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Ellipsoïde — Wikipédia
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Wikipédia" autocapitalize="sentences" title="Rechercher sur Wikipédia [f]" accesskey="f"> <span class="search-box-icon-overlay"><span class="minerva-icon minerva-icon--search"></span> </span> </div><button id="searchIcon" class="cdx-button cdx-button--size-large cdx-button--icon-only cdx-button--weight-quiet skin-minerva-search-trigger"> <span class="minerva-icon minerva-icon--search"></span> <span>Rechercher</span> </button> </form> <nav class="minerva-user-navigation" aria-label="Navigation utilisateur"> </nav> </div> </header> <main id="content" class="mw-body"> <div class="banner-container"> <div id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Ellipsoïde</span></h1> <div class="tagline"> surface du second degré de l'espace euclidien à trois dimensions </div> </div> <nav class="page-actions-menu"> <ul id="p-views" 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open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <p>En <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Math%C3%A9matiques?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathématiques">mathématiques</a>, et plus précisément en <a href="https://fr-m-wikipedia-org.translate.goog/wiki/G%C3%A9om%C3%A9trie_euclidienne?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Géométrie euclidienne">géométrie euclidienne</a>, un <b>ellipsoïde</b> est une <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Surface_(g%C3%A9om%C3%A9trie_analytique)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Surface (géométrie analytique)">surface</a> du second degré de l'<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Espace_euclidien?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Espace euclidien">espace euclidien</a> à trois dimensions. Il fait donc partie des <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Quadrique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Quadrique">quadriques</a>, avec pour caractéristique principale de ne pas posséder de <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Point_%C3%A0_l%27infini?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Point à l'infini">point à l'infini</a>.</p> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Fichier:Ellipsoide.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ellipsoide.png/330px-Ellipsoide.png" decoding="async" width="330" height="267" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ellipsoide.png/495px-Ellipsoide.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/b/b4/Ellipsoide.png 2x" data-file-width="497" data-file-height="402"></a> <figcaption> Ellipsoïde avec <span class="texhtml">(<i>a</i>, <i>b</i>, <i>c</i>) = (4, 2, 1)</span>. </figcaption> </figure> <p>L'ellipsoïde admet un centre et au moins trois plans de <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Sym%C3%A9trie_(transformation_g%C3%A9om%C3%A9trique)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Symétrie (transformation géométrique)">symétrie</a>. L'<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Intersection_(math%C3%A9matiques)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Intersection (mathématiques)">intersection</a> d'un ellipsoïde avec un plan est une <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipse_(math%C3%A9matiques)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellipse (mathématiques)">ellipse</a>, un point ou l'<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ensemble_vide?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ensemble vide">ensemble vide</a>.</p> <p>L'équation d'un ellipsoïde centré à l'origine d'un <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Base_orthonorm%C3%A9e?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Base orthonormée">repère orthonormé</a> et aligné avec les axes du repère est de la forme</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over 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 {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/303573ace88047e7932c90d55d2c691d44ce2d39" 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 {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1}"></span> </dd> </dl> <p>où <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span> et <span class="texhtml mvar" style="font-style:italic;">c</span>, appelés <b>demi-axes</b> de l'ellipsoïde, sont des paramètres strictement positifs.</p> <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="fr" dir="ltr"> <h2 id="mw-toc-heading">Sommaire</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Les_ellipso%C3%AFdes_dans_l'histoire_des_sciences"><span class="tocnumber">1</span> <span class="toctext">Les ellipsoïdes dans l'histoire des sciences</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#La_forme_de_la_Terre_(et_des_plan%C3%A8tes)"><span class="tocnumber">1.1</span> <span class="toctext">La forme de la Terre (et des planètes)</span></a></li> <li class="toclevel-2 tocsection-3"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Les_gouttes_d'eau_(et_autres_objets_macroscopiques)"><span class="tocnumber">1.2</span> <span class="toctext">Les gouttes d'eau (et autres objets macroscopiques)</span></a></li> <li class="toclevel-2 tocsection-4"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Les_noyaux_atomiques"><span class="tocnumber">1.3</span> <span class="toctext">Les noyaux atomiques</span></a></li> </ul></li> <li class="toclevel-1 tocsection-5"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Param%C3%A9trisations"><span class="tocnumber">2</span> <span class="toctext">Paramétrisations</span></a> <ul> <li class="toclevel-2 tocsection-6"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%C3%89quation_g%C3%A9n%C3%A9ralis%C3%A9e"><span class="tocnumber">2.1</span> <span class="toctext">Équation généralisée</span></a></li> <li class="toclevel-2 tocsection-7"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Coordonn%C3%A9es_sph%C3%A9riques"><span class="tocnumber">2.2</span> <span class="toctext">Coordonnées sphériques</span></a></li> <li class="toclevel-2 tocsection-8"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Coordonn%C3%A9es_indic%C3%A9es"><span class="tocnumber">2.3</span> <span class="toctext">Coordonnées indicées</span></a></li> <li class="toclevel-2 tocsection-9"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Coordonn%C3%A9es_polaires_r%C3%A9duites"><span class="tocnumber">2.4</span> <span class="toctext">Coordonnées polaires réduites</span></a> <ul> <li class="toclevel-3 tocsection-10"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Espace_projectif"><span class="tocnumber">2.4.1</span> <span class="toctext">Espace projectif</span></a></li> </ul></li> </ul></li> <li class="toclevel-1 tocsection-11"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Cas_particuliers"><span class="tocnumber">3</span> <span class="toctext">Cas particuliers</span></a> <ul> <li class="toclevel-2 tocsection-12"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Ellipso%C3%AFde_triaxial"><span class="tocnumber">3.1</span> <span class="toctext">Ellipsoïde triaxial</span></a></li> <li class="toclevel-2 tocsection-13"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Ellipso%C3%AFde_de_r%C3%A9volution"><span class="tocnumber">3.2</span> <span class="toctext">Ellipsoïde de révolution</span></a></li> <li class="toclevel-2 tocsection-14"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Ellipso%C3%AFde_r%C3%A9ciproque"><span class="tocnumber">3.3</span> <span class="toctext">Ellipsoïde réciproque</span></a></li> </ul></li> <li class="toclevel-1 tocsection-15"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Propri%C3%A9t%C3%A9s_de_base"><span class="tocnumber">4</span> <span class="toctext">Propriétés de base</span></a> <ul> <li class="toclevel-2 tocsection-16"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Volume"><span class="tocnumber">4.1</span> <span class="toctext">Volume</span></a></li> <li class="toclevel-2 tocsection-17"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Excentricit%C3%A9"><span class="tocnumber">4.2</span> <span class="toctext">Excentricité</span></a></li> <li class="toclevel-2 tocsection-18"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Aire"><span class="tocnumber">4.3</span> <span class="toctext">Aire</span></a></li> </ul></li> <li class="toclevel-1 tocsection-19"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Propri%C3%A9t%C3%A9s_d%C3%A9velopp%C3%A9es"><span class="tocnumber">5</span> <span class="toctext">Propriétés développées</span></a> <ul> <li class="toclevel-2 tocsection-20"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Notations"><span class="tocnumber">5.1</span> <span class="toctext">Notations</span></a></li> <li class="toclevel-2 tocsection-21"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Moments_de_volume"><span class="tocnumber">5.2</span> <span class="toctext">Moments de volume</span></a></li> <li class="toclevel-2 tocsection-22"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Moments_angulaires"><span class="tocnumber">5.3</span> <span class="toctext">Moments angulaires</span></a></li> <li class="toclevel-2 tocsection-23"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Moments_de_surface"><span class="tocnumber">5.4</span> <span class="toctext">Moments de surface</span></a></li> <li class="toclevel-2 tocsection-24"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%C3%89nergie_gravitationnelle_et/ou_coulombienne"><span class="tocnumber">5.5</span> <span class="toctext">Énergie gravitationnelle et/ou coulombienne</span></a></li> </ul></li> <li class="toclevel-1 tocsection-25"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Applications_et_exemples"><span class="tocnumber">6</span> <span class="toctext">Applications et exemples</span></a> <ul> <li class="toclevel-2 tocsection-26"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Exemples"><span class="tocnumber">6.1</span> <span class="toctext">Exemples</span></a></li> </ul></li> <li class="toclevel-1 tocsection-27"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Notes_et_r%C3%A9f%C3%A9rences"><span class="tocnumber">7</span> <span class="toctext">Notes et références</span></a></li> <li class="toclevel-1 tocsection-28"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Voir_aussi"><span class="tocnumber">8</span> <span class="toctext">Voir aussi</span></a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Les_ellipsoïdes_dans_l'histoire_des_sciences"><span id="Les_ellipso.C3.AFdes_dans_l.27histoire_des_sciences"></span>Les ellipsoïdes dans l'<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Histoire_des_sciences?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Histoire des sciences">histoire des sciences</a></h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Les ellipsoïdes dans l'histoire des sciences" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>Depuis la fin du <abbr class="abbr" title="17ᵉ siècle"><span class="romain">XVII</span><sup style="font-size:72%">e</sup></abbr> siècle, les propriétés des ellipsoïdes ont fait l'objet d'intenses études par les mathématiciens et les physiciens en raison de leurs applications en <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Cosmologie?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cosmologie">physique céleste</a>, en <a href="https://fr-m-wikipedia-org.translate.goog/wiki/M%C3%A9canique_des_fluides?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mécanique des fluides">mécanique des fluides</a> et plus récemment en <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Physique_nucl%C3%A9aire?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Physique nucléaire">physique nucléaire</a><sup id="cite_ref-1" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>.</p> <p>La thématique générale est l'étude de la forme d'équilibre des objets déformables en rotation. Selon les forces internes ou externes s'exerçant sur ces objets et leurs éventuels mouvements internes (écoulements, <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Tourbillon_(physique)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tourbillon (physique)">vortex</a>), diverses formes d'équilibre et leur stabilité ont été étudiées par les plus grands mathématiciens. Ces formes à l'équilibre peuvent être des <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde_de_r%C3%A9volution?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellipsoïde de révolution">ellipsoïdes de révolution</a>, mais leur stabilité nécessite la connaissance des propriétés des ellipsoïdes triaxiaux<sup id="cite_ref-:0_2-0" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>.</p> <div class="mw-heading mw-heading3"> <h3 id="La_forme_de_la_Terre_(et_des_planètes)"><span id="La_forme_de_la_Terre_.28et_des_plan.C3.A8tes.29"></span>La forme de la Terre (et des planètes)</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : La forme de la Terre (et des planètes)" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"> <div class="bandeau-cell bandeau-icone-css loupe"> Article détaillé : <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Figure_de_la_Terre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Figure de la Terre">Figure de la Terre</a>. </div> </div> <p>La recherche de la <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Figure_de_la_Terre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Figure de la Terre">forme de la Terre</a>, initiée par Newton est l'archétype de l'étude des corps déformables en rotation uniforme, dont la cohésion est assurée par les forces internes de <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Gravitation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gravitation">gravitation</a>, en l'absence de forces externes. Les résultats (proche de l'équilibre) de Newton furent développés par <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Colin_Maclaurin?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Colin Maclaurin">Maclaurin</a> (1742) pour le calcul aux grandes vitesses de rotation. <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Charles_Gustave_Jacob_Jacobi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Charles Gustave Jacob Jacobi">Jacobi</a> (1834) a montré qu'au-delà d'un certain régime critique, les ellipsoïdes triaxiaux peuvent être des figures d'équilibre. Plus tard, dans les séquences de formes apparaissant à des vitesses de rotation de plus en plus élevées, des <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Th%C3%A9orie_des_bifurcations?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Théorie des bifurcations">bifurcations</a> ont été étudiées par <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Johann_Peter_Gustav_Lejeune_Dirichlet?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Johann Peter Gustav Lejeune Dirichlet">Dirichlet</a>, <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Richard_Dedekind?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Richard Dedekind">Dedekind</a>. <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Bernhard_Riemann?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bernhard Riemann">Riemann</a> (1860) a généralisé les études au cas où la rotation n'est pas uniforme et où des mouvements internes (<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Tourbillon_(physique)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tourbillon (physique)">vortex</a>) sont pris en compte. <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Henri_Poincar%C3%A9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Henri Poincaré">Poincaré</a> et <a href="https://fr-m-wikipedia-org.translate.goog/wiki/%C3%89lie_Cartan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Élie Cartan">Cartan</a> ont démontré qu'à partir de fréquences de rotation critiques, des bifurcations vers des formes non ellipsoïdales apparaissent (<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Fission_nucl%C3%A9aire?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fission nucléaire">fission</a>, formes asymétriques en poire).</p> <div class="mw-heading mw-heading3"> <h3 id="Les_gouttes_d'eau_(et_autres_objets_macroscopiques)"><span id="Les_gouttes_d.27eau_.28et_autres_objets_macroscopiques.29"></span>Les gouttes d'eau (et autres objets macroscopiques)</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Les gouttes d'eau (et autres objets macroscopiques)" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Les gouttes d'eau (ou tout autre liquide) sont des objets dont la cohésion est assurée par des forces de surface (tension <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Tension_superficielle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tension superficielle">superficielle</a>). Mises en rotation, elles subissent des bifurcations entre différentes séquences de formes d'équilibre<sup id="cite_ref-3" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-:2_4-0" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:2-4"><span class="cite_crochet">[</span>4<span class="cite_crochet">]</span></a></sup>, parmi lesquelles des ellipsoïdes de révolution ou des ellipsoïdes triaxiaux. Ces figures sont difficilement accessibles à l'expérience sur Terre, en raison de l'influence externe de la pesanteur. Il a fallu attendre les années 1990, avec des expériences en apesanteur<sup id="cite_ref-5" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-5"><span class="cite_crochet">[</span>5<span class="cite_crochet">]</span></a></sup> pour valider les calculs théoriques.</p> <div class="mw-heading mw-heading3"> <h3 id="Les_noyaux_atomiques">Les noyaux atomiques</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Les noyaux atomiques" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"> <div class="bandeau-cell bandeau-icone-css loupe"> Article détaillé : <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Mod%C3%A8le_de_la_goutte_liquide?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modèle de la goutte liquide">Modèle de la goutte liquide</a>. </div> </div> <p>Dans les années 1930, <a href="https://fr-m-wikipedia-org.translate.goog/wiki/George_Gamow?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="George Gamow">G.Gamow</a>, puis <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Niels_Bohr?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Niels Bohr">N.Bohr</a> et <a href="https://fr-m-wikipedia-org.translate.goog/wiki/John_Wheeler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="John Wheeler">JA.Wheeler</a>, pour modéliser le phénomène de <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Fission_nucl%C3%A9aire?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fission nucléaire">fission nucléaire</a> récemment découvert, ont développé le <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Mod%C3%A8le_de_la_goutte_liquide?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modèle de la goutte liquide">modèle de la goutte liquide</a> du noyau atomique<sup id="cite_ref-6" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-6"><span class="cite_crochet">[</span>6<span class="cite_crochet">]</span></a></sup>. Dans ce modèle, les formes d'équilibre résultent de la compétition entre l'<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Force_nucl%C3%A9aire?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Force nucléaire">interaction nucléaire</a> de courte portée et attractive (générant l'analogue d'une tension superficielle), l'<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Interaction_coulombienne?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Interaction coulombienne">interaction coulombienne</a> à longue portée et répulsive et le cas échéant les <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Force_centrifuge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Force centrifuge">forces centrifuges</a> liées à la rotation. Plus tard, l'expérience et la théorie on montré que de nombreux noyaux ont des formes ellipsoïdales dans leur état fondamental<sup id="cite_ref-7" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-7"><span class="cite_crochet">[</span>7<span class="cite_crochet">]</span></a></sup>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Paramétrisations"><span id="Param.C3.A9trisations"></span>Paramétrisations</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Paramétrisations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>Selon le contexte et les propriétés que l'on veut étudier, différentes paramétrisations sont utilisées pour décrire un ellipsoïde.