevisionId":247956334,"wgRevisionId":247956334,"wgArticleId":93501,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":[ "Geometrie"],"wgPageViewLanguage":"de","wgPageContentLanguage":"de","wgPageContentModel":"wikitext","wgRelevantPageName":"Ellipsoid","wgRelevantArticleId":93501,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgFlaggedRevsParams":{"tags":{"accuracy":{"levels":1}}},"wgStableRevisionId":247956334,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"de","pageLanguageDir":"ltr","pageVariantFallbacks":"de"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":30000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":true,"wgVector2022LanguageInHeader":false,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId": "Q190046","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.citeRef":"ready","ext.gadget.defaultPlainlinks":"ready","ext.gadget.dewikiCommonHide":"ready","ext.gadget.dewikiCommonLayout":"ready","ext.gadget.dewikiCommonStyle":"ready","ext.gadget.NavFrame":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.styles.legacy":"ready","ext.flaggedRevs.basic":"ready","mediawiki.codex.messagebox.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","codex-search-styles":"ready","ext.uls.interlanguage":"ready", "wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.legacy.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.flaggedRevs.advanced","ext.gadget.createNewSection","ext.gadget.WikiMiniAtlas","ext.gadget.OpenStreetMap","ext.gadget.CommonsDirekt","ext.gadget.donateLink","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.compactlinks","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.checkUser.clientHints","ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=de&modules=codex-search-styles%7Cext.cite.styles%7Cext.flaggedRevs.basic%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cmediawiki.codex.messagebox.styles%7Cskins.vector.styles.legacy%7Cwikibase.client.init&only=styles&skin=vector"> <script async="" src="/w/load.php?lang=de&modules=startup&only=scripts&raw=1&skin=vector"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=de&modules=ext.gadget.NavFrame%2CciteRef%2CdefaultPlainlinks%2CdewikiCommonHide%2CdewikiCommonLayout%2CdewikiCommonStyle&only=styles&skin=vector"> <link rel="stylesheet" href="/w/load.php?lang=de&modules=site.styles&only=styles&skin=vector"> <meta name="generator" content="MediaWiki 1.44.0-wmf.6"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Ellipsoide.svg/1200px-Ellipsoide.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="917"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Ellipsoide.svg/800px-Ellipsoide.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="611"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Ellipsoide.svg/640px-Ellipsoide.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="489"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Ellipsoid – Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//de.m.wikipedia.org/wiki/Ellipsoid"> <link rel="alternate" type="application/x-wiki" title="Seite bearbeiten" href="/w/index.php?title=Ellipsoid&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (de)"> <link rel="EditURI" type="application/rsd+xml" href="//de.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://de.wikipedia.org/wiki/Ellipsoid"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de"> <link rel="alternate" type="application/atom+xml" title="Atom-Feed für „Wikipedia“" href="/w/index.php?title=Spezial:Letzte_%C3%84nderungen&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin-vector-legacy mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Ellipsoid rootpage-Ellipsoid skin-vector action-view"><div id="mw-page-base" class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice"></div> <div class="mw-indicators"> </div> <h1 id="firstHeading" class="firstHeading mw-first-heading">Ellipsoid</h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">aus Wikipedia, der freien Enzyklopädie</div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Zur Navigation springen</a> <a class="mw-jump-link" href="#searchInput">Zur Suche springen</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/Datei:Ellipsoide.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Ellipsoide.svg/400px-Ellipsoide.svg.png" decoding="async" width="400" height="306" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Ellipsoide.svg/600px-Ellipsoide.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Ellipsoide.svg/800px-Ellipsoide.svg.png 2x" data-file-width="891" data-file-height="681" /></a><figcaption><a href="/wiki/Kugel" title="Kugel">Kugel</a> (oben), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d366942ea55e335b92439f564940a422e749648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=4}">, <a href="/wiki/Rotationsellipsoid" title="Rotationsellipsoid">Rotationsellipsoid</a> (unten links), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b=5,\ c=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mtext> </mtext> <mi>c</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b=5,\ c=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17ddd8142276a563574c845a2de5f8c6d280ff21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.469ex; height:2.509ex;" alt="{\displaystyle a=b=5,\ c=3}">, triaxiales Ellipsoid (unten rechts), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=4{,}5,\ b=6,\ c=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>,</mo> <mtext> </mtext> <mi>b</mi> <mo>=</mo> <mn>6</mn> <mo>,</mo> <mtext> </mtext> <mi>c</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=4{,}5,\ b=6,\ c=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45f6987365d2e75c17b95eefc43041a4679eb02c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.055ex; height:2.509ex;" alt="{\displaystyle a=4{,}5,\ b=6,\ c=3}"></figcaption></figure> Ein Ellipsoid ist die <a href="/wiki/3-dimensional" class="mw-redirect" title="3-dimensional">dreidimensionale</a> Entsprechung einer <a href="/wiki/Ellipse" title="Ellipse">Ellipse</a>. So wie sich eine Ellipse als <a href="/wiki/Affine_Abbildung" title="Affine Abbildung">affines</a> Bild des <a href="/wiki/Einheitskreis" title="Einheitskreis">Einheitskreises</a> auffassen lässt, gilt: <ul><li>Ein Ellipsoid (als <a href="/wiki/Fl%C3%A4che_(Mathematik)" title="Fläche (Mathematik)">Fläche</a>) ist ein affines Bild der <a href="/wiki/Einheitskugel" title="Einheitskugel">Einheitskugel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}+z^{2}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}+z^{2}=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26790310fffdf30f449c78609788054d278dea7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.332ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}+z^{2}=1.}"></li></ul> Die einfachsten affinen Abbildungen sind die Skalierungen der <a href="/wiki/Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinaten</a>. Sie liefern Ellipsoide mit <a href="/wiki/Gleichung" title="Gleichung">Gleichungen</a> <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{abc}\colon \quad {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1,\quad a,b,c>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msub> <mo>:</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{abc}\colon \quad {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1,\quad a,b,c>0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d73bf669bebc2b6da640db47302adc9747c3b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:40.351ex; height:6.009ex;" alt="{\displaystyle E_{abc}\colon \quad {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1,\quad a,b,c>0.}"></li></ul> So ein Ellipsoid ist <a href="/wiki/Punktsymmetrisch" class="mw-redirect" title="Punktsymmetrisch">punktsymmetrisch</a> zum Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c6fd55d5621fd95ca93549660fbb355fd9bd22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,0,0)}">, dem Mittelpunkt des Ellipsoids. Die Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"> sind analog zu einer Ellipse die Halbachsen des Ellipsoids und die <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkte</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\pm a,0,0),(0,\pm b,0),(0,0,\pm c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>±</mo> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mi>b</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\pm a,0,0),(0,\pm b,0),(0,0,\pm c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9764c4bf1cd4a9138b39e965fe92de8eb16aa05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.333ex; height:2.843ex;" alt="{\displaystyle (\pm a,0,0),(0,\pm b,0),(0,0,\pm c)}"> seine 6 <a href="/wiki/Scheitelpunkt" title="Scheitelpunkt">Scheitelpunkte</a>. <ul><li>Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b=c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41dfe5f5f74cb7a2c79093352406fd1d19123882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.431ex; height:2.176ex;" alt="{\displaystyle a=b=c}"> ist, ist das Ellipsoid eine <a href="/wiki/Kugel" title="Kugel">Kugel</a>.</li> <li>Falls genau zwei Halbachsen übereinstimmen, ist das Ellipsoid ein prolates oder oblates <a href="/wiki/Rotationsellipsoid" title="Rotationsellipsoid">Rotationsellipsoid</a>.</li> <li>Falls die 3 Halbachsen alle verschieden sind, heißt das Ellipsoid triaxial oder <a href="/wiki/Dreiachsiges_Ellipsoid" title="Dreiachsiges Ellipsoid">dreiachsig</a>.</li></ul> Alle Ellipsoide <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{abc}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{abc}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b96d78d2f98308cceced7b04e70afcb0261c7f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.234ex; height:2.509ex;" alt="{\displaystyle E_{abc}}"> sind <a href="/wiki/Symmetrie_(Geometrie)#Spiegelsymmetrie" title="Symmetrie (Geometrie)">symmetrisch</a> zu jeder der drei Koordinatenebenen. Beim Rotationsellipsoid kommt noch die <a href="/wiki/Rotationssymmetrie" class="mw-redirect" title="Rotationssymmetrie">Rotationssymmetrie</a> bezüglich der <a href="/wiki/Rotationsachse" title="Rotationsachse">Rotationsachse</a> hinzu. Eine Kugel ist zu jeder <a href="/wiki/Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> durch den Mittelpunkt symmetrisch. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Jupiter-Ellipsoid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Jupiter-Ellipsoid.png/220px-Jupiter-Ellipsoid.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Jupiter-Ellipsoid.png/330px-Jupiter-Ellipsoid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Jupiter-Ellipsoid.png/440px-Jupiter-Ellipsoid.png 2x" data-file-width="860" data-file-height="860" /></a><figcaption>Jupiters Durchmesser von Pol zu Pol ist deutlich kleiner als am Äquator (zum Vergleich roter Kreis).</figcaption></figure> Angenäherte Beispiele für Rotationsellipsoide sind der <a href="/wiki/Rugbyball" title="Rugbyball">Rugbyball</a> und abgeplattete <a href="/wiki/Rotation_(Physik)" title="Rotation (Physik)">rotierende</a> <a href="/wiki/Himmelsk%C3%B6rper" class="mw-redirect" title="Himmelskörper">Himmelskörper</a>, etwa die <a href="/wiki/Erdellipsoid" class="mw-redirect" title="Erdellipsoid">Erde</a> oder andere <a href="/wiki/Planet" title="Planet">Planeten</a> (<a href="/wiki/Jupiter_(Planet)" title="Jupiter (Planet)">Jupiter</a>), <a href="/wiki/Sonne" title="Sonne">Sonnen</a> oder <a href="/wiki/Galaxie" title="Galaxie">Galaxien</a>. <a href="/wiki/Elliptische_Galaxie" title="Elliptische Galaxie">Elliptische Galaxien</a> und <a href="/wiki/Zwergplanet" title="Zwergplanet">Zwergplaneten</a> (z. B. <a href="/wiki/(136108)_Haumea" title="(136108) Haumea">(136108) Haumea</a>) können auch triaxial sein. In der <a href="/wiki/Lineare_Optimierung" title="Lineare Optimierung">Linearen Optimierung</a> werden Ellipsoide in der <a href="/wiki/Ellipsoid-Methode" class="mw-redirect" title="Ellipsoid-Methode">Ellipsoid-Methode</a> verwendet. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Parameterdarstellung">1 Parameterdarstellung</a></li> <li class="toclevel-1 tocsection-2"><a href="#Volumen">2 Volumen</a></li> <li class="toclevel-1 tocsection-3"><a href="#Oberfläche">3 Oberfläche</a> <ul> <li class="toclevel-2 tocsection-4"><a href="#Oberfläche_eines_Rotationsellipsoids">3.1 Oberfläche eines Rotationsellipsoids</a></li> <li class="toclevel-2 tocsection-5"><a href="#Oberfläche_eines_triaxialen_Ellipsoids">3.2 Oberfläche eines triaxialen Ellipsoids</a></li> </ul> </li> <li class="toclevel-1 tocsection-6"><a href="#Anwendungsbeispiel_zu_den_Formeln">4 Anwendungsbeispiel zu den Formeln</a></li> <li class="toclevel-1 tocsection-7"><a href="#Ebene_Schnitte">5 Ebene Schnitte</a> <ul> <li class="toclevel-2 tocsection-8"><a href="#Eigenschaften">5.1 Eigenschaften</a></li> <li class="toclevel-2 tocsection-9"><a href="#Bestimmung_einer_Schnittellipse">5.2 Bestimmung einer Schnittellipse</a></li> </ul> </li> <li class="toclevel-1 tocsection-10"><a href="#Fadenkonstruktion">6 Fadenkonstruktion</a></li> <li class="toclevel-1 tocsection-11"><a href="#Ellipsoid_in_beliebiger_Lage">7 Ellipsoid in beliebiger Lage</a> <ul> <li class="toclevel-2 tocsection-12"><a href="#Parameterdarstellung_2">7.1 Parameterdarstellung</a></li> <li class="toclevel-2 tocsection-13"><a href="#Ellipsoid_als_Quadrik">7.2 Ellipsoid als Quadrik</a></li> </ul> </li> <li class="toclevel-1 tocsection-14"><a href="#Ellipsoid_in_der_projektiven_Geometrie">8 Ellipsoid in der projektiven Geometrie</a></li> <li class="toclevel-1 tocsection-15"><a href="#Siehe_auch">9 Siehe auch</a></li> <li class="toclevel-1 tocsection-16"><a href="#Weblinks">10 Weblinks</a></li> <li class="toclevel-1 tocsection-17"><a href="#Einzelnachweise">11 Einzelnachweise</a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Parameterdarstellung">Parameterdarstellung</h2>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=1" title="Abschnitt bearbeiten: Parameterdarstellung" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Parameterdarstellung">Quelltext bearbeiten</a>]</div> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Parameterdarstellung" title="Parameterdarstellung">Parameterdarstellung</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Datei:Kugelkoord-def.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Kugelkoord-def.svg/300px-Kugelkoord-def.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Kugelkoord-def.svg/450px-Kugelkoord-def.