Ellipsoïde de révolution

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open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <div class="bandeau-container metadata homonymie hatnote"> <div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"> <span class="noviewer" typeof="mw:File"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Aide:Homonymie?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Aide:Homonymie"><img alt="Page d’aide sur l’homonymie" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/20px-Logo_disambig.svg.png" decoding="async" width="20" height="15" 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/a/a9/Logo_disambig.svg/30px-Logo_disambig.svg.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/40px-Logo_disambig.svg.png 2x" data-file-width="512" data-file-height="375"></a></span> </div> <div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p>Pour les articles homonymes, voir <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>.</p> </div> </div> <div class="bandeau-container metadata homonymie hatnote"> <div class="bandeau-cell bandeau-icone-css general" style="display:table-cell;padding-right:0.5em"> <p>Pour un article plus général, voir <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" title="Ellipsoïde">Ellipsoïde</a>.</p> </div> </div> <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>, un <b>ellipsoïde de révolution</b>, ou <b>sphéroïde,</b> est une <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> obtenue par <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 dans l'espace</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 de l'un de ses <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Axe_(math%C3%A9matiques)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Axe (mathématiques)">axes</a> de symétrie. Comme tout <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" title="Ellipsoïde">ellipsoïde</a>, il s'agit d'une <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Surface_(g%C3%A9om%C3%A9trie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Surface (géométrie)">surface</a> <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">quadrique</a>, c'est-à-dire qu'elle est décrite par une équation de degré 2 en chaque coordonnée dans un <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Rep%C3%A8re_cart%C3%A9sien?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Repère cartésien">repère cartésien</a>.</p> <div class="thumb tright"> <div class="thumbinner" style="width:350px;"> <div style="border:1px solid #ccc; padding:0.5em; text-align:center; color:inherit;"> <span typeof="mw:File"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Fichier:ProlateSpheroid.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/8/88/ProlateSpheroid.png/100px-ProlateSpheroid.png" decoding="async" width="100" height="163" 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/8/88/ProlateSpheroid.png/150px-ProlateSpheroid.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/8/88/ProlateSpheroid.png/200px-ProlateSpheroid.png 2x" data-file-width="252" data-file-height="411"></a></span> <span typeof="mw:File"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Fichier:OblateSpheroid.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/b5/OblateSpheroid.PNG/150px-OblateSpheroid.PNG" decoding="async" width="150" height="127" 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/b5/OblateSpheroid.PNG/225px-OblateSpheroid.PNG 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/b/b5/OblateSpheroid.PNG/300px-OblateSpheroid.PNG 2x" data-file-width="503" data-file-height="426"></a></span> </div> <div class="thumbcaption"> Ellipsoïdes allongé (oblong ou prolate), et aplati (oblate). </div> </div> </div> <p>L'expression peut aussi parfois désigner le volume borné délimité par cette surface, notamment pour décrire des objets physiques tels que la <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Terre_(plan%C3%A8te)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Terre (planète)">Terre</a> ou des <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>.</p> <p>Un ellipsoïde de révolution peut être :</p> <ul> <li>allongé (ou <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wiktionary.org/wiki/oblong%23fr" class="extiw" title="wikt:oblong">oblong</a> ou <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wiktionary.org/wiki/prolate%23fr" class="extiw" title="wikt:prolate">prolate</a>) si l'axe de rotation est l'axe principal (ou grand axe) de l'ellipse, ce qui lui donne une forme de <span class="page_h"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Ballon_de_rugby?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-disambig" title="Ballon de rugby">ballon de rugby</a></span> ;</li> <li>aplati (<a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wiktionary.org/wiki/oblate%23fr" class="extiw" title="wikt:oblate">oblate</a>) dans le cas contraire (par exemple 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">surface de la Terre</a>, approximativement) ;</li> <li><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">sphérique</a>, dans le cas particulier où l'ellipse génératrice est un cercle.</li> </ul> <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_de_r%C3%A9volution?