</p> <div class="mw-heading mw-heading3"> <h3 id="Équation_généralisée"><span id=".C3.89quation_g.C3.A9n.C3.A9ralis.C3.A9e"></span>Équation généralisée</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Équation généralisée" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Dans un repère cartésien en trois dimensions, l'équation d'une <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Quadrique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Quadrique">surface quadratique</a> est<span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ^{\mathsf {T}}A\,\mathbf {x} +B^{\mathsf {T}}\mathbf {x} +C=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mi> A </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> + </mo> <msup> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> + </mo> <mi> C </mi> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {x} ^{\mathsf {T}}A\,\mathbf {x} +B^{\mathsf {T}}\mathbf {x} +C=0} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9741d3938936b818dbecdb6803e9a73666aa49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.538ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} ^{\mathsf {T}}A\,\mathbf {x} +B^{\mathsf {T}}\mathbf {x} +C=0}"> </noscript><span class="lazy-image-placeholder" style="width: 22.538ex;height: 2.843ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9741d3938936b818dbecdb6803e9a73666aa49" data-alt="{\displaystyle \mathbf {x} ^{\mathsf {T}}A\,\mathbf {x} +B^{\mathsf {T}}\mathbf {x} +C=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span> où la matrice <span class="texhtml mvar" style="font-style:italic;">A</span> est, par construction, une <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Matrice_sym%C3%A9trique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matrice symétrique">matrice symétrique</a> réelle. D'après le <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Th%C3%A9or%C3%A8me_spectral?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Théorème spectral">théorème spectral</a>, elle est <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Diagonalisable?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Diagonalisable">diagonalisable</a> et ses <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Valeurs_propres?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Valeurs propres">valeurs propres</a> sont toutes réelles. Si ces trois valeurs propres sont strictement positives (ou strictement négatives), c'est-à-dire que <span class="texhtml mvar" style="font-style:italic;">A</span> est de signature (3, 0) (ou (0, 3)), cette équation définit une quadratique type ellipsoïde. À condition éventuellement de changer tous les coefficients de l'équation par leur opposé, la matrice <span class="texhtml mvar" style="font-style:italic;">A</span> est alors <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Matrice_d%C3%A9finie_positive?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matrice définie positive">définie positive</a>. Le déterminant de <span class="texhtml mvar" style="font-style:italic;">A</span> n'étant pas nul, la quadratique possède un centre dont les coordonnées sont<span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =-{\frac {1}{2}}A^{-1}B,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo> = </mo> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msup> <mi> B </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} =-{\frac {1}{2}}A^{-1}B,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716e749e3ba1a752a63c2a16d5f3a32b9a573866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.803ex; height:5.176ex;" alt="{\displaystyle \mathbf {v} =-{\frac {1}{2}}A^{-1}B,}"> </noscript><span class="lazy-image-placeholder" style="width: 14.803ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716e749e3ba1a752a63c2a16d5f3a32b9a573866" data-alt="{\displaystyle \mathbf {v} =-{\frac {1}{2}}A^{-1}B,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span>et son équation s'écrit sous la forme :<span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {x} -\mathbf {v} \right)^{\mathsf {T}}A\left(\mathbf {x} -\mathbf {v} \right)=k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mi> A </mi> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\mathbf {x} -\mathbf {v} \right)^{\mathsf {T}}A\left(\mathbf {x} -\mathbf {v} \right)=k} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abbbfb06b9804e439e2544301b0178ed0d15bfa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.735ex; height:3.343ex;" alt="{\displaystyle \left(\mathbf {x} -\mathbf {v} \right)^{\mathsf {T}}A\left(\mathbf {x} -\mathbf {v} \right)=k}"> </noscript><span class="lazy-image-placeholder" style="width: 22.735ex;height: 3.343ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abbbfb06b9804e439e2544301b0178ed0d15bfa9" data-alt="{\displaystyle \left(\mathbf {x} -\mathbf {v} \right)^{\mathsf {T}}A\left(\mathbf {x} -\mathbf {v} \right)=k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span>avec<span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {1}{4}}B^{\mathsf {T}}A^{-1}B-C.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <msup> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msup> <mi> B </mi> <mo> −<!-- − --> </mo> <mi> C </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k={\frac {1}{4}}B^{\mathsf {T}}A^{-1}B-C.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c003784b464cd82e036abb83643da96a36e4e77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.517ex; height:5.176ex;" alt="{\displaystyle k={\frac {1}{4}}B^{\mathsf {T}}A^{-1}B-C.}"> </noscript><span class="lazy-image-placeholder" style="width: 20.517ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c003784b464cd82e036abb83643da96a36e4e77" data-alt="{\displaystyle k={\frac {1}{4}}B^{\mathsf {T}}A^{-1}B-C.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span></p> <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;"> Démonstration </div> <div class="NavContent" style="padding-bottom:0.4em"> <p>Puisque d'une part <span class="texhtml mvar" style="font-style:italic;">A</span> est symétrique et que d'autre part tout <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Scalaire_(math%C3%A9matiques)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Scalaire (mathématiques)">scalaire</a> est égal à sa <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Matrice_transpos%C3%A9e?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matrice transposée">matrice transposée</a>, <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} ^{\mathsf {T}}A=\mathbf {v} ^{\mathsf {T}}A^{\mathsf {T}}=\left(A\mathbf {v} \right)^{\mathsf {T}}\quad {\rm {et}}\quad \mathbf {x} ^{\mathsf {T}}A\mathbf {v} =\left(\mathbf {x} ^{\mathsf {T}}A\mathbf {v} \right)^{\mathsf {T}}=\left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {x} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mi> A </mi> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mo> = </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> e </mi> <mi mathvariant="normal"> t </mi> </mrow> </mrow> <mspace width="1em"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo> = </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mo> = </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} ^{\mathsf {T}}A=\mathbf {v} ^{\mathsf {T}}A^{\mathsf {T}}=\left(A\mathbf {v} \right)^{\mathsf {T}}\quad {\rm {et}}\quad \mathbf {x} ^{\mathsf {T}}A\mathbf {v} =\left(\mathbf {x} ^{\mathsf {T}}A\mathbf {v} \right)^{\mathsf {T}}=\left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {x} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/274a2f6c45ae3ef906a56a72b4f9299f24f95b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:58.694ex; height:3.843ex;" alt="{\displaystyle \mathbf {v} ^{\mathsf {T}}A=\mathbf {v} ^{\mathsf {T}}A^{\mathsf {T}}=\left(A\mathbf {v} \right)^{\mathsf {T}}\quad {\rm {et}}\quad \mathbf {x} ^{\mathsf {T}}A\mathbf {v} =\left(\mathbf {x} ^{\mathsf {T}}A\mathbf {v} \right)^{\mathsf {T}}=\left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {x} }"> </noscript><span class="lazy-image-placeholder" style="width: 58.694ex;height: 3.843ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/274a2f6c45ae3ef906a56a72b4f9299f24f95b02" data-alt="{\displaystyle \mathbf {v} ^{\mathsf {T}}A=\mathbf {v} ^{\mathsf {T}}A^{\mathsf {T}}=\left(A\mathbf {v} \right)^{\mathsf {T}}\quad {\rm {et}}\quad \mathbf {x} ^{\mathsf {T}}A\mathbf {v} =\left(\mathbf {x} ^{\mathsf {T}}A\mathbf {v} \right)^{\mathsf {T}}=\left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {x} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span> donc <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left(\mathbf {x} -\mathbf {v} \right)^{\mathsf {T}}A\left(\mathbf {x} -\mathbf {v} \right)-k&=\mathbf {x} ^{\mathsf {T}}A\mathbf {x} -\left(\mathbf {v} ^{\mathsf {T}}A\mathbf {x} +\mathbf {x} ^{\mathsf {T}}A\mathbf {v} \right)+\mathbf {v} ^{\mathsf {T}}A\mathbf {v} -k\\&=\mathbf {x} ^{\mathsf {T}}A\mathbf {x} -2\left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {x} +\left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {v} -k.\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mi> A </mi> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> −<!-- − --> </mo> <mi> k </mi> </mtd> <mtd> <mi></mi> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> −<!-- − --> </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> + </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo> −<!-- − --> </mo> <mi> k </mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> −<!-- − --> </mo> <mn> 2 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo> −<!-- − --> </mo> <mi> k </mi> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\left(\mathbf {x} -\mathbf {v} \right)^{\mathsf {T}}A\left(\mathbf {x} -\mathbf {v} \right)-k&=\mathbf {x} ^{\mathsf {T}}A\mathbf {x} -\left(\mathbf {v} ^{\mathsf {T}}A\mathbf {x} +\mathbf {x} ^{\mathsf {T}}A\mathbf {v} \right)+\mathbf {v} ^{\mathsf {T}}A\mathbf {v} -k\\&=\mathbf {x} ^{\mathsf {T}}A\mathbf {x} -2\left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {x} +\left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {v} -k.\end{aligned}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d580cbb26c22d2bf5417d0aa69b96f74237df6cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:64.695ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}\left(\mathbf {x} -\mathbf {v} \right)^{\mathsf {T}}A\left(\mathbf {x} -\mathbf {v} \right)-k&=\mathbf {x} ^{\mathsf {T}}A\mathbf {x} -\left(\mathbf {v} ^{\mathsf {T}}A\mathbf {x} +\mathbf {x} ^{\mathsf {T}}A\mathbf {v} \right)+\mathbf {v} ^{\mathsf {T}}A\mathbf {v} -k\\&=\mathbf {x} ^{\mathsf {T}}A\mathbf {x} -2\left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {x} +\left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {v} -k.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 64.695ex;height: 7.176ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d580cbb26c22d2bf5417d0aa69b96f74237df6cd" data-alt="{\displaystyle {\begin{aligned}\left(\mathbf {x} -\mathbf {v} \right)^{\mathsf {T}}A\left(\mathbf {x} -\mathbf {v} \right)-k&=\mathbf {x} ^{\mathsf {T}}A\mathbf {x} -\left(\mathbf {v} ^{\mathsf {T}}A\mathbf {x} +\mathbf {x} ^{\mathsf {T}}A\mathbf {v} \right)+\mathbf {v} ^{\mathsf {T}}A\mathbf {v} -k\\&=\mathbf {x} ^{\mathsf {T}}A\mathbf {x} -2\left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {x} +\left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {v} -k.\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span> Or <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(A\mathbf {v} \right)^{\mathsf {T}}=-{\frac {1}{2}}B^{\mathsf {T}}\quad {\rm {et}}\quad \left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {v} -k=C}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo> ( </mo> <mrow> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mo> = </mo> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <msup> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> e </mi> <mi mathvariant="normal"> t </mi> </mrow> </mrow> <mspace width="1em"></mspace> <msup> <mrow> <mo> ( </mo> <mrow> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo> −<!-- − --> </mo> <mi> k </mi> <mo> = </mo> <mi> C </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(A\mathbf {v} \right)^{\mathsf {T}}=-{\frac {1}{2}}B^{\mathsf {T}}\quad {\rm {et}}\quad \left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {v} -k=C} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78aeb8c86dce00548de4548b53f39ef7af73cc03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.56ex; height:5.176ex;" alt="{\displaystyle \left(A\mathbf {v} \right)^{\mathsf {T}}=-{\frac {1}{2}}B^{\mathsf {T}}\quad {\rm {et}}\quad \left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {v} -k=C}"> </noscript><span class="lazy-image-placeholder" style="width: 39.56ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78aeb8c86dce00548de4548b53f39ef7af73cc03" data-alt="{\displaystyle \left(A\mathbf {v} \right)^{\mathsf {T}}=-{\frac {1}{2}}B^{\mathsf {T}}\quad {\rm {et}}\quad \left(A\mathbf {v} \right)^{\mathsf {T}}\mathbf {v} -k=C}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span> si (et seulement si) <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =-{\frac {1}{2}}A^{-1}B\quad {\rm {et}}\quad k={\frac {1}{4}}B^{\mathsf {T}}A^{-1}B-C.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo> = </mo> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msup> <mi> B </mi> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> e </mi> <mi mathvariant="normal"> t </mi> </mrow> </mrow> <mspace width="1em"></mspace> <mi> k </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <msup> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msup> <mi> B </mi> <mo> −<!-- − --> </mo> <mi> C </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} =-{\frac {1}{2}}A^{-1}B\quad {\rm {et}}\quad k={\frac {1}{4}}B^{\mathsf {T}}A^{-1}B-C.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ef089faea6f0eede8fe3b91a048c69d7519427" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.256ex; height:5.176ex;" alt="{\displaystyle \mathbf {v} =-{\frac {1}{2}}A^{-1}B\quad {\rm {et}}\quad k={\frac {1}{4}}B^{\mathsf {T}}A^{-1}B-C.}"> </noscript><span class="lazy-image-placeholder" style="width: 41.256ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ef089faea6f0eede8fe3b91a048c69d7519427" data-alt="{\displaystyle \mathbf {v} =-{\frac {1}{2}}A^{-1}B\quad {\rm {et}}\quad k={\frac {1}{4}}B^{\mathsf {T}}A^{-1}B-C.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span></p> </div> <div class="clear" style="clear:both;"></div> </div> <p>Si <span class="texhtml"><i>k</i></span> est strictement positif, l'ellipsoïde (centré en <span class="texhtml">v</span> et arbitrairement orienté) est alors l'ensemble des points <span class="texhtml">x</span> vérifiant l'équation :</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {x} -\mathbf {v} \right)^{\mathsf {T}}A_{1}\left(\mathbf {x} -\mathbf {v} \right)=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\mathbf {x} -\mathbf {v} \right)^{\mathsf {T}}A_{1}\left(\mathbf {x} -\mathbf {v} \right)=1} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed3c1ce54e24019e2ec78a84c4748259744bb1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.74ex; height:3.343ex;" alt="{\displaystyle \left(\mathbf {x} -\mathbf {v} \right)^{\mathsf {T}}A_{1}\left(\mathbf {x} -\mathbf {v} \right)=1}"> </noscript><span class="lazy-image-placeholder" style="width: 23.74ex;height: 3.343ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed3c1ce54e24019e2ec78a84c4748259744bb1f" data-alt="{\displaystyle \left(\mathbf {x} -\mathbf {v} \right)^{\mathsf {T}}A_{1}\left(\mathbf {x} -\mathbf {v} \right)=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>où <span class="texhtml"><i>A</i><sub>1</sub></span> est réelle, <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Matrice_d%C3%A9finie_positive?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matrice définie positive">définie positive</a>.</p> <p>De plus, les vecteurs propres de <span class="texhtml"><i>A</i><sub>1</sub></span> définissent les axes de l'ellipsoïde et les valeurs propres de <span class="texhtml"><i>A</i><sub>1</sub></span> sont égales à l'inverse du carré des demi-axes (c'est-à-dire <span class="texhtml">1/<i>a</i><sup>2</sup></span>, <span class="texhtml">1/<i>b</i><sup>2</sup></span> et <span class="texhtml">1/<i>c</i><sup>2</sup></span>)<sup id="cite_ref-8" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-8"><span class="cite_crochet">[</span>8<span class="cite_crochet">]</span></a></sup>. Les <a href="https://fr-m-wikipedia-org.translate.goog/wiki/D%C3%A9composition_en_valeurs_singuli%C3%A8res?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Décomposition en valeurs singulières">valeurs singulières</a> de <span class="texhtml"><i>A</i><sub>1</sub></span>, étant égales aux valeurs propres, sont donc égales à l'inverse du carré des demi-axes.</p> <div class="mw-heading mw-heading3"> <h3 id="Coordonnées_sphériques"><span id="Coordonn.C3.A9es_sph.C3.A9riques"></span>Coordonnées sphériques</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Coordonnées sphériques" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Un ellipsoïde peut être paramétré de différentes manières. Une des possibilités, en choisissant l'axe <span class="texhtml mvar" style="font-style:italic;">z</span>, est la suivante :<span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}x=a\cos \theta \cos \phi \\y=b\cos \theta \sin \phi \\z=c\sin \theta \end{cases}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> { </mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi> x </mi> <mo> = </mo> <mi> a </mi> <mi> cos </mi> <mo> <!-- --> </mo> <mi> θ<!-- θ --> </mi> <mi> cos </mi> <mo> <!-- --> </mo> <mi> ϕ<!-- ϕ --> </mi> </mtd> </mtr> <mtr> <mtd> <mi> y </mi> <mo> = </mo> <mi> b </mi> <mi> cos </mi> <mo> <!-- --> </mo> <mi> θ<!-- θ --> </mi> <mi> sin </mi> <mo> <!-- --> </mo> <mi> ϕ<!-- ϕ --> </mi> </mtd> </mtr> <mtr> <mtd> <mi> z </mi> <mo> = </mo> <mi> c </mi> <mi> sin </mi> <mo> <!-- --> </mo> <mi> θ<!-- θ --> </mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{cases}x=a\cos \theta \cos \phi \\y=b\cos \theta \sin \phi \\z=c\sin \theta \end{cases}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d33da5e5c5c6d74c85adf9bdc9a03213bd98e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:18.722ex; height:8.509ex;" alt="{\displaystyle {\begin{cases}x=a\cos \theta \cos \phi \\y=b\cos \theta \sin \phi \\z=c\sin \theta \end{cases}}}"> </noscript><span class="lazy-image-placeholder" style="width: 18.722ex;height: 8.509ex;vertical-align: -3.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d33da5e5c5c6d74c85adf9bdc9a03213bd98e6" data-alt="{\displaystyle {\begin{cases}x=a\cos \theta \cos \phi \\y=b\cos \theta \sin \phi \\z=c\sin \theta \end{cases}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span>où<span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi /2\leq \theta \leq \pi /2\quad {\rm {et}}\quad -\pi \leq \phi \leq \pi .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> −<!-- − --> </mo> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo> ≤<!-- ≤ --> </mo> <mi> θ<!-- θ --> </mi> <mo> ≤<!-- ≤ --> </mo> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> e </mi> <mi mathvariant="normal"> t </mi> </mrow> </mrow> <mspace width="1em"></mspace> <mo> −<!-- − --> </mo> <mi> π<!-- π --> </mi> <mo> ≤<!-- ≤ --> </mo> <mi> ϕ<!-- ϕ --> </mi> <mo> ≤<!-- ≤ --> </mo> <mi> π<!-- π --> </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -\pi /2\leq \theta \leq \pi /2\quad {\rm {et}}\quad -\pi \leq \phi \leq \pi .} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a9413e8ff2cbf1f81e5904319a2df0c0a352eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.725ex; height:2.843ex;" alt="{\displaystyle -\pi /2\leq \theta \leq \pi /2\quad {\rm {et}}\quad -\pi \leq \phi \leq \pi .}"> </noscript><span class="lazy-image-placeholder" style="width: 36.725ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a9413e8ff2cbf1f81e5904319a2df0c0a352eb" data-alt="{\displaystyle -\pi /2\leq \theta \leq \pi /2\quad {\rm {et}}\quad -\pi \leq \phi \leq \pi .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span>Les paramètres peuvent être vus comme des <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Coordonn%C3%A9es_sph%C3%A9riques?