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Kugelkoord-def.svg/600px-Kugelkoord-def.svg.png 2x" data-file-width="261" data-file-height="261" /></a><figcaption><a href="/wiki/Kugelkoordinaten" title="Kugelkoordinaten">Kugelkoordinaten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r,\theta ,\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,\theta ,\varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/265c0d2c72d017d4a5f3648b6f78bcbf4af63119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.727ex; height:2.676ex;" alt="{\displaystyle r,\theta ,\varphi }"> eines Punktes und kartesisches <a href="/wiki/Koordinatensystem" title="Koordinatensystem">Koordinatensystem</a></figcaption></figure> Die Punkte auf der <a href="/wiki/Einheitskugel" title="Einheitskugel">Einheitskugel</a> können wie folgt parametrisiert werden (siehe <a href="/wiki/Kugelkoordinaten" title="Kugelkoordinaten">Kugelkoordinaten</a>): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{cll}x&=&\sin \theta \cdot \cos \varphi \\y&=&\sin \theta \cdot \sin \varphi \\z&=&\cos \theta \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{cll}x&=&\sin \theta \cdot \cos \varphi \\y&=&\sin \theta \cdot \sin \varphi \\z&=&\cos \theta \end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3099f6bf3df441b2ece586a1de9ddd6d7981c999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:19.565ex; height:9.509ex;" alt="{\displaystyle {\begin{array}{cll}x&=&\sin \theta \cdot \cos \varphi \\y&=&\sin \theta \cdot \sin \varphi \\z&=&\cos \theta \end{array}}}"></dd></dl> Für den <a href="/wiki/Winkel" title="Winkel">Winkel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> (von der z-Achse aus gemessen) gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \theta \leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>θ</mi> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \theta \leq \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f85d6a6521db7f3738e48ea472c8d88f0c4e0c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.782ex; height:2.343ex;" alt="{\displaystyle 0\leq \theta \leq \pi }">. Für den Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> (von der x-Achse aus gemessen) gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \varphi <2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>φ</mi> <mo><</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \varphi <2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07ebffe3181ca98c6933cf63cb31d5247fab3f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.374ex; height:2.676ex;" alt="{\displaystyle 0\leq \varphi <2\pi }">. Skaliert man die einzelnen <a href="/wiki/Koordinatensystem" title="Koordinatensystem">Koordinaten</a> mit den <a href="/wiki/Faktor_(Mathematik)" class="mw-redirect" title="Faktor (Mathematik)">Faktoren</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}">, so ergibt sich eine Parameterdarstellung des Ellipsoids <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{abc}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{abc}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b96d78d2f98308cceced7b04e70afcb0261c7f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.234ex; height:2.509ex;" alt="{\displaystyle E_{abc}}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{cll}x&=&a\cdot \sin \theta \cdot \cos \varphi \\y&=&b\cdot \sin \theta \cdot \sin \varphi \\z&=&c\cdot \cos \theta \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>a</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>b</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>c</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{cll}x&=&a\cdot \sin \theta \cdot \cos \varphi \\y&=&b\cdot \sin \theta \cdot \sin \varphi \\z&=&c\cdot \cos \theta \end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efab44922dddc89496fdd92b23c9635a323dd9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:22.474ex; height:9.509ex;" alt="{\displaystyle {\begin{array}{cll}x&=&a\cdot \sin \theta \cdot \cos \varphi \\y&=&b\cdot \sin \theta \cdot \sin \varphi \\z&=&c\cdot \cos \theta \end{array}}}"></dd></dl> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \theta \leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>θ</mi> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \theta \leq \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f85d6a6521db7f3738e48ea472c8d88f0c4e0c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.782ex; height:2.343ex;" alt="{\displaystyle 0\leq \theta \leq \pi }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \varphi <2\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>φ</mi> <mo><</mo> <mn>2</mn> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \varphi <2\pi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc4ca60655cbea1accc4e20641f1bf82b37293a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.021ex; height:2.676ex;" alt="{\displaystyle 0\leq \varphi <2\pi .}"> <div class="mw-heading mw-heading2"><h2 id="Volumen">Volumen</h2>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=2" title="Abschnitt bearbeiten: Volumen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Volumen">Quelltext bearbeiten</a>]</div> Das <a href="/wiki/Volumen" title="Volumen">Volumen</a> des Ellipsoids <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{abc}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{abc}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b96d78d2f98308cceced7b04e70afcb0261c7f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.234ex; height:2.509ex;" alt="{\displaystyle E_{abc}}"> ist <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\frac {4}{3}}\pi abc.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {4}{3}}\pi abc.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c97b4d4165be7ec43988fe62a95c818b0abf0eb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.097ex; height:5.176ex;" alt="{\displaystyle V={\frac {4}{3}}\pi abc.}"></dd></dl> Eine <a href="/wiki/Kugel" title="Kugel">Kugel</a> mit <a href="/wiki/Radius" title="Radius">Radius</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> hat das Volumen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\tfrac {4}{3}}\pi r^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\tfrac {4}{3}}\pi r^{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13191b552f419d7ade7f65f2047398c17f515991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.626ex; height:3.676ex;" alt="{\displaystyle V={\tfrac {4}{3}}\pi r^{3}.}"> <dl><dt>Herleitung</dt></dl> Der Schnitt des Ellipsoids <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{abc}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{abc}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b96d78d2f98308cceced7b04e70afcb0261c7f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.234ex; height:2.509ex;" alt="{\displaystyle E_{abc}}"> mit einer <a href="/wiki/Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> in der <a href="/wiki/H%C3%B6he_(Geometrie)" title="Höhe (Geometrie)">Höhe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> ist die <a href="/wiki/Ellipse" title="Ellipse">Ellipse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1-{\frac {z^{2}}{c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1-{\frac {z^{2}}{c^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9eb518cf3a62ec823d0e1f16414ee267917844d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.193ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1-{\frac {z^{2}}{c^{2}}}}"> mit den Halbachsen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a'=a{\sqrt {1-{\frac {z^{2}}{c^{2}}}}},\ b'=b{\sqrt {1-{\frac {z^{2}}{c^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo>,</mo> <mtext> </mtext> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a'=a{\sqrt {1-{\frac {z^{2}}{c^{2}}}}},\ b'=b{\sqrt {1-{\frac {z^{2}}{c^{2}}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ee71de0ee0a9ba432831472087f611acf01239" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.25ex; height:7.509ex;" alt="{\displaystyle a'=a{\sqrt {1-{\frac {z^{2}}{c^{2}}}}},\ b'=b{\sqrt {1-{\frac {z^{2}}{c^{2}}}}}}">.</dd></dl> Der <a href="/wiki/Ellipse#Flächeninhalt" title="Ellipse">Flächeninhalt</a> dieser Ellipse ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(z)=\pi a'b'=\pi ab(1-{\frac {z^{2}}{c^{2}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>π</mi> <msup> <mi>a</mi> <mo>′</mo> </msup> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>π</mi> <mi>a</mi> <mi>b</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(z)=\pi a'b'=\pi ab(1-{\frac {z^{2}}{c^{2}}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef32171f83f62d393b43bc51f574c745d7daebc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.118ex; height:6.009ex;" alt="{\displaystyle A(z)=\pi a'b'=\pi ab(1-{\frac {z^{2}}{c^{2}}})}">. Das Volumen ergibt sich dann aus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-c}^{c}A(z)\ \mathrm {d} z=\pi ab\int _{-c}^{c}(1-{\frac {z^{2}}{c^{2}}})\ \mathrm {d} z={\frac {4}{3}}\pi abc.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mi>A</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> <mo>=</mo> <mi>π</mi> <mi>a</mi> <mi>b</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-c}^{c}A(z)\ \mathrm {d} z=\pi ab\int _{-c}^{c}(1-{\frac {z^{2}}{c^{2}}})\ \mathrm {d} z={\frac {4}{3}}\pi abc.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09db04d6c06b44ef5c02ba4906f68f1f77868a26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.129ex; height:6.343ex;" alt="{\displaystyle \int _{-c}^{c}A(z)\ \mathrm {d} z=\pi ab\int _{-c}^{c}(1-{\frac {z^{2}}{c^{2}}})\ \mathrm {d} z={\frac {4}{3}}\pi abc.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Oberfläche">Oberfläche</h2>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=3" title="Abschnitt bearbeiten: Oberfläche" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Oberfläche">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Oberfläche_eines_Rotationsellipsoids">Oberfläche eines Rotationsellipsoids</h3>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=4" title="Abschnitt bearbeiten: Oberfläche eines Rotationsellipsoids" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Oberfläche eines Rotationsellipsoids">Quelltext bearbeiten</a>]</div> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Rotationsellipsoid" title="Rotationsellipsoid">Rotationsellipsoid</a></div> Die <a href="/wiki/Fl%C3%A4che_(Mathematik)" title="Fläche (Mathematik)">Oberfläche</a> eines <a href="/wiki/Abplattung" title="Abplattung">abgeplatteten</a> Rotationsellipsoids <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{aac}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>a</mi> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{aac}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f81c04c72eb2b6fb87773a5b86672fc11f0346e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.399ex; height:2.509ex;" alt="{\displaystyle E_{aac}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/555821d1966fad5b635b19d54e36d88b3623b991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.335ex; height:1.843ex;" alt="{\displaystyle a>c}"> ist <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=2\pi a\left(a+{\frac {c^{2}}{\sqrt {a^{2}-c^{2}}}}\,\operatorname {arsinh} \left({\frac {\sqrt {a^{2}-c^{2}}}{c}}\right)\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>arsinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2\pi a\left(a+{\frac {c^{2}}{\sqrt {a^{2}-c^{2}}}}\,\operatorname {arsinh} \left({\frac {\sqrt {a^{2}-c^{2}}}{c}}\right)\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a6e585d39737c217123d383cfd1a9554367762" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:49.107ex; height:7.509ex;" alt="{\displaystyle A=2\pi a\left(a+{\frac {c^{2}}{\sqrt {a^{2}-c^{2}}}}\,\operatorname {arsinh} \left({\frac {\sqrt {a^{2}-c^{2}}}{c}}\right)\right),}"></dd></dl> die des <a href="/wiki/Verl%C3%A4ngertes_Ellipsoid" title="Verlängertes Ellipsoid">verlängerten Ellipsoids</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c>a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8035c5151ae8e735de542149a111325a3a47a98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.335ex; height:1.843ex;" alt="{\displaystyle c>a}">) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=2\pi a\left(a+{\frac {c^{2}}{\sqrt {c^{2}-a^{2}}}}\,\operatorname {arcsin} \left({\frac {\sqrt {c^{2}-a^{2}}}{c}}\right)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msqrt> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2\pi a\left(a+{\frac {c^{2}}{\sqrt {c^{2}-a^{2}}}}\,\operatorname {arcsin} \left({\frac {\sqrt {c^{2}-a^{2}}}{c}}\right)\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed3af84635ef8dbf61fa61e20fb2d2a9fdebad5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.847ex; height:7.509ex;" alt="{\displaystyle A=2\pi a\left(a+{\frac {c^{2}}{\sqrt {c^{2}-a^{2}}}}\,\operatorname {arcsin} \left({\frac {\sqrt {c^{2}-a^{2}}}{c}}\right)\right).}"></dd></dl> Eine <a href="/wiki/Kugel" title="Kugel">Kugel</a> mit <a href="/wiki/Radius" title="Radius">Radius</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> hat die Oberfläche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=4\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=4\pi r^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca925588619ce34da35b2c2ffb5b267e313d50ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.439ex; height:2.676ex;" alt="{\displaystyle A=4\pi r^{2}}">. <div class="mw-heading mw-heading3"><h3 id="Oberfläche_eines_triaxialen_Ellipsoids">Oberfläche eines triaxialen Ellipsoids</h3>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=5" title="Abschnitt bearbeiten: Oberfläche eines triaxialen Ellipsoids" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Oberfläche eines triaxialen Ellipsoids">Quelltext bearbeiten</a>]</div> Die <a href="/wiki/Fl%C3%A4che_(Mathematik)" title="Fläche (Mathematik)">Oberfläche</a> eines triaxialen Ellipsoids lässt sich nicht mit Hilfe von <a href="/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktionen</a> ausdrücken, die man als <a href="/wiki/Elementare_Funktion" title="Elementare Funktion">elementar</a> ansieht, wie z. B. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arsinh} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arsinh</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arsinh} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e4506c4572caf5a4700de12122e9690f1f9d3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.222ex; height:2.176ex;" alt="{\displaystyle \operatorname {arsinh} }"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627fcadc0495786c9167c5487a1a795bb1940edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.962ex; height:2.176ex;" alt="{\displaystyle \arcsin }"> oben beim <a href="/wiki/Rotationsellipsoid" title="Rotationsellipsoid">Rotationsellipsoid</a>. Die Flächenberechnung gelang <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Adrien-Marie Legendre</a> mit Hilfe der <a href="/wiki/Elliptisches_Integral" class="mw-redirect" title="Elliptisches Integral">elliptischen Integrale</a>. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b>c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> <mo>></mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b>c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3756bee44d7cb7e6221499eedb579fb848d2e5ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.431ex; height:2.176ex;" alt="{\displaystyle a>b>c}">. Schreibt man <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {a}{b}}{\frac {\sqrt {b^{2}-c^{2}}}{\sqrt {a^{2}-c^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {a}{b}}{\frac {\sqrt {b^{2}-c^{2}}}{\sqrt {a^{2}-c^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab15a153ae4e254e313be10367879b547624654d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.721ex; height:7.509ex;" alt="{\displaystyle k={\frac {a}{b}}{\frac {\sqrt {b^{2}-c^{2}}}{\sqrt {a^{2}-c^{2}}}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\arcsin {\frac {\sqrt {a^{2}-c^{2}}}{a}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\arcsin {\frac {\sqrt {a^{2}-c^{2}}}{a}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62ccef22cc8837b73dfe71c565679ff7138106c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.96ex; height:6.176ex;" alt="{\displaystyle \varphi =\arcsin {\frac {\sqrt {a^{2}-c^{2}}}{a}},}"></dd></dl> so lauten die <a href="/wiki/Integralrechnung" title="Integralrechnung">Integrale</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(k,\varphi )=\int _{0}^{\sin \varphi }{\sqrt {\frac {1-k^{2}x^{2}}{1-x^{2}}}}\ \mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(k,\varphi )=\int _{0}^{\sin \varphi }{\sqrt {\frac {1-k^{2}x^{2}}{1-x^{2}}}}\ \mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c050589407f3e034a3bf81c35e1dfb5b14af71f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.028ex; height:7.676ex;" alt="{\displaystyle E(k,\varphi )=\int _{0}^{\sin \varphi }{\sqrt {\frac {1-k^{2}x^{2}}{1-x^{2}}}}\ \mathrm {d} x}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(k,\varphi )=\int _{0}^{\sin \varphi }{\frac {1}{{\sqrt {1-x^{2}}}{\sqrt {1-k^{2}x^{2}}}}}\ \mathrm {d} x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k,\varphi )=\int _{0}^{\sin \varphi }{\frac {1}{{\sqrt {1-x^{2}}}{\sqrt {1-k^{2}x^{2}}}}}\ \mathrm {d} x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07605aedf65ee548cde98378b5490ba6cd21e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.351ex; height:7.009ex;" alt="{\displaystyle F(k,\varphi )=\int _{0}^{\sin \varphi }{\frac {1}{{\sqrt {1-x^{2}}}{\sqrt {1-k^{2}x^{2}}}}}\ \mathrm {d} x.