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Propri%C3%A9t%C3%A9s"><span class="tocnumber">1</span> <span class="toctext">Propriétés</span></a> <ul> <li class="toclevel-2 tocsection-2"><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#Param%C3%A9trisation"><span class="tocnumber">1.1</span> <span class="toctext">Paramétrisation</span></a></li> <li class="toclevel-2 tocsection-3"><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#%C3%89quation_cart%C3%A9sienne"><span class="tocnumber">1.2</span> <span class="toctext">Équation cartésienne</span></a></li> <li class="toclevel-2 tocsection-4"><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#Volume_int%C3%A9rieur"><span class="tocnumber">1.3</span> <span class="toctext">Volume intérieur</span></a></li> <li class="toclevel-2 tocsection-5"><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#Aire"><span class="tocnumber">1.4</span> <span class="toctext">Aire</span></a></li> </ul></li> <li class="toclevel-1 tocsection-6"><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#Applications"><span class="tocnumber">2</span> <span class="toctext">Applications</span></a></li> <li class="toclevel-1 tocsection-7"><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#Liens_externes"><span class="tocnumber">3</span> <span class="toctext">Liens externes</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="Propriétés"><span id="Propri.C3.A9t.C3.A9s"></span>Propriétés</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde_de_r%C3%A9volution&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Propriétés" 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"> <div class="mw-heading mw-heading3"> <h3 id="Paramétrisation"><span id="Param.C3.A9trisation"></span>Paramétrisation</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde_de_r%C3%A9volution&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Paramétrisation" 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 plan de coupe contenant l'axe de rotation, la trace de l'ellipsoïde est une ellipse paramétrée en <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Coordonn%C3%A9es_cylindriques?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Coordonnées cylindriques">coordonnées cylindriques</a> par un angle au centre <span class="texhtml mvar" style="font-style:italic;">θ</span> variant entre <span class="texhtml">0</span> et <span class="texhtml">2π</span> sous 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 {\begin{cases}r(\theta )=q\cos \theta \\z(\theta )=p\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> r </mi> <mo stretchy="false"> ( </mo> <mi> θ </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> q </mi> <mi> cos </mi> <mo> ⁡ </mo> <mi> θ </mi> </mtd> </mtr> <mtr> <mtd> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> θ </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> p </mi> <mi> sin </mi> <mo> ⁡ </mo> <mi> 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}r(\theta )=q\cos \theta \\z(\theta )=p\sin \theta \end{cases}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2cfcfc0ef5050371ca8b47e908641d375fb54aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.587ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}r(\theta )=q\cos \theta \\z(\theta )=p\sin \theta \end{cases}}}"> </noscript><span class="lazy-image-placeholder" style="width: 15.587ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2cfcfc0ef5050371ca8b47e908641d375fb54aa" data-alt="{\displaystyle {\begin{cases}r(\theta )=q\cos \theta \\z(\theta )=p\sin \theta \end{cases}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>où <span class="texhtml mvar" style="font-style:italic;">p</span> est le rayon polaire (longueur du demi-axe de rotation) et <span class="texhtml mvar" style="font-style:italic;">q</span> le rayon équatorial de l'ellipsoïde.</p> <p>L'ellipsoïde de révolution est donc paramétré en coordonnées cartésiennes dans 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 orthonormal</a> approprié par : <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(\theta ,\phi )=q\cos \theta \cos \phi \\y(\theta ,\phi )=q\cos \theta \sin \phi \\z(\theta ,\phi )=p\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 stretchy="false"> ( </mo> <mi> θ </mi> <mo> , </mo> <mi> ϕ </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> q </mi> <mi> cos </mi> <mo> ⁡ </mo> <mi> θ </mi> <mi> cos </mi> <mo> ⁡ </mo> <mi> ϕ </mi> </mtd> </mtr> <mtr> <mtd> <mi> y </mi> <mo stretchy="false"> ( </mo> <mi> θ </mi> <mo> , </mo> <mi> ϕ </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> q </mi> <mi> cos </mi> <mo> ⁡ </mo> <mi> θ </mi> <mi> sin </mi> <mo> ⁡ </mo> <mi> ϕ </mi> </mtd> </mtr> <mtr> <mtd> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> θ </mi> <mo> , </mo> <mi> ϕ </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> p </mi> <mi> sin </mi> <mo> ⁡ </mo> <mi> 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(\theta ,\phi )=q\cos \theta \cos \phi \\y(\theta ,\phi )=q\cos \theta \sin \phi \\z(\theta ,\phi )=p\sin \theta \end{cases}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9227679f10673cfd44ab86e54c5643628f2c73c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:23.881ex; height:8.843ex;" alt="{\displaystyle {\begin{cases}x(\theta ,\phi )=q\cos \theta \cos \phi \\y(\theta ,\phi )=q\cos \theta \sin \phi \\z(\theta ,\phi )=p\sin \theta \end{cases}}}"> </noscript><span class="lazy-image-placeholder" style="width: 23.881ex;height: 8.843ex;vertical-align: -3.