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Coordonnées sphériques">coordonnées sphériques</a>. Pour un <span class="texhtml mvar" style="font-style:italic;">θ</span> constant, nous obtenons une ellipse qui est l'intersection de l'ellipsoïde et d'un plan <span class="texhtml mvar" style="font-style:italic;">z = k</span>. Le paramètre <span class="texhtml mvar" style="font-style:italic;">ϕ</span> correspond alors à l'<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Anomalie_excentrique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Anomalie excentrique">anomalie excentrique</a> de cette ellipse. Seuls les ellipsoïdes de révolution possèdent une unique définition de la <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Latitude?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Latitudes_cartographiques" title="Latitude">latitude réduite</a>.</p> <div class="mw-heading mw-heading3"> <h3 id="Coordonnées_indicées"><span id="Coordonn.C3.A9es_indic.C3.A9es"></span>Coordonnées indicées</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Coordonnées indicées" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Les propriétés d'un ellipsoïde sont invariantes par permutation des indices (voir infra), suivant Chandrasekhar, on peut définir un ellipsoïde :</p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle \sum _{i=1}^{3}\left({\frac {x_{i}}{a_{i}}}\right)^{2}=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mn> 1 </mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \displaystyle \sum _{i=1}^{3}\left({\frac {x_{i}}{a_{i}}}\right)^{2}=1} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65ff0496e18095ef11f44b0018468917b5169c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.444ex; height:7.176ex;" alt="{\displaystyle \displaystyle \sum _{i=1}^{3}\left({\frac {x_{i}}{a_{i}}}\right)^{2}=1}"> </noscript><span class="lazy-image-placeholder" style="width: 15.444ex;height: 7.176ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65ff0496e18095ef11f44b0018468917b5169c65" data-alt="{\displaystyle \displaystyle \sum _{i=1}^{3}\left({\frac {x_{i}}{a_{i}}}\right)^{2}=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, en ordonnant les demi-axes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\geq a_{2}\geq a_{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> ≥<!-- ≥ --> </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> ≥<!-- ≥ --> </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a_{1}\geq a_{2}\geq a_{3}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a769d6865df281fbd77fa6a20cac8648caad8a9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.049ex; height:2.343ex;" alt="{\displaystyle a_{1}\geq a_{2}\geq a_{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 13.049ex;height: 2.343ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a769d6865df281fbd77fa6a20cac8648caad8a9f" data-alt="{\displaystyle a_{1}\geq a_{2}\geq a_{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> <div class="mw-heading mw-heading3"> <h3 id="Coordonnées_polaires_réduites"><span id="Coordonn.C3.A9es_polaires_r.C3.A9duites"></span>Coordonnées polaires réduites</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Coordonnées polaires réduites" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Tout point d'une surface dans un espace à 3 dimensions est défini par 2 paramètres. La paramétrisation dite d'Hill-Wheeler<sup id="cite_ref-9" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-9"><span class="cite_crochet">[</span>9<span class="cite_crochet">]</span></a></sup> définit les trois demi-axes :<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}=R_{0}\exp[{\delta \,\cos(\gamma -k{\frac {2\pi }{3}})}]\;,k=1..3\;;0\leq \gamma \leq \pi /3}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mi> exp </mi> <mo> <!-- --> </mo> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> δ<!-- δ --> </mi> <mspace width="thinmathspace"></mspace> <mi> cos </mi> <mo> <!-- --> </mo> <mo stretchy="false"> ( </mo> <mi> γ<!-- γ --> </mi> <mo> −<!-- − --> </mo> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mi> π<!-- π --> </mi> </mrow> <mn> 3 </mn> </mfrac> </mrow> <mo stretchy="false"> ) </mo> </mrow> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> , </mo> <mi> k </mi> <mo> = </mo> <mn> 1..3 </mn> <mspace width="thickmathspace"></mspace> <mo> ; </mo> <mn> 0 </mn> <mo> ≤<!-- ≤ --> </mo> <mi> γ<!-- γ --> </mi> <mo> ≤<!-- ≤ --> </mo> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 3 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a_{k}=R_{0}\exp[{\delta \,\cos(\gamma -k{\frac {2\pi }{3}})}]\;,k=1..3\;;0\leq \gamma \leq \pi /3} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803e71f670c95ceda53e023b37c7d0dedbdb8413" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.421ex; height:5.176ex;" alt="{\displaystyle a_{k}=R_{0}\exp[{\delta \,\cos(\gamma -k{\frac {2\pi }{3}})}]\;,k=1..3\;;0\leq \gamma \leq \pi /3}"> </noscript><span class="lazy-image-placeholder" style="width: 52.421ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803e71f670c95ceda53e023b37c7d0dedbdb8413" data-alt="{\displaystyle a_{k}=R_{0}\exp[{\delta \,\cos(\gamma -k{\frac {2\pi }{3}})}]\;,k=1..3\;;0\leq \gamma \leq \pi /3}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span>où <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> δ<!-- δ --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \delta } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 2.343ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" data-alt="{\displaystyle \delta }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> est lié à l'élongation de l'ellipsoïde (excentricité principale) et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> γ<!-- γ --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \gamma } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"> </noscript><span class="lazy-image-placeholder" style="width: 1.262ex;height: 2.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" data-alt="{\displaystyle \gamma }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> à son asymétrie (de prolate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> γ<!-- γ --> </mi> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \gamma =0} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5e84cac32e896a80a89f8cd1917c2defcf4108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.676ex;" alt="{\displaystyle \gamma =0}"> </noscript><span class="lazy-image-placeholder" style="width: 5.523ex;height: 2.676ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5e84cac32e896a80a89f8cd1917c2defcf4108" data-alt="{\displaystyle \gamma =0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> à oblate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =\pi /3}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> γ<!-- γ --> </mi> <mo> = </mo> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 3 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \gamma =\pi /3} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3633b9f5425850448fe5e3847780fe361da581eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.018ex; height:2.843ex;" alt="{\displaystyle \gamma =\pi /3}"> </noscript><span class="lazy-image-placeholder" style="width: 8.018ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3633b9f5425850448fe5e3847780fe361da581eb" data-alt="{\displaystyle \gamma =\pi /3}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>). La symétrie par permutation des axes permet de limiter le domaine de variation de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> γ<!-- γ --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \gamma } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"> </noscript><span class="lazy-image-placeholder" style="width: 1.262ex;height: 2.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" data-alt="{\displaystyle \gamma }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> de 0 à 60°.</p> <div class="mw-heading mw-heading4"> <h4 id="Espace_projectif">Espace projectif</h4><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Espace projectif" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p><span class="need_ref" title="Ce passage nécessite une référence (demandé le mars 2014)." style="cursor:help;">En <a href="https://fr-m-wikipedia-org.translate.goog/wiki/G%C3%A9om%C3%A9trie_projective?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Géométrie projective">géométrie projective</a></span><sup class="need_ref_tag" style="padding-left:2px;"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Aide:R%C3%A9f%C3%A9rence_n%C3%A9cessaire?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Aide:Référence nécessaire">[réf. nécessaire]</a></sup>, l'équation d'un ellipsoïde imaginaire est de la forme</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}+1=0.}"><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> <mo> = </mo> <mn> 0. </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=0.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df7973ae162ef948f9158cf11ea227c2f8440eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.843ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}+1=0.}"> </noscript><span class="lazy-image-placeholder" style="width: 23.843ex;height: 6.009ex;vertical-align: -2.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df7973ae162ef948f9158cf11ea227c2f8440eee" data-alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}+1=0.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>L'équation genre <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Conique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Classification_projective_des_coniques_r%C3%A9elles" title="Conique">ellipsoïde, cône imaginaire</a> :</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=0.}"><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> 0. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=0.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1671b7d0a49c95393fe7784fa7bbfb850886205c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.84ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=0.}"> </noscript><span class="lazy-image-placeholder" style="width: 19.84ex;height: 6.009ex;vertical-align: -2.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1671b7d0a49c95393fe7784fa7bbfb850886205c" data-alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=0.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Cas_particuliers">Cas particuliers</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Cas particuliers" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <div class="mw-heading mw-heading3"> <h3 id="Ellipsoïde_triaxial"><span id="Ellipso.C3.AFde_triaxial"></span>Ellipsoïde triaxial</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Ellipsoïde triaxial" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Un ellipsoïde est triaxial (ou scalène) si ses trois demi-axes sont différents.</p> <div class="mw-heading mw-heading3"> <h3 id="Ellipsoïde_de_révolution"><span id="Ellipso.C3.AFde_de_r.C3.A9volution"></span>Ellipsoïde de révolution</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=13&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Ellipsoïde de révolution" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"> <div class="bandeau-cell bandeau-icone-css loupe"> Article détaillé : <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde_de_r%C3%A9volution?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellipsoïde de révolution">Ellipsoïde de révolution</a>. </div> </div> <p>Dans le cas où seuls deux demi-axes sont égaux, l'ellipsoïde peut être engendré par la <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Rotation_dans_l%27espace?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rotation dans l'espace">rotation</a> d'une <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipse_(math%C3%A9matiques)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellipse (mathématiques)">ellipse</a> autour d'un de ses axes. Il s'agit d'un <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde_de_r%C3%A9volution?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellipsoïde de révolution">ellipsoïde de révolution</a>, parfois appelé <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Sph%C3%A9ro%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sphéroïde">sphéroïde</a>, permettant d'obtenir les <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Miroir_(optique)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Miroir (optique)">miroirs elliptiques</a> des projecteurs de cinéma et les <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ballon_de_rugby_%C3%A0_XIII?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ballon de rugby à XIII">ballons de rugby</a>. On montre aussi que cette surface est optimale pour les <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Dirigeable?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Dirigeable">dirigeables</a>.</p> <p>En prenant <span class="texhtml"><i>a = b</i></span>, l'équation s'écrit :</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}-1=0.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <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> 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> <mn> 0. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}-1=0.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32a2a2a0ca7e7fdedb57f175f867b8575d95b156" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.007ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}-1=0.}"> </noscript><span class="lazy-image-placeholder" style="width: 23.007ex;height: 6.009ex;vertical-align: -2.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32a2a2a0ca7e7fdedb57f175f867b8575d95b156" data-alt="{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}-1=0.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>On obtient un ellipsoïde de révolution d'axe <span class="texhtml"><i>Oz</i></span>. En effet, les sections par les plans <span class="texhtml"><i>z = k</i></span> sont des cercles d'axe <span class="texhtml"><i>Oz</i></span>.</p> <ul> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b<c}"><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></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d70a96ad4f65ae8f76c297346246f2c0a128db5c" 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}"> </noscript><span class="lazy-image-placeholder" style="width: 9.431ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d70a96ad4f65ae8f76c297346246f2c0a128db5c" data-alt="{\displaystyle a=b<c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, l’ellipsoïde est dit <i>prolate</i> (c'est-à-dire allongé).</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b=c}"><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></span> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 9.431ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41dfe5f5f74cb7a2c79093352406fd1d19123882" data-alt="{\displaystyle a=b=c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, l’ellipsoïde est une <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Sph%C3%A8re?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sphère">sphère</a>.</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b>c}"><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></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4e2c770be564fcc272802932f2139280fb5720" 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}"> </noscript><span class="lazy-image-placeholder" style="width: 9.431ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4e2c770be564fcc272802932f2139280fb5720" data-alt="{\displaystyle a=b>c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, l’ellipsoïde est dit <i>oblate</i> (c'est-à-dire aplati).</li> </ul> <p>La <a href="https://fr-m-wikipedia-org.translate.goog/wiki/M%C3%A9ridienne_(homonymie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-disambig" title="Méridienne (homonymie)">méridienne</a> dans le plan <span class="texhtml"><i>xOz</i></span> que l'on obtient avec <span class="texhtml"><i>y</i> = 0</span> est l'ellipse d'équation :</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}-1=0.}"><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> 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> <mn> 0. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}-1=0.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8da28fe78ed67d7ac58b2552f4b526842635b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.952ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}-1=0.}"> </noscript><span class="lazy-image-placeholder" style="width: 17.952ex;height: 6.009ex;vertical-align: -2.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8da28fe78ed67d7ac58b2552f4b526842635b0" data-alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}-1=0.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>On remarque que l'on passe de l'équation de la méridienne à l'équation de la <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Surface_de_r%C3%A9volution?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Surface de révolution">surface de révolution</a> en remplaçant <span class="texhtml"><i>x</i><sup>2</sup></span> par <span class="texhtml"><i>x</i><sup>2</sup> + <i>y</i><sup>2</sup></span>.</p> <div class="mw-heading mw-heading3"> <h3 id="Ellipsoïde_réciproque"><span id="Ellipso.C3.AFde_r.C3.A9ciproque"></span>Ellipsoïde réciproque</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=14&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Ellipsoïde réciproque" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>L'inversion géométrique d'un ellipsoïde de volume unité, définit l'ellipsoïde réciproque<sup id="cite_ref-:1_10-0" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:1-10"><span class="cite_crochet">[</span>10<span class="cite_crochet">]</span></a></sup></p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\displaystyle {x^{2} \over 1/a^{2}}+{y^{2} \over 1/b^{2}}+{z^{2} \over 1/c^{2}}=1}\;;}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mrow> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> = </mo> <mn> 1 </mn> </mstyle> </mrow> <mspace width="thickmathspace"></mspace> <mo> ; </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\displaystyle {x^{2} \over 1/a^{2}}+{y^{2} \over 1/b^{2}}+{z^{2} \over 1/c^{2}}=1}\;;} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4c12bfe8d5d9aaa4550ca20d94578c68965268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.114ex; height:6.509ex;" alt="{\displaystyle {\displaystyle {x^{2} \over 1/a^{2}}+{y^{2} \over 1/b^{2}}+{z^{2} \over 1/c^{2}}=1}\;;}"> </noscript><span class="lazy-image-placeholder" style="width: 27.114ex;height: 6.509ex;vertical-align: -2.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4c12bfe8d5d9aaa4550ca20d94578c68965268" data-alt="{\displaystyle {\displaystyle {x^{2} \over 1/a^{2}}+{y^{2} \over 1/b^{2}}+{z^{2} \over 1/c^{2}}=1}\;;}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>qui fait correspondre à un ellipsoïde prolate un ellipsoïde oblate (et vice versa) ; les propriétés des deux ellipsoïdes sont intrinsèquement liées (voir infra).</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Propriétés_de_base"><span id="Propri.C3.A9t.C3.A9s_de_base"></span>Propriétés de base</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=15&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Propriétés de base" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <div class="mw-heading mw-heading3"> <h3 id="Volume">Volume</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=16&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Volume" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Le <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Volume?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Volume">volume</a> de l'espace délimité par un ellipsoïde est égal à : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\frac {4}{3}}\pi abc={\frac {4\pi }{3}}{\sqrt {\det \left({A_{1}}^{-1}\right)}}.}"><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<!-- π --> </mi> <mi> a </mi> <mi> b </mi> <mi> c </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 4 </mn> <mi> π<!