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Spheroid - Wikipedia
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1.0.2 (change every time you update a partial) --> <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <p class="mw-empty-elt"></p> <table class="wikitable" align="right"> <caption> Spheroids with vertical rotational axes </caption> <tbody> <tr> <td colspan="3"><span typeof="mw:File"><a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Spheroids.svg?_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/e/e6/Spheroids.svg/360px-Spheroids.svg.png" decoding="async" width="360" height="160" 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/e/e6/Spheroids.svg/540px-Spheroids.svg.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Spheroids.svg/720px-Spheroids.svg.png 2x" data-file-width="900" data-file-height="400"></a></span></td> </tr> <tr style="text-align: center"> <th colspan="2" width="100"><i>oblate</i></th> <th><i>prolate</i></th> </tr> </tbody> </table> <p>A <b>spheroid</b>, also known as an <b>ellipsoid of revolution</b> or <b>rotational ellipsoid</b>, is a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Quadric?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Quadric">quadric</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Surface_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Surface (mathematics)">surface</a> obtained by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Surface_of_revolution?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Surface of revolution">rotating</a> an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ellipse?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellipse">ellipse</a> about one of its principal axes; in other words, an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ellipsoid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellipsoid">ellipsoid</a> with two equal <a href="https://en-m-wikipedia-org.translate.goog/wiki/Semi-diameter?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Semi-diameter">semi-diameters</a>. A spheroid has <a href="https://en-m-wikipedia-org.translate.goog/wiki/Circular_symmetry?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Circular symmetry">circular symmetry</a>.</p> <p>If the ellipse is rotated about its <a href="https://en-m-wikipedia-org.translate.goog/wiki/Major_axis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Major axis">major axis</a>, the result is a <i><b>prolate spheroid</b></i>, elongated like a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rugby_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rugby ball">rugby ball</a>. The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ball_(gridiron_football)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ball (gridiron football)">American football</a> is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its <a href="https://en-m-wikipedia-org.translate.goog/wiki/Minor_axis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Minor axis">minor axis</a>, the result is an <i><b>oblate spheroid</b></i>, flattened like a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Lentil?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lentil">lentil</a> or a plain <a href="https://en-m-wikipedia-org.translate.goog/wiki/M%26M%27s?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="M&M's">M&M</a>. If the generating ellipse is a circle, the result is a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sphere?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sphere">sphere</a>.</p> <p>Due to the combined effects of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gravity?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gravity">gravity</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rotation_of_the_Earth?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Rotation of the Earth">rotation</a>, the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Figure_of_the_Earth?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Figure of the Earth">figure of the Earth</a> (and of all <a href="https://en-m-wikipedia-org.translate.goog/wiki/Planet?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Planet">planets</a>) is not quite a sphere, but instead is slightly <a href="https://en-m-wikipedia-org.translate.goog/wiki/Flattening?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Flattening">flattened</a> in the direction of its axis of rotation. For that reason, in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cartography?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cartography">cartography</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Geodesy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geodesy">geodesy</a> the Earth is often approximated by an oblate spheroid, known as the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Reference_ellipsoid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Reference ellipsoid">reference ellipsoid</a>, instead of a sphere. The current <a href="https://en-m-wikipedia-org.translate.goog/wiki/World_Geodetic_System?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="World Geodetic System">World Geodetic System</a> model uses a spheroid whose radius is 6,378.137 km (3,963.191 mi) at the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Equator?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Equator">Equator</a> and 6,356.752 km (3,949.903 mi) at the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Geographical_pole?