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9227679f10673cfd44ab86e54c5643628f2c73c9" data-alt="{\displaystyle {\begin{cases}x(\theta ,\phi )=q\cos \theta \cos \phi \\y(\theta ,\phi )=q\cos \theta \sin \phi \\z(\theta ,\phi )=p\sin \theta \end{cases}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span> où l'angle de rotation <span class="texhtml mvar" style="font-style:italic;">ϕ</span> varie entre <span class="texhtml">0</span> et <span class="texhtml">π</span>.</p> <p>Cette paramétrisation n'est pas unique. Une paramétrisation équivalente, mais qui rend justice à la symétrie de révolution autour de l'axe <span class="texhtml">O<i>z</i></span> et à la symétrie par rapport au plan <span class="texhtml"><i>x</i>O<i>y</i></span>, prend <span class="texhtml mvar" style="font-style:italic;">θ</span> compris entre −<span class="texhtml"><span style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">π</span><span style="position:absolute;left:-10000px;top:auto;width:1px;height:1px;overflow:hidden">/</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span></span> et +<span class="texhtml"><span style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">π</span><span style="position:absolute;left:-10000px;top:auto;width:1px;height:1px;overflow:hidden">/</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span></span>, et <span class="texhtml mvar" style="font-style:italic;">ϕ</span> entre <span class="texhtml">0</span> et <span class="texhtml">2π</span> ou −<span class="texhtml">π</span> et +<span class="texhtml">π</span>.</p> <div class="mw-heading mw-heading3"> <h3 id="Équation_cartésienne"><span id=".C3.89quation_cart.C3.A9sienne"></span>Équation cartésienne</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde_de_r%C3%A9volution&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Équation cartésienne" 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>La paramétrisation proposée ci-dessus fournit l'<a href="https://fr-m-wikipedia-org.translate.goog/wiki/%C3%89quation_cart%C3%A9sienne?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Équation cartésienne">équation cartésienne</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 {\frac {x^{2}}{q^{2}}}+{\frac {y^{2}}{q^{2}}}+{\frac {z^{2}}{p^{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> q </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> q </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> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> = </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {x^{2}}{q^{2}}}+{\frac {y^{2}}{q^{2}}}+{\frac {z^{2}}{p^{2}}}=1} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acbcff7b8fa3d294c90c71f6896ccaf2006f13d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.272ex; height:6.343ex;" alt="{\displaystyle {\frac {x^{2}}{q^{2}}}+{\frac {y^{2}}{q^{2}}}+{\frac {z^{2}}{p^{2}}}=1}"> </noscript><span class="lazy-image-placeholder" style="width: 19.272ex;height: 6.343ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acbcff7b8fa3d294c90c71f6896ccaf2006f13d2" data-alt="{\displaystyle {\frac {x^{2}}{q^{2}}}+{\frac {y^{2}}{q^{2}}}+{\frac {z^{2}}{p^{2}}}=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span>qui montre que l'ellipsoïde de révolution est une surface <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">quadrique</a>.</p> <p>Avec ces notations, un ellipsoïde de révolution apparaît comme l'image d'une sphère de rayon <span class="texhtml mvar" style="font-style:italic;">q</span> par une <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Affinit%C3%A9_(math%C3%A9matiques)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Affinité (mathématiques)">affinité</a> de rapport <span class="texhtml"><i>p</i>/<i>q</i></span> parallèlement à l'axe de rotation.</p> <div class="mw-heading mw-heading3"> <h3 id="Volume_intérieur"><span id="Volume_int.C3.A9rieur"></span>Volume intérieur</h3><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde_de_r%C3%A9volution&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Volume intérieur" 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 volume intérieur délimité par un ellipsoïde de révolution s'obtient comme cas particulier du <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" title="Ellipsoïde">volume d'un ellipsoïde quelconque</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 V={\frac {4}{3}}\pi pq^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> V </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 4 </mn> <mn> 3 </mn> </mfrac> </mrow> <mi> π </mi> <mi> p </mi> <msup> <mi> q </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V={\frac {4}{3}}\pi pq^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b8c086d1f576d60d66c04e69233c3aa0f63f540" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.519ex; height:5.176ex;" alt="{\displaystyle V={\frac {4}{3}}\pi pq^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.519ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b8c086d1f576d60d66c04e69233c3aa0f63f540" data-alt="{\displaystyle V={\frac {4}{3}}\pi pq^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span> où <span class="texhtml mvar" style="font-style:italic;">p</span> est le rayon polaire et <span class="texhtml mvar" style="font-style:italic;">q</span> le rayon à l'équateur.</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_de_r%C3%A9volution&action=edit&section=5&_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'aire d'un ellipsoïde de révolution est donnée par deux formules différentes selon que l'axe de symétrie de l'ellipse utilisé pour la rotation est son grand axe ou son <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Petit_axe?