-- π --> </mi> </mrow> <mn> 3 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo movablelimits="true" form="prefix"> det </mo> <mrow> <mo> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msup> <mo> ) </mo> </mrow> </msqrt> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V={\frac {4}{3}}\pi abc={\frac {4\pi }{3}}{\sqrt {\det \left({A_{1}}^{-1}\right)}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b6f0d087f7f4cb901f37845555b042a98bdaee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.727ex; height:5.176ex;" alt="{\displaystyle V={\frac {4}{3}}\pi abc={\frac {4\pi }{3}}{\sqrt {\det \left({A_{1}}^{-1}\right)}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 31.727ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b6f0d087f7f4cb901f37845555b042a98bdaee" data-alt="{\displaystyle V={\frac {4}{3}}\pi abc={\frac {4\pi }{3}}{\sqrt {\det \left({A_{1}}^{-1}\right)}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span></p> <p>Cette formule donne le <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Boule_(solide)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Formulaire" title="Boule (solide)">volume d'une boule</a> de rayon <span class="texhtml mvar" style="font-style:italic;">a</span> dans le cas où les trois demi-axes sont de la même longueur.</p> <p>Les volumes du plus grand <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Parall%C3%A9l%C3%A9pip%C3%A8de_rectangle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Parallélépipède rectangle">parallélépipède rectangle</a> inscrit et du plus petit <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Parall%C3%A9l%C3%A9pip%C3%A8de_rectangle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Parallélépipède rectangle">parallélépipède rectangle</a> circonscrit sont donnés par les formules suivantes : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\max }={\frac {8}{3{\sqrt {3}}}}abc\quad {\rm {et}}\quad V_{\min }=8abc.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix"> max </mo> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 8 </mn> <mrow> <mn> 3 </mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi> a </mi> <mi> b </mi> <mi> c </mi> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> e </mi> <mi mathvariant="normal"> t </mi> </mrow> </mrow> <mspace width="1em"></mspace> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix"> min </mo> </mrow> </msub> <mo> = </mo> <mn> 8 </mn> <mi> a </mi> <mi> b </mi> <mi> c </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V_{\max }={\frac {8}{3{\sqrt {3}}}}abc\quad {\rm {et}}\quad V_{\min }=8abc.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2de61e1a951c2bf6cebafe8e77e062d21dde2b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:35.128ex; height:6.176ex;" alt="{\displaystyle V_{\max }={\frac {8}{3{\sqrt {3}}}}abc\quad {\rm {et}}\quad V_{\min }=8abc.}"> </noscript><span class="lazy-image-placeholder" style="width: 35.128ex;height: 6.176ex;vertical-align: -2.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2de61e1a951c2bf6cebafe8e77e062d21dde2b5" data-alt="{\displaystyle V_{\max }={\frac {8}{3{\sqrt {3}}}}abc\quad {\rm {et}}\quad V_{\min }=8abc.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span></p> <div class="mw-heading mw-heading3"> <h3 id="Excentricité"><span id="Excentricit.C3.A9"></span>Excentricité</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=17&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Excentricité" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Si <span class="texhtml"><i>a</i> ≥ <i>b</i> ≥ <i>c</i></span> (c'est-à-dire si <span class="texhtml mvar" style="font-style:italic;">a</span> est la longueur du plus grand demi-axe et <span class="texhtml mvar" style="font-style:italic;">c</span> est la longueur du plus petit demi-axe), on définit l'excentricité principale de l'ellipsoïde par la relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e={\sqrt {\frac {a^{2}-c^{2}}{a^{2}}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> e </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <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> </mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e={\sqrt {\frac {a^{2}-c^{2}}{a^{2}}}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6506e6e0901829218111c90eb62270dcf90b2d68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.527ex; height:7.509ex;" alt="{\displaystyle e={\sqrt {\frac {a^{2}-c^{2}}{a^{2}}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.527ex;height: 7.509ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6506e6e0901829218111c90eb62270dcf90b2d68" data-alt="{\displaystyle e={\sqrt {\frac {a^{2}-c^{2}}{a^{2}}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <div class="mw-heading mw-heading3"> <h3 id="Aire">Aire</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=18&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Aire" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>L'<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Aire_(g%C3%A9om%C3%A9trie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Aire (géométrie)">aire</a> d'un ellipsoïde quelconque est donnée par la formule<sup id="cite_ref-11" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-11"><span class="cite_crochet">[</span>11<span class="cite_crochet">]</span></a></sup> (voir infra) :</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}=2\pi c^{2}+{\frac {2\pi ab}{\sin(\phi )}}\left(E(\phi ,k)\sin ^{2}(\phi )+F(\phi ,k)\cos ^{2}(\phi )\right),}"><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"> A </mi> </mrow> </mrow> <mo> = </mo> <mn> 2 </mn> <mi> π<!-- 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<!-- π --> </mi> <mi> a </mi> <mi> b </mi> </mrow> <mrow> <mi> sin </mi> <mo> <!-- --> </mo> <mo stretchy="false"> ( </mo> <mi> ϕ<!-- ϕ --> </mi> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <mi> E </mi> <mo stretchy="false"> ( </mo> <mi> ϕ<!-- ϕ --> </mi> <mo> , </mo> <mi> k </mi> <mo stretchy="false"> ) </mo> <msup> <mi> sin </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> <!-- --> </mo> <mo stretchy="false"> ( </mo> <mi> ϕ<!-- ϕ --> </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> ϕ<!-- ϕ --> </mi> <mo> , </mo> <mi> k </mi> <mo stretchy="false"> ) </mo> <msup> <mi> cos </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> <!-- --> </mo> <mo stretchy="false"> ( </mo> <mi> ϕ<!-- ϕ --> </mi> <mo stretchy="false"> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathcal {A}}=2\pi c^{2}+{\frac {2\pi ab}{\sin(\phi )}}\left(E(\phi ,k)\sin ^{2}(\phi )+F(\phi ,k)\cos ^{2}(\phi )\right),} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/917230af0e28fc313cfef2a5f852f5fcf6f912ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:55.311ex; height:6.176ex;" alt="{\displaystyle {\mathcal {A}}=2\pi c^{2}+{\frac {2\pi ab}{\sin(\phi )}}\left(E(\phi ,k)\sin ^{2}(\phi )+F(\phi ,k)\cos ^{2}(\phi )\right),}"> </noscript><span class="lazy-image-placeholder" style="width: 55.311ex;height: 6.176ex;vertical-align: -2.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/917230af0e28fc313cfef2a5f852f5fcf6f912ba" data-alt="{\displaystyle {\mathcal {A}}=2\pi c^{2}+{\frac {2\pi ab}{\sin(\phi )}}\left(E(\phi ,k)\sin ^{2}(\phi )+F(\phi ,k)\cos ^{2}(\phi )\right),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>où</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\phi )={\frac {c}{a}},\qquad k^{2}={\frac {a^{2}\left(b^{2}-c^{2}\right)}{b^{2}\left(a^{2}-c^{2}\right)}},\qquad a\geq b\geq c,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> cos </mi> <mo> <!-- --> </mo> <mo stretchy="false"> ( </mo> <mi> ϕ<!-- ϕ --> </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> c </mi> <mi> a </mi> </mfrac> </mrow> <mo> , </mo> <mspace width="2em"></mspace> <msup> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mo> ( </mo> <mrow> <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> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mo> ( </mo> <mrow> <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> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> , </mo> <mspace width="2em"></mspace> <mi> a </mi> <mo> ≥<!-- ≥ --> </mo> <mi> b </mi> <mo> ≥<!-- ≥ --> </mo> <mi> c </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \cos(\phi )={\frac {c}{a}},\qquad k^{2}={\frac {a^{2}\left(b^{2}-c^{2}\right)}{b^{2}\left(a^{2}-c^{2}\right)}},\qquad a\geq b\geq c,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa265a647b202fca7c85fc924651b68a3f61f202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.861ex; height:6.843ex;" alt="{\displaystyle \cos(\phi )={\frac {c}{a}},\qquad k^{2}={\frac {a^{2}\left(b^{2}-c^{2}\right)}{b^{2}\left(a^{2}-c^{2}\right)}},\qquad a\geq b\geq c,}"> </noscript><span class="lazy-image-placeholder" style="width: 50.861ex;height: 6.843ex;vertical-align: -2.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa265a647b202fca7c85fc924651b68a3f61f202" data-alt="{\displaystyle \cos(\phi )={\frac {c}{a}},\qquad k^{2}={\frac {a^{2}\left(b^{2}-c^{2}\right)}{b^{2}\left(a^{2}-c^{2}\right)}},\qquad a\geq b\geq c,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>et où <span class="texhtml"><i>F</i>(<i>ϕ</i> , <i>k</i>)</span> et <span class="texhtml"><i>E</i>(<i>ϕ</i> , <i>k</i>)</span> sont les <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Int%C3%A9grale_elliptique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Intégrale elliptique">intégrales elliptiques</a> incomplètes de première et deuxième espèce respectivement (<span title="Voir la section Notations"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Notations">voir <i>infra</i></a></span>).</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Propriétés_développées"><span id="Propri.C3.A9t.C3.A9s_d.C3.A9velopp.C3.A9es"></span>Propriétés développées</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=19&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Propriétés développées" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <p>Cette section est basée sur le livre de référence de Chandrasekhar<sup id="cite_ref-:0_2-1" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>.</p> <p>L'utilisation des ellipsoïdes dans la modélisation des systèmes physiques nécessite le calcul de grandeurs, notamment pour les comparer avec celles de la sphère de volume identique (que l'on supposera de volume unité pour simplifier les notations) ; ce sont notamment les moments de volume, les moments de surface et les moments angulaires :</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x_{i}^{n}\rangle _{\tau }=(3/4\pi )\iiint x_{i}^{n}d\tau \;,\langle x_{i}^{n}\rangle _{\sigma }=(1/4\pi )\iint x_{i}^{n}d\sigma \;,\langle x_{i}^{n}\rangle _{\omega }=(1/4\pi )\iint x_{i}^{n}d\omega \;;}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> τ<!-- τ --> </mi> </mrow> </msub> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 3 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 4 </mn> <mi> π<!-- π --> </mi> <mo stretchy="false"> ) </mo> <mo> ∭<!-- ∭ --> </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msubsup> <mi> d </mi> <mi> τ<!-- τ --> </mi> <mspace width="thickmathspace"></mspace> <mo> , </mo> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> σ<!-- σ --> </mi> </mrow> </msub> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 4 </mn> <mi> π<!-- π --> </mi> <mo stretchy="false"> ) </mo> <mo> ∬<!-- ∬ --> </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msubsup> <mi> d </mi> <mi> σ<!-- σ --> </mi> <mspace width="thickmathspace"></mspace> <mo> , </mo> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> ω<!-- ω --> </mi> </mrow> </msub> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 4 </mn> <mi> π<!-- π --> </mi> <mo stretchy="false"> ) </mo> <mo> ∬<!-- ∬ --> </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msubsup> <mi> d </mi> <mi> ω<!-- ω --> </mi> <mspace width="thickmathspace"></mspace> <mo> ; </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle x_{i}^{n}\rangle _{\tau }=(3/4\pi )\iiint x_{i}^{n}d\tau \;,\langle x_{i}^{n}\rangle _{\sigma }=(1/4\pi )\iint x_{i}^{n}d\sigma \;,\langle x_{i}^{n}\rangle _{\omega }=(1/4\pi )\iint x_{i}^{n}d\omega \;;} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf0b54587e098d08cea86d359967c47f35f7ddb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:79.445ex; height:5.676ex;" alt="{\displaystyle \langle x_{i}^{n}\rangle _{\tau }=(3/4\pi )\iiint x_{i}^{n}d\tau \;,\langle x_{i}^{n}\rangle _{\sigma }=(1/4\pi )\iint x_{i}^{n}d\sigma \;,\langle x_{i}^{n}\rangle _{\omega }=(1/4\pi )\iint x_{i}^{n}d\omega \;;}"> </noscript><span class="lazy-image-placeholder" style="width: 79.445ex;height: 5.676ex;vertical-align: -2.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf0b54587e098d08cea86d359967c47f35f7ddb9" data-alt="{\displaystyle \langle x_{i}^{n}\rangle _{\tau }=(3/4\pi )\iiint x_{i}^{n}d\tau \;,\langle x_{i}^{n}\rangle _{\sigma }=(1/4\pi )\iint x_{i}^{n}d\sigma \;,\langle x_{i}^{n}\rangle _{\omega }=(1/4\pi )\iint x_{i}^{n}d\omega \;;}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>où <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau \,,d\sigma \,{\text{et}}\,d\tau \,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mi> τ<!-- τ --> </mi> <mspace width="thinmathspace"></mspace> <mo> , </mo> <mi> d </mi> <mi> σ<!-- σ --> </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> et </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> τ<!-- τ --> </mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle d\tau \,,d\sigma \,{\text{et}}\,d\tau \,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2448d3f51c7ae0442ec17721e99bbf6add2a78c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.901ex; height:2.509ex;" alt="{\displaystyle d\tau \,,d\sigma \,{\text{et}}\,d\tau \,}"> </noscript><span class="lazy-image-placeholder" style="width: 11.901ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2448d3f51c7ae0442ec17721e99bbf6add2a78c" data-alt="{\displaystyle d\tau \,,d\sigma \,{\text{et}}\,d\tau \,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>sont respectivement les éléments de volume, de surface et d'angle solide ; les moments sont normalisés à l'unité pour une sphère avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n=0} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=0}"> </noscript><span class="lazy-image-placeholder" style="width: 5.656ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae" data-alt="{\displaystyle n=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> <div class="mw-heading mw-heading3"> <h3 id="Notations">Notations</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=20&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Notations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Pour unifier et simplifier les formules on utilisera les coordonnées indicées (voir supra) en ordonnant les demi-axes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\geq a_{2}\geq a_{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> ≥<!-- ≥ --> </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> ≥<!-- ≥ --> </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a_{1}\geq a_{2}\geq a_{3}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a769d6865df281fbd77fa6a20cac8648caad8a9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.049ex; height:2.343ex;" alt="{\displaystyle a_{1}\geq a_{2}\geq a_{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 13.049ex;height: 2.343ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a769d6865df281fbd77fa6a20cac8648caad8a9f" data-alt="{\displaystyle a_{1}\geq a_{2}\geq a_{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. Les trois excentricités de l'ellipsoïde sont</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1}^{2}=1-\left({\frac {a_{3}}{a_{1}}}\right)^{2}\;,e_{2}^{2}=1-\left({\frac {a_{2}}{a_{1}}}\right)^{2}\;{\text{et}}\;e_{3}^{2}=1-\left({\frac {a_{3}}{a_{2}}}\right)^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> = </mo> <mn> 1 </mn> <mo> −<!-- − --> </mo> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mspace width="thickmathspace"></mspace> <mo> , </mo> <msubsup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> = </mo> <mn> 1 </mn> <mo> −<!-- − --> </mo> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> et </mtext> </mrow> <mspace width="thickmathspace"></mspace> <msubsup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> = </mo> <mn> 1 </mn> <mo> −<!-- − --> </mo> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e_{1}^{2}=1-\left({\frac {a_{3}}{a_{1}}}\right)^{2}\;,e_{2}^{2}=1-\left({\frac {a_{2}}{a_{1}}}\right)^{2}\;{\text{et}}\;e_{3}^{2}=1-\left({\frac {a_{3}}{a_{2}}}\right)^{2}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b20d73bba09f37ff1f248cd7c5d778db675c88c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.057ex; height:6.509ex;" alt="{\displaystyle e_{1}^{2}=1-\left({\frac {a_{3}}{a_{1}}}\right)^{2}\;,e_{2}^{2}=1-\left({\frac {a_{2}}{a_{1}}}\right)^{2}\;{\text{et}}\;e_{3}^{2}=1-\left({\frac {a_{3}}{a_{2}}}\right)^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 56.057ex;height: 6.509ex;vertical-align: -2.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b20d73bba09f37ff1f248cd7c5d778db675c88c" data-alt="{\displaystyle e_{1}^{2}=1-\left({\frac {a_{3}}{a_{1}}}\right)^{2}\;,e_{2}^{2}=1-\left({\frac {a_{2}}{a_{1}}}\right)^{2}\;{\text{et}}\;e_{3}^{2}=1-\left({\frac {a_{3}}{a_{2}}}\right)^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Elles ne sont pas indépendantes et seuls interviennent dans les résultats : leurs ratios <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=e_{2}/e_{1}{\text{ et }}k'=e_{3}/e_{1}\;;}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> <mo> = </mo> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> et </mtext> </mrow> <msup> <mi> k </mi> <mo> ′ </mo> </msup> <mo> = </mo> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mspace width="thickmathspace"></mspace> <mo> ; </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k=e_{2}/e_{1}{\text{ et }}k'=e_{3}/e_{1}\;;} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd892da9816e975580381befa13d07081b420c7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.57ex; height:3.009ex;" alt="{\displaystyle k=e_{2}/e_{1}{\text{ et }}k'=e_{3}/e_{1}\;;}"> </noscript><span class="lazy-image-placeholder" style="width: 24.57ex;height: 3.009ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd892da9816e975580381befa13d07081b420c7a" data-alt="{\displaystyle k=e_{2}/e_{1}{\text{ et }}k'=e_{3}/e_{1}\;;}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> ainsi qu' un angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ϕ<!-- ϕ --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \phi } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"> </noscript><span class="lazy-image-placeholder" style="width: 1.385ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" data-alt="{\displaystyle \phi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, défini par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \phi =e_{1}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> sin </mi> <mo> <!-- --> </mo> <mi> ϕ<!-- ϕ --> </mi> <mo> = </mo> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sin \phi =e_{1}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9af4c82849212d68753f50ba32bbaab8ec087f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.511ex; height:2.509ex;" alt="{\displaystyle \sin \phi =e_{1}.}"> </noscript><span class="lazy-image-placeholder" style="width: 10.511ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9af4c82849212d68753f50ba32bbaab8ec087f2" data-alt="{\displaystyle \sin \phi =e_{1}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>La plupart des propriétés des ellipsoïdes s'expriment en fonction des <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Int%C3%A9grale_elliptique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Intégrale elliptique">intégrales elliptiques</a> de première et de seconde espèce :</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\displaystyle F(\phi ,k)=\int _{0}^{\phi }{\frac {\mathrm {d} \theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}},\quad E(\phi ,k)=\int _{0}^{\phi }{\sqrt {1-k^{2}\sin ^{2}\theta }}~\mathrm {d} \theta .}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> ϕ<!-- ϕ --> </mi> <mo> , </mo> <mi> k </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> ϕ<!-- ϕ --> </mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> θ<!-- θ --> </mi> </mrow> <msqrt> <mn> 1 </mn> <mo> −<!-- − --> </mo> <msup> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> sin </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> <!