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geographical pole">poles</a>.</p> <p>The word <i>spheroid</i> originally meant "an approximately spherical body", admitting irregularities even beyond the bi- or tri-axial ellipsoidal shape; that is how the term is used in some older papers on geodesy (for example, referring to truncated spherical harmonic expansions of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Earth%27s_gravity?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Earth's gravity">Earth's gravity</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Geopotential_model?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Geopotential model">geopotential model</a>).<sup id="cite_ref-1" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></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="en" dir="ltr"> <h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Equation"><span class="tocnumber">1</span> <span class="toctext">Equation</span></a></li> <li class="toclevel-1 tocsection-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Properties"><span class="tocnumber">2</span> <span class="toctext">Properties</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Circumference"><span class="tocnumber">2.1</span> <span class="toctext">Circumference</span></a></li> <li class="toclevel-2 tocsection-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Area"><span class="tocnumber">2.2</span> <span class="toctext">Area</span></a></li> <li class="toclevel-2 tocsection-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Volume"><span class="tocnumber">2.3</span> <span class="toctext">Volume</span></a></li> <li class="toclevel-2 tocsection-6"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Curvature"><span class="tocnumber">2.4</span> <span class="toctext">Curvature</span></a></li> <li class="toclevel-2 tocsection-7"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Aspect_ratio"><span class="tocnumber">2.5</span> <span class="toctext">Aspect ratio</span></a></li> </ul></li> <li class="toclevel-1 tocsection-8"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Occurrence_and_applications"><span class="tocnumber">3</span> <span class="toctext">Occurrence and applications</span></a> <ul> <li class="toclevel-2 tocsection-9"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Oblate_spheroids"><span class="tocnumber">3.1</span> <span class="toctext">Oblate spheroids</span></a></li> <li class="toclevel-2 tocsection-10"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Prolate_spheroids"><span class="tocnumber">3.2</span> <span class="toctext">Prolate spheroids</span></a></li> <li class="toclevel-2 tocsection-11"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Dynamical_properties"><span class="tocnumber">3.3</span> <span class="toctext">Dynamical properties</span></a></li> </ul></li> <li class="toclevel-1 tocsection-12"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#See_also"><span class="tocnumber">4</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-13"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#References"><span class="tocnumber">5</span> <span class="toctext">References</span></a></li> <li class="toclevel-1 tocsection-14"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#External_links"><span class="tocnumber">6</span> <span class="toctext">External links</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="Equation">Equation</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Equation" 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>edit</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <figure typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Ellipsoid-rot-ax.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/5/53/Ellipsoid-rot-ax.svg/350px-Ellipsoid-rot-ax.svg.png" decoding="async" width="350" height="196" class="mw-file-element" data-file-width="289" data-file-height="162"> </noscript><span class="lazy-image-placeholder" style="width: 350px;height: 196px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Ellipsoid-rot-ax.svg/350px-Ellipsoid-rot-ax.svg.png" data-width="350" data-height="196" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Ellipsoid-rot-ax.svg/525px-Ellipsoid-rot-ax.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Ellipsoid-rot-ax.svg/700px-Ellipsoid-rot-ax.svg.png 2x" data-class="mw-file-element"> </span></a> <figcaption> The assignment of semi-axes on a spheroid. It is oblate if <span class="texhtml"><i>c</i> < <i>a</i></span> (left) and prolate if <span class="texhtml"><i>c</i> > <i>a</i></span> (right). </figcaption> </figure> <p>The equation of a tri-axial ellipsoid centred at the origin with semi-axes <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span> and <span class="texhtml mvar" style="font-style:italic;">c</span> aligned along the coordinate axes is</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.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> = </mo> <mn> 1. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d7de3306b9584a6696d085680460270c1de893" 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}}}=1.