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Petit axe">petit axe</a>. Pour lever les ambiguïtés, les notations choisies sont les notations usuelles pour les ellipses : la demi-longueur du grand axe est notée <span class="texhtml mvar" style="font-style:italic;">a</span>, celle du petit axe est notée <span class="texhtml mvar" style="font-style:italic;">b</span>, l'<a href="https://fr-m-wikipedia-org.translate.goog/wiki/Excentricit%C3%A9_(math%C3%A9matiques)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Excentricité (mathématiques)">excentricité</a> <span class="texhtml mvar" style="font-style:italic;">e</span> étant donnée par la formule :<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 e={\frac {\sqrt {a^{2}-b^{2}}}{a}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> e </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> <mi> a </mi> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e={\frac {\sqrt {a^{2}-b^{2}}}{a}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bed5a1479d6ab45360ff6812fcb935fadb507ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.165ex; height:6.176ex;" alt="{\displaystyle e={\frac {\sqrt {a^{2}-b^{2}}}{a}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 15.165ex;height: 6.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bed5a1479d6ab45360ff6812fcb935fadb507ca" data-alt="{\displaystyle e={\frac {\sqrt {a^{2}-b^{2}}}{a}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span></p> <ul> <li>Si <span class="texhtml"><i>a = b</i></span>, l'aire se calcule avec la formule 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 A=4\pi R^{2},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <mo> = </mo> <mn> 4 </mn> <mi> π </mi> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A=4\pi R^{2},} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71eae019bece6f79e0c361486e1990d9a061f833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.801ex; height:3.009ex;" alt="{\displaystyle A=4\pi R^{2},}"> </noscript><span class="lazy-image-placeholder" style="width: 10.801ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71eae019bece6f79e0c361486e1990d9a061f833" data-alt="{\displaystyle A=4\pi R^{2},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span>où <span class="texhtml"><i>R = a = b</i></span>.</li> <li>Lorsque l'axe de rotation est le petit axe, l'ellipsoïde est aplati, son rayon polaire étant strictement inférieur à son rayon équatorial, et l'aire est donnée par la formule :<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 A=2\pi a^{2}+{\frac {\pi b^{2}}{e}}\ln {\frac {1+e}{1-e}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <mo> = </mo> <mn> 2 </mn> <mi> π </mi> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> π </mi> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mi> e </mi> </mfrac> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> e </mi> </mrow> <mrow> <mn> 1 </mn> <mo> − </mo> <mi> e </mi> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A=2\pi a^{2}+{\frac {\pi b^{2}}{e}}\ln {\frac {1+e}{1-e}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04c41e3418ae37864322c32647718b0a550e87f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.963ex; height:5.843ex;" alt="{\displaystyle A=2\pi a^{2}+{\frac {\pi b^{2}}{e}}\ln {\frac {1+e}{1-e}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 25.963ex;height: 5.843ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04c41e3418ae37864322c32647718b0a550e87f3" data-alt="{\displaystyle A=2\pi a^{2}+{\frac {\pi b^{2}}{e}}\ln {\frac {1+e}{1-e}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span></li> <li>Lorsque l'axe de rotation est le grand axe, l'ellipsoïde est allongé, son rayon polaire étant strictement supérieur à son rayon équatorial, et l'aire est donnée par la formule :<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 A=2\pi b^{2}+2\pi ab{\frac {\arcsin e}{e}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <mo> = </mo> <mn> 2 </mn> <mi> π </mi> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mn> 2 </mn> <mi> π </mi> <mi> a </mi> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> arcsin </mi> <mo> ⁡ </mo> <mi> e </mi> </mrow> <mi> e </mi> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A=2\pi b^{2}+2\pi ab{\frac {\arcsin e}{e}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424186157422eaf43ceba43729b056165652b038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.866ex; height:5.176ex;" alt="{\displaystyle A=2\pi b^{2}+2\pi ab{\frac {\arcsin e}{e}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 25.866ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424186157422eaf43ceba43729b056165652b038" data-alt="{\displaystyle A=2\pi b^{2}+2\pi ab{\frac {\arcsin e}{e}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span></li> </ul> <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><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#Propri%C3%A9t%C3%A9s_m%C3%A9triques" title="Surface de révolution">L'aire est donnée par la formule</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 A=4\pi \int _{0}^{\pi /2}q\cos \theta {\sqrt {q^{2}\sin ^{2}\theta +p^{2}\cos ^{2}\theta }}\,\mathrm {d} \theta }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <mo> = </mo> <mn> 4 </mn> <mi> π </mi> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> π </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msubsup> <mi> q </mi> <mi> cos </mi> <mo> ⁡ </mo> <mi> θ </mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi> q </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 </mi> <mo> + </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> cos </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> ⁡ </mo> <mi> θ </mi> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> θ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A=4\pi \int _{0}^{\pi /2}q\cos \theta {\sqrt {q^{2}\sin ^{2}\theta +p^{2}\cos ^{2}\theta }}\,\mathrm {d} \theta } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155a7f66e6a75da3ff16b133547bc847a90a613a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.534ex; height:6.343ex;" alt="{\displaystyle A=4\pi \int _{0}^{\pi /2}q\cos \theta {\sqrt {q^{2}\sin ^{2}\theta +p^{2}\cos ^{2}\theta }}\,\mathrm {d} \theta }"> </noscript><span class="lazy-image-placeholder" style="width: 43.534ex;height: 6.343ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155a7f66e6a75da3ff16b133547bc847a90a613a" data-alt="{\displaystyle A=4\pi \int _{0}^{\pi /2}q\cos \theta {\sqrt {q^{2}\sin ^{2}\theta +p^{2}\cos ^{2}\theta }}\,\mathrm {d} \theta }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span> donc à l'aide du <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Int%C3%A9gration_par_changement_de_variable?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Intégration par changement de variable">changement de variable</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=\sin \theta }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> <mo> = </mo> <mi> sin </mi> <mo> ⁡ </mo> <mi> θ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u=\sin \theta } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1407a0e30600e7193647754c4cbab511708e159d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.761ex; height:2.176ex;" alt="{\displaystyle u=\sin \theta }"> </noscript><span class="lazy-image-placeholder" style="width: 8.761ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1407a0e30600e7193647754c4cbab511708e159d" data-alt="{\displaystyle u=\sin \theta }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> 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 \mathrm {d} u=\cos \theta \mathrm {d} \theta }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> u </mi> <mo> = </mo> <mi> cos </mi> <mo> ⁡ </mo> <mi> θ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> θ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathrm {d} u=\cos \theta \mathrm {d} \theta } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b24a4138af9e21a3214cb8ae2a874c712b4892e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.692ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} u=\cos \theta \mathrm {d} \theta }"> </noscript><span class="lazy-image-placeholder" style="width: 12.692ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b24a4138af9e21a3214cb8ae2a874c712b4892e9" data-alt="{\displaystyle \mathrm {d} u=\cos \theta \mathrm {d} \theta }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <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 A=4\pi q\int _{0}^{1}\!\!{\sqrt {q^{2}u^{2}+p^{2}(1-u^{2})}}\ \mathrm {d} u=4\pi q\int _{0}^{1}\!\!{\sqrt {(q^{2}-p^{2})u^{2}+p^{2}}}\ \mathrm {d} u.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <mo> = </mo> <mn> 4 </mn> <mi> π </mi> <mi> q </mi> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msubsup> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi> q </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </msqrt> </mrow> <mtext>   </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> u </mi> <mo> = </mo> <mn> 4 </mn> <mi> π </mi> <mi> q </mi> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msubsup> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false"> ( </mo> <msup> <mi> q </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mtext>   </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> u </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A=4\pi q\int _{0}^{1}\!\!{\sqrt {q^{2}u^{2}+p^{2}(1-u^{2})}}\ \mathrm {d} u=4\pi q\int _{0}^{1}\!\!{\sqrt {(q^{2}-p^{2})u^{2}+p^{2}}}\ \mathrm {d} u.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17f3191c544ed771f1194d2a01d1a46276a7a939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:68.038ex; height:6.176ex;" alt="{\displaystyle A=4\pi q\int _{0}^{1}\!\!{\sqrt {q^{2}u^{2}+p^{2}(1-u^{2})}}\ \mathrm {d} u=4\pi q\int _{0}^{1}\!\!{\sqrt {(q^{2}-p^{2})u^{2}+p^{2}}}\ \mathrm {d} u.}"> </noscript><span class="lazy-image-placeholder" style="width: 68.038ex;height: 6.176ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17f3191c544ed771f1194d2a01d1a46276a7a939" data-alt="{\displaystyle A=4\pi q\int _{0}^{1}\!\!{\sqrt {q^{2}u^{2}+p^{2}(1-u^{2})}}\ \mathrm {d} u=4\pi q\int _{0}^{1}\!\!{\sqrt {(q^{2}-p^{2})u^{2}+p^{2}}}\ \mathrm {d} u.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span> La suite des calculs dépend du signe de la différence <span class="texhtml"><i>q</i><sup>2</sup> – <i>p</i><sup>2</sup></span> pour appliquer les formules des <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Primitives_de_fonctions_irrationnelles?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Primitives de fonctions irrationnelles">primitives de fonctions irrationnelles</a>.</p> <ul> <li>Si <span class="texhtml"><i>q</i> > <i>p</i></span> : avec les égalités <span class="texhtml"><i>q = a</i></span> et <span class="texhtml"><i>p = b</i></span>, l'intégrale s'écrit :<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 \int _{0}^{1}\!\!