}"></dd></dl> Die Oberfläche hat mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> nach Legendre<a href="#cite_note-1">[1]</a> den Wert <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=2\pi c^{2}+{\frac {2\pi b}{\sqrt {a^{2}-c^{2}}}}\left(c^{2}F(k,\varphi )+(a^{2}-c^{2})E(k,\varphi )\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>b</mi> </mrow> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2\pi c^{2}+{\frac {2\pi b}{\sqrt {a^{2}-c^{2}}}}\left(c^{2}F(k,\varphi )+(a^{2}-c^{2})E(k,\varphi )\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296382cb03c25dee35a4861d297f1577fa65960b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.696ex; height:6.676ex;" alt="{\displaystyle A=2\pi c^{2}+{\frac {2\pi b}{\sqrt {a^{2}-c^{2}}}}\left(c^{2}F(k,\varphi )+(a^{2}-c^{2})E(k,\varphi )\right).}"></dd></dl> Werden die <a href="/wiki/Ausdruck_(Mathematik)" class="mw-redirect" title="Ausdruck (Mathematik)">Ausdrücke</a> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> sowie die <a href="/wiki/Substitution_(Mathematik)" title="Substitution (Mathematik)">Substitutionen</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u:={\frac {\sqrt {a^{2}-c^{2}}}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u:={\frac {\sqrt {a^{2}-c^{2}}}{a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b86ff872f7d16d06038a2249beefe63057b290b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.42ex; height:6.176ex;" alt="{\displaystyle u:={\frac {\sqrt {a^{2}-c^{2}}}{a}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v:={\frac {\sqrt {b^{2}-c^{2}}}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v:={\frac {\sqrt {b^{2}-c^{2}}}{b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04b884b8027c8ddf0c0c08bf5a5258051d8ba819" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.986ex; height:6.343ex;" alt="{\displaystyle v:={\frac {\sqrt {b^{2}-c^{2}}}{b}}}"></dd></dl> in die <a href="/wiki/Gleichung" title="Gleichung">Gleichung</a> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> eingesetzt, so ergibt sich die Schreibweise <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=2\pi c^{2}+2\pi ab\int _{0}^{1}{\frac {1-u^{2}v^{2}x^{2}}{{\sqrt {1-u^{2}x^{2}}}{\sqrt {1-v^{2}x^{2}}}}}\ \mathrm {d} x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>a</mi> <mi>b</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2\pi c^{2}+2\pi ab\int _{0}^{1}{\frac {1-u^{2}v^{2}x^{2}}{{\sqrt {1-u^{2}x^{2}}}{\sqrt {1-v^{2}x^{2}}}}}\ \mathrm {d} x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/706e63a9f32ab8946f60927631f336c019e5b451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.925ex; height:7.009ex;" alt="{\displaystyle A=2\pi c^{2}+2\pi ab\int _{0}^{1}{\frac {1-u^{2}v^{2}x^{2}}{{\sqrt {1-u^{2}x^{2}}}{\sqrt {1-v^{2}x^{2}}}}}\ \mathrm {d} x.}"></dd></dl> Von Knud Thomsen stammt die integralfreie <a href="/wiki/N%C3%A4herungsformel" class="mw-redirect" title="Näherungsformel">Näherungsformel</a><a href="#cite_note-2">[2]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\approx 4\pi \left({\frac {(ab)^{\frac {8}{5}}+(ac)^{\frac {8}{5}}+(bc)^{\frac {8}{5}}}{3}}\right)^{\frac {5}{8}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≈</mo> <mn>4</mn> <mi>π</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>5</mn> </mfrac> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>5</mn> </mfrac> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>8</mn> </mfrac> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\approx 4\pi \left({\frac {(ab)^{\frac {8}{5}}+(ac)^{\frac {8}{5}}+(bc)^{\frac {8}{5}}}{3}}\right)^{\frac {5}{8}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2c7153b99394f3d947433f9024676333615dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:37.406ex; height:10.009ex;" alt="{\displaystyle A\approx 4\pi \left({\frac {(ab)^{\frac {8}{5}}+(ac)^{\frac {8}{5}}+(bc)^{\frac {8}{5}}}{3}}\right)^{\frac {5}{8}}.}"></dd></dl> Die maximale Abweichung vom exakten Resultat beträgt weniger als 1,2 %. Im Grenzfall eines vollständig plattgedrückten Ellipsoids <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(c\to 0\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(c\to 0\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de1877930faca6cf84da308a14a9a13dafad7f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.593ex; height:2.843ex;" alt="{\displaystyle \left(c\to 0\right)}"> streben alle drei angegebenen <a href="/wiki/Formel" title="Formel">Formeln</a> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> gegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi ab,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>a</mi> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi ab,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be44ba692c0e6f70f1873225c138984551e17ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.369ex; height:2.509ex;" alt="{\displaystyle 2\pi ab,}"> den doppelten Wert des <a href="/wiki/Fl%C3%A4cheninhalt" title="Flächeninhalt">Flächeninhalts</a> einer <a href="/wiki/Ellipse" title="Ellipse">Ellipse</a> mit den Halbachsen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">. <div class="mw-heading mw-heading2"><h2 id="Anwendungsbeispiel_zu_den_Formeln">Anwendungsbeispiel zu den Formeln</h2>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=6" title="Abschnitt bearbeiten: Anwendungsbeispiel zu den Formeln" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Anwendungsbeispiel zu den Formeln">Quelltext bearbeiten</a>]</div> Der Planet Jupiter ist wegen der durch die schnelle Rotation wirkenden <a href="/wiki/Zentrifugalkraft" title="Zentrifugalkraft">Zentrifugalkräfte</a> an den Polen deutlich flacher als am Äquator und hat annähernd die Form eines <a href="/wiki/Rotationsellipsoid" title="Rotationsellipsoid">Rotationsellipsoids</a>. Der Jupiter hat den Äquatordurchmesser 142984 km und den Poldurchmesser 133708 km. Also gilt für die Halbachsen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b=71492\ \mathrm {km} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>=</mo> <mn>71492</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b=71492\ \mathrm {km} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e858a072a95f2d728e2680a05414f04495e49bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.981ex; height:2.176ex;" alt="{\displaystyle a=b=71492\ \mathrm {km} }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=66854\ \mathrm {km} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>66854</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=66854\ \mathrm {km} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d9cad00b7aed82ff773c62e6cb58dcdfa05964a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.662ex; height:2.176ex;" alt="{\displaystyle c=66854\ \mathrm {km} }">. Die <a href="/wiki/Masse_(Physik)" title="Masse (Physik)">Masse</a> des Jupiter beträgt etwa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1{,}899\cdot 10^{27}\ \mathrm {kg} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1,899</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>27</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">g</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1{,}899\cdot 10^{27}\ \mathrm {kg} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc31cae6ce29df57053217773cc30c8fe186e9d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.147ex; height:3.009ex;" alt="{\displaystyle 1{,}899\cdot 10^{27}\ \mathrm {kg} }">. Daraus ergibt sich mithilfe der oben genannten <a href="/wiki/Mathematische_Formel" class="mw-redirect" title="Mathematische Formel">Formeln</a> für das <a href="/wiki/Volumen" title="Volumen">Volumen</a>, die mittlere <a href="/wiki/Dichte" title="Dichte">Dichte</a> und die <a href="/wiki/Fl%C3%A4che_(Mathematik)" title="Fläche (Mathematik)">Oberfläche</a>: <ul><li>Volumen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\frac {4}{3}}\cdot \pi \cdot a\cdot b\cdot c\approx 1{,}4313\cdot 10^{15}\ \mathrm {km^{3}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>π</mi> <mo>⋅</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo>≈</mo> <mn>1,431</mn> <mn>3</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <msup> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {4}{3}}\cdot \pi \cdot a\cdot b\cdot c\approx 1{,}4313\cdot 10^{15}\ \mathrm {km^{3}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88f11e3600e53be80d00f666ff67f7eb53e2035f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.403ex; height:5.176ex;" alt="{\displaystyle V={\frac {4}{3}}\cdot \pi \cdot a\cdot b\cdot c\approx 1{,}4313\cdot 10^{15}\ \mathrm {km^{3}} }"></li></ul> <dl><dd>Das ist etwa 1321-mal so viel wie das <a href="/wiki/Volumen" title="Volumen">Volumen</a> der Erde.</dd></dl> <ul><li>Mittlere Dichte: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\frac {m}{V}}={\frac {1{,}899\cdot 10^{27}\ \mathrm {kg} }{1{,}4313\cdot 10^{15}\ \mathrm {km^{3}} }}={\frac {1{,}899\cdot 10^{27}\ \mathrm {kg} }{1{,}4313\cdot 10^{24}\ \mathrm {m^{3}} }}\approx 1327\ \mathrm {kg} /\mathrm {m^{3}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>V</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1,899</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>27</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">g</mi> </mrow> </mrow> <mrow> <mn>1,431</mn> <mn>3</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <msup> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1,899</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>27</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">g</mi> </mrow> </mrow> <mrow> <mn>1,431</mn> <mn>3</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>1327</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {m}{V}}={\frac {1{,}899\cdot 10^{27}\ \mathrm {kg} }{1{,}4313\cdot 10^{15}\ \mathrm {km^{3}} }}={\frac {1{,}899\cdot 10^{27}\ \mathrm {kg} }{1{,}4313\cdot 10^{24}\ \mathrm {m^{3}} }}\approx 1327\ \mathrm {kg} /\mathrm {m^{3}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a867517f6fb3f6cd3a882fc252379e47884fe8c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:62.965ex; height:6.843ex;" alt="{\displaystyle \rho ={\frac {m}{V}}={\frac {1{,}899\cdot 10^{27}\ \mathrm {kg} }{1{,}4313\cdot 10^{15}\ \mathrm {km^{3}} }}={\frac {1{,}899\cdot 10^{27}\ \mathrm {kg} }{1{,}4313\cdot 10^{24}\ \mathrm {m^{3}} }}\approx 1327\ \mathrm {kg} /\mathrm {m^{3}} }"></li></ul> <dl><dd>Der Jupiter hat also insgesamt eine etwas höhere <a href="/wiki/Dichte" title="Dichte">Dichte</a> als Wasser unter <a href="/wiki/Standardbedingungen" title="Standardbedingungen">Standardbedingungen</a>.</dd></dl> <ul><li>Oberfläche: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\approx 4\cdot \pi \cdot \left({\frac {(a\cdot b)^{\frac {8}{5}}+(a\cdot c)^{\frac {8}{5}}+(b\cdot c)^{\frac {8}{5}}}{3}}\right)^{\frac {5}{8}}\approx 6{,}15\cdot 10^{10}\ \mathrm {km^{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≈</mo> <mn>4</mn> <mo>⋅</mo> <mi>π</mi> <mo>⋅</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>5</mn> </mfrac> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>5</mn> </mfrac> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>8</mn> </mfrac> </mrow> </msup> <mo>≈</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>15</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <msup> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\approx 4\cdot \pi \cdot \left({\frac {(a\cdot b)^{\frac {8}{5}}+(a\cdot c)^{\frac {8}{5}}+(b\cdot c)^{\frac {8}{5}}}{3}}\right)^{\frac {5}{8}}\approx 6{,}15\cdot 10^{10}\ \mathrm {km^{2}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8abac1030293e4a2aad3742c59640062ea86d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:63.066ex; height:10.009ex;" alt="{\displaystyle A\approx 4\cdot \pi \cdot \left({\frac {(a\cdot b)^{\frac {8}{5}}+(a\cdot c)^{\frac {8}{5}}+(b\cdot c)^{\frac {8}{5}}}{3}}\right)^{\frac {5}{8}}\approx 6{,}15\cdot 10^{10}\ \mathrm {km^{2}} }"></li></ul> <dl><dd>Das ist etwa 121-mal so viel wie die <a href="/wiki/Fl%C3%A4che_(Mathematik)" title="Fläche (Mathematik)">Oberfläche</a> der Erde.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Ebene_Schnitte">Ebene Schnitte</h2>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=7" title="Abschnitt bearbeiten: Ebene Schnitte" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Ebene Schnitte">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Eigenschaften">Eigenschaften</h3>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=8" title="Abschnitt bearbeiten: Eigenschaften" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Eigenschaften">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Ellipsoid-ebener-Schnitt.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Ellipsoid-ebener-Schnitt.svg/330px-Ellipsoid-ebener-Schnitt.svg.png" decoding="async" width="330" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Ellipsoid-ebener-Schnitt.svg/495px-Ellipsoid-ebener-Schnitt.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/Ellipsoid-ebener-Schnitt.svg/660px-Ellipsoid-ebener-Schnitt.svg.png 2x" data-file-width="636" data-file-height="253" /></a><figcaption>Ebener Schnitt eines Ellipsoids</figcaption></figure> Der Schnitt eines Ellipsoids mit einer <a href="/wiki/Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> ist <ul><li>eine <a href="/wiki/Ellipse" title="Ellipse">Ellipse</a>, falls er wenigstens zwei Punkte enthält,</li> <li>ein <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</a>, falls die Ebene eine <a href="/wiki/Tangentialebene" title="Tangentialebene">Tangentialebene</a> ist,</li> <li>andernfalls <a href="/wiki/Leere_Menge" title="Leere Menge">leer</a>.</li></ul> Der erste Fall folgt aus der Tatsache, dass eine Ebene eine <a href="/wiki/Kugel" title="Kugel">Kugel</a> in einem <a href="/wiki/Kreis" title="Kreis">Kreis</a> schneidet und ein Kreis bei einer <a href="/wiki/Affine_Abbildung" title="Affine Abbildung">affinen Abbildung</a> in eine Ellipse übergeht. Dass einige der Schnittellipsen Kreise sind, ist bei einem <a href="/wiki/Rotationsellipsoid" title="Rotationsellipsoid">Rotationsellipsoid</a> offensichtlich: Alle ebenen Schnitte, die wenigstens 2 Punkte enthalten und deren Ebenen senkrecht zur <a href="/wiki/Rotationsachse" title="Rotationsachse">Rotationsachse</a> sind, sind Kreise. Dass aber auch jedes 3-achsige Ellipsoid Kreise enthält, ist nicht offensichtlich und wird im Artikel <a href="/wiki/Kreisschnittebene" title="Kreisschnittebene">Kreisschnittebene</a> erklärt. Der <a href="/wiki/Umrisskonstruktion" title="Umrisskonstruktion">wahre Umriss</a> eines beliebigen Ellipsoids ist sowohl bei <a href="/wiki/Parallelprojektion" title="Parallelprojektion">Parallelprojektion</a> als auch bei <a href="/wiki/Zentralprojektion" title="Zentralprojektion">Zentralprojektion</a> ein ebener Schnitt, also eine Ellipse (siehe Bilder). <div class="mw-heading mw-heading3"><h3 id="Bestimmung_einer_Schnittellipse">Bestimmung einer Schnittellipse</h3>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=9" title="Abschnitt bearbeiten: Bestimmung einer Schnittellipse" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Bestimmung einer Schnittellipse">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Ellipso-eb-beisp.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Ellipso-eb-beisp.svg/220px-Ellipso-eb-beisp.svg.png" decoding="async" width="220" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Ellipso-eb-beisp.svg/330px-Ellipso-eb-beisp.