}"> </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/45d7de3306b9584a6696d085680460270c1de893" data-alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>The equation of a spheroid with <span class="texhtml mvar" style="font-style:italic;">z</span> as the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Symmetry_axis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Symmetry axis">symmetry axis</a> is given by setting <span class="texhtml"><i>a</i> = <i>b</i></span>:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}=1.}"><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> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}=1.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d8c624552f8ba8c310df2750f4aed1a0111a22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.004ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}=1.}"> </noscript><span class="lazy-image-placeholder" style="width: 19.004ex;height: 6.009ex;vertical-align: -2.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d8c624552f8ba8c310df2750f4aed1a0111a22" data-alt="{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}=1.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>The semi-axis <span class="texhtml mvar" style="font-style:italic;">a</span> is the equatorial radius of the spheroid, and <span class="texhtml mvar" style="font-style:italic;">c</span> is the distance from centre to pole along the symmetry axis. There are two possible cases:</p> <ul> <li><span class="texhtml"><i>c</i> < <i>a</i></span>: oblate spheroid</li> <li><span class="texhtml"><i>c</i> > <i>a</i></span>: prolate spheroid</li> </ul> <p>The case of <span class="texhtml"><i>a</i> = <i>c</i></span> reduces to a sphere.</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="Properties">Properties</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Properties" 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>edit</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <div class="mw-heading mw-heading3"> <h3 id="Circumference">Circumference</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Circumference" 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>edit</span> </a> </span> </div> <p><span class="anchor" id="Equatorial_circumference"></span>The equatorial circumference of a spheroid is measured around its <a href="https://en-m-wikipedia-org.translate.goog/wiki/Equator?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Equator">equator</a> and is given as:</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 C_{\text{e}}=2\pi a}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> C </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> e </mtext> </mrow> </msub> <mo> = </mo> <mn> 2 </mn> <mi> π<!-- π --> </mi> <mi> a </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle C_{\text{e}}=2\pi a} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1483aa7effe5a29680ef8bdf377645e35254e354" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.447ex; height:2.509ex;" alt="{\displaystyle C_{\text{e}}=2\pi a}"> </noscript><span class="lazy-image-placeholder" style="width: 9.447ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1483aa7effe5a29680ef8bdf377645e35254e354" data-alt="{\displaystyle C_{\text{e}}=2\pi a}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p><span class="anchor" id="Meridional_circumference"></span><span class="anchor" id="Polar_circumference"></span>The meridional or polar circumference of a spheroid is measured through its <a href="https://en-m-wikipedia-org.translate.goog/wiki/Semi-major_and_semi-minor_axes?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Semi-major and semi-minor axes">poles</a> and is given as: <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 C_{\text{p}}\,=\,4a\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> C </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> p </mtext> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo> = </mo> <mspace width="thinmathspace"></mspace> <mn> 4 </mn> <mi> a </mi> <msubsup> <mo> ∫<!-- ∫ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 1 </mn> <mo> −<!-- − --> </mo> <msup> <mi> e </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> <mi> d </mi> <mi> θ<!-- θ --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle C_{\text{p}}\,=\,4a\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/315dee6ac2bae8461795a2f6730ad2f0d644aa2c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.255ex; height:6.343ex;" alt="{\displaystyle C_{\text{p}}\,=\,4a\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta }"> </noscript><span class="lazy-image-placeholder" style="width: 32.255ex;height: 6.343ex;vertical-align: -2.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/315dee6ac2bae8461795a2f6730ad2f0d644aa2c" data-alt="{\displaystyle C_{\text{p}}\,=\,4a\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta }" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> <span class="anchor" id="Volumetric_circumference"></span>The volumetric circumference of a spheroid is the circumference of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sphere?