-- --> </mo> <mi> θ<!-- θ --> </mi> </msqrt> </mfrac> </mrow> <mo> , </mo> <mspace width="1em"></mspace> <mi> E </mi> <mo stretchy="false"> ( </mo> <mi> ϕ<!-- ϕ --> </mi> <mo> , </mo> <mi> k </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> ϕ<!-- ϕ --> </mi> </mrow> </msubsup> <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> sin </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> <!-- --> </mo> <mi> θ<!-- θ --> </mi> </msqrt> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> θ<!-- θ --> </mi> <mo> . </mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\displaystyle F(\phi ,k)=\int _{0}^{\phi }{\frac {\mathrm {d} \theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}},\quad E(\phi ,k)=\int _{0}^{\phi }{\sqrt {1-k^{2}\sin ^{2}\theta }}~\mathrm {d} \theta .}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/566ac8fa3ae8c6ccc0ac59b4098777e45b4354e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.256ex; height:7.176ex;" alt="{\displaystyle {\displaystyle F(\phi ,k)=\int _{0}^{\phi }{\frac {\mathrm {d} \theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}},\quad E(\phi ,k)=\int _{0}^{\phi }{\sqrt {1-k^{2}\sin ^{2}\theta }}~\mathrm {d} \theta .}}"> </noscript><span class="lazy-image-placeholder" style="width: 65.256ex;height: 7.176ex;vertical-align: -3.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/566ac8fa3ae8c6ccc0ac59b4098777e45b4354e6" data-alt="{\displaystyle {\displaystyle F(\phi ,k)=\int _{0}^{\phi }{\frac {\mathrm {d} \theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}},\quad E(\phi ,k)=\int _{0}^{\phi }{\sqrt {1-k^{2}\sin ^{2}\theta }}~\mathrm {d} \theta .}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <div class="mw-heading mw-heading3"> <h3 id="Moments_de_volume">Moments de volume</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=21&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Moments de volume" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Les moments de volumes sont les plus simples, par changement de variable on obtient :<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x_{i}^{n}\rangle _{\tau }=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> τ<!-- τ --> </mi> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle x_{i}^{n}\rangle _{\tau }=0} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cffb685519d83702995c59ce71a61b9dfa2dad71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.701ex; height:3.009ex;" alt="{\displaystyle \langle x_{i}^{n}\rangle _{\tau }=0}"> </noscript><span class="lazy-image-placeholder" style="width: 9.701ex;height: 3.009ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cffb685519d83702995c59ce71a61b9dfa2dad71" data-alt="{\displaystyle \langle x_{i}^{n}\rangle _{\tau }=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> si <i>n</i> est impair, et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x_{i}^{n}\rangle _{\tau }={\frac {3}{(n+1)(n+3)}}a_{i}^{n}\;}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> τ<!-- τ --> </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 3 </mn> <mrow> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mn> 3 </mn> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <msubsup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msubsup> <mspace width="thickmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle x_{i}^{n}\rangle _{\tau }={\frac {3}{(n+1)(n+3)}}a_{i}^{n}\;} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21c5a2a5037ba0945c82d398bac8fbaad516569" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.881ex; height:6.009ex;" alt="{\displaystyle \langle x_{i}^{n}\rangle _{\tau }={\frac {3}{(n+1)(n+3)}}a_{i}^{n}\;}"> </noscript><span class="lazy-image-placeholder" style="width: 26.881ex;height: 6.009ex;vertical-align: -2.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21c5a2a5037ba0945c82d398bac8fbaad516569" data-alt="{\displaystyle \langle x_{i}^{n}\rangle _{\tau }={\frac {3}{(n+1)(n+3)}}a_{i}^{n}\;}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> si <i>n</i> est pair. Les <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Moment_d%27inertie?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Moment d'inertie">moments d'inertie</a> (<i>n</i>=2) sont particulièrement importants pour l'étude des ellipsoïdes en rotation ; le moment d'inertie d'un ellipsoïde de densité uniforme par rapport à un axe de ses axes de symétrie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a_{i}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.029ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" data-alt="{\displaystyle a_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> s'écrit donc :</p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{a_{i}}={\frac {m}{5}}\sum _{k=1,\neq i}^{3}a_{k}^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> m </mi> <mn> 5 </mn> </mfrac> </mrow> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mo> ≠<!-- ≠ --> </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <msubsup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{a_{i}}={\frac {m}{5}}\sum _{k=1,\neq i}^{3}a_{k}^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bcecaef343deb5d0a43ca6f872c431c4398fbea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:17.078ex; height:7.676ex;" alt="{\displaystyle I_{a_{i}}={\frac {m}{5}}\sum _{k=1,\neq i}^{3}a_{k}^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 17.078ex;height: 7.676ex;vertical-align: -3.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bcecaef343deb5d0a43ca6f872c431c4398fbea" data-alt="{\displaystyle I_{a_{i}}={\frac {m}{5}}\sum _{k=1,\neq i}^{3}a_{k}^{2}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span>.</p> <p>Soit avec les notations non indicées, avec <i>a</i> sur l'axe <i>Ox</i> , avec <i>b</i> sur l'axe <i>Oy</i> et <i>c</i> sur l'axe <i>Oz</i> :</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{xx}={\frac {m}{5}}(b^{2}+c^{2})\;,I_{yy}={\frac {m}{5}}(a^{2}+c^{2})\;,I_{zz}={\frac {m}{5}}(a^{2}+b^{2}).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> x </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> m </mi> <mn> 5 </mn> </mfrac> </mrow> <mo stretchy="false"> ( </mo> <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> <mo stretchy="false"> ) </mo> <mspace width="thickmathspace"></mspace> <mo> , </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> y </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> m </mi> <mn> 5 </mn> </mfrac> </mrow> <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> <mspace width="thickmathspace"></mspace> <mo> , </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> <mi> z </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> m </mi> <mn> 5 </mn> </mfrac> </mrow> <mo stretchy="false"> ( </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> <mo stretchy="false"> ) </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{xx}={\frac {m}{5}}(b^{2}+c^{2})\;,I_{yy}={\frac {m}{5}}(a^{2}+c^{2})\;,I_{zz}={\frac {m}{5}}(a^{2}+b^{2}).} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07e333d9ef865343e30602ccd9bc4aa353296fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:57.492ex; height:4.676ex;" alt="{\displaystyle I_{xx}={\frac {m}{5}}(b^{2}+c^{2})\;,I_{yy}={\frac {m}{5}}(a^{2}+c^{2})\;,I_{zz}={\frac {m}{5}}(a^{2}+b^{2}).}"> </noscript><span class="lazy-image-placeholder" style="width: 57.492ex;height: 4.676ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07e333d9ef865343e30602ccd9bc4aa353296fd" data-alt="{\displaystyle I_{xx}={\frac {m}{5}}(b^{2}+c^{2})\;,I_{yy}={\frac {m}{5}}(a^{2}+c^{2})\;,I_{zz}={\frac {m}{5}}(a^{2}+b^{2}).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Les moments d'inertie non diagonaux sont tous nuls dans ce système d'axe :</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{xy}=I_{yx}=I_{xz}=I_{zx}=I_{yz}=I_{zy}=0.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> y </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> x </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> z </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> <mi> x </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> z </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> <mi> y </mi> </mrow> </msub> <mo> = </mo> <mn> 0. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{xy}=I_{yx}=I_{xz}=I_{zx}=I_{yz}=I_{zy}=0.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4226476cfe52bb8beee104ec57dbdbc0ae5a3cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.039ex; height:2.843ex;" alt="{\displaystyle I_{xy}=I_{yx}=I_{xz}=I_{zx}=I_{yz}=I_{zy}=0.}"> </noscript><span class="lazy-image-placeholder" style="width: 38.039ex;height: 2.843ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4226476cfe52bb8beee104ec57dbdbc0ae5a3cb" data-alt="{\displaystyle I_{xy}=I_{yx}=I_{xz}=I_{zx}=I_{yz}=I_{zy}=0.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <div class="mw-heading mw-heading3"> <h3 id="Moments_angulaires">Moments angulaires</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=22&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Moments angulaires" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Ces <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Moment_(math%C3%A9matiques)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Moment (mathématiques)">moments</a> sont des propriétés statiques des ellipsoïdes ; ils diffèrent des <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Moment_cin%C3%A9tique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Moment cinétique">moments angulaires</a> (ou moments cinétiques) des solides en rotation. Les moments d'ordre 2 sont liés par l'équation de l'ellipsoïde :<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{3}\langle x_{i}^{2}\rangle _{\omega }/a_{i}^{2}=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> ω<!-- ω --> </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <msubsup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> = </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sum _{i=1}^{3}\langle x_{i}^{2}\rangle _{\omega }/a_{i}^{2}=1} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d1d797f38958adf23c47c96cbd0223814d7bd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.51ex; height:7.176ex;" alt="{\displaystyle \sum _{i=1}^{3}\langle x_{i}^{2}\rangle _{\omega }/a_{i}^{2}=1}"> </noscript><span class="lazy-image-placeholder" style="width: 16.51ex;height: 7.176ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d1d797f38958adf23c47c96cbd0223814d7bd3" data-alt="{\displaystyle \sum _{i=1}^{3}\langle x_{i}^{2}\rangle _{\omega }/a_{i}^{2}=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>; ils se déduisent de la relation :<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle r^{2}\rangle _{\omega }={\frac {a_{2}a_{3}}{e_{1}}}F(\phi ,k)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> ω<!-- ω --> </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mrow> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mfrac> </mrow> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> ϕ<!-- ϕ --> </mi> <mo> , </mo> <mi> k </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle r^{2}\rangle _{\omega }={\frac {a_{2}a_{3}}{e_{1}}}F(\phi ,k)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3211fbc2f8e497ea27de93997ee25c05395e74" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.85ex; height:5.009ex;" alt="{\displaystyle \langle r^{2}\rangle _{\omega }={\frac {a_{2}a_{3}}{e_{1}}}F(\phi ,k)}"> </noscript><span class="lazy-image-placeholder" style="width: 20.85ex;height: 5.009ex;vertical-align: -2.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3211fbc2f8e497ea27de93997ee25c05395e74" data-alt="{\displaystyle \langle r^{2}\rangle _{\omega }={\frac {a_{2}a_{3}}{e_{1}}}F(\phi ,k)}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span>par le <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Th%C3%A9or%C3%A8me_d%27Euler_(fonctions_de_plusieurs_variables)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Théorème d'Euler (fonctions de plusieurs variables)">théorème d'Euler</a> sur les fonctions homogènes, on en déduit : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x_{i}^{2}\rangle _{\omega }=\langle r_{i}^{2}\rangle _{\omega }-a_{i}{\frac {\partial {\langle r_{i}^{2}\rangle _{\omega }}}{\partial {a_{i}}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> ω<!-- ω --> </mi> </mrow> </msub> <mo> = </mo> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> ω<!-- ω --> </mi> </mrow> </msub> <mo> −<!-- − --> </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> ω<!-- ω --> </mi> </mrow> </msub> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle x_{i}^{2}\rangle _{\omega }=\langle r_{i}^{2}\rangle _{\omega }-a_{i}{\frac {\partial {\langle r_{i}^{2}\rangle _{\omega }}}{\partial {a_{i}}}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e8bffc41449c7d356132e5d12b7d1bab54602ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.551ex; height:6.343ex;" alt="{\displaystyle \langle x_{i}^{2}\rangle _{\omega }=\langle r_{i}^{2}\rangle _{\omega }-a_{i}{\frac {\partial {\langle r_{i}^{2}\rangle _{\omega }}}{\partial {a_{i}}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 26.551ex;height: 6.343ex;vertical-align: -2.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e8bffc41449c7d356132e5d12b7d1bab54602ab" data-alt="{\displaystyle \langle x_{i}^{2}\rangle _{\omega }=\langle r_{i}^{2}\rangle _{\omega }-a_{i}{\frac {\partial {\langle r_{i}^{2}\rangle _{\omega }}}{\partial {a_{i}}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>Les moments angulaires d'un ellipsoïde sont fortement liés aux moments de surface de l'ellipsoïde réciproque<sup id="cite_ref-:1_10-1" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:1-10"><span class="cite_crochet">[</span>10<span class="cite_crochet">]</span></a></sup> et interviennent dans le calcul de l'énergie potentielle (voir infra) des ellipsoïdes homogènes.</p> <div class="mw-heading mw-heading3"> <h3 id="Moments_de_surface">Moments de surface</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=23&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Moments de surface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Ils permettent d'analyser les déviations de la surface de l'ellipsoïde par rapport à celle de la sphère de même volume. Les plus utiles sont liés aux trois composantes de la normale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}}"><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> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {n}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49569db585c1b6306d5ffd91161775f67235fae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.343ex;" alt="{\displaystyle {\vec {n}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 2.343ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49569db585c1b6306d5ffd91161775f67235fae0" data-alt="{\displaystyle {\vec {n}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> à la surface : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}=\langle n_{i}^{2}\rangle _{\sigma }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msubsup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> σ<!-- σ --> </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{i}=\langle n_{i}^{2}\rangle _{\sigma }} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9067e62feed7de6d3547bd49e70db5f629e72d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.072ex; height:3.176ex;" alt="{\displaystyle A_{i}=\langle n_{i}^{2}\rangle _{\sigma }}"> </noscript><span class="lazy-image-placeholder" style="width: 11.072ex;height: 3.176ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9067e62feed7de6d3547bd49e70db5f629e72d7" data-alt="{\displaystyle A_{i}=\langle n_{i}^{2}\rangle _{\sigma }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> <p>On en déduit l'aire de l'ellipsoïde (cf. supra les définitions de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ϕ<!-- ϕ --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \phi } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"> </noscript><span class="lazy-image-placeholder" style="width: 1.385ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" data-alt="{\displaystyle \phi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> de <i>k'</i> ):<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=4\pi (A_{1}+A_{2}+A_{3})={\frac {2\pi a_{1}a_{2}}{e_{1}}}\left[(1-e_{1}^{2})F(\phi ,k')]+e_{1}^{2}E(\phi ,k')+e_{1}(1-e_{1}^{2})^{1/2}(1-e_{3}^{2})^{1/2}\right]\;;}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <mo> = </mo> <mn> 4 </mn> <mi> π<!-- π --> </mi> <mo stretchy="false"> ( </mo> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mi> π<!-- π --> </mi> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mrow> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo> [ </mo> <mrow> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> −<!-- − --> </mo> <msubsup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo stretchy="false"> ) </mo> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> ϕ<!-- ϕ --> </mi> <mo> , </mo> <msup> <mi> k </mi> <mo> ′ </mo> </msup> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mo> + </mo> <msubsup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mi> E </mi> <mo stretchy="false"> ( </mo> <mi> ϕ<!-- ϕ --> </mi> <mo> , </mo> <msup> <mi> k </mi> <mo> ′ </mo> </msup> <mo stretchy="false"> ) </mo> <mo> + </mo> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> −<!-- − --> </mo> <msubsup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> −<!-- − --> </mo> <msubsup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ] </mo> </mrow> <mspace width="thickmathspace"></mspace> <mo> ; </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A=4\pi (A_{1}+A_{2}+A_{3})={\frac {2\pi a_{1}a_{2}}{e_{1}}}\left[(1-e_{1}^{2})F(\phi ,k')]+e_{1}^{2}E(\phi ,k')+e_{1}(1-e_{1}^{2})^{1/2}(1-e_{3}^{2})^{1/2}\right]\;;} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd0ee9ea07a333336b15de76d317830d2b0f2dc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:94.091ex; height:5.509ex;" alt="{\displaystyle A=4\pi (A_{1}+A_{2}+A_{3})={\frac {2\pi a_{1}a_{2}}{e_{1}}}\left[(1-e_{1}^{2})F(\phi ,k')]+e_{1}^{2}E(\phi ,k')+e_{1}(1-e_{1}^{2})^{1/2}(1-e_{3}^{2})^{1/2}\right]\;;}"> </noscript><span class="lazy-image-placeholder" style="width: 94.091ex;height: 5.509ex;vertical-align: -2.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd0ee9ea07a333336b15de76d317830d2b0f2dc" data-alt="{\displaystyle A=4\pi (A_{1}+A_{2}+A_{3})={\frac {2\pi a_{1}a_{2}}{e_{1}}}\left[(1-e_{1}^{2})F(\phi ,k')]+e_{1}^{2}E(\phi ,k')+e_{1}(1-e_{1}^{2})^{1/2}(1-e_{3}^{2})^{1/2}\right]\;;}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span></p> <p>expression équivalente à celle ci-dessus en fonction des demi-axes <i>a</i>, <i>b</i> et <i>c</i>.</p> <div class="mw-heading mw-heading3"> <h3 id="Énergie_gravitationnelle_et/ou_coulombienne"><span id=".C3.89nergie_gravitationnelle_et.2Fou_coulombienne"></span>Énergie gravitationnelle et/ou coulombienne</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=24&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Énergie gravitationnelle et/ou coulombienne" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Pour les ellipsoïdes dont la cohésion est assurée par la <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Gravitation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gravitation">gravité</a> ou pour les ellipsoïdes uniformément <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Potentiel_%C3%A9lectrostatique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Potentiel électrostatique">chargés</a>, il est nécessaire d'évaluer le <a href="https://fr-m-wikipedia-org.translate.goog/wiki/%C3%89nergie_potentielle_de_pesanteur?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Énergie potentielle de pesanteur">potentiel</a> en un point par l'intégrale de volume :</p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V({\textbf {r}})=\iiint {\frac {d{\textbf {r'}}}{\|{\textbf {r}}-{\textbf {r'}}\|}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> V </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold"> r </mtext> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mo> ∭<!