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Ellipso-eb-beisp.svg/440px-Ellipso-eb-beisp.svg.png 2x" data-file-width="260" data-file-height="244" /></a><figcaption>Ebener Schnitt eines Ellipsoids</figcaption></figure> Gegeben: Ellipsoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68f83fa62cd61acf1ca6e1a1ab024fc7bf317df1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.193ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1}"> und eine Ebene mit der <a href="/wiki/Gleichung" title="Gleichung">Gleichung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{x}x+n_{y}y+n_{z}z=d,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>z</mi> <mo>=</mo> <mi>d</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{x}x+n_{y}y+n_{z}z=d,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dbeb37285f4bab5c5682e8662241b825fd7f28c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.623ex; height:2.843ex;" alt="{\displaystyle n_{x}x+n_{y}y+n_{z}z=d,}"> die das Ellipsoid in einer Ellipse schneidet. Gesucht: Drei <a href="/wiki/Vektor" title="Vektor">Vektoren</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ae526b00782f0150985cd3fa027d82369c0abd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.719ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{0}}"> (<a href="/wiki/Mittelpunkt" title="Mittelpunkt">Mittelpunkt</a>) und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{1},\;{\vec {f}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{1},\;{\vec {f}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50eaca08529d3507b9cb4a288f2447b22d735f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.117ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{1},\;{\vec {f}}_{2}}"> (konjugierte Vektoren) so, dass die Schnittellipse durch die <a href="/wiki/Parameterdarstellung" title="Parameterdarstellung">Parameterdarstellung</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}={\vec {f}}_{0}+{\vec {f}}_{1}\cos t+{\vec {f}}_{2}\sin t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\vec {f}}_{0}+{\vec {f}}_{1}\cos t+{\vec {f}}_{2}\sin t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f436e646b8234d29b0bfa1d3da4d6af584b7aec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.46ex; height:3.509ex;" alt="{\displaystyle {\vec {x}}={\vec {f}}_{0}+{\vec {f}}_{1}\cos t+{\vec {f}}_{2}\sin t}"></dd></dl> beschrieben werden kann (siehe <a href="/wiki/Ellipse#Ellipse_als_affines_Bild_des_Einheitskreises" title="Ellipse">Ellipse</a>). <figure typeof="mw:File/Thumb"><a href="/wiki/Datei:Ellipso-eb-ku.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Ellipso-eb-ku.svg/300px-Ellipso-eb-ku.svg.png" decoding="async" width="300" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Ellipso-eb-ku.svg/450px-Ellipso-eb-ku.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Ellipso-eb-ku.svg/600px-Ellipso-eb-ku.svg.png 2x" data-file-width="331" data-file-height="176" /></a><figcaption>Ebener Schnitt der <a href="/wiki/Einheitskugel" title="Einheitskugel">Einheitskugel</a></figcaption></figure> Lösung: Die Skalierung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u={\frac {x}{a}}\ ,\ v={\frac {y}{b}}\,\ w={\frac {z}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>a</mi> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>b</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mtext> </mtext> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\frac {x}{a}}\ ,\ v={\frac {y}{b}}\,\ w={\frac {z}{c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ad3f9f48f20c6f1741318fd4a59f78aef42c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.662ex; height:5.009ex;" alt="{\displaystyle u={\frac {x}{a}}\ ,\ v={\frac {y}{b}}\,\ w={\frac {z}{c}}}"> führt das Ellipsoid in die <a href="/wiki/Einheitskugel" title="Einheitskugel">Einheitskugel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{2}+v^{2}+w^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{2}+v^{2}+w^{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e817edbffc7ac3943e53331623518824c6985827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.226ex; height:2.843ex;" alt="{\displaystyle u^{2}+v^{2}+w^{2}=1}"> und die gegebene <a href="/wiki/Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> in die Ebene mit der Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{x}au+n_{y}bv+n_{z}cw=d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>a</mi> <mi>u</mi> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>b</mi> <mi>v</mi> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>c</mi> <mi>w</mi> <mo>=</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{x}au+n_{y}bv+n_{z}cw=d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e0bff6e1123556199b205ace41f7c2aa8875fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.758ex; height:2.843ex;" alt="{\displaystyle n_{x}au+n_{y}bv+n_{z}cw=d}"> über. Die <a href="/wiki/Hesse-Normalform" class="mw-redirect" title="Hesse-Normalform">Hesse-Normalform</a> der neuen Ebene sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{u}u+m_{v}v+m_{w}w=\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mi>u</mi> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mi>v</mi> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mi>w</mi> <mo>=</mo> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{u}u+m_{v}v+m_{w}w=\delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be81b2303334464ce977ec96cb9f383781175df2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.682ex; height:2.676ex;" alt="{\displaystyle m_{u}u+m_{v}v+m_{w}w=\delta }"> mit dem <a href="/wiki/Normaleneinheitsvektor" class="mw-redirect" title="Normaleneinheitsvektor">Normaleneinheitsvektor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {m}}=(m_{u},m_{v},m_{w})^{T}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {m}}=(m_{u},m_{v},m_{w})^{T}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24adf2cfcc1a98fe2de024244e2359892c3c742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.784ex; height:3.176ex;" alt="{\displaystyle {\vec {m}}=(m_{u},m_{v},m_{w})^{T}.}"> Dann ist der <a href="/wiki/Mittelpunkt" title="Mittelpunkt">Mittelpunkt</a> des Schnittkreises <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{0}=\delta \;{\vec {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>δ</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{0}=\delta \;{\vec {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/499c361276f3892ce36455affdf7b0ce4729558f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.11ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{0}=\delta \;{\vec {m}}}"> und dessen <a href="/wiki/Radius" title="Radius">Radius</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\sqrt {1-\delta ^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\sqrt {1-\delta ^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb923d1700dbfc63d1be0c67dc2df2d29211e62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.382ex; height:3.509ex;" alt="{\displaystyle \rho ={\sqrt {1-\delta ^{2}}}.}"> Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{w}=\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{w}=\pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6cb60d3fdd3c0213f7b4ed87e532feaddcfbefd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.518ex; height:2.509ex;" alt="{\displaystyle m_{w}=\pm 1}"> ist, sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{1}=(\rho ,0,0)^{T},\ {\vec {e}}_{2}=(0,\rho ,0)^{T}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>ρ</mi> <mo>,</mo> <mn>0</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{1}=(\rho ,0,0)^{T},\ {\vec {e}}_{2}=(0,\rho ,0)^{T}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b216df8df7fd5d6a0e24beb40fe7408d310aa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.6ex; height:3.176ex;" alt="{\displaystyle {\vec {e}}_{1}=(\rho ,0,0)^{T},\ {\vec {e}}_{2}=(0,\rho ,0)^{T}.}"> (Die Ebene ist horizontal!) Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{w}\neq \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo>≠</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{w}\neq \pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d8da044a7e99b18966263ec8b4b24bb9fa44b70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.518ex; height:2.676ex;" alt="{\displaystyle m_{w}\neq \pm 1}"> ist, sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{1}=\rho \,{\frac {(m_{v},-m_{u},0)^{T}}{\sqrt {m_{u}^{2}+m_{v}^{2}}}},\ {\vec {e}}_{2}={\vec {m}}\times {\vec {e}}_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>,</mo> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>,</mo> <mn>0</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mrow> <msqrt> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{1}=\rho \,{\frac {(m_{v},-m_{u},0)^{T}}{\sqrt {m_{u}^{2}+m_{v}^{2}}}},\ {\vec {e}}_{2}={\vec {m}}\times {\vec {e}}_{1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5f9bbc471dca0b52e006382d448eb6d9784de24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.116ex; height:7.176ex;" alt="{\displaystyle {\vec {e}}_{1}=\rho \,{\frac {(m_{v},-m_{u},0)^{T}}{\sqrt {m_{u}^{2}+m_{v}^{2}}}},\ {\vec {e}}_{2}={\vec {m}}\times {\vec {e}}_{1}.}"> Die <a href="/wiki/Vektor" title="Vektor">Vektoren</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{1},{\vec {e}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{1},{\vec {e}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a03096cb1626c143ad3bd9473c694b2ac889ca85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.589ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{1},{\vec {e}}_{2}}"> sind in jedem Fall zwei in der Schnittebene liegende orthogonale Vektoren der Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"> (Kreisradius), d. h., der Schnittkreis wird durch die Parameterdarstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}={\vec {e}}_{0}+{\vec {e}}_{1}\cos t+{\vec {e}}_{2}\sin t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}={\vec {e}}_{0}+{\vec {e}}_{1}\cos t+{\vec {e}}_{2}\sin t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/028ef0ef0f3579f4a4ebcedc013c49cc1809e797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.135ex; height:2.676ex;" alt="{\displaystyle {\vec {u}}={\vec {e}}_{0}+{\vec {e}}_{1}\cos t+{\vec {e}}_{2}\sin t}"> beschrieben. Macht man nun die obige Skalierung (<a href="/wiki/Affine_Abbildung" title="Affine Abbildung">affine Abbildung</a>) rückgängig, so wird die Einheitskugel wieder zum gegebenen Ellipsoid und man erhält aus den Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{0},{\vec {e}}_{1},{\vec {e}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{0},{\vec {e}}_{1},{\vec {e}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12fdd38bfc6ef7a56ee7f64b21cb60eca51be7da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.9ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{0},{\vec {e}}_{1},{\vec {e}}_{2}}"> die gesuchten Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{0},{\vec {f}}_{1},{\vec {f}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{0},{\vec {f}}_{1},{\vec {f}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0583853158e69e51d160d215326363e456e6b23a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.225ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{0},{\vec {f}}_{1},{\vec {f}}_{2}}">, mit denen man die Schnittellipse beschreiben kann. Wie man daraus die <a href="/wiki/Scheitelpunkt" title="Scheitelpunkt">Scheitelpunkte</a> der Ellipse und damit ihre Halbachsen bestimmt, wird unter <a href="/wiki/Ellipse#Ellipse_als_affines_Bild_des_Einheitskreises" title="Ellipse">Ellipse</a> erklärt. Beispiel: Die Bilder gehören zu dem Beispiel mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=4,\;b=5,\;c=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>b</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>c</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=4,\;b=5,\;c=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda8dffad60eb97206db03615335762ea957c189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.375ex; height:2.509ex;" alt="{\displaystyle a=4,\;b=5,\;c=3}"> und der Schnittebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y+z=5.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>5.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y+z=5.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe89fdc93db66dbe99012fa4c6d4d0416003c66e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.162ex; height:2.509ex;" alt="{\displaystyle x+y+z=5.}"> Das Bild des Ellipsoidschnittes ist eine senkrechte <a href="/wiki/Parallelprojektion" title="Parallelprojektion">Parallelprojektion</a> auf eine Ebene parallel zur Schnittebene, d. h., die Ellipse erscheint bis auf eine uniforme Skalierung in wahrer Gestalt. Man beachte, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ae526b00782f0150985cd3fa027d82369c0abd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.719ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{0}}"> hier im Gegensatz zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686ab9780c489d02208a20dfdccb4c0365f415e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{0}}"> nicht auf der Schnittebene senkrecht steht. Die <a href="/wiki/Vektor" title="Vektor">Vektoren</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{1},\;{\vec {f}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{1},\;{\vec {f}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50eaca08529d3507b9cb4a288f2447b22d735f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.117ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{1},\;{\vec {f}}_{2}}"> sind hier im Gegensatz zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{1},\;{\vec {e}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{1},\;{\vec {e}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc72c0f6efc243a3fd4ef49c8b81c62a2b7cb95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.234ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{1},\;{\vec {e}}_{2}}">nicht <a href="/wiki/Orthogonalit%C3%A4t" title="Orthogonalität">orthogonal</a>. <div class="mw-heading mw-heading2"><h2 id="Fadenkonstruktion">Fadenkonstruktion</h2>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=10" title="Abschnitt bearbeiten: Fadenkonstruktion" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Fadenkonstruktion">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Ellipse-gaertner-k.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Ellipse-gaertner-k.svg/220px-Ellipse-gaertner-k.svg.png" decoding="async" width="220" height="178" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Ellipse-gaertner-k.svg/330px-Ellipse-gaertner-k.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Ellipse-gaertner-k.svg/440px-Ellipse-gaertner-k.svg.png 2x" data-file-width="293" data-file-height="237" /></a><figcaption>Fadenkonstruktion einer <a href="/wiki/Ellipse" title="Ellipse">Ellipse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |S_{1}S_{2}|=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |S_{1}S_{2}|=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ab5be77b33d1dd06305901fb6143105556d8fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.705ex; height:2.843ex;" alt="{\displaystyle |S_{1}S_{2}|=}"> Länge des Fades (rot)</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Fokalks-ellipsoid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Fokalks-ellipsoid.svg/260px-Fokalks-ellipsoid.svg.png" decoding="async" width="260" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Fokalks-ellipsoid.svg/390px-Fokalks-ellipsoid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Fokalks-ellipsoid.svg/520px-Fokalks-ellipsoid.svg.png 2x" data-file-width="331" data-file-height="299" /></a><figcaption>Fadenkonstruktion eines Ellipsoids</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Fokalks-ellipsoid-xyz.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Fokalks-ellipsoid-xyz.svg/260px-Fokalks-ellipsoid-xyz.svg.png" decoding="async" width="260" height="425" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Fokalks-ellipsoid-xyz.svg/390px-Fokalks-ellipsoid-xyz.