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sphere">sphere</a> of equal volume as the spheroid and is given as:</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 C_{\text{v}}=2{\sqrt[{3}]{a^{2}c}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> C </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> v </mtext> </mrow> </msub> <mo> = </mo> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> c </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle C_{\text{v}}=2{\sqrt[{3}]{a^{2}c}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271e21c03f9e1f8e341ed781367399f38cf3ec78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.638ex; height:3.343ex;" alt="{\displaystyle C_{\text{v}}=2{\sqrt[{3}]{a^{2}c}}}"> </noscript><span class="lazy-image-placeholder" style="width: 12.638ex;height: 3.343ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271e21c03f9e1f8e341ed781367399f38cf3ec78" data-alt="{\displaystyle C_{\text{v}}=2{\sqrt[{3}]{a^{2}c}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <div class="mw-heading mw-heading3"> <h3 id="Area">Area</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Area" 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>edit</span> </a> </span> </div> <p>An oblate spheroid with <span class="texhtml"><i>c</i> < <i>a</i></span> has <a href="https://en-m-wikipedia-org.translate.goog/wiki/Surface_area?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Surface area">surface area</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 S_{\text{oblate}}=2\pi a^{2}\left(1+{\frac {1-e^{2}}{e}}\operatorname {arctanh} e\right)=2\pi a^{2}+\pi {\frac {c^{2}}{e}}\ln \left({\frac {1+e}{1-e}}\right)\qquad {\mbox{where}}\quad e^{2}=1-{\frac {c^{2}}{a^{2}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> S </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> oblate </mtext> </mrow> </msub> <mo> = </mo> <mn> 2 </mn> <mi> π<!-- π --> </mi> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 1 </mn> <mo> −<!-- − --> </mo> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mi> e </mi> </mfrac> </mrow> <mi> arctanh </mi> <mo> <!-- --> </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mn> 2 </mn> <mi> π<!-- π --> </mi> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> e </mi> </mfrac> </mrow> <mi> ln </mi> <mo> <!-- --> </mo> <mrow> <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> </mrow> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> where </mtext> </mstyle> </mrow> <mspace width="1em"></mspace> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mn> 1 </mn> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> c </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> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S_{\text{oblate}}=2\pi a^{2}\left(1+{\frac {1-e^{2}}{e}}\operatorname {arctanh} e\right)=2\pi a^{2}+\pi {\frac {c^{2}}{e}}\ln \left({\frac {1+e}{1-e}}\right)\qquad {\mbox{where}}\quad e^{2}=1-{\frac {c^{2}}{a^{2}}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb9d2abe25be0cefe53cb6adcd18510249d017dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:90.861ex; height:6.343ex;" alt="{\displaystyle S_{\text{oblate}}=2\pi a^{2}\left(1+{\frac {1-e^{2}}{e}}\operatorname {arctanh} e\right)=2\pi a^{2}+\pi {\frac {c^{2}}{e}}\ln \left({\frac {1+e}{1-e}}\right)\qquad {\mbox{where}}\quad e^{2}=1-{\frac {c^{2}}{a^{2}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 90.861ex;height: 6.343ex;vertical-align: -2.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb9d2abe25be0cefe53cb6adcd18510249d017dd" data-alt="{\displaystyle S_{\text{oblate}}=2\pi a^{2}\left(1+{\frac {1-e^{2}}{e}}\operatorname {arctanh} e\right)=2\pi a^{2}+\pi {\frac {c^{2}}{e}}\ln \left({\frac {1+e}{1-e}}\right)\qquad {\mbox{where}}\quad e^{2}=1-{\frac {c^{2}}{a^{2}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>The oblate spheroid is generated by rotation about the <span class="texhtml mvar" style="font-style:italic;">z</span>-axis of an ellipse with semi-major axis <span class="texhtml mvar" style="font-style:italic;">a</span> and semi-minor axis <span class="texhtml mvar" style="font-style:italic;">c</span>, therefore <span class="texhtml mvar" style="font-style:italic;">e</span> may be identified as the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Eccentricity_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Eccentricity (mathematics)">eccentricity</a>. (See <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ellipse?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellipse">ellipse</a>.)<sup id="cite_ref-2" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></p> <p>A prolate spheroid with <span class="texhtml"><i>c</i> > <i>a</i></span> has surface area</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 S_{\text{prolate}}=2\pi a^{2}\left(1+{\frac {c}{ae}}\arcsin \,e\right)\qquad {\mbox{where}}\quad e^{2}=1-{\frac {a^{2}}{c^{2}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> S </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> prolate </mtext> </mrow> </msub> <mo> = </mo> <mn> 2 </mn> <mi> π<!-- π --> </mi> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> c </mi> <mrow> <mi> a </mi> <mi> e </mi> </mrow> </mfrac> </mrow> <mi> arcsin </mi> <mspace width="thinmathspace"></mspace> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> where </mtext> </mstyle> </mrow> <mspace width="1em"></mspace> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mn> 1 </mn> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> a </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> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S_{\text{prolate}}=2\pi a^{2}\left(1+{\frac {c}{ae}}\arcsin \,e\right)\qquad {\mbox{where}}\quad e^{2}=1-{\frac {a^{2}}{c^{2}}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/001766593e0cd530ca4873dc97e5bf7a1851d5cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:59.393ex; height:6.009ex;" alt="{\displaystyle S_{\text{prolate}}=2\pi a^{2}\left(1+{\frac {c}{ae}}\arcsin \,e\right)\qquad {\mbox{where}}\quad e^{2}=1-{\frac {a^{2}}{c^{2}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 59.393ex;height: 6.009ex;vertical-align: -2.