-- ∭ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold"> r' </mtext> </mrow> </mrow> </mrow> <mrow> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold"> r </mtext> </mrow> </mrow> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold"> r' </mtext> </mrow> </mrow> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V({\textbf {r}})=\iiint {\frac {d{\textbf {r'}}}{\|{\textbf {r}}-{\textbf {r'}}\|}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a120167a9cc433a5d5afe843ccbe02a4b7481c60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.732ex; height:6.176ex;" alt="{\displaystyle V({\textbf {r}})=\iiint {\frac {d{\textbf {r'}}}{\|{\textbf {r}}-{\textbf {r'}}\|}}}"> </noscript><span class="lazy-image-placeholder" style="width: 21.732ex;height: 6.176ex;vertical-align: -2.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a120167a9cc433a5d5afe843ccbe02a4b7481c60" data-alt="{\displaystyle V({\textbf {r}})=\iiint {\frac {d{\textbf {r'}}}{\|{\textbf {r}}-{\textbf {r'}}\|}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> <p><a href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Oliver_Kellogg&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Oliver Kellogg (page inexistante)">Oliver Kellogg</a> <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Oliver_Kellogg" class="extiw" title="en:Oliver Kellogg"><span class="indicateur-langue" title="Article en anglais : « Oliver Kellogg »">(en)</span></a> a démontré que le potentiel peut s'exprimer en fonction des moments angulaires (cf. supra)<sup id="cite_ref-12" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-12"><span class="cite_crochet">[</span>12<span class="cite_crochet">]</span></a></sup> :</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V({\textbf {r}})=2\pi \sum _{i=1}^{3}\langle x_{i}^{2}\rangle _{\omega }\left(1-{\frac {x_{i}^{2}}{a_{i}^{2}}}\right)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> V </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold"> r </mtext> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 2 </mn> <mi> π<!-- π --> </mi> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> ω<!-- ω --> </mi> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <msubsup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V({\textbf {r}})=2\pi \sum _{i=1}^{3}\langle x_{i}^{2}\rangle _{\omega }\left(1-{\frac {x_{i}^{2}}{a_{i}^{2}}}\right)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83750a046ed584078777f9c24d39b057302ab992" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.773ex; height:7.509ex;" alt="{\displaystyle V({\textbf {r}})=2\pi \sum _{i=1}^{3}\langle x_{i}^{2}\rangle _{\omega }\left(1-{\frac {x_{i}^{2}}{a_{i}^{2}}}\right)}"> </noscript><span class="lazy-image-placeholder" style="width: 30.773ex;height: 7.509ex;vertical-align: -3.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83750a046ed584078777f9c24d39b057302ab992" data-alt="{\displaystyle V({\textbf {r}})=2\pi \sum _{i=1}^{3}\langle x_{i}^{2}\rangle _{\omega }\left(1-{\frac {x_{i}^{2}}{a_{i}^{2}}}\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </dd> </dl> <p>Carlson a démontré plus précisément que l'énergie potentielle totale d'un ellipsoïde se déduit simplement de l'énergie potentielle d'une sphère équivalente<sup id="cite_ref-13" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-13"><span class="cite_crochet">[</span>13<span class="cite_crochet">]</span></a></sup> :</p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {V}}(a_{1},a_{2},a_{3})={\overline {V}}(1,1,1)<r^{2}>_{\omega }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> V </mi> <mo accent="false"> ¯<!-- ¯ --> </mo> </mover> </mrow> <mo stretchy="false"> ( </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> V </mi> <mo accent="false"> ¯<!-- ¯ --> </mo> </mover> </mrow> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> < </mo> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msub> <mo> > </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> ω<!-- ω --> </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {V}}(a_{1},a_{2},a_{3})={\overline {V}}(1,1,1)<r^{2}>_{\omega }} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3162cfebf2ed3d4e176df442c564910cd3d87046" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.468ex; height:3.509ex;" alt="{\displaystyle {\overline {V}}(a_{1},a_{2},a_{3})={\overline {V}}(1,1,1)<r^{2}>_{\omega }}"> </noscript><span class="lazy-image-placeholder" style="width: 34.468ex;height: 3.509ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3162cfebf2ed3d4e176df442c564910cd3d87046" data-alt="{\displaystyle {\overline {V}}(a_{1},a_{2},a_{3})={\overline {V}}(1,1,1)<r^{2}>_{\omega }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>,</p> <p>expression valable même pour les ellipsoïdes inhomogènes mais dont les équipotentielles sont des ellipsoïdes concentriques. La valeur moyenne de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V({\textbf {r}})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> V </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold"> r </mtext> </mrow> </mrow> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V({\textbf {r}})} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a18b0f3ed8277b13edc0a78a55ddfcdff37338c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.699ex; height:2.843ex;" alt="{\displaystyle V({\textbf {r}})}"> </noscript><span class="lazy-image-placeholder" style="width: 4.699ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a18b0f3ed8277b13edc0a78a55ddfcdff37338c" data-alt="{\displaystyle V({\textbf {r}})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> dans tout l'ellipsoïde est donc :</p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {V}}={\frac {8\pi }{5}}\langle r^{2}\rangle _{\omega }={\frac {8\pi }{5}}a_{2}a_{3}e_{1}^{-1}F(\phi ,k).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> V </mi> <mo accent="false"> ¯<!-- ¯ --> </mo> </mover> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 8 </mn> <mi> π<!-- π --> </mi> </mrow> <mn> 5 </mn> </mfrac> </mrow> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> ω<!-- ω --> </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 8 </mn> <mi> π<!-- π --> </mi> </mrow> <mn> 5 </mn> </mfrac> </mrow> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <msubsup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msubsup> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> ϕ<!-- ϕ --> </mi> <mo> , </mo> <mi> k </mi> <mo stretchy="false"> ) </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {V}}={\frac {8\pi }{5}}\langle r^{2}\rangle _{\omega }={\frac {8\pi }{5}}a_{2}a_{3}e_{1}^{-1}F(\phi ,k).} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c81437c91227adc0c2b97f8f1db25fa3a2cdba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.02ex; height:5.176ex;" alt="{\displaystyle {\overline {V}}={\frac {8\pi }{5}}\langle r^{2}\rangle _{\omega }={\frac {8\pi }{5}}a_{2}a_{3}e_{1}^{-1}F(\phi ,k).}"> </noscript><span class="lazy-image-placeholder" style="width: 36.02ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c81437c91227adc0c2b97f8f1db25fa3a2cdba" data-alt="{\displaystyle {\overline {V}}={\frac {8\pi }{5}}\langle r^{2}\rangle _{\omega }={\frac {8\pi }{5}}a_{2}a_{3}e_{1}^{-1}F(\phi ,k).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span></p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Applications_et_exemples">Applications et exemples</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=25&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Applications et exemples" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> <p>Les propriétés des ellipsoïdes ont été étudiées depuis des siècles dans la recherche des formes que prennent les systèmes en <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Rotation_(physique)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rotation (physique)">rotation</a>, <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Isaac_Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Isaac Newton">Newton</a> s'est intéressé en particulier à l'aplatissement des pôles (<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Mod%C3%A8le_ellipso%C3%AFdal_de_la_Terre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modèle ellipsoïdal de la Terre">modèle ellipsoïdal de la Terre</a>). Les mathématiciens les plus brillants ont contribué à l'étude <i>des figures ellipsoïdales d'équilibre</i> des systèmes en rotation<sup id="cite_ref-:0_2-2" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup> que l'on rencontre dans la nature.</p> <p>Les <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Objet_c%C3%A9leste?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Objet céleste">corps célestes</a> dont la cohésion est assurée par la <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Gravitation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gravitation">gravitation</a> ; en rotation autour d'un axe, les formes d'équilibre minimisent la somme de l'énergie potentielle (cf. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {V}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> V </mi> <mo accent="false"> ¯<!-- ¯ --> </mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {V}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38cfbd5e453955544af57ce8845a6474944393f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.183ex; height:3.009ex;" alt="{\displaystyle {\overline {V}}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.183ex;height: 3.009ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38cfbd5e453955544af57ce8845a6474944393f1" data-alt="{\displaystyle {\overline {V}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> ci- dessus) et de l'énergie cinétique de rotation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}I\,\Omega ^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mi> I </mi> <mspace width="thinmathspace"></mspace> <msup> <mi mathvariant="normal"> Ω<!-- Ω --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {1}{2}}I\,\Omega ^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/284cdc808d2deb435ed38a2e7c8fd4adb0a207b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.29ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}I\,\Omega ^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.29ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/284cdc808d2deb435ed38a2e7c8fd4adb0a207b9" data-alt="{\displaystyle {\frac {1}{2}}I\,\Omega ^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>(qui est fonction du moment d'inertie). Aux faibles vitesses de rotation, la figure d'équilibre est un ellipsoïde oblate ; à partir d'une vitesse de rotation critique<sup id="cite_ref-:0_2-3" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>, se produisent des bifurcations vers d'autres figures pour la plupart non ellipsoïdales (voire figure).</p> <figure typeof="mw:File/Thumb"> <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Fichier:Jacobi-ellipsoid-dimensions-2.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Jacobi-ellipsoid-dimensions-2.svg/440px-Jacobi-ellipsoid-dimensions-2.svg.png" decoding="async" width="440" height="330" class="mw-file-element" data-file-width="640" data-file-height="480"> </noscript><span class="lazy-image-placeholder" style="width: 440px;height: 330px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Jacobi-ellipsoid-dimensions-2.svg/440px-Jacobi-ellipsoid-dimensions-2.svg.png" data-width="440" data-height="330" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Jacobi-ellipsoid-dimensions-2.svg/660px-Jacobi-ellipsoid-dimensions-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Jacobi-ellipsoid-dimensions-2.svg/880px-Jacobi-ellipsoid-dimensions-2.svg.png 2x" data-class="mw-file-element"> </span></a> <figcaption> Formes ellipsoïdales d'équilibre d'un corps céleste en rotation </figcaption> </figure> <p>La figure ci-contre illustre les formes ellipsoïdales d'équilibre d'un corps céleste en rotation selon les modèles respectifs de Maclaurin et Jacobi : l'abscisse est proportionnelle à la vitesse de rotation angulaire, l'ordonnée donne les longueurs des demi-axes des ellipsoïdes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\geq b\geq 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\geq b\geq c} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a82e6284fedb32392e7cd3a5d0284907898bb0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.431ex; height:2.343ex;" alt="{\displaystyle a\geq b\geq c}"> </noscript><span class="lazy-image-placeholder" style="width: 9.431ex;height: 2.343ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a82e6284fedb32392e7cd3a5d0284907898bb0a" data-alt="{\displaystyle a\geq b\geq c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> ; l'axe de rotation est selon <i>c.</i></p> <p>Aux faibles vitesses de rotation (abscisse ≤ 0,3), les deux modèles prédisent un aplatissement progressif (forme oblate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b\geq 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\geq c} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ffb56ad890665d00740e9f349f5bbcf0c513f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.431ex; height:2.343ex;" alt="{\displaystyle a=b\geq c}"> </noscript><span class="lazy-image-placeholder" style="width: 9.431ex;height: 2.343ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ffb56ad890665d00740e9f349f5bbcf0c513f2" data-alt="{\displaystyle a=b\geq c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>). Pour les vitesses de rotation plus élevées (abscisse > 0,3), le modèle de Jacobi (lignes continues) diffère de celui Maclaurin (lignes pointillées) ; il prévoit une bifurcation vers une forme triaxiale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b>c}"><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></span> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 9.431ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3756bee44d7cb7e6221499eedb579fb848d2e5ac" data-alt="{\displaystyle a>b>c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, qui converge vers une forme prolate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b=c}"><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></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bee403f55d1baf1a567507b184d2b5dd39b6aa7d" 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}"> </noscript><span class="lazy-image-placeholder" style="width: 9.431ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bee403f55d1baf1a567507b184d2b5dd39b6aa7d" data-alt="{\displaystyle a>b=c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> . Le modèle de Jacobi est plus réaliste car il prend en compte les mouvements internes (fluide visqueux).</p> <p>Les deux modèles se limitent aux formes ellipsoïdales ; aux plus hautes vitesses de rotation des bifurcations apparaissent vers d'autres formes d'équilibre, comme par exemple des formes à deux lobes ou plus (formes de « poire » ou de « cacahuète ») ou des <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Tore?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tore">tores</a>.</p> <p>Ont été étudiés aussi les <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Fluide_incompressible?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fluide incompressible">fluides</a> dont la cohésion est assurée par la <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Tension_superficielle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tension superficielle">tension superficielle</a> : par exemple les gouttes d'eau en <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Impesanteur?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Impesanteur">impesanteur</a><sup id="cite_ref-:2_4-1" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:2-4"><span class="cite_crochet">[</span>4<span class="cite_crochet">]</span></a></sup>. L'énergie potentielle à prendre en compte est la tension superficielle directement proportionnelle à l'aire de l'ellipsoïde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {V}}=A\times T}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> V </mi> <mo accent="false"> ¯<!-- ¯ --> </mo> </mover> </mrow> <mo> = </mo> <mi> A </mi> <mo> ×<!-- × --> </mo> <mi> T </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {V}}=A\times T} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac23ae0fe00ba3c0932e9b868655e61826c6a06c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.501ex; height:3.009ex;" alt="{\displaystyle {\overline {V}}=A\times T}"> </noscript><span class="lazy-image-placeholder" style="width: 11.501ex;height: 3.009ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac23ae0fe00ba3c0932e9b868655e61826c6a06c" data-alt="{\displaystyle {\overline {V}}=A\times T}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, où <i><b>T</b></i> dépend de la nature du liquide (<i>T</i> = 8 × 10<sup>−3</sup> <abbr class="abbr" title="joule par mètre carré">J m<sup>−2</sup></abbr> pour l'eau à température ordinaire). Comme pour les planètes, les gouttes d'eau ont des formes oblates aux faibles vitesses de rotation, qui bifurquent vers d'autres formes aux vitesses plus élevées.</p> <p>Les <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Noyau_atomique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Noyau atomique">noyaux atomiques</a> peuvent être déformés<sup id="cite_ref-14" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-14"><span class="cite_crochet">[</span>14<span class="cite_crochet">]</span></a></sup> : soit par une <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Brisure_spontan%C3%A9e_de_sym%C3%A9trie?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Brisure spontanée de symétrie">brisure spontanée de symétrie</a> dans leur état fondamental, soit lorsque leur énergie coulombienne (répulsive) atteint une valeur critique, soit enfin par rotation induite dans les <a href="https://fr-m-wikipedia-org.translate.goog/wiki/R%C3%A9actions_nucl%C3%A9aires_avec_des_ions_lourds?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Réactions nucléaires avec des ions lourds">réactions nucléaires</a>. En première approximation, on peut considérer la <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Mati%C3%A8re_nucl%C3%A9aire_(physique)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matière nucléaire (physique)">matière nucléaire</a> comme incompressible ; dans le <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Mod%C3%A8le_de_la_goutte_liquide?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modèle de la goutte liquide">modèle de la goutte liquide</a>, l'énergie potentielle dépendante de la forme est donc dominée par la compétition entre une énergie de surface attractive et une énergie coulombienne répulsive. Dans leur état fondamental, la plupart des déformations des noyaux lourds sont de type ellipsoïdal ; pour les noyaux très lourds ou en rotation, ces formes deviennent instables ce qui conduit au phénomène de <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Fission_nucl%C3%A9aire?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fission nucléaire">fission</a> (induite ou spontanée).</p> <div class="mw-heading mw-heading3"> <h3 id="Exemples">Exemples</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=26&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Exemples" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <p>Certains ellipsoïdes ont des propriétés spécifiques liées à leur domaine d'application :</p> <ul> <li><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde_de_Fresnel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellipsoïde de Fresnel">Ellipsoïde de Fresnel</a> dans la propagation des ondes mécaniques ou électromagnétiques</li> <li><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde_de_Bessel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellipsoïde de Bessel">Ellipsoïde de Bessel</a> en topographie</li> <li><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/index_ellipsoid" class="extiw" title="en:index ellipsoid">Ellipsoïde diélectrique</a> pour la propagation de la lumière dans les milieux non-isotropes</li> <li><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde_de_Poinsot?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Ellipsoïde de Poinsot">Ellipsoïde de Poinsot</a> en mécanique du solide</li> <li><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Mod%C3%A8le_ellipso%C3%AFdal_de_la_Terre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modèle ellipsoïdal de la Terre">Modèle ellipsoïdal de la Terre</a> (ou ellipsoïde normale) dans l'étude de la <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Forme_de_la_Terre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Forme de la Terre">forme de la Terre</a> et des planètes ou étoiles.