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Fokalks-ellipsoid-xyz.svg/520px-Fokalks-ellipsoid-xyz.svg.png 2x" data-file-width="295" data-file-height="482" /></a><figcaption>Fadenkonstruktion: Bestimmung der Halbachsen</figcaption></figure> Die Fadenkonstruktion eines Ellipsoids ist eine Übertragung der Idee der <a href="/wiki/Ellipse#Gärtnerkonstruktion" title="Ellipse">Gärtnerkonstruktion</a> einer <a href="/wiki/Ellipse" title="Ellipse">Ellipse</a> (siehe Abbildung). Eine Fadenkonstruktion eines <a href="/wiki/Rotationsellipsoid" title="Rotationsellipsoid">Rotationsellipsoids</a> ergibt sich durch Konstruktion der Meridian-Ellipsen mit Hilfe eines Fadens. Punkte eines 3-achsigen Ellipsoids mit Hilfe eines gespannten Fadens zu konstruieren ist etwas komplizierter. <a href="/wiki/Wolfgang_Boehm_(Mathematiker)" title="Wolfgang Boehm (Mathematiker)">Wolfgang Boehm</a> schreibt in dem Artikel Die Fadenkonstruktion der Flächen zweiter Ordnung<a href="#cite_note-3">[3]</a> die Grundidee der Fadenkonstruktion eines Ellipsoids dem schottischen Physiker <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> (1868) zu. <a href="/wiki/Otto_Staude" title="Otto Staude">Otto Staude</a> hat in Arbeiten 1882, 1886, 1898<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a> die Fadenkonstruktion dann auf <a href="/wiki/Quadrik" title="Quadrik">Quadriken</a> verallgemeinert. Die Fadenkonstruktion für Ellipsoide und <a href="/wiki/Hyperboloid" title="Hyperboloid">Hyperboloide</a> wird auch in dem Buch Anschauliche Geometrie<a href="#cite_note-7">[7]</a> von <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> und <a href="/wiki/Stefan_Cohn-Vossen" title="Stefan Cohn-Vossen">Stefan Cohn-Vossen</a> beschrieben. Auch <a href="/wiki/Sebastian_Finsterwalder" title="Sebastian Finsterwalder">Sebastian Finsterwalder</a> beschäftigte sich 1886 mit diesem Thema.<a href="#cite_note-8">[8]</a> <dl><dt>Konstruktionsschritte</dt> <dd>(1) Man wähle eine <a href="/wiki/Ellipse" title="Ellipse">Ellipse</a> und eine <a href="/wiki/Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbel</a>, die ein Paar von <a href="/wiki/Fokalkegelschnitt" title="Fokalkegelschnitt">Fokalkegelschnitten</a> bilden: <dl><dd>Ellipse: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(\varphi )=(a\cos \varphi ,b\sin \varphi ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(\varphi )=(a\cos \varphi ,b\sin \varphi ,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4bb14d1e11ea240333124043a370499b4255728" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.026ex; height:2.843ex;" alt="{\displaystyle E(\varphi )=(a\cos \varphi ,b\sin \varphi ,0)}"> und</dd> <dd>Hyperbel: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(\psi )=(c\cosh \psi ,0,b\sinh \psi )\quad ,\ c^{2}=a^{2}-b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mi>cosh</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>b</mi> <mi>sinh</mi> <mo>⁡</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mo>,</mo> <mtext> </mtext> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(\psi )=(c\cosh \psi ,0,b\sinh \psi )\quad ,\ c^{2}=a^{2}-b^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6445d0f14cbb270759dbcaa95056793c7db2db93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.928ex; height:3.176ex;" alt="{\displaystyle H(\psi )=(c\cosh \psi ,0,b\sinh \psi )\quad ,\ c^{2}=a^{2}-b^{2}}"></dd></dl></dd> <dd>mit den <a href="/wiki/Scheitelpunkt" title="Scheitelpunkt">Scheitelpunkten</a> und <a href="/wiki/Brennpunkt_(Geometrie)" title="Brennpunkt (Geometrie)">Brennpunkten</a> der Ellipse <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}=(a,0,0),\ F_{1}=(c,0,0),\ F_{2}=(-c,0,0),\ S_{2}=(-a,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>c</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}=(a,0,0),\ F_{1}=(c,0,0),\ F_{2}=(-c,0,0),\ S_{2}=(-a,0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77564e7b919fe60a480a1de44b3480cc03996669" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.191ex; height:2.843ex;" alt="{\displaystyle S_{1}=(a,0,0),\ F_{1}=(c,0,0),\ F_{2}=(-c,0,0),\ S_{2}=(-a,0,0)}"></dd></dl></dd> <dd>und einen Faden (in der Abbildung Bild rot) der Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}">.</dd> <dd>(2) Man befestige das eine Ende des Fadens im Scheitelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}"> und das andere Ende im Brennpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd17e0779153d765b40ebef91533489b87b2e37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{2}}">. Der Faden wird in einem <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> so gespannt gehalten, dass der Faden von hinten auf der Hyperbel und von vorn auf der Ellipse gleiten kann (siehe Abbildung). Der Faden geht über denjenigen Hyperbelpunkt, mit dem die Entfernung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}"> über einen Hyperbelpunkt minimal wird. Analoges gilt für den Fadenteil von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd17e0779153d765b40ebef91533489b87b2e37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{2}}"> über einen Ellipsenpunkt.</dd> <dd>(3) Wählt man den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> so, dass er positive y- und z-<a href="/wiki/Koordinatensystem" title="Koordinatensystem">Koordinaten</a> hat, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> ein Punkt des Ellipsoids mit der <a href="/wiki/Gleichung" title="Gleichung">Gleichung</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{2}}{r_{x}^{2}}}+{\frac {y^{2}}{r_{y}^{2}}}+{\frac {z^{2}}{r_{z}^{2}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}}{r_{x}^{2}}}+{\frac {y^{2}}{r_{y}^{2}}}+{\frac {z^{2}}{r_{z}^{2}}}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e500c7e7f2326ce8d813647ade49e42b0a7ecbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.193ex; height:6.676ex;" alt="{\displaystyle {\frac {x^{2}}{r_{x}^{2}}}+{\frac {y^{2}}{r_{y}^{2}}}+{\frac {z^{2}}{r_{z}^{2}}}=1}"> und</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{x}={\frac {1}{2}}(l-a+c)\ ,\quad r_{y}={\sqrt {r_{x}^{2}-c^{2}}}\ ,\quad r_{z}={\sqrt {r_{x}^{2}-a^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>l</mi> <mo>−</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{x}={\frac {1}{2}}(l-a+c)\ ,\quad r_{y}={\sqrt {r_{x}^{2}-c^{2}}}\ ,\quad r_{z}={\sqrt {r_{x}^{2}-a^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c897d5ca525a5927fb32dd362e4f32d83da9250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.72ex; height:5.176ex;" alt="{\displaystyle r_{x}={\frac {1}{2}}(l-a+c)\ ,\quad r_{y}={\sqrt {r_{x}^{2}-c^{2}}}\ ,\quad r_{z}={\sqrt {r_{x}^{2}-a^{2}}}.}"></dd></dl></dd> <dd>(4) Die restlichen Punkte des Ellipsoids erhält man durch geeignetes Umspannen des Fadens an den Fokalkegelschnitten.</dd></dl> Die Gleichungen für die Halbachsen des erzeugten Ellipsoids ergeben sich, wenn man den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> in die beiden <a href="/wiki/Scheitelpunkt" title="Scheitelpunkt">Scheitelpunkte</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=(0,r_{y},0),\ Z=(0,0,r_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mi>Z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=(0,r_{y},0),\ Z=(0,0,r_{z})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a74350d97ed31d0b935c18bd38c323af027fa2ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.818ex; height:3.009ex;" alt="{\displaystyle Y=(0,r_{y},0),\ Z=(0,0,r_{z})}"> fallen lässt: Aus der unteren Zeichnung erkennt man, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1},F_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1},F_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddbaefad7d000285a069031a623cdc454e4b79e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.132ex; height:2.509ex;" alt="{\displaystyle F_{1},F_{2}}"> auch die <a href="/wiki/Brennpunkt_(Geometrie)" title="Brennpunkt (Geometrie)">Brennpunkte</a> der Äquatorellipse sind. D. h.: Die Äquatorellipse ist <a href="/wiki/Konfokale_Kegelschnitte" title="Konfokale Kegelschnitte">konfokal</a> zur gegebenen Fokalellipse. Also ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l=2r_{x}+(a-c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=2r_{x}+(a-c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfbf92c9f5a47d8b4560c1f6d71a5dae0c739f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.902ex; height:2.843ex;" alt="{\displaystyle l=2r_{x}+(a-c)}">, woraus sich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{x}={\frac {1}{2}}(l-a+c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>l</mi> <mo>−</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{x}={\frac {1}{2}}(l-a+c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b1c38da7d9fec7a7fde6b247eef62810a066b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.738ex; height:5.176ex;" alt="{\displaystyle r_{x}={\frac {1}{2}}(l-a+c)}"> ergibt. Ferner erkennt man, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{y}^{2}=r_{x}^{2}-c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{y}^{2}=r_{x}^{2}-c^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b5f783ce41cf0779274a4257d52c3e92c9e129b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.324ex; height:3.343ex;" alt="{\displaystyle r_{y}^{2}=r_{x}^{2}-c^{2}}"> ist. Aus der oberen Zeichnung ergibt sich: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1},S_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1},S_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05289e5891f97d54bdc946771362aedd6bb76917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.992ex; height:2.509ex;" alt="{\displaystyle S_{1},S_{2}}"> sind die Brennpunkte der Ellipse in der x-z-<a href="/wiki/Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> und es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{z}^{2}=r_{x}^{2}-a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{z}^{2}=r_{x}^{2}-a^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/404bb9a4d9bf81152882db483b3e8e573192f5b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.547ex; height:3.009ex;" alt="{\displaystyle r_{z}^{2}=r_{x}^{2}-a^{2}}">. Umkehrung: Möchte man ein durch seine Gleichung gegebenes 3-achsiges Ellipsoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}"> mit den Halbachsen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{x},r_{y},r_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{x},r_{y},r_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c32b2530cd4e81bdce43b0aab48beb71faed0c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.437ex; height:2.343ex;" alt="{\displaystyle r_{x},r_{y},r_{z}}"> konstruieren, so lassen sich aus den Gleichungen im Schritt (3) die für die Fadenkonstruktion nötigen Parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,l}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55e5e377020203e03a8c19bc70bd84e6eb3fc8d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.989ex; height:2.509ex;" alt="{\displaystyle a,b,l}"> berechnen. Für die folgenden Überlegungen wichtig sind die Gleichungen <dl><dd>(5)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\ r_{x}^{2}-r_{y}^{2}=c^{2},\quad r_{x}^{2}-r_{z}^{2}=a^{2},\quad r_{y}^{2}-r_{z}^{2}=a^{2}-c^{2}=b^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mtext> </mtext> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\ r_{x}^{2}-r_{y}^{2}=c^{2},\quad r_{x}^{2}-r_{z}^{2}=a^{2},\quad r_{y}^{2}-r_{z}^{2}=a^{2}-c^{2}=b^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db58a40933193d1dc9168374d0d6eb6277ff6286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:56.584ex; height:3.343ex;" alt="{\displaystyle :\ r_{x}^{2}-r_{y}^{2}=c^{2},\quad r_{x}^{2}-r_{z}^{2}=a^{2},\quad r_{y}^{2}-r_{z}^{2}=a^{2}-c^{2}=b^{2}.}"></dd></dl> Konfokale Ellipsoide: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {\overline {E}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> <mo accent="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {\overline {E}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eda78acf40a35f89df13b66d0407d3b940acab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.48ex; height:3.009ex;" alt="{\displaystyle {\mathcal {\overline {E}}}}"> ein zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}"> <a href="/wiki/Konfokale_Quadriken" class="mw-redirect" title="Konfokale Quadriken">konfokales Ellipsoid</a> mit den Quadraten der Halbachsen <dl><dd>(6)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\ {\overline {r}}_{x}^{2}=r_{x}^{2}-\lambda ,\quad {\overline {r}}_{y}^{2}=r_{y}^{2}-\lambda ,\quad {\overline {r}}_{z}^{2}=r_{z}^{2}-\lambda ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mtext> </mtext> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mi>λ</mi> <mo>,</mo> <mspace width="1em" /> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mi>λ</mi> <mo>,</mo> <mspace width="1em" /> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mi>λ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\ {\overline {r}}_{x}^{2}=r_{x}^{2}-\lambda ,\quad {\overline {r}}_{y}^{2}=r_{y}^{2}-\lambda ,\quad {\overline {r}}_{z}^{2}=r_{z}^{2}-\lambda ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a67e686937096ddfda97a0a58737c78dde604454" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:44.314ex; height:3.676ex;" alt="{\displaystyle :\ {\overline {r}}_{x}^{2}=r_{x}^{2}-\lambda ,\quad {\overline {r}}_{y}^{2}=r_{y}^{2}-\lambda ,\quad {\overline {r}}_{z}^{2}=r_{z}^{2}-\lambda ,}"></dd></dl> so erkennt man aus den vorigen Gleichungen, dass die zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {\overline {E}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> <mo accent="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {\overline {E}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eda78acf40a35f89df13b66d0407d3b940acab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.48ex; height:3.009ex;" alt="{\displaystyle {\mathcal {\overline {E}}}}"> gehörigen Fokalkegelschnitte für die Fadenerzeugung dieselben Halbachsen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"> wie die von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}"> besitzen. Deshalb fasst man – analog der Rolle der <a href="/wiki/Brennpunkt_(Geometrie)" title="Brennpunkt (Geometrie)">Brennpunkte</a> bei der Fadenerzeugung einer Ellipse – die Fokalkegelschnitte eines 3-achsigen Ellipsoids als deren unendlich viele Brennpunkte auf und nennt sie Fokalkurven des Ellipsoids.<a href="#cite_note-9">[9]</a> Auch die Umkehrung ist richtig: Wählt man einen zweiten Faden der Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {l}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>l</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {l}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/334e4ddd1db99bab51300576cbdc346a2f033c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.808ex; height:3.009ex;" alt="{\displaystyle {\overline {l}}}"> und setzt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =r_{x}^{2}-{\overline {r}}_{x}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =r_{x}^{2}-{\overline {r}}_{x}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a65ecfcd4f2649ea29fd40dc6a471e1c32d873f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.851ex; height:3.343ex;" alt="{\displaystyle \lambda =r_{x}^{2}-{\overline {r}}_{x}^{2}}">, so gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {r}}_{y}^{2}=r_{y}^{2}-\lambda ,\ {\overline {r}}_{z}^{2}=r_{z}^{2}-\lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mi>λ</mi> <mo>,</mo> <mtext> </mtext> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mi>λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {r}}_{y}^{2}=r_{y}^{2}-\lambda ,\ {\overline {r}}_{z}^{2}=r_{z}^{2}-\lambda .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74644faf45c5610bdcb0d0b7d1c81c7b7c4a0d2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.491ex; height:3.676ex;" alt="{\displaystyle {\overline {r}}_{y}^{2}=r_{y}^{2}-\lambda ,\ {\overline {r}}_{z}^{2}=r_{z}^{2}-\lambda .}"> D. h.: Die beiden Ellipsoide sind konfokal. Grenzfall <a href="/wiki/Rotationsellipsoid" title="Rotationsellipsoid">Rotationsellipsoid</a>: Im Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1beb3f1b1ad87e99791ba713839204a88b27239a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.335ex; height:1.676ex;" alt="{\displaystyle a=c}"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}=F_{1},\;S_{2}=F_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}=F_{1},\;S_{2}=F_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb1ec6f346234c478e40598a35e4f245a1ad1587" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.932ex; height:2.509ex;" alt="{\displaystyle S_{1}=F_{1},\;S_{2}=F_{2}}">, d. h., die Fokalellipse artet in eine Strecke und die <a href="/wiki/Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Hyperbel</a> in zwei <a href="/wiki/Strahl_(Geometrie)" title="Strahl (Geometrie)">Strahlen</a> auf der x-Achse aus. Das Ellipsoid ist dann ein Rotationsellipsoid mit der x-Achse als <a href="/wiki/Rotationsachse" title="Rotationsachse">Rotationsachse</a>. Es ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{x}={\tfrac {l}{2}},\;r_{y}=r_{z}={\sqrt {r_{x}^{2}-c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>l</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{x}={\tfrac {l}{2}},\;r_{y}=r_{z}={\sqrt {r_{x}^{2}-c^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/544f97db5787e4f07d691e555f0c9aa44a33db5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:28.448ex; height:4.843ex;" alt="{\displaystyle r_{x}={\tfrac {l}{2}},\;r_{y}=r_{z}={\sqrt {r_{x}^{2}-c^{2}}}}">. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Ellipsoid-pk-zk.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Ellipsoid-pk-zk.svg/330px-Ellipsoid-pk-zk.svg.png" decoding="async" width="330" height="288" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Ellipsoid-pk-zk.svg/495px-Ellipsoid-pk-zk.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Ellipsoid-pk-zk.svg/660px-Ellipsoid-pk-zk.svg.png 2x" data-file-width="777" data-file-height="678" /></a><figcaption>Unten: <a href="/wiki/Parallelprojektion" title="Parallelprojektion">Parallelprojektion</a> und <a href="/wiki/Zentralprojektion" title="Zentralprojektion">Zentralprojektion</a> eines 3-achsigen Ellipsoids, wo der scheinbare Umriss ein <a href="/wiki/Kreis" title="Kreis">Kreis</a> ist</figcaption></figure> Eigenschaften der Fokalhyperbel: Betrachtet man ein Ellipsoid von einem außerhalb gelegenen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> auf der zugehörigen Fokalhyperbel aus, so erscheint der Umriss des Ellipsoids als <a href="/wiki/Kreis" title="Kreis">Kreis</a>. Oder, anders ausgedrückt: Die <a href="/wiki/Tangente" title="Tangente">Tangenten</a> des Ellipsoids durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> bilden einen senkrechten <a href="/wiki/Kreiskegel" class="mw-redirect" title="Kreiskegel">Kreiskegel</a>, dessen Rotationsachse Tangente in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> an die Hyperbel ist.<a href="#cite_note-10">[10]</a><a href="#cite_note-11">[11]</a> Lässt man den Augpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> ins Unendliche laufen, entsteht die Ansicht einer senkrechten <a href="/wiki/Parallelprojektion" title="Parallelprojektion">Parallelprojektion</a> mit einer <a href="/wiki/Asymptote" title="Asymptote">Asymptote</a> der Fokalhyperbel als Projektionsrichtung. Die <a href="/wiki/Umrisskonstruktion" title="Umrisskonstruktion">wahre Umrisskurve</a> auf dem Ellipsoid ist im Allgemeinen kein Kreis. In der Abbildung ist unten links eine Parallelprojektion eines 3-achsigen Ellipsoids (Halbachsen: 60, 40, 30) in Richtung einer Asymptote und unten rechts eine <a href="/wiki/Zentralprojektion" title="Zentralprojektion">Zentralprojektion</a> mit Zentrum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> auf der Fokalhyperbel und Hauptpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> auf der Tangente an die Hyperbel in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> dargestellt. In beiden Projektionen sind die scheinbaren Umrisse Kreise. Links ist das Bild des <a href="/wiki/Koordinatenursprung" class="mw-redirect" title="Koordinatenursprung">Koordinatenursprungs</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"> der <a href="/wiki/Mittelpunkt" title="Mittelpunkt">Mittelpunkt</a> des Umrisskreises, rechts ist der Hauptpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> der Mittelpunkt. Die Fokalhyperbel eines Ellipsoids schneidet das Ellipsoid in seinen vier <a href="/wiki/Nabelpunkt" class="mw-redirect" title="Nabelpunkt">Nabelpunkten</a>.<a href="#cite_note-12">[12]</a> Eigenschaft der Fokalellipse: Die Fokalellipse mit ihrem Inneren kann als <a href="/wiki/Grenzfl%C3%A4che" title="Grenzfläche">Grenzfläche</a> der durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"> bestimmten Schar von konfokalen Ellipsoide für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{z}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{z}\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06e0481a91dddd073574ecb497b7df55497b0225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.827ex; height:2.509ex;" alt="{\displaystyle r_{z}\to 0}"> als unendlich dünnes Ellipsoid angesehen werden. Es ist dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{x}=a,\;r_{y}=b,\;l=3a-c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>b</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>l</mi> <mo>=</mo> <mn>3</mn> <mi>a</mi> <mo>−</mo> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{x}=a,\;r_{y}=b,\;l=3a-c.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8ddebb473ee1ba0e915985a4d29d7ed28b3edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.78ex; height:2.843ex;" alt="{\displaystyle r_{x}=a,\;r_{y}=b,\;l=3a-c.}"> <div class="mw-heading mw-heading2"><h2 id="Ellipsoid_in_beliebiger_Lage">Ellipsoid in beliebiger Lage</h2>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=11" title="Abschnitt bearbeiten: Ellipsoid in beliebiger Lage" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Ellipsoid in beliebiger Lage">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Ellipsoid-affin.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Ellipsoid-affin.svg/330px-Ellipsoid-affin.svg.png" decoding="async" width="330" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Ellipsoid-affin.svg/495px-Ellipsoid-affin.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Ellipsoid-affin.svg/660px-Ellipsoid-affin.svg.png 2x" data-file-width="366" data-file-height="157" /></a><figcaption>Ellipsoid als <a href="/wiki/Affine_Abbildung" title="Affine Abbildung">affines</a> Bild der <a href="/wiki/Einheitskugel" title="Einheitskugel">Einheitskugel</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Parameterdarstellung_2">Parameterdarstellung</h3>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=12" title="Abschnitt bearbeiten: Parameterdarstellung" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Parameterdarstellung">Quelltext bearbeiten</a>]</div> Eine <a href="/wiki/Affine_Abbildung" title="Affine Abbildung">affine Abbildung</a> lässt sich durch eine <a href="/wiki/Parallelverschiebung" title="Parallelverschiebung">Parallelverschiebung</a> um <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ae526b00782f0150985cd3fa027d82369c0abd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.719ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{0}}"> und eine <a href="/wiki/Regul%C3%A4re_Matrix" title="Reguläre Matrix">reguläre</a> 3×3-<a href="/wiki/Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> beschreiben: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}\mapsto {\vec {f}}_{0}+A{\vec {x}}={\vec {f}}_{0}+x{\vec {f}}_{1}+y{\vec {f}}_{2}+z{\vec {f}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">↦</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>z</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}\mapsto {\vec {f}}_{0}+A{\vec {x}}={\vec {f}}_{0}+x{\vec {f}}_{1}+y{\vec {f}}_{2}+z{\vec {f}}_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5352c98396678906bce3d9b56b3c1f7acc7b433c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.645ex; height:3.509ex;" alt="{\displaystyle {\vec {x}}\mapsto {\vec {f}}_{0}+A{\vec {x}}={\vec {f}}_{0}+x{\vec {f}}_{1}+y{\vec {f}}_{2}+z{\vec {f}}_{3}}">,</dd></dl> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca9be64134bc95772a521100b1190af3d2fee43f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.225ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}}"> die Spaltenvektoren der Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> sind. Die <a href="/wiki/Parameterdarstellung" title="Parameterdarstellung">Parameterdarstellung</a> eines beliebigen Ellipsoids ergibt sich aus der obigen Parameterdarstellung der <a href="/wiki/Einheitskugel" title="Einheitskugel">Einheitskugel</a> und der Beschreibung einer affinen Abbildung: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}(\theta ,\varphi )={\vec {f}}_{0}+{\vec {f}}_{1}\cos \theta \cos \varphi +{\vec {f}}_{2}\cos \theta \sin \varphi +{\vec {f}}_{3}\sin \theta ,\quad -\pi /2\leq \theta \leq \pi /2,\ 0\leq \varphi <2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> <mspace width="1em" /> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>≤</mo> <mi>θ</mi> <mo>≤</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo>≤</mo> <mi>φ</mi> <mo><</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}(\theta ,\varphi )={\vec {f}}_{0}+{\vec {f}}_{1}\cos \theta \cos \varphi +{\vec {f}}_{2}\cos \theta \sin \varphi +{\vec {f}}_{3}\sin \theta ,\quad -\pi /2\leq \theta \leq \pi /2,\ 0\leq \varphi <2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74b842756cf08dad99864ec536d631f31579692e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:87.261ex; height:3.509ex;" alt="{\displaystyle {\vec {x}}(\theta ,\varphi )={\vec {f}}_{0}+{\vec {f}}_{1}\cos \theta \cos \varphi +{\vec {f}}_{2}\cos \theta \sin \varphi +{\vec {f}}_{3}\sin \theta ,\quad -\pi /2\leq \theta \leq \pi /2,\ 0\leq \varphi <2\pi }"></dd></dl> Umgekehrt gilt: Wählt man einen <a href="/wiki/Vektor" title="Vektor">Vektor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ae526b00782f0150985cd3fa027d82369c0abd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.719ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{0}}"> beliebig und die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca9be64134bc95772a521100b1190af3d2fee43f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.225ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}}"> beliebig, aber <a href="/wiki/Linear_unabh%C3%A4ngig" class="mw-redirect" title="Linear unabhängig">linear unabhängig</a>, so beschreibt die obige <a href="/wiki/Parameterdarstellung" title="Parameterdarstellung">Parameterdarstellung</a> in jedem Fall ein Ellipsoid. Bilden die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca9be64134bc95772a521100b1190af3d2fee43f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.225ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}}"> ein <a href="/wiki/Orthogonalsystem" title="Orthogonalsystem">Orthogonalsystem</a>, so sind die <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkte</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{0}\pm {\vec {f}}_{i},\ i=1,2,3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>±</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{0}\pm {\vec {f}}_{i},\ i=1,2,3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282a154dceb23ce03fabf9df906342c293819f6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.095ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{0}\pm {\vec {f}}_{i},\ i=1,2,3}"> die <a href="/wiki/Scheitelpunkt" title="Scheitelpunkt">Scheitelpunkte</a> des Ellipsoids und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {f}}_{1}|,|{\vec {f}}_{2}|,|{\vec {f}}_{3}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {f}}_{1}|,|{\vec {f}}_{2}|,|{\vec {f}}_{3}|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/996d4d7abf22008638853b19f251e7f9b84d5938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.106ex; height:3.509ex;" alt="{\displaystyle |{\vec {f}}_{1}|,|{\vec {f}}_{2}|,|{\vec {f}}_{3}|}"> die zugehörigen Halbachsen. Ein <a href="/wiki/Normalenvektor" title="Normalenvektor">Normalenvektor</a> im <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}(\theta ,\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}(\theta ,\varphi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49378df5a954a5038c081aecd3acddd20b80cc72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.784ex; height:2.843ex;" alt="{\displaystyle {\vec {x}}(\theta ,\varphi )}"> ist <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}(\theta ,\varphi )={\vec {f}}_{2}\times {\vec {f}}_{3}\cos \theta \cos \varphi +{\vec {f}}_{3}\times {\vec {f}}_{1}\cos \theta \sin \varphi +{\vec {f}}_{1}\times {\vec {f}}_{2}\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {n}}(\theta ,\varphi )={\vec {f}}_{2}\times {\vec {f}}_{3}\cos \theta \cos \varphi +{\vec {f}}_{3}\times {\vec {f}}_{1}\cos \theta \sin \varphi +{\vec {f}}_{1}\times {\vec {f}}_{2}\sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f9dbecc78969ba810bac73f8a80e05fa2d8d7db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:65.69ex; height:3.509ex;" alt="{\displaystyle {\vec {n}}(\theta ,\varphi )={\vec {f}}_{2}\times {\vec {f}}_{3}\cos \theta \cos \varphi +{\vec {f}}_{3}\times {\vec {f}}_{1}\cos \theta \sin \varphi +{\vec {f}}_{1}\times {\vec {f}}_{2}\sin \theta }"></dd></dl> Zu einer Parameterdarstellung eines beliebigen Ellipsoids lässt sich auch eine <a href="/wiki/Implizite_Fl%C3%A4che" title="Implizite Fläche">implizite</a> Beschreibung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y,z)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x,y,z)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0796f292c93e738b5c7c46a9aea6023ceb2d8ec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.452ex; height:2.843ex;" alt="{\displaystyle F(x,y,z)=0}"> angeben. Für ein Ellipsoid mit <a href="/wiki/Mittelpunkt" title="Mittelpunkt">Mittelpunkt</a> im <a href="/wiki/Koordinatenursprung" class="mw-redirect" title="Koordinatenursprung">Koordinatenursprung</a>, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{0}=(0,0,0)^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{0}=(0,0,0)^{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803dc776afa0e7e9cb99129d0f6b7a9dd98f70b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.571ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{0}=(0,0,0)^{T}}">, ist <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y,z)=\operatorname {det} ({\vec {x}},{\vec {f}}_{2},{\vec {f}}_{3})^{2}\;+\;\operatorname {det} ({\vec {f}}_{1},{\vec {x}},{\vec {f}}_{3})^{2}\;+\;\operatorname {det} ({\vec {f}}_{1},{\vec {f}}_{2},{\vec {x}})^{2}\;-\;\operatorname {det} ({\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3})^{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>det</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> <mi>det</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> <mi>det</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> <mi>det</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x,y,z)=\operatorname {det} ({\vec {x}},{\vec {f}}_{2},{\vec {f}}_{3})^{2}\;+\;\operatorname {det} ({\vec {f}}_{1},{\vec {x}},{\vec {f}}_{3})^{2}\;+\;\operatorname {det} ({\vec {f}}_{1},{\vec {f}}_{2},{\vec {x}})^{2}\;-\;\operatorname {det} ({\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3})^{2}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80684c980982334fd3f44976ec5ca5990a525556" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:90.047ex; height:3.509ex;" alt="{\displaystyle F(x,y,z)=\operatorname {det} ({\vec {x}},{\vec {f}}_{2},{\vec {f}}_{3})^{2}\;+\;\operatorname {det} ({\vec {f}}_{1},{\vec {x}},{\vec {f}}_{3})^{2}\;+\;\operatorname {det} ({\vec {f}}_{1},{\vec {f}}_{2},{\vec {x}})^{2}\;-\;\operatorname {det} ({\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3})^{2}=0}"></dd></dl> eine implizite Darstellung.