{\sqrt {b^{2}+(a^{2}-b^{2})u^{2}}}\ \mathrm {d} u={\frac {a}{2}}+{\frac {b^{2}}{2{\sqrt {a^{2}-b^{2}}}}}\operatorname {artanh} {\frac {\sqrt {a^{2}-b^{2}}}{a}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msubsup> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <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> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mtext>   </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> u </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> a </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi> artanh </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> <mi> a </mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{0}^{1}\!\!{\sqrt {b^{2}+(a^{2}-b^{2})u^{2}}}\ \mathrm {d} u={\frac {a}{2}}+{\frac {b^{2}}{2{\sqrt {a^{2}-b^{2}}}}}\operatorname {artanh} {\frac {\sqrt {a^{2}-b^{2}}}{a}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a22555eb6b61bf40fcc7055f75e09f6a86a18472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.259ex; height:7.509ex;" alt="{\displaystyle \int _{0}^{1}\!\!{\sqrt {b^{2}+(a^{2}-b^{2})u^{2}}}\ \mathrm {d} u={\frac {a}{2}}+{\frac {b^{2}}{2{\sqrt {a^{2}-b^{2}}}}}\operatorname {artanh} {\frac {\sqrt {a^{2}-b^{2}}}{a}}}"> </noscript><span class="lazy-image-placeholder" style="width: 62.259ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a22555eb6b61bf40fcc7055f75e09f6a86a18472" data-alt="{\displaystyle \int _{0}^{1}\!\!{\sqrt {b^{2}+(a^{2}-b^{2})u^{2}}}\ \mathrm {d} u={\frac {a}{2}}+{\frac {b^{2}}{2{\sqrt {a^{2}-b^{2}}}}}\operatorname {artanh} {\frac {\sqrt {a^{2}-b^{2}}}{a}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span>donc l'aire se réécrit :<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 A=2\pi a^{2}+2\pi b^{2}{\frac {\operatorname {artanh} e}{e}}=2\pi a^{2}+{\frac {\pi b^{2}}{e}}\ln {\frac {1+e}{1-e}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <mo> = </mo> <mn> 2 </mn> <mi> π </mi> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mn> 2 </mn> <mi> π </mi> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> artanh </mi> <mo> ⁡ </mo> <mi> e </mi> </mrow> <mi> e </mi> </mfrac> </mrow> <mo> = </mo> <mn> 2 </mn> <mi> π </mi> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> π </mi> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mi> e </mi> </mfrac> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> e </mi> </mrow> <mrow> <mn> 1 </mn> <mo> − </mo> <mi> e </mi> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A=2\pi a^{2}+2\pi b^{2}{\frac {\operatorname {artanh} e}{e}}=2\pi a^{2}+{\frac {\pi b^{2}}{e}}\ln {\frac {1+e}{1-e}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c95ae457114ea3149cdcf71165be7e1d47cd93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.26ex; height:5.843ex;" alt="{\displaystyle A=2\pi a^{2}+2\pi b^{2}{\frac {\operatorname {artanh} e}{e}}=2\pi a^{2}+{\frac {\pi b^{2}}{e}}\ln {\frac {1+e}{1-e}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 50.26ex;height: 5.843ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c95ae457114ea3149cdcf71165be7e1d47cd93" data-alt="{\displaystyle A=2\pi a^{2}+2\pi b^{2}{\frac {\operatorname {artanh} e}{e}}=2\pi a^{2}+{\frac {\pi b^{2}}{e}}\ln {\frac {1+e}{1-e}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span></li> <li>Si <span class="texhtml"><i>q</i> < <i>p</i></span> : avec les égalités <span class="texhtml"><i>p = a</i></span> et <span class="texhtml"><i>q = b</i></span>, l'intégrale s'écrit :<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 \int _{0}^{1}\!\!{\sqrt {a^{2}-(a^{2}-b^{2})u^{2}}}\ \mathrm {d} u={\frac {b}{2}}+{\frac {a^{2}}{2{\sqrt {a^{2}-b^{2}}}}}\arcsin {\frac {\sqrt {a^{2}-b^{2}}}{a}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msubsup> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <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> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mtext>   </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> u </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> b </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi> arcsin </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> <mi> a </mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{0}^{1}\!\!{\sqrt {a^{2}-(a^{2}-b^{2})u^{2}}}\ \mathrm {d} u={\frac {b}{2}}+{\frac {a^{2}}{2{\sqrt {a^{2}-b^{2}}}}}\arcsin {\frac {\sqrt {a^{2}-b^{2}}}{a}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7dc987fd994e5393e8b5664c12993aa2d3e2cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:61.66ex; height:7.509ex;" alt="{\displaystyle \int _{0}^{1}\!\!{\sqrt {a^{2}-(a^{2}-b^{2})u^{2}}}\ \mathrm {d} u={\frac {b}{2}}+{\frac {a^{2}}{2{\sqrt {a^{2}-b^{2}}}}}\arcsin {\frac {\sqrt {a^{2}-b^{2}}}{a}}}"> </noscript><span class="lazy-image-placeholder" style="width: 61.66ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7dc987fd994e5393e8b5664c12993aa2d3e2cb" data-alt="{\displaystyle \int _{0}^{1}\!\!