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/001766593e0cd530ca4873dc97e5bf7a1851d5cb" data-alt="{\displaystyle S_{\text{prolate}}=2\pi a^{2}\left(1+{\frac {c}{ae}}\arcsin \,e\right)\qquad {\mbox{where}}\quad e^{2}=1-{\frac {a^{2}}{c^{2}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>The prolate spheroid is generated by rotation about the <span class="texhtml mvar" style="font-style:italic;">z</span>-axis of an ellipse with semi-major axis <span class="texhtml mvar" style="font-style:italic;">c</span> and semi-minor axis <span class="texhtml mvar" style="font-style:italic;">a</span>; therefore, <span class="texhtml mvar" style="font-style:italic;">e</span> may again be identified as the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Eccentricity_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Eccentricity (mathematics)">eccentricity</a>. (See <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ellipse?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellipse">ellipse</a>.) <sup id="cite_ref-3" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></p> <p>These formulas are identical in the sense that the formula for <span class="texhtml"><i>S</i><sub>oblate</sub></span> can be used to calculate the surface area of a prolate spheroid and vice versa. However, <span class="texhtml mvar" style="font-style:italic;">e</span> then becomes <a href="https://en-m-wikipedia-org.translate.goog/wiki/Imaginary_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Imaginary number">imaginary</a> and can no longer directly be identified with the eccentricity. Both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse.</p> <div class="mw-heading mw-heading3"> <h3 id="Volume">Volume</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit 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>edit</span> </a> </span> </div> <p>The volume inside a spheroid (of any kind) is</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 {\tfrac {4}{3}}\pi a^{2}c\approx 4.19a^{2}c.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 4 </mn> <mn> 3 </mn> </mfrac> </mstyle> </mrow> <mi> π<!-- π --> </mi> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> c </mi> <mo> ≈<!-- ≈ --> </mo> <mn> 4.19 </mn> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> c </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\tfrac {4}{3}}\pi a^{2}c\approx 4.19a^{2}c.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adfd95321f83ce76128022d3c7b66b93c2d4d77f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.451ex; height:3.676ex;" alt="{\displaystyle {\tfrac {4}{3}}\pi a^{2}c\approx 4.19a^{2}c.}"> </noscript><span class="lazy-image-placeholder" style="width: 17.451ex;height: 3.676ex;vertical-align: -1.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adfd95321f83ce76128022d3c7b66b93c2d4d77f" data-alt="{\displaystyle {\tfrac {4}{3}}\pi a^{2}c\approx 4.19a^{2}c.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>If <span class="texhtml"><i>A</i> = 2<i>a</i></span> is the equatorial diameter, and <span class="texhtml"><i>C</i> = 2<i>c</i></span> is the polar diameter, the volume is</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 {\tfrac {\pi }{6}}A^{2}C\approx 0.523A^{2}C.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi> π<!-- π --> </mi> <mn> 6 </mn> </mfrac> </mstyle> </mrow> <msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> C </mi> <mo> ≈<!-- ≈ --> </mo> <mn> 0.523 </mn> <msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> C </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\tfrac {\pi }{6}}A^{2}C\approx 0.523A^{2}C.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae1486c6e58fd9fc5243a78574d9cfec7ea41dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.947ex; height:3.676ex;" alt="{\displaystyle {\tfrac {\pi }{6}}A^{2}C\approx 0.523A^{2}C.}"> </noscript><span class="lazy-image-placeholder" style="width: 19.947ex;height: 3.676ex;vertical-align: -1.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae1486c6e58fd9fc5243a78574d9cfec7ea41dc" data-alt="{\displaystyle {\tfrac {\pi }{6}}A^{2}C\approx 0.523A^{2}C.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <div class="mw-heading mw-heading3"> <h3 id="Curvature">Curvature</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Curvature" 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>edit</span> </a> </span> </div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style> <div role="note" class="hatnote navigation-not-searchable"> See also: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Radius_of_the_Earth?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Radii_of_curvature" class="mw-redirect" title="Radius of the Earth">Radius of the Earth § Radii of curvature</a> </div> <p>Let a spheroid be parameterized as</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 {\boldsymbol {\sigma }}(\beta ,\lambda )=(a\cos \beta \cos \lambda ,a\cos \beta \sin \lambda ,c\sin \beta ),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ<!-- σ --> </mi> </mrow> <mo stretchy="false"> ( </mo> <mi> β<!-- β --> </mi> <mo> , </mo> <mi> λ<!-- λ --> </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mo stretchy="false"> ( </mo> <mi> a </mi> <mi> cos </mi> <mo> <!-- --> </mo> <mi> β<!-- β --> </mi> <mi> cos </mi> <mo> <!-- --> </mo> <mi> λ<!