<a href="#cite_note-13">[13]</a> Bemerkung: Das durch obige Parameterdarstellung beschriebene Ellipsoid ist in dem eventuell schiefen <a href="/wiki/Koordinatensystem" title="Koordinatensystem">Koordinatensystem</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ae526b00782f0150985cd3fa027d82369c0abd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.719ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{0}}"> (<a href="/wiki/Koordinatenursprung" class="mw-redirect" title="Koordinatenursprung">Koordinatenursprung</a>), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca9be64134bc95772a521100b1190af3d2fee43f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.225ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}}"> (<a href="/wiki/Basisvektor" class="mw-redirect" title="Basisvektor">Basisvektoren</a>) die Einheitskugel. <div class="mw-heading mw-heading3"><h3 id="Ellipsoid_als_Quadrik">Ellipsoid als Quadrik</h3>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=13" title="Abschnitt bearbeiten: Ellipsoid als Quadrik" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Ellipsoid als Quadrik">Quelltext bearbeiten</a>]</div> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Quadrik" title="Quadrik">Quadrik</a></div> Ein beliebiges Ellipsoid mit <a href="/wiki/Mittelpunkt" title="Mittelpunkt">Mittelpunkt</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {f}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ae526b00782f0150985cd3fa027d82369c0abd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.719ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}_{0}}"> lässt sich als <a href="/wiki/L%C3%B6sungsmenge" title="Lösungsmenge">Lösungsmenge</a> einer <a href="/wiki/Gleichung" title="Gleichung">Gleichung</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {x}}-{\vec {f}}_{0})^{\top }A\,({\vec {x}}-{\vec {f}}_{0})=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi>A</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {x}}-{\vec {f}}_{0})^{\top }A\,({\vec {x}}-{\vec {f}}_{0})=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f5f59795ec2297800706f0c2510f19bf2cd8e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.299ex; height:3.509ex;" alt="{\displaystyle ({\vec {x}}-{\vec {f}}_{0})^{\top }A\,({\vec {x}}-{\vec {f}}_{0})=1}"></dd></dl> schreiben, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> eine <a href="/wiki/Positiv_definite_Matrix" class="mw-redirect" title="Positiv definite Matrix">positiv definite Matrix</a> ist. Die <a href="/wiki/Eigenvektor" class="mw-redirect" title="Eigenvektor">Eigenvektoren</a> der <a href="/wiki/Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> bestimmen die Hauptachsenrichtungen des Ellipsoids und die <a href="/wiki/Eigenwerte" class="mw-redirect" title="Eigenwerte">Eigenwerte</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> sind die <a href="/wiki/Kehrwert" title="Kehrwert">Kehrwerte</a> der Quadrate der Halbachsen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbecdccf102d1e32bb76f6079b283352aec4df73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.563ex; height:2.676ex;" alt="{\displaystyle a^{-2}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{-2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51f7411d1c9b892dd0d025b35701da226d74dc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.33ex; height:2.676ex;" alt="{\displaystyle b^{-2}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{-2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2f059b44c0119a3fd80d157590d2cd16fc751b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.34ex; height:2.676ex;" alt="{\displaystyle c^{-2}}">.<a href="#cite_note-14">[14]</a> <div class="mw-heading mw-heading2"><h2 id="Ellipsoid_in_der_projektiven_Geometrie">Ellipsoid in der projektiven Geometrie</h2>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=14" title="Abschnitt bearbeiten: Ellipsoid in der projektiven Geometrie" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Ellipsoid in der projektiven Geometrie">Quelltext bearbeiten</a>]</div> Schließt man den <a href="/wiki/3-dimensional" class="mw-redirect" title="3-dimensional">3-dimensionalen</a> <a href="/wiki/Affiner_Raum" title="Affiner Raum">affinen Raum</a> und die einzelnen <a href="/wiki/Quadrik" title="Quadrik">Quadriken</a> <a href="/wiki/Projektiver_Raum" title="Projektiver Raum">projektiv</a> durch eine <a href="/wiki/Fernebene" class="mw-redirect" title="Fernebene">Fernebene</a> bzw. Fernpunkte ab, so sind die folgenden Quadriken projektiv äquivalent, d. h., es gibt jeweils eine projektive <a href="/wiki/Kollineation" title="Kollineation">Kollineation</a>, die die eine Quadrik in die andere überführt: <ul><li>Ellipsoid, elliptisches <a href="/wiki/Paraboloid" title="Paraboloid">Paraboloid</a> und 2-schaliges <a href="/wiki/Hyperboloid" title="Hyperboloid">Hyperboloid</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=15" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Siehe auch">Quelltext bearbeiten</a>]</div> <ul><li><a href="/wiki/Referenzellipsoid" title="Referenzellipsoid">Referenzellipsoid</a></li> <li><a href="/wiki/Tr%C3%A4gheitsellipsoid" title="Trägheitsellipsoid">Trägheitsellipsoid</a></li> <li><a href="/wiki/Indexellipsoid" title="Indexellipsoid">Indexellipsoid</a></li> <li><a href="/wiki/Hom%C3%B6oid" title="Homöoid">Homöoid</a></li> <li><a href="/wiki/Fokaloid" title="Fokaloid">Fokaloid</a></li> <li><a href="/wiki/Konfokale_Quadriken" class="mw-redirect" title="Konfokale Quadriken">Konfokale Quadriken</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=16" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=16" title="Quellcode des Abschnitts bearbeiten: Weblinks">Quelltext bearbeiten</a>]</div> <ul><li><a rel="nofollow" class="external text" href="https://keisan.casio.com/has10/SpecExec.cgi?path=05000000.Mathematics%2f01000300.Volume%20and%20surface%20area%2f13000600.Volume%20of%20an%20ellipsoid%2fdefault.xml&charset=utf-8">Online-Berechnung von Volumen und Oberfläche eines Ellipsoids</a> (englisch)</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150203063110/http://pi.physik.uni-bonn.de/~dieckman/SurfaceEllipsoid/SurfEll.html">Herleitung der Formel für die Oberfläche eines Ellipsoids.</a> Juli 2003, archiviert vom <style data-mw-deduplicate="TemplateStyles:r250917974">.mw-parser-output .dewiki-iconexternal>a{background-position:center right!important;background-repeat:no-repeat!important}body.skin-minerva .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/a/a4/OOjs_UI_icon_external-link-ltr-progressive.svg")!important;background-size:10px!important;padding-right:13px!important}body.skin-timeless .mw-parser-output .dewiki-iconexternal>a,body.skin-monobook .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/3/30/MediaWiki_external_link_icon.svg")!important;padding-right:13px!important}body.skin-vector .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/9/96/Link-external-small-ltr-progressive.svg")!important;background-size:0.857em!important;padding-right:1em!important}</style><a class="external text" href="https://redirecter.toolforge.org/?url=http%3A%2F%2Fpi.physik.uni-bonn.de%2F%7Edieckman%2FSurfaceEllipsoid%2FSurfEll.html">Original</a> am 3. Februar 2015; abgerufen am 3. Februar 2015 (englisch).<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AEllipsoid&rft.title=Herleitung+der+Formel+f%C3%BCr+die+Oberfl%C3%A4che+eines+Ellipsoids&rft.description=Herleitung+der+Formel+f%C3%BCr+die+Oberfl%C3%A4che+eines+Ellipsoids&rft.identifier=https%3A%2F%2Fweb.archive.org%2Fweb%2F20150203063110%2Fhttp%3A%2F%2Fpi.physik.uni-bonn.de%2F%7Edieckman%2FSurfaceEllipsoid%2FSurfEll.html&rft.date=2003-07&rft.source=http://pi.physik.uni-bonn.de/~dieckman/SurfaceEllipsoid/SurfEll.html&rft.language=en"> </li> <li><a rel="nofollow" class="external text" href="http://www.mathematische-basteleien.de/ellipsoid.html">Mathematische Basteleien: Ellipsoid</a></li> <li><a href="/wiki/Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Ellipsoid.html">Ellipsoid.</a> In: <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> (englisch).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2>[<a href="/w/index.php?title=Ellipsoid&veaction=edit&section=17" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Ellipsoid&action=edit&section=17" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise">Quelltext bearbeiten</a>]</div> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> Adrien-Marie Legendre: Traite des fonctions elliptiques et des intégrales <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euleriennes</a>, Bd. 1. Hugard-Courier, Paris 1825, S. 357. </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> Suzanne M. Kresta, Arthur W. Etchells III, David S. Dickey, Victor A. Atiemo-Obeng (Hrsg.): Advances in Industrial Mixing: A Companion to the Handbook of Industrial Mixing. John Wiley & Sons, 11. März 2016, <a href="/wiki/Spezial:ISBN-Suche/9780470523827" class="internal mw-magiclink-isbn">ISBN 978-0-470-52382-7</a>, <a rel="nofollow" class="external text" href="https://books.google.de/books?id=fia8CwAAQBAJ&pg=PA524#v=onepage">Seite 524 unten</a> in der Google-Buchsuche. </li> <li id="cite_note-3"><a href="#cite_ref-3">↑</a> W. Böhm: Die Fadenkonstruktion der Flächen zweiter Ordnung, Mathemat. Nachrichten 13, 1955, S. 151. </li> <li id="cite_note-4"><a href="#cite_ref-4">↑</a> O. Staude: Ueber Fadenconstructionen des Ellipsoides. Math. Ann. 20, 147–184 (1882). </li> <li id="cite_note-5"><a href="#cite_ref-5">↑</a> O. Staude: Ueber neue Focaleigenschaften der Flächen 2. Grades. Math. Ann. 27, 253–271 (1886). </li> <li id="cite_note-6"><a href="#cite_ref-6">↑</a> O. Staude: Die algebraischen Grundlagen der Focaleigenschaften der Flächen 2. Ordnung. Math. Ann. 50, 398–428 (1898). </li> <li id="cite_note-7"><a href="#cite_ref-7">↑</a> D. Hilbert, S. Cohn-Vossen: Anschauliche Geometrie. Springer-Verlag, 2013, <a href="/wiki/Spezial:ISBN-Suche/3662366851" class="internal mw-magiclink-isbn">ISBN 3662366851</a>, S. 18. </li> <li id="cite_note-8"><a href="#cite_ref-8">↑</a> S. Finsterwalder: Über die Fadenconstruction des Ellipsoides. Mathematische Annalen Bd. 26, 1886, S. 546–556. </li> <li id="cite_note-9"><a href="#cite_ref-9">↑</a> O. Hesse: Analytische Geometrie des Raumes. Teubner, Leipzig 1861, S. 287. </li> <li id="cite_note-10"><a href="#cite_ref-10">↑</a> D. Hilbert, S. Cohn-Vossen: Anschauliche Geometrie. S. 22. </li> <li id="cite_note-11"><a href="#cite_ref-11">↑</a> O. Hesse: Analytische Geometrie des Raumes. S. 301. </li> <li id="cite_note-12"><a href="#cite_ref-12">↑</a> W. Blaschke: Analytische Geometrie. S. 125. </li> <li id="cite_note-13"><a href="#cite_ref-13">↑</a> <a rel="nofollow" class="external text" href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf">Computerunterstützte Darstellende und Konstruktive Geometrie.</a> Uni Darmstadt (PDF; 3,4 MB), S. 88. </li> <li id="cite_note-14"><a href="#cite_ref-14">↑</a> <a rel="nofollow" class="external text" href="http://see.stanford.edu/materials/lsoeldsee263/15-symm.pdf">Symmetric matrices, quadratic forms, matrix norm, and SVD.</a> </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Ellipsoid&oldid=247956334">https://de.wikipedia.org/w/index.php?title=Ellipsoid&oldid=247956334</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorie</a>: <ul><li><a href="/wiki/Kategorie:Geometrie" title="Kategorie:Geometrie">Geometrie</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > Meine Werkzeuge </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item">Nicht angemeldet</li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n">Diskussionsseite</a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y">Beiträge</a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Ellipsoid" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. Es ist jedoch nicht zwingend erforderlich.">Benutzerkonto erstellen</a></li><li id="pt-login" class="mw-list-item"><a href="/w/index.php?title=Spezial:Anmelden&returnto=Ellipsoid" title="Anmelden ist zwar keine Pflicht, wird aber gerne gesehen. [o]" accesskey="o">Anmelden</a></li> </ul> </div> </nav> <div id="left-navigation"> <nav id="p-namespaces" class="mw-portlet mw-portlet-namespaces vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-namespaces-label" > <h3 id="p-namespaces-label" class="vector-menu-heading " > Namensräume </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected mw-list-item"><a href="/wiki/Ellipsoid" title="Seiteninhalt anzeigen [c]" accesskey="c">Artikel</a></li><li id="ca-talk" class="mw-list-item"><a href="/wiki/Diskussion:Ellipsoid" rel="discussion" title="Diskussion zum Seiteninhalt [t]" accesskey="t">Diskussion</a></li> </ul> </div> </nav> <nav id="p-variants" class="mw-portlet mw-portlet-variants emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-variants-label" > <input type="checkbox" id="p-variants-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-variants" class="vector-menu-checkbox" aria-labelledby="p-variants-label" > <label id="p-variants-label" class="vector-menu-heading " > Deutsch </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> </div> <div id="right-navigation"> <nav id="p-views" class="mw-portlet mw-portlet-views vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-views-label" > <h3 id="p-views-label" class="vector-menu-heading " > Ansichten </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected mw-list-item"><a href="/wiki/Ellipsoid">Lesen</a></li><li id="ca-ve-edit" class="mw-list-item"><a href="/w/index.php?title=Ellipsoid&veaction=edit" title="Diese Seite mit dem VisualEditor bearbeiten [v]" accesskey="v">Bearbeiten</a></li><li id="ca-edit" class="collapsible mw-list-item"><a href="/w/index.php?title=Ellipsoid&action=edit" title="Den Quelltext dieser Seite bearbeiten [e]" accesskey="e">Quelltext bearbeiten</a></li><li id="ca-history" class="mw-list-item"><a href="/w/index.php?title=Ellipsoid&action=history" title="Frühere Versionen dieser Seite [h]" accesskey="h">Versionsgeschichte</a></li> </ul> </div> </nav> <nav id="p-cactions" class="mw-portlet mw-portlet-cactions emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-cactions-label" title="Weitere Optionen" > <input type="checkbox" id="p-cactions-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-cactions" class="vector-menu-checkbox" aria-labelledby="p-cactions-label" > <label id="p-cactions-label" class="vector-menu-heading " > Weitere </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <h3 >Suche</h3> <form action="/w/index.php" id="searchform" class="vector-search-box-form"> <div id="simpleSearch" class="vector-search-box-inner" data-search-loc="header-navigation"> <input class="vector-search-box-input" type="search" name="search" placeholder="Wikipedia durchsuchen" aria-label="Wikipedia durchsuchen" autocapitalize="sentences" title="Durchsuche die Wikipedia [f]" accesskey="f" id="searchInput" > <input type="hidden" name="title" value="Spezial:Suche"> <input id="mw-searchButton" class="searchButton mw-fallbackSearchButton" type="submit" name="fulltext" title="Suche nach Seiten, die diesen Text enthalten" value="Suchen"> <input id="searchButton" class="searchButton" type="submit" name="go" title="Gehe direkt zu der Seite mit genau diesem Namen, falls sie vorhanden ist." value="Artikel"> </div> </form> </div> </div> </div> <div id="mw-panel" class="vector-legacy-sidebar"> <div id="p-logo" role="banner"> <a class="mw-wiki-logo" href="/wiki/Wikipedia:Hauptseite" title="Hauptseite"></a> </div> <nav id="p-navigation" class="mw-portlet mw-portlet-navigation vector-menu-portal portal vector-menu" aria-labelledby="p-navigation-label" > <h3 id="p-navigation-label" class="vector-menu-heading " > Navigation </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Wikipedia:Hauptseite" title="Hauptseite besuchen [z]" accesskey="z">Hauptseite</a></li><li id="n-topics" class="mw-list-item"><a href="/wiki/Portal:Wikipedia_nach_Themen">Themenportale</a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Spezial:Zuf%C3%A4llige_Seite" title="Zufällige Seite aufrufen [x]" accesskey="x">Zufälliger Artikel</a></li> </ul> </div> </nav> <nav id="p-Mitmachen" class="mw-portlet mw-portlet-Mitmachen vector-menu-portal portal vector-menu" aria-labelledby="p-Mitmachen-label" > <h3 id="p-Mitmachen-label" class="vector-menu-heading " > Mitmachen </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-Artikel-verbessern" class="mw-list-item"><a href="/wiki/Wikipedia:Beteiligen">Artikel verbessern</a></li><li id="n-Neuerartikel" class="mw-list-item"><a href="/wiki/Hilfe:Neuen_Artikel_anlegen">Neuen Artikel anlegen</a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Autorenportal" title="Info-Zentrum über Beteiligungsmöglichkeiten">Autorenportal</a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Hilfe:%C3%9Cbersicht" title="Übersicht über Hilfeseiten">Hilfe</a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Spezial:Letzte_%C3%84nderungen" title="Liste