{\sqrt {a^{2}-(a^{2}-b^{2})u^{2}}}\ \mathrm {d} u={\frac {b}{2}}+{\frac {a^{2}}{2{\sqrt {a^{2}-b^{2}}}}}\arcsin {\frac {\sqrt {a^{2}-b^{2}}}{a}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span>donc l'aire se réécrit :<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 A=2\pi b^{2}+2\pi ab{\frac {\arcsin e}{e}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <mo> = </mo> <mn> 2 </mn> <mi> π </mi> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mn> 2 </mn> <mi> π </mi> <mi> a </mi> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> arcsin </mi> <mo> ⁡ </mo> <mi> e </mi> </mrow> <mi> e </mi> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A=2\pi b^{2}+2\pi ab{\frac {\arcsin e}{e}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424186157422eaf43ceba43729b056165652b038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.866ex; height:5.176ex;" alt="{\displaystyle A=2\pi b^{2}+2\pi ab{\frac {\arcsin e}{e}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 25.866ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424186157422eaf43ceba43729b056165652b038" data-alt="{\displaystyle A=2\pi b^{2}+2\pi ab{\frac {\arcsin e}{e}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span></li> </ul> </div> <div class="clear" style="clear:both;"></div> </div> </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="Applications">Applications</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde_de_r%C3%A9volution&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Applications" 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>Plusieurs exemples d'ellipsoïdes de révolution apparaissent en <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Physique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Physique">physique</a>. Par exemple, une masse fluide soumise à sa propre <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Attraction_gravitationnelle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Attraction gravitationnelle">attraction gravitationnelle</a> et en rotation sur elle-même forme un ellipsoïde aplati. Un autre exemple est donné par la déformation de la Terre et surtout du niveau des océans en un ellipsoïde allongé sous l'action d'un champ gravitationnel extérieur, donnant lieu au phénomène des <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Mar%C3%A9e?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Marée">marées</a>.</p> </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="Liens_externes">Liens externes</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Ellipso%C3%AFde_de_r%C3%A9volution&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Liens externes" 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"> <ul> <li><span class="ouvrage">« <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathcurve.com/surfaces/ellipsoid/ellipsoidrevol.shtml"><cite style="font-style:normal;">Ellipsoïde de révolution</cite></a> », sur <span class="italique">MathCurve</span></span></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" 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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> <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:Physique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Portail de la physique"> <noscript> <img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/24px-Circle-icons-physics-logo.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-src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/24px-Circle-icons-physics-logo.svg.png" data-alt="icône décorative" data-width="24" 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href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/wiki/Rotationsellipsoid" title="Rotationsellipsoid – allemand" lang="de" hreflang="de" data-title="Rotationsellipsoid" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</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/Spheroid" title="Spheroid – anglais" lang="en" hreflang="en" data-title="Spheroid" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</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/Esferoide" title="Esferoide – espagnol" lang="es" hreflang="es" data-title="Esferoide" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</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/Esferoide" title="Esferoide – basque" lang="eu" hreflang="eu" data-title="Esferoide" 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/%25DA%25A9%25D8%25B1%25D9%2587%25E2%2580%258C%25D9%2588%25D8%25A7%25D8%25B1" title="کره‌وار – persan" lang="fa" hreflang="fa" data-title="کره‌وار" 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/Py%25C3%25B6r%25C3%25A4hdysellipsoidi" title="Pyörähdysellipsoidi – finnois" lang="fi" hreflang="fi" data-title="Pyörähdysellipsoidi" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</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%25A1%25D7%25A4%25D7%25A8%25D7%2595%25D7%2590%25D7%2599%25D7%2593" title="ספרואיד – hébreu" lang="he" hreflang="he" data-title="ספרואיד" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</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/Szferoid" title="Szferoid – hongrois" lang="hu" hreflang="hu" data-title="Szferoid" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</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/Sferoid" title="Sferoid – indonésien" lang="id" hreflang="id" data-title="Sferoid" 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-io mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://io.wikipedia.org/wiki/Sferoido" title="Sferoido – ido" lang="io" hreflang="io" data-title="Sferoido" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</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/Sferoide" title="Sferoide – italien" lang="it" hreflang="it" data-title="Sferoide" 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/%25E5%259B%259E%25E8%25BB%25A2%25E6%25A5%2595%25E5%2586%2586%25E4%25BD%2593" title="回転楕円体 – japonais" lang="ja" hreflang="ja" data-title="回転楕円体" 