</li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=27&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Notes et références" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> <div style="font-size:85%; padding-left:1.6em; margin:0.3em 0;"> <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé <span class="plainlinks">« <a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Ellipsoid?oldid%3D598762912">Ellipsoid</a> » <small>(<a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Ellipsoid?action%3Dhistory">voir la liste des auteurs</a>)</small></span>. </div> <div class="references-small decimal" style=""> <div class="mw-references-wrap mw-references-columns"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-1">↑</a> </span><span class="reference-text">Pour un rappel historique détaillé, voir <span class="ouvrage" id="Chandrasekhar1967"><span class="ouvrage" id="Subrahmanyan_Chandrasekhar1967"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Subrahmanyan_Chandrasekhar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Subrahmanyan Chandrasekhar">Subrahmanyan Chandrasekhar</a>, « <cite style="font-style:normal" lang="en">Ellipsoidal figures of equilibrium - Historical account</cite> », <i><span class="lang-en" lang="en"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Communications_on_Pure_and_Applied_Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Communications on Pure and Applied Mathematics">Comm. Pure Appl. Math.</a></span></i>, <abbr class="abbr" title="volume">vol.</abbr> XX, <time>1967</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">251-265</span> <small style="line-height:1em;">(<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Digital_Object_Identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://dx.doi.org/10.1002/cpa.3160200203">10.1002/cpa.3160200203</a></span>, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20150701043441/http://people.ucsc.edu/~igarrick/EART290/chandrasekhar_1967.pdf">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Ellipsoidal+figures+of+equilibrium+-+Historical+account&rft.jtitle=Comm.+Pure+Appl.+Math.&rft.aulast=Chandrasekhar&rft.aufirst=Subrahmanyan&rft.date=1967&rft.volume=XX&rft.pages=251-265&rft_id=info%3Adoi%2F10.1002%2Fcpa.3160200203&rft_id=https%3A%2F%2Fweb.archive.org%2Fweb%2F20150701043441%2Fhttp%3A%2F%2Fpeople.ucsc.edu%2F~igarrick%2FEART290%2Fchandrasekhar_1967.pdf&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AEllipso%C3%AFde"></span></span></span>.</span></li> <li id="cite_note-:0-2"><span class="mw-cite-backlink noprint">↑ <sup><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:0_2-0">a</a> <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:0_2-1">b</a> <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:0_2-2">c</a> et <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:0_2-3">d</a></sup> </span><span class="reference-text"><span class="ouvrage" id="Chandrasekhar1969"><span class="ouvrage" id="Subrahmanyan_Chandrasekhar1969"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Subrahmanyan Chandrasekhar, <cite class="italique" lang="en">Ellipsoidal Figures of Equilibrium</cite>, New Haven (USA), Yale University Press, <time>1969</time>, 253 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="https://fr-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Standard Book Number">ISBN</a> <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/0-486-65258-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Spécial:Ouvrages de référence/0-486-65258-0"><span class="nowrap">0-486-65258-0</span></a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ellipsoidal+Figures+of+Equilibrium&rft.place=New+Haven+%28USA%29&rft.pub=Yale+University+Press&rft.aulast=Chandrasekhar&rft.aufirst=Subrahmanyan&rft.date=1969&rft.tpages=253&rft.isbn=0-486-65258-0&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AEllipso%C3%AFde"></span></span></span>.</span></li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-3">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Chandrasekhar1965"><span class="ouvrage" id="S._Chandrasekhar1965"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> S. <span class="nom_auteur">Chandrasekhar</span>, « <cite style="font-style:normal" lang="en">The stability of a rotating liquid drop</cite> », <i><span class="lang-en" lang="en"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Proceedings_of_the_Royal_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Proceedings of the Royal Society">Proc. R. Soc. A</a></span></i>, <abbr class="abbr" title="volume">vol.</abbr> 286, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 1404, <time>1965</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">1-26</span> <small style="line-height:1em;">(<a href="https://fr-m-wikipedia-org.translate.goog/wiki/JSTOR?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="JSTOR">JSTOR</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://jstor.org/stable/2415184">2415184</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+stability+of+a+rotating+liquid+drop&rft.jtitle=Proc.+R.+Soc.+A&rft.issue=1404&rft.aulast=Chandrasekhar&rft.aufirst=S.&rft.date=1965&rft.volume=286&rft.pages=1-26&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AEllipso%C3%AFde"></span></span></span>.</span></li> <li id="cite_note-:2-4"><span class="mw-cite-backlink noprint">↑ <sup><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:2_4-0">a</a> et <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:2_4-1">b</a></sup> </span><span class="reference-text"><span class="ouvrage" id="BrownScrivenLighthill1980"><span class="ouvrage" id="R._A._BrownL._E._ScrivenMichael_James_Lighthill1980"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> R. A. <span class="nom_auteur">Brown</span>, L. E. <span class="nom_auteur">Scriven</span> et Michael James <span class="nom_auteur">Lighthill</span>, « <cite style="font-style:normal" lang="en">The shape and stability of rotating liquid drops</cite> », <i><span class="lang-en" lang="en">Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences</span></i>, <abbr class="abbr" title="volume">vol.</abbr> 371, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 1746, <time class="nowrap" datetime="1980-06-30" data-sort-value="1980-06-30">30 juin 1980</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">331-357</span> <small style="line-height:1em;">(<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Digital_Object_Identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://dx.doi.org/10.1098/rspa.1980.0084">10.1098/rspa.1980.0084</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+shape+and+stability+of+rotating+liquid+drops&rft.jtitle=Proceedings+of+the+Royal+Society+of+London.+A.+Mathematical+and+Physical+Sciences&rft.issue=1746&rft.aulast=Brown&rft.aufirst=R.+A.&rft.au=Scriven%2C+L.+E.&rft.au=Lighthill%2C+Michael+James&rft.date=1980-06-30&rft.volume=371&rft.pages=331-357&rft_id=info%3Adoi%2F10.1098%2Frspa.1980.0084&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AEllipso%C3%AFde"></span></span></span>.</span></li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-5">↑</a> </span><span class="reference-text"><span class="ouvrage" id="WangAnilkumarLeeLin1994"><span class="ouvrage" id="T._G._WangA._V._AnilkumarC._P._LeeK._C._Lin1994"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> T. G. <span class="nom_auteur">Wang</span>, A. V. <span class="nom_auteur">Anilkumar</span>, C. P. <span class="nom_auteur">Lee</span> et K. C. <span class="nom_auteur">Lin</span>, « <cite style="font-style:normal" lang="en">Bifurcation of rotating liquid drops: results from USML-1 experiments in Space</cite> », <i><span class="lang-en" lang="en"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Journal_of_Fluid_Mechanics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Journal of Fluid Mechanics">Journal of Fluid Mechanics</a></span></i>, <abbr class="abbr" title="volume">vol.</abbr> 276, <time class="nowrap" datetime="1994-10" data-sort-value="1994-10">octobre 1994</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">389-403</span> <small style="line-height:1em;">(<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Digital_Object_Identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://dx.doi.org/10.1017/S0022112094002612">10.1017/S0022112094002612</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Bifurcation+of+rotating+liquid+drops%3A+results+from+USML-1+experiments+in+Space&rft.jtitle=Journal+of+Fluid+Mechanics&rft.aulast=Wang&rft.aufirst=T.+G.&rft.au=Anilkumar%2C+A.+V.&rft.au=Lee%2C+C.+P.&rft.au=Lin%2C+K.+C.&rft.date=1994-10&rft.volume=276&rft.pages=389-403&rft_id=info%3Adoi%2F10.1017%2FS0022112094002612&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AEllipso%C3%AFde"></span></span></span>.</span></li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-6">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Fernandez2006"><span class="ouvrage" id="Bernard_Fernandez2006">Bernard <span class="nom_auteur">Fernandez</span>, <cite class="italique">De l'atome au noyau : une approche historique de la physique atomique et de la physique nucléaire</cite>, Ellipses, <time>2006</time> <small style="line-height:1em;">(<a href="https://fr-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Standard Book Number">ISBN</a> <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-2-7298-2784-7?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Spécial:Ouvrages de référence/978-2-7298-2784-7"><span class="nowrap">978-2-7298-2784-7</span></a>, <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Online_Computer_Library_Center?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Online Computer Library Center">OCLC</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://worldcat.org/fr/title/69665126">69665126</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=De+l%27atome+au+noyau+%3A+une+approche+historique+de+la+physique+atomique+et+de+la+physique+nucl%C3%A9aire&rft.pub=Ellipses&rft.aulast=Fernandez&rft.aufirst=Bernard&rft.date=2006&rft.isbn=978-2-7298-2784-7&rft_id=info%3Aoclcnum%2F69665126&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AEllipso%C3%AFde"></span></span></span>.</span></li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-7">↑</a> </span><span class="reference-text"><span class="ouvrage" id="2016">« <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.ganil-spiral2.eu/fr/2016/04/22/une-histoire-de-formes-de-noyaux-vieille-de-50-ans-enfin-resolue/"><cite style="font-style:normal;">Une histoire de formes de noyaux vieille de 50 ans … enfin résolue ?</cite></a> », sur <span class="italique">Ganil.spiral2.eu</span>, <time class="nowrap" datetime="2016-04" data-sort-value="2016-04">avril 2016</time> <small style="line-height:1em;">(consulté le <time class="nowrap" datetime="2022-07-27" data-sort-value="2022-07-27">27 juillet 2022</time>)</small></span>.</span></li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-8">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Boyd2012"><span class="ouvrage" id=":Stephen_P._Boyd2012"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Stephen_P._Boyd&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Stephen P. Boyd (page inexistante)">Stephen P. Boyd</a> <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Stephen_P._Boyd" class="extiw" title="en:Stephen P. Boyd"><span class="indicateur-langue" title="Article en anglais : « Stephen P. Boyd »">(en)</span></a>, « <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://stanford.edu/class/ee263/lectures/symm.pdf"><cite style="font-style:normal;" lang="en">Lecture 15. Symmetric matrices, quadratic forms, matrix norm, and SVD</cite></a> », sur <span class="italique"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Universit%C3%A9_Stanford?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Université Stanford">Université Stanford</a></span>, automne 2012-2013</span></span>.</span></li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-9">↑</a> </span><span class="reference-text"><span class="ouvrage" id="HillWheeler1953"><span class="ouvrage" id="David_Lawrence_HillJohn_Archibald_Wheeler1953"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> David Lawrence <span class="nom_auteur">Hill</span> et John Archibald <span class="nom_auteur">Wheeler</span>, « <cite style="font-style:normal" lang="en">Nuclear constitution and the interpretation of fission phenomena</cite> », <i><span class="lang-en" lang="en"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Physical_Review?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Physical Review">Physical Review</a></span></i>, <abbr class="abbr" title="volume">vol.</abbr> 89, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 5, <time class="nowrap" datetime="1953-03-01" data-sort-value="1953-03-01"><abbr class="abbr" title="premier">1<sup>er</sup></abbr> mars 1953</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">1102-1145</span> <small style="line-height:1em;">(<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Digital_Object_Identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://dx.doi.org/10.1103/PhysRev.89.1102">10.1103/PhysRev.89.1102</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Nuclear+constitution+and+the+interpretation+of+fission+phenomena&rft.jtitle=Physical+Review&rft.issue=5&rft.aulast=Hill&rft.aufirst=David+Lawrence&rft.au=Wheeler%2C+John+Archibald&rft.date=1953-03-01&rft.volume=89&rft.pages=1102-1145&rft_id=info%3Adoi%2F10.1103%2FPhysRev.89.1102&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AEllipso%C3%AFde"></span></span></span>.</span></li> <li id="cite_note-:1-10"><span class="mw-cite-backlink noprint">↑ <sup><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:1_10-0">a</a> et <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:1_10-1">b</a></sup> </span><span class="reference-text"><span class="ouvrage" id="RemaudRoyer1981"><span class="ouvrage" id="B._RemaudG._Royer1981"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> B. <span class="nom_auteur">Remaud</span> et G. <span class="nom_auteur">Royer</span>, « <cite style="font-style:normal" lang="en">On the energy dependences of ellipsoidal leptodermous systems</cite> », <i><span class="lang-en" lang="en"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Journal_of_Physics_A?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Journal of Physics A">Journal of Physics A</a>: Mathematical and General</span></i>, <abbr class="abbr" title="volume">vol.</abbr> 14, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 11, <time>1981</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">2897-2910</span> <small style="line-height:1em;">(<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Digital_Object_Identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://dx.doi.org/10.1088/0305-4470/14/11/013">10.1088/0305-4470/14/11/013</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+the+energy+dependences+of+ellipsoidal+leptodermous+systems&rft.jtitle=Journal+of+Physics+A%3A+Mathematical+and+General&rft.issue=11&rft.aulast=Remaud&rft.aufirst=B.&rft.au=Royer%2C+G.&rft.date=1981&rft.volume=14&rft.pages=2897-2910&rft_id=info%3Adoi%2F10.1088%2F0305-4470%2F14%2F11%2F013&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AEllipso%C3%AFde"></span></span></span>.</span></li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-11">↑</a> </span><span class="reference-text"><span class="ouvrage"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> « <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://dlmf.nist.gov/19.33"><cite style="font-style:normal;" lang="en">DLMF: § 19.33</cite></a> », sur <span class="italique">dlmf.<a href="https://fr-m-wikipedia-org.translate.goog/wiki/NIST?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="NIST">nist</a>.gov</span></span>.</span></li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-12">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Kellogg1929"><span class="ouvrage" id="O._D._Kellogg1929">O. D. <span class="nom_auteur">Kellogg</span>, <cite class="italique">Foundations of Potential theory</cite>, Springer, <time>1929</time> <small style="line-height:1em;">(<a href="https://fr-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Standard Book Number">ISBN</a> <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-0-486-60144-1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Spécial:Ouvrages de référence/978-0-486-60144-1"><span class="nowrap">978-0-486-60144-1</span></a> et <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-1-4437-2153-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Spécial:Ouvrages de référence/978-1-4437-2153-0"><span class="nowrap">978-1-4437-2153-0</span></a>, <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Online_Computer_Library_Center?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Online Computer Library Center">OCLC</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://worldcat.org/fr/title/826373">826373</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Potential+theory&rft.pub=Springer&rft.aulast=Kellogg&rft.aufirst=O.+D.&rft.date=1929&rft.isbn=978-0-486-60144-1&rft_id=info%3Aoclcnum%2F826373&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AEllipso%C3%AFde"></span></span></span>.</span></li> <li id="cite_note-13"><span class="mw-cite-backlink"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-13">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Carlson1961"><span class="ouvrage" id="B._C._Carlson1961"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> B. C. <span class="nom_auteur">Carlson</span>, « <cite style="font-style:normal" lang="en">Ellipsoidal distributions of charge or mass</cite> », <i><span class="lang-en" lang="en"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Journal_of_Mathematical_Physics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Journal of Mathematical Physics">J. Math. Phys.</a></span></i>, <abbr class="abbr" title="volume">vol.</abbr> 2, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 3, <time>1961</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">441-450</span> <small style="line-height:1em;">(<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Digital_Object_Identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://dx.doi.org/10.1063/1.1703729">10.1063/1.1703729</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Ellipsoidal+distributions+of+charge+or+mass&rft.jtitle=J.+Math.+Phys.&rft.issue=3&rft.aulast=Carlson&rft.aufirst=B.+C.&rft.date=1961&rft.volume=2&rft.pages=441-450&rft_id=info%3Adoi%2F10.1063%2F1.1703729&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AEllipso%C3%AFde"></span></span></span>.</span></li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-14">↑</a> </span><span class="reference-text"><span class="ouvrage" id="BohrB.R.Mottelson1998"><span class="ouvrage" id="A._BohrB.R.Mottelson1998"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> A. <span class="nom_auteur">Bohr</span> et B.R.Mottelson, <cite class="italique" lang="en">Nuclear structure</cite>, <abbr class="abbr" title="volume">vol.</abbr> 2 - Nuclear deformations, World Scientific, <time>1998</time> <small style="line-height:1em;">(<a href="https://fr-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Standard Book Number">ISBN</a> <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-981-02-3197-2?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Spécial:Ouvrages de référence/978-981-02-3197-2"><span class="nowrap">978-981-02-3197-2</span></a>, <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Online_Computer_Library_Center?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Online Computer Library Center">OCLC</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://worldcat.org/fr/title/38112813">38112813</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nuclear+structure&rft.pub=World+Scientific&rft.aulast=Bohr&rft.aufirst=A.&rft.au=B.R.Mottelson&rft.date=1998&rft.volume=2+-+Nuclear+deformations&rft.isbn=978-981-02-3197-2&rft_id=info%3Aoclcnum%2F38112813&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AEllipso%C3%AFde"></span></span></span>.