der letzten Änderungen in Wikipedia [r]" accesskey="r">Letzte Änderungen</a></li><li id="n-contact" class="mw-list-item"><a href="/wiki/Wikipedia:Kontakt" title="Kontaktmöglichkeiten">Kontakt</a></li><li id="n-sitesupport" class="mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=de.wikipedia.org&uselang=de" title="Unterstütze uns">Spenden</a></li> </ul> </div> </nav> <nav id="p-tb" class="mw-portlet mw-portlet-tb vector-menu-portal portal vector-menu" aria-labelledby="p-tb-label" > <h3 id="p-tb-label" class="vector-menu-heading " > Werkzeuge </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Spezial:Linkliste/Ellipsoid" title="Liste aller Seiten, die hierher verlinken [j]" accesskey="j">Links auf diese Seite</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Spezial:%C3%84nderungen_an_verlinkten_Seiten/Ellipsoid" rel="nofollow" title="Letzte Änderungen an Seiten, die von hier verlinkt sind [k]" accesskey="k">Änderungen an verlinkten Seiten</a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Spezial:Spezialseiten" title="Liste aller Spezialseiten [q]" accesskey="q">Spezialseiten</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Ellipsoid&oldid=247956334" title="Dauerhafter Link zu dieser Seitenversion">Permanenter Link</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Ellipsoid&action=info" title="Weitere Informationen über diese Seite">Seiteninformationen</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Spezial:Zitierhilfe&page=Ellipsoid&id=247956334&wpFormIdentifier=titleform" title="Hinweise, wie diese Seite zitiert werden kann">Artikel zitieren</a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Spezial:URL-K%C3%BCrzung&url=https%3A%2F%2Fde.wikipedia.org%2Fwiki%2FEllipsoid">Kurzlink</a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Spezial:QrKodu&url=https%3A%2F%2Fde.wikipedia.org%2Fwiki%2FEllipsoid">QR-Code herunterladen</a></li> </ul> </div> </nav> <nav id="p-coll-print_export" class="mw-portlet mw-portlet-coll-print_export vector-menu-portal portal vector-menu" aria-labelledby="p-coll-print_export-label" > <h3 id="p-coll-print_export-label" class="vector-menu-heading " > Drucken/exportieren </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Spezial:DownloadAsPdf&page=Ellipsoid&action=show-download-screen">Als PDF herunterladen</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Ellipsoid&printable=yes" title="Druckansicht dieser Seite [p]" accesskey="p">Druckversion</a></li> </ul> </div> </nav> <nav id="p-wikibase-otherprojects" class="mw-portlet mw-portlet-wikibase-otherprojects vector-menu-portal portal vector-menu" aria-labelledby="p-wikibase-otherprojects-label" > <h3 id="p-wikibase-otherprojects-label" class="vector-menu-heading " > In anderen Projekten </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Ellipsoids" hreflang="en">Commons</a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q190046" title="Link zum verbundenen Objekt im Datenrepositorium [g]" accesskey="g">Wikidata-Datenobjekt</a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > In anderen Sprachen </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B3%D8%B7%D8%AD_%D9%86%D8%A7%D9%82%D8%B5%D9%8A" title="سطح ناقصي – Arabisch" lang="ar" hreflang="ar" data-title="سطح ناقصي" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Elipsoide_de_revoluci%C3%B3n" title="Elipsoide de revolución – Asturisch" lang="ast" hreflang="ast" data-title="Elipsoide de revolución" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – Aserbaidschanisch" lang="az" hreflang="az" data-title="Ellipsoid" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaidschanisch" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%AD%D0%BB%D1%96%D0%BF%D1%81%D0%BE%D1%96%D0%B4" title="Эліпсоід – Belarussisch" lang="be" hreflang="be" data-title="Эліпсоід" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%BB%D0%B8%D0%BF%D1%81%D0%BE%D0%B8%D0%B4" title="Елипсоид – Bulgarisch" lang="bg" hreflang="bg" data-title="Елипсоид" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/El%C2%B7lipsoide" title="El·lipsoide – Katalanisch" lang="ca" hreflang="ca" data-title="El·lipsoide" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Elipsoid" title="Elipsoid – Tschechisch" lang="cs" hreflang="cs" data-title="Elipsoid" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%81%D0%BE%D0%B8%D0%B4" title="Эллипсоид – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Эллипсоид" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Elipsoid" title="Elipsoid – Walisisch" lang="cy" hreflang="cy" data-title="Elipsoid" data-language-autonym="Cymraeg" data-language-local-name="Walisisch" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ellipsoide" title="Ellipsoide – Dänisch" lang="da" hreflang="da" data-title="Ellipsoide" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CE%BB%CE%BB%CE%B5%CE%B9%CF%88%CE%BF%CE%B5%CE%B9%CE%B4%CE%AE" title="Ελλειψοειδή – Griechisch" lang="el" hreflang="el" data-title="Ελλειψοειδή" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – Englisch" lang="en" hreflang="en" data-title="Ellipsoid" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Elipsoido" title="Elipsoido – Esperanto" lang="eo" hreflang="eo" data-title="Elipsoido" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Elipsoide" title="Elipsoide – Spanisch" lang="es" hreflang="es" data-title="Elipsoide" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – Estnisch" lang="et" hreflang="et" data-title="Ellipsoid" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Elipsoide" title="Elipsoide – Baskisch" lang="eu" hreflang="eu" data-title="Elipsoide" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A8%DB%8C%D8%B6%DB%8C%E2%80%8C%DA%AF%D9%88%D9%86" title="بیضی‌گون – Persisch" lang="fa" hreflang="fa" data-title="بیضی‌گون" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ellipsoidi" title="Ellipsoidi – Finnisch" lang="fi" hreflang="fi" data-title="Ellipsoidi" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Ellipso%C3%AFde" title="Ellipsoïde – Französisch" lang="fr" hreflang="fr" data-title="Ellipsoïde" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/%C3%89ileaps%C3%B3ideach" title="Éileapsóideach – Irisch" lang="ga" hreflang="ga" data-title="Éileapsóideach" data-language-autonym="Gaeilge" data-language-local-name="Irisch" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%99%D7%A4%D7%A1%D7%95%D7%90%D7%99%D7%93" title="אליפסואיד – Hebräisch" lang="he" hreflang="he" data-title="אליפסואיד" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%80%E0%A4%B0%E0%A5%8D%E0%A4%98%E0%A4%B5%E0%A5%83%E0%A4%A4%E0%A5%8D%E0%A4%A4%E0%A4%BE%E0%A4%AD" title="दीर्घवृत्ताभ – Hindi" lang="hi" hreflang="hi" data-title="दीर्घवृत्ताभ" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Ellipszoid" title="Ellipszoid – Ungarisch" lang="hu" hreflang="hu" data-title="Ellipszoid" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B7%D5%AC%D5%AB%D5%BA%D5%BD%D5%B8%D5%AB%D5%A4" title="Էլիպսոիդ – Armenisch" lang="hy" hreflang="hy" data-title="Էլիպսոիդ" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Elipsoid" title="Elipsoid – Indonesisch" lang="id" hreflang="id" data-title="Elipsoid" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Sporvala" title="Sporvala – Isländisch" lang="is" hreflang="is" data-title="Sporvala" data-language-autonym="Íslenska" data-language-local-name="Isländisch" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Ellissoide" title="Ellissoide – Italienisch" lang="it" hreflang="it" data-title="Ellissoide" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%A5%95%E5%86%86%E4%BD%93" title="楕円体 – Japanisch" lang="ja" hreflang="ja" data-title="楕円体" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%94%E1%83%9A%E1%83%98%E1%83%A4%E1%83%A1%E1%83%9D%E1%83%98%E1%83%93%E1%83%98" title="ელიფსოიდი – Georgisch" lang="ka" hreflang="ka" data-title="ელიფსოიდი" data-language-autonym="ქართული" data-language-local-name="Georgisch" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%81%D0%BE%D0%B8%D0%B4" title="Эллипсоид – Kasachisch" lang="kk" hreflang="kk" data-title="Эллипсоид" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%83%80%EC%9B%90%EB%A9%B4" title="타원면 – Koreanisch" lang="ko" hreflang="ko" data-title="타원면" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%81%D0%BE%D0%B8%D0%B4" title="Эллипсоид – Kirgisisch" lang="ky" hreflang="ky" data-title="Эллипсоид" data-language-autonym="Кыргызча" data-language-local-name="Kirgisisch" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – Luxemburgisch" lang="lb" hreflang="lb" data-title="Ellipsoid" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxemburgisch" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Elipso%C4%ABds" title="Elipsoīds – Lettisch" lang="lv" hreflang="lv" data-title="Elipsoīds" data-language-autonym="Latviešu" data-language-local-name="Lettisch" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ellipso%C3%AFde" title="Ellipsoïde – Niederländisch" lang="nl" hreflang="nl" data-title="Ellipsoïde" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ellipsoide" title="Ellipsoide – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Ellipsoide" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Ellipsoide" title="Ellipsoide – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Ellipsoide" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Elipsoida" title="Elipsoida – Polnisch" lang="pl" hreflang="pl" data-title="Elipsoida" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Elipsoide" title="Elipsoide – Portugiesisch" lang="pt" hreflang="pt" data-title="Elipsoide" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Elipsoid" title="Elipsoid – Rumänisch" lang="ro" hreflang="ro" data-title="Elipsoid" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%81%D0%BE%D0%B8%D0%B4" title="Эллипсоид – Russisch" lang="ru" hreflang="ru" data-title="Эллипсоид" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Ellipsoid" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Elipsoid" title="Elipsoid – Slowakisch" lang="sk" hreflang="sk" data-title="Elipsoid" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Elipsoid" title="Elipsoid – Slowenisch" lang="sl" hreflang="sl" data-title="Elipsoid" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Elipsoidi" title="Elipsoidi – Albanisch" lang="sq" hreflang="sq" data-title="Elipsoidi" data-language-autonym="Shqip" data-language-local-name="Albanisch" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – Schwedisch" lang="sv" hreflang="sv" data-title="Ellipsoid" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%80%E0%AE%B3%E0%AF%8D%E0%AE%B5%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%A3%E0%AF%8D%E0%AE%AE%E0%AE%AE%E0%AF%8D" title="நீள்வட்டத்திண்மம் – Tamil" lang="ta" hreflang="ta" data-title="நீள்வட்டத்திண்மம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A3%E0%B8%87%E0%B8%A3%E0%B8%B5" title="ทรงรี – Thailändisch" lang="th" hreflang="th" data-title="ทรงรี" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Elipsoit" title="Elipsoit – Türkisch" lang="tr" hreflang="tr" data-title="Elipsoit" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%BB%D1%96%D0%BF%D1%81%D0%BE%D1%97%D0%B4" title="Еліпсоїд – Ukrainisch" lang="uk" hreflang="uk" data-title="Еліпсоїд" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – Usbekisch" lang="uz" hreflang="uz" data-title="Ellipsoid" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Usbekisch" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – Vietnamesisch" lang="vi" hreflang="vi" data-title="Ellipsoid" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%A4%AD%E7%90%83" title="椭球 – Chinesisch" lang="zh" hreflang="zh" data-title="椭球" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%A9%A2%E7%90%83" title="橢球 – Kantonesisch" lang="yue" hreflang="yue" data-title="橢球" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target">粵語</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q190046#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen bearbeiten" class="wbc-editpage">Links bearbeiten</a></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Diese Seite wurde zuletzt am 22. August 2024 um 16:49 Uhr bearbeitet.</li> <li id="footer-info-copyright"><div id="footer-info-copyright-stats" class="noprint"><a rel="nofollow" class="external text" href="https://pageviews.wmcloud.org/?pages=Ellipsoid&project=de.wikipedia.org">Abrufstatistik</a> · <a rel="nofollow" class="external text" href="https://xtools.wmcloud.org/authorship/de.wikipedia.org/Ellipsoid?uselang=de">Autoren</a> </div><div id="footer-info-copyright-separator"> </div><div id="footer-info-copyright-info"> Der Text ist unter der Lizenz <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">„Creative-Commons Namensnennung – Weitergabe unter gleichen Bedingungen“</a> verfügbar; Informationen zu den Urhebern und zum Lizenzstatus eingebundener Mediendateien (etwa Bilder oder Videos) können im Regelfall durch Anklicken dieser abgerufen werden. Möglicherweise unterliegen die Inhalte jeweils zusätzlichen Bedingungen. Durch die Nutzung dieser Website erklären Sie sich mit den <a class="external text" href="https://foundation.wikimedia.org/wiki/Policy:Terms_of_Use/de">Nutzungsbedingungen</a> und der <a class="external text" href="https://foundation.wikimedia.org/wiki/Policy:Privacy_policy/de">Datenschutzrichtlinie</a> einverstanden. Wikipedia® ist eine eingetragene Marke der Wikimedia Foundation Inc.</div></li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy/de">Datenschutz</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:%C3%9Cber_Wikipedia">Über Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:Impressum">Impressum</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Verhaltenskodex</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Entwickler</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/de.wikipedia.org">Statistiken</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Stellungnahme zu Cookies</a></li> <li id="footer-places-mobileview"><a href="//de.m.wikipedia.org/w/index.php?title=Ellipsoid&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile Ansicht</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> <script>(RLQ=window.RLQ||[]).push(function(){mw.log.warn("This page is using the deprecated ResourceLoader module \"codex-search-styles\".\n[1.43] Use a CodexModule with codexComponents to set your specific components used: https://www.mediawiki.org/wiki/Codex#Using_a_limited_subset_of_components");mw.config.set({"wgHostname":"mw-web.codfw.main-59b954b7fb-xdpbd","wgBackendResponseTime":149,"wgPageParseReport":{"limitreport":{"cputime":"0.247","walltime":"0.457","ppvisitednodes":{"value":1832,"limit":1000000},"postexpandincludesize":{"value":6833,"limit":2097152},"templateargumentsize":{"value":2730,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":4,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":13224,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 129.866 1 -total"," 67.39% 87.516 1 Vorlage:Internetquelle"," 12.68% 16.468 1 Vorlage:Referrer"," 11.26% 14.624 1 Vorlage:IconExternal"," 10.73% 13.931 1 Vorlage:MathWorld"," 10.57% 13.727 2 Vorlage:FormatDate"," 7.14% 9.271 1 Vorlage:Google_Buch"," 4.60% 5.971 3 Vorlage:Hauptartikel"," 4.14% 5.372 1 Vorlage:Str_len"," 1.67% 2.169 1 Vorlage:Str_find"]},"scribunto":{"limitreport-timeusage":{"value":"0.021","limit":"10.000"},"limitreport-memusage":{"value":2034137,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-7c4dcdbb87-c4pkv","timestamp":"20241203121229","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Ellipsoid","url":"https:\/\/de.wikipedia.org\/wiki\/Ellipsoid","sameAs":"http:\/\/www.wikidata.org\/entity\/Q190046","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q190046","author":{"@type":"Organization","name":"Autoren der Wikimedia-Projekte"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-12-31T18:20:12Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/3\/33\/Ellipsoide.svg","headline":"dreidimensionale Entsprechung einer Ellipse"}</script> </body> </html>

CINXE.COM

Ellipsoid – Wikipedia