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-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%259A%258C%25EC%25A0%2584%25ED%2583%2580%25EC%259B%2590%25EB%25A9%25B4" title="회전타원면 – coréen" lang="ko" hreflang="ko" data-title="회전타원면" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li> <li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mk.wikipedia.org/wiki/%25D0%25A1%25D1%2584%25D0%25B5%25D1%2580%25D0%25BE%25D0%25B8%25D0%25B4" title="Сфероид – macédonien" lang="mk" hreflang="mk" data-title="Сфероид" data-language-autonym="Македонски" data-language-local-name="macédonien" class="interlanguage-link-target"><span>Македонски</span></a></li> <li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ml.wikipedia.org/wiki/%25E0%25B4%2597%25E0%25B5%258B%25E0%25B4%25B3%25E0%25B4%25BE%25E0%25B4%25AD%25E0%25B4%2582" title="ഗോളാഭം – malayalam" lang="ml" hreflang="ml" data-title="ഗോളാഭം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li> <li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ms.wikipedia.org/wiki/Sferoid" title="Sferoid – malais" lang="ms" hreflang="ms" data-title="Sferoid" data-language-autonym="Bahasa Melayu" data-language-local-name="malais" class="interlanguage-link-target"><span>Bahasa Melayu</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/Sfero%25C3%25AFde" title="Sferoïde – néerlandais" lang="nl" hreflang="nl" data-title="Sferoïde" 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/Sf%25C3%25A6roide" title="Sfæroide – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Sfæroide" 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/Rotasjonsellipsoide" title="Rotasjonsellipsoide – norvégien bokmål" lang="nb" hreflang="nb" data-title="Rotasjonsellipsoide" 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_obrotowa" title="Elipsoida obrotowa – polonais" lang="pl" hreflang="pl" data-title="Elipsoida obrotowa" 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/Esferoide" title="Esferoide – portugais" lang="pt" hreflang="pt" data-title="Esferoide" 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/Sferoid" title="Sferoid – roumain" lang="ro" hreflang="ro" data-title="Sferoid" 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_%25D0%25B2%25D1%2580%25D0%25B0%25D1%2589%25D0%25B5%25D0%25BD%25D0%25B8%25D1%258F" title="Эллипсоид вращения – russe" lang="ru" hreflang="ru" data-title="Эллипсоид вращения" 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/Spheroid" title="Spheroid – Simple English" lang="en-simple" hreflang="en-simple" data-title="Spheroid" 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-sl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sl.wikipedia.org/wiki/Sferoid" title="Sferoid – slovène" lang="sl" hreflang="sl" data-title="Sferoid" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li> <li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sr.wikipedia.org/wiki/%25D0%25A1%25D1%2584%25D0%25B5%25D1%2580%25D0%25BE%25D0%25B8%25D0%25B4" title="Сфероид – serbe" lang="sr" hreflang="sr" data-title="Сфероид" data-language-autonym="Српски / srpski" data-language-local-name="serbe" class="interlanguage-link-target"><span>Српски / srpski</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/Rotationsellipsoid" title="Rotationsellipsoid – suédois" lang="sv" hreflang="sv" data-title="Rotationsellipsoid" 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%2595%25E0%25AF%258B%25E0%25AE%25B3%25E0%25AE%25B5%25E0%25AF%2581%25E0%25AE%25B0%25E0%25AF%2581" title="கோளவுரு – tamoul" lang="ta" hreflang="ta" data-title="கோளவுரு" 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%2584%25E0%25B8%25A5%25E0%25B9%2589%25E0%25B8%25B2%25E0%25B8%25A2%25E0%25B8%2597%25E0%25B8%25A3%25E0%25B8%2587%25E0%25B8%2581%25E0%25B8%25A5%25E0%25B8%25A1" title="ทรงคล้ายทรงกลม – thaï" lang="th" hreflang="th" data-title="ทรงคล้ายทรงกลม" 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/Sferoit" title="Sferoit – turc" lang="tr" hreflang="tr" data-title="Sferoit" 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%25A1%25D1%2584%25D0%25B5%25D1%2580%25D0%25BE%25D1%2597%25D0%25B4" title="Сфероїд – ukrainien" lang="uk" hreflang="uk" data-title="Сфероїд" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</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/Ph%25E1%25BB%258Fng_c%25E1%25BA%25A7u" title="Phỏng cầu – vietnamien" lang="vi" hreflang="vi" data-title="Phỏng cầu" 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-yi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://yi.wikipedia.org/wiki/%25D7%25A1%25D7%25A4%25D7%25A2%25D7%25A8%25D7%2595%25D7%2599%25D7%2593" title="ספערויד – yiddish" lang="yi" hreflang="yi" data-title="ספערויד" data-language-autonym="ייִדיש" data-language-local-name="yiddish" class="interlanguage-link-target"><span>ייִדיש</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/%25E9%25A1%259E%25E7%2590%2583%25E9%259D%25A2" title="類球面 – chinois" lang="zh" hreflang="zh" data-title="類球面" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</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" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod">La dernière modification de cette page a été faite le 18 décembre 2023 à 12:27.</li> <li id="footer-info-copyright">Le contenu est disponible sous licence <a class="external" rel="nofollow" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://creativecommons.org/licenses/by-sa/4.0/deed.fr">CC BY-SA 4.0</a> sauf mention contraire.</li> 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Ellipsoïde de révolution — Wikipédia