</span></li> </ol> </div> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=edit&section=28&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Voir aussi" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> <style data-mw-deduplicate="TemplateStyles:r194021218">.mw-parser-output .autres-projets>.titre{text-align:center;margin:0.2em 0}.mw-parser-output .autres-projets>ul{margin:0;padding:0}.mw-parser-output .autres-projets>ul>li{list-style:none;margin:0.2em 0;text-indent:0;padding-left:24px;min-height:20px;text-align:left;display:block}.mw-parser-output .autres-projets>ul>li>a{font-style:italic}@media(max-width:720px){.mw-parser-output .autres-projets{float:none}}</style> <div class="autres-projets boite-grise boite-a-droite noprint js-interprojets"> <p class="titre">Sur les autres projets Wikimedia :</p> <ul class="noarchive plainlinks"> <li class="commons"><a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://commons.wikimedia.org/wiki/Category:Ellipsoids?uselang%3Dfr">Ellipsoïde</a>, sur <span class="project">Wikimedia Commons</span></li> <li class="wiktionary"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wiktionary.org/wiki/ellipso%25C3%25AFde" class="extiw" title="wikt:ellipsoïde">ellipsoïde</a>, <span class="nowrap">sur le <span class="project">Wiktionnaire</span></span></li> </ul> </div> <ul> <li><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ellipso%C3%AFde_d%27inertie?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Ellipsoïde d'inertie">Ellipsoïde d'inertie</a></li> <li><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Th%C3%A9orie_des_figures_d%27%C3%A9quilibre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Théorie des figures d'équilibre">Théorie des figures d'équilibre</a></li> <li><a href="https://fr-m-wikipedia-org.translate.goog/wiki/M%C3%A9thode_de_l%27ellipso%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Méthode de l'ellipsoïde">Méthode de l'ellipsoïde</a></li> <li><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Syst%C3%A8me_de_coordonn%C3%A9es_(cartographie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Système de coordonnées (cartographie)">Système de coordonnées (cartographie)</a></li> </ul> <div class="navbox-container" style="clear:both;"> </div> <ul id="bandeau-portail" class="bandeau-portail"> <li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Portail:G%C3%A9om%C3%A9trie?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Portail de la géométrie"> <noscript> <img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Circle-icons-rulertriangle.svg/24px-Circle-icons-rulertriangle.svg.png" decoding="async" width="24" height="24" class="mw-file-element" data-file-width="512" data-file-height="512"> </noscript><span class="lazy-image-placeholder" style="width: 24px;height: 24px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Circle-icons-rulertriangle.svg/24px-Circle-icons-rulertriangle.svg.png" data-alt="icône décorative" data-width="24" data-height="24" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Circle-icons-rulertriangle.svg/36px-Circle-icons-rulertriangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Circle-icons-rulertriangle.svg/48px-Circle-icons-rulertriangle.svg.png 2x" data-class="mw-file-element"> </span></a></span></span> <span class="bandeau-portail-texte"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Portail:G%C3%A9om%C3%A9trie?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Portail:Géométrie">Portail de la géométrie</a></span> </span></li> </ul><!-- NewPP limit report Parsed by 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Rendering was triggered because: api-parse --> </section> </div><!-- MobileFormatter took 0.033 seconds --><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --> <noscript> <img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=mobile&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div class="printfooter" data-nosnippet=""> Ce document provient de « <a dir="ltr" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/w/index.php?title%3DEllipso%C3%AFde%26oldid%3D220500782">https://fr.wikipedia.org/w/index.php?title=Ellipsoïde&oldid=220500782</a> ». </div> </div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"><a class="last-modified-bar" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde&action=history&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB"> <div class="post-content last-modified-bar__content"><span class="minerva-icon minerva-icon-size-medium minerva-icon--modified-history"></span> <span class="last-modified-bar__text modified-enhancement" data-user-name="Vega" data-user-gender="male" data-timestamp="1732229405"> <span>Dernière modification le 21 novembre 2024, à 23:50</span> </span> <span class="minerva-icon minerva-icon-size-small minerva-icon--expand"></span> </div></a> <div class="post-content footer-content"> <div id="mw-data-after-content"> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Langues</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ar.wikipedia.org/wiki/%25D8%25B3%25D8%25B7%25D8%25AD_%25D9%2586%25D8%25A7%25D9%2582%25D8%25B5%25D9%258A" title="سطح ناقصي – arabe" lang="ar" hreflang="ar" data-title="سطح ناقصي" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li> <li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ast.wikipedia.org/wiki/Elipsoide_de_revoluci%25C3%25B3n" title="Elipsoide de revolución – asturien" lang="ast" hreflang="ast" data-title="Elipsoide de revolución" data-language-autonym="Asturianu" data-language-local-name="asturien" class="interlanguage-link-target"><span>Asturianu</span></a></li> <li class="interlanguage-link interwiki-az mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://az.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – azerbaïdjanais" lang="az" hreflang="az" data-title="Ellipsoid" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaïdjanais" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li> <li class="interlanguage-link interwiki-be mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://be.wikipedia.org/wiki/%25D0%25AD%25D0%25BB%25D1%2596%25D0%25BF%25D1%2581%25D0%25BE%25D1%2596%25D0%25B4" title="Эліпсоід – biélorusse" lang="be" hreflang="be" data-title="Эліпсоід" data-language-autonym="Беларуская" data-language-local-name="biélorusse" class="interlanguage-link-target"><span>Беларуская</span></a></li> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bg.wikipedia.org/wiki/%25D0%2595%25D0%25BB%25D0%25B8%25D0%25BF%25D1%2581%25D0%25BE%25D0%25B8%25D0%25B4" title="Елипсоид – bulgare" lang="bg" hreflang="bg" data-title="Елипсоид" data-language-autonym="Български" data-language-local-name="bulgare" class="interlanguage-link-target"><span>Български</span></a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/El%25C2%25B7lipsoide" title="El·lipsoide – catalan" lang="ca" hreflang="ca" data-title="El·lipsoide" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/wiki/Elipsoid" title="Elipsoid – tchèque" lang="cs" hreflang="cs" data-title="Elipsoid" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li> <li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cv.wikipedia.org/wiki/%25D0%25AD%25D0%25BB%25D0%25BB%25D0%25B8%25D0%25BF%25D1%2581%25D0%25BE%25D0%25B8%25D0%25B4" title="Эллипсоид – tchouvache" lang="cv" hreflang="cv" data-title="Эллипсоид" data-language-autonym="Чӑвашла" data-language-local-name="tchouvache" class="interlanguage-link-target"><span>Чӑвашла</span></a></li> <li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cy.wikipedia.org/wiki/Elipsoid" title="Elipsoid – gallois" lang="cy" hreflang="cy" data-title="Elipsoid" data-language-autonym="Cymraeg" data-language-local-name="gallois" class="interlanguage-link-target"><span>Cymraeg</span></a></li> <li class="interlanguage-link interwiki-da mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://da.wikipedia.org/wiki/Ellipsoide" title="Ellipsoide – danois" lang="da" hreflang="da" data-title="Ellipsoide" data-language-autonym="Dansk" data-language-local-name="danois" class="interlanguage-link-target"><span>Dansk</span></a></li> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – allemand" lang="de" hreflang="de" data-title="Ellipsoid" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li> <li class="interlanguage-link interwiki-el mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://el.wikipedia.org/wiki/%25CE%2595%25CE%25BB%25CE%25BB%25CE%25B5%25CE%25B9%25CF%2588%25CE%25BF%25CE%25B5%25CE%25B9%25CE%25B4%25CE%25AE" title="Ελλειψοειδή – grec" lang="el" hreflang="el" data-title="Ελλειψοειδή" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – anglais" lang="en" hreflang="en" data-title="Ellipsoid" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li> <li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eo.wikipedia.org/wiki/Elipsoido" title="Elipsoido – espéranto" lang="eo" hreflang="eo" data-title="Elipsoido" data-language-autonym="Esperanto" data-language-local-name="espéranto" class="interlanguage-link-target"><span>Esperanto</span></a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Elipsoide" title="Elipsoide – espagnol" lang="es" hreflang="es" data-title="Elipsoide" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li> <li class="interlanguage-link interwiki-et mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – estonien" lang="et" hreflang="et" data-title="Ellipsoid" data-language-autonym="Eesti" data-language-local-name="estonien" class="interlanguage-link-target"><span>Eesti</span></a></li> <li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eu.wikipedia.org/wiki/Elipsoide" title="Elipsoide – basque" lang="eu" hreflang="eu" data-title="Elipsoide" data-language-autonym="Euskara" data-language-local-name="basque" class="interlanguage-link-target"><span>Euskara</span></a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25D8%25A8%25DB%258C%25D8%25B6%25DB%258C%25E2%2580%258C%25DA%25AF%25D9%2588%25D9%2586" title="بیضیگون – persan" lang="fa" hreflang="fa" data-title="بیضیگون" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</span></a></li> <li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fi.wikipedia.org/wiki/Ellipsoidi" title="Ellipsoidi – finnois" lang="fi" hreflang="fi" data-title="Ellipsoidi" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</span></a></li> <li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ga.wikipedia.org/wiki/%25C3%2589ileaps%25C3%25B3ideach" title="Éileapsóideach – irlandais" lang="ga" hreflang="ga" data-title="Éileapsóideach" data-language-autonym="Gaeilge" data-language-local-name="irlandais" class="interlanguage-link-target"><span>Gaeilge</span></a></li> <li class="interlanguage-link interwiki-he mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%2590%25D7%259C%25D7%2599%25D7%25A4%25D7%25A1%25D7%2595%25D7%2590%25D7%2599%25D7%2593" title="אליפסואיד – hébreu" lang="he" hreflang="he" data-title="אליפסואיד" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</span></a></li> <li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hi.wikipedia.org/wiki/%25E0%25A4%25A6%25E0%25A5%2580%25E0%25A4%25B0%25E0%25A5%258D%25E0%25A4%2598%25E0%25A4%25B5%25E0%25A5%2583%25E0%25A4%25A4%25E0%25A5%258D%25E0%25A4%25A4%25E0%25A4%25BE%25E0%25A4%25AD" title="दीर्घवृत्ताभ – hindi" lang="hi" hreflang="hi" data-title="दीर्घवृत्ताभ" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li> <li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hu.wikipedia.org/wiki/Ellipszoid" title="Ellipszoid – hongrois" lang="hu" hreflang="hu" data-title="Ellipszoid" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</span></a></li> <li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hy.wikipedia.org/wiki/%25D4%25B7%25D5%25AC%25D5%25AB%25D5%25BA%25D5%25BD%25D5%25B8%25D5%25AB%25D5%25A4" title="Էլիպսոիդ – arménien" lang="hy" hreflang="hy" data-title="Էլիպսոիդ" data-language-autonym="Հայերեն" data-language-local-name="arménien" class="interlanguage-link-target"><span>Հայերեն</span></a></li> <li class="interlanguage-link interwiki-id mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://id.wikipedia.org/wiki/Elipsoid" title="Elipsoid – indonésien" lang="id" hreflang="id" data-title="Elipsoid" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li> <li class="interlanguage-link interwiki-is mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://is.wikipedia.org/wiki/Sporvala" title="Sporvala – islandais" lang="is" hreflang="is" data-title="Sporvala" data-language-autonym="Íslenska" data-language-local-name="islandais" class="interlanguage-link-target"><span>Íslenska</span></a></li> <li class="interlanguage-link interwiki-it mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://it.wikipedia.org/wiki/Ellissoide" title="Ellissoide – italien" lang="it" hreflang="it" data-title="Ellissoide" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target"><span>Italiano</span></a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E6%25A5%2595%25E5%2586%2586%25E4%25BD%2593" title="楕円体 – japonais" lang="ja" hreflang="ja" data-title="楕円体" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li> <li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ka.wikipedia.org/wiki/%25E1%2583%2594%25E1%2583%259A%25E1%2583%2598%25E1%2583%25A4%25E1%2583%25A1%25E1%2583%259D%25E1%2583%2598%25E1%2583%2593%25E1%2583%2598" title="ელიფსოიდი – géorgien" lang="ka" hreflang="ka" data-title="ელიფსოიდი" data-language-autonym="ქართული" data-language-local-name="géorgien" class="interlanguage-link-target"><span>ქართული</span></a></li> <li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://kk.wikipedia.org/wiki/%25D0%25AD%25D0%25BB%25D0%25BB%25D0%25B8%25D0%25BF%25D1%2581%25D0%25BE%25D0%25B8%25D0%25B4" title="Эллипсоид – kazakh" lang="kk" hreflang="kk" data-title="Эллипсоид" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li> <li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ko.wikipedia.org/wiki/%25ED%2583%2580%25EC%259B%2590%25EB%25A9%25B4" title="타원면 – coréen" lang="ko" hreflang="ko" data-title="타원면" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li> <li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ky.wikipedia.org/wiki/%25D0%25AD%25D0%25BB%25D0%25BB%25D0%25B8%25D0%25BF%25D1%2581%25D0%25BE%25D0%25B8%25D0%25B4" title="Эллипсоид – kirghize" lang="ky" hreflang="ky" data-title="Эллипсоид" data-language-autonym="Кыргызча" data-language-local-name="kirghize" class="interlanguage-link-target"><span>Кыргызча</span></a></li> <li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lb.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – luxembourgeois" lang="lb" hreflang="lb" data-title="Ellipsoid" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxembourgeois" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li> <li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lv.wikipedia.org/wiki/Elipso%25C4%25ABds" title="Elipsoīds – letton" lang="lv" hreflang="lv" data-title="Elipsoīds" data-language-autonym="Latviešu" data-language-local-name="letton" class="interlanguage-link-target"><span>Latviešu</span></a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Ellipso%25C3%25AFde" title="Ellipsoïde – néerlandais" lang="nl" hreflang="nl" data-title="Ellipsoïde" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nn.wikipedia.org/wiki/Ellipsoide" title="Ellipsoide – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Ellipsoide" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li> <li class="interlanguage-link interwiki-no mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://no.wikipedia.org/wiki/Ellipsoide" title="Ellipsoide – norvégien bokmål" lang="nb" hreflang="nb" data-title="Ellipsoide" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li> <li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pl.wikipedia.org/wiki/Elipsoida" title="Elipsoida – polonais" lang="pl" hreflang="pl" data-title="Elipsoida" data-language-autonym="Polski" data-language-local-name="polonais" class="interlanguage-link-target"><span>Polski</span></a></li> <li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/Elipsoide" title="Elipsoide – portugais" lang="pt" hreflang="pt" data-title="Elipsoide" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li> <li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ro.wikipedia.org/wiki/Elipsoid" title="Elipsoid – roumain" lang="ro" hreflang="ro" data-title="Elipsoid" data-language-autonym="Română" data-language-local-name="roumain" class="interlanguage-link-target"><span>Română</span></a></li> <li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%25AD%25D0%25BB%25D0%25BB%25D0%25B8%25D0%25BF%25D1%2581%25D0%25BE%25D0%25B8%25D0%25B4" title="Эллипсоид – russe" lang="ru" hreflang="ru" data-title="Эллипсоид" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li> <li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://simple.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – Simple English" lang="en-simple" hreflang="en-simple" data-title="Ellipsoid" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li> <li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sk.wikipedia.org/wiki/Elipsoid" title="Elipsoid – slovaque" lang="sk" hreflang="sk" data-title="Elipsoid" data-language-autonym="Slovenčina" data-language-local-name="slovaque" class="interlanguage-link-target"><span>Slovenčina</span></a></li> <li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sl.wikipedia.org/wiki/Elipsoid" title="Elipsoid – slovène" lang="sl" hreflang="sl" data-title="Elipsoid" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li> <li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sq.wikipedia.org/wiki/Elipsoidi" title="Elipsoidi – albanais" lang="sq" hreflang="sq" data-title="Elipsoidi" data-language-autonym="Shqip" data-language-local-name="albanais" class="interlanguage-link-target"><span>Shqip</span></a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – suédois" lang="sv" hreflang="sv" data-title="Ellipsoid" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target"><span>Svenska</span></a></li> <li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ta.wikipedia.org/wiki/%25E0%25AE%25A8%25E0%25AF%2580%25E0%25AE%25B3%25E0%25AF%258D%25E0%25AE%25B5%25E0%25AE%259F%25E0%25AF%258D%25E0%25AE%259F%25E0%25AE%25A4%25E0%25AF%258D%25E0%25AE%25A4%25E0%25AE%25BF%25E0%25AE%25A3%25E0%25AF%258D%25E0%25AE%25AE%25E0%25AE%25AE%25E0%25AF%258D" title="நீள்வட்டத்திண்மம் – tamoul" lang="ta" hreflang="ta" data-title="நீள்வட்டத்திண்மம்" data-language-autonym="தமிழ்" data-language-local-name="tamoul" class="interlanguage-link-target"><span>தமிழ்</span></a></li> <li class="interlanguage-link interwiki-th mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://th.wikipedia.org/wiki/%25E0%25B8%2597%25E0%25B8%25A3%25E0%25B8%2587%25E0%25B8%25A3%25E0%25B8%25B5" title="ทรงรี – thaï" lang="th" hreflang="th" data-title="ทรงรี" data-language-autonym="ไทย" data-language-local-name="thaï" class="interlanguage-link-target"><span>ไทย</span></a></li> <li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/wiki/Elipsoit" title="Elipsoit – turc" lang="tr" hreflang="tr" data-title="Elipsoit" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%2595%25D0%25BB%25D1%2596%25D0%25BF%25D1%2581%25D0%25BE%25D1%2597%25D0%25B4" title="Еліпсоїд – ukrainien" lang="uk" hreflang="uk" data-title="Еліпсоїд" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li> <li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uz.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – ouzbek" lang="uz" hreflang="uz" data-title="Ellipsoid" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ouzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li> <li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vi.wikipedia.org/wiki/Ellipsoid" title="Ellipsoid – vietnamien" lang="vi" hreflang="vi" data-title="Ellipsoid" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamien" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li> <li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh.wikipedia.org/wiki/%25E6%25A4%25AD%25E7%2590%2583" title="椭球 – chinois" lang="zh" hreflang="zh" data-title="椭球" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li> <li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh-yue.wikipedia.org/wiki/%25E6%25A9%25A2%25E7%2590%2583" title="橢球 – cantonais" lang="yue" hreflang="yue" data-title="橢球" data-language-autonym="粵語" data-language-local-name="cantonais" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> </section> </div> <div class="minerva-footer-logo"> <img src="/static/images/mobile/copyright/wikipedia-wordmark-fr.svg" alt="Wikipédia" width="119" height="18" style="width: 7.4375em; height: 1.125em;"> </div> <ul id="footer-info" 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