-- λ --> </mi> <mo> , </mo> <mi> a </mi> <mi> cos </mi> <mo> <!-- --> </mo> <mi> β<!-- β --> </mi> <mi> sin </mi> <mo> <!-- --> </mo> <mi> λ<!-- λ --> </mi> <mo> , </mo> <mi> c </mi> <mi> sin </mi> <mo> <!-- --> </mo> <mi> β<!-- β --> </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\sigma }}(\beta ,\lambda )=(a\cos \beta \cos \lambda ,a\cos \beta \sin \lambda ,c\sin \beta ),} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0280001175902c166d332fb24a1e45f45c9ecc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.836ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\sigma }}(\beta ,\lambda )=(a\cos \beta \cos \lambda ,a\cos \beta \sin \lambda ,c\sin \beta ),}"> </noscript><span class="lazy-image-placeholder" style="width: 43.836ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0280001175902c166d332fb24a1e45f45c9ecc9" data-alt="{\displaystyle {\boldsymbol {\sigma }}(\beta ,\lambda )=(a\cos \beta \cos \lambda ,a\cos \beta \sin \lambda ,c\sin \beta ),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>where <span class="texhtml mvar" style="font-style:italic;">β</span> is the <i>reduced latitude</i> or <i><a href="https://en-m-wikipedia-org.translate.goog/wiki/Parametric_latitude?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Parametric latitude">parametric latitude</a></i>, <span class="texhtml mvar" style="font-style:italic;">λ</span> is the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Longitude?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Longitude">longitude</a>, and <span class="texhtml">−<style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac"><span class="tion"><span class="num">π</span><span class="sr-only">/</span><span class="den">2</span></span></span> < <i>β</i> < + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num">π</span><span class="sr-only">/</span><span class="den">2</span></span></span></span> and <span class="texhtml">−π < <i>λ</i> < +π</span>. Then, the spheroid's <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gaussian_curvature?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gaussian curvature">Gaussian curvature</a> is</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 K(\beta ,\lambda )={\frac {c^{2}}{\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{2}}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> K </mi> <mo stretchy="false"> ( </mo> <mi> β<!-- β --> </mi> <mo> , </mo> <mi> λ<!-- λ --> </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> −<!-- − --> </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <msup> <mi> cos </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> <!-- --> </mo> <mi> β<!-- β --> </mi> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle K(\beta ,\lambda )={\frac {c^{2}}{\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{2}}},} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c370fcbbf3522726e32f8f75329ec65d91b3f61a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.432ex; height:7.009ex;" alt="{\displaystyle K(\beta ,\lambda )={\frac {c^{2}}{\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{2}}},}"> </noscript><span class="lazy-image-placeholder" style="width: 35.432ex;height: 7.009ex;vertical-align: -3.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c370fcbbf3522726e32f8f75329ec65d91b3f61a" data-alt="{\displaystyle K(\beta ,\lambda )={\frac {c^{2}}{\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{2}}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>and its <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mean_curvature?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mean curvature">mean curvature</a> is</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 H(\beta ,\lambda )={\frac {c\left(2a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)}{2a\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{\frac {3}{2}}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> H </mi> <mo stretchy="false"> ( </mo> <mi> β<!-- β --> </mi> <mo> , </mo> <mi> λ<!-- λ --> </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> c </mi> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> −<!-- − --> </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <msup> <mi> cos </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> <!-- --> </mo> <mi> β<!-- β --> </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mi> a </mi> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> −<!-- − --> </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <msup> <mi> cos </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> <!-- --> </mo> <mi> β<!-- β --> </mi> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle H(\beta ,\lambda )={\frac {c\left(2a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)}{2a\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{\frac {3}{2}}}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4090feb27a03974194c29f6b3e0a83599017e5fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:38.504ex; height:8.009ex;" alt="{\displaystyle H(\beta ,\lambda )={\frac {c\left(2a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)}{2a\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{\frac {3}{2}}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 38.504ex;height: 8.009ex;vertical-align: -3.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4090feb27a03974194c29f6b3e0a83599017e5fb" data-alt="{\displaystyle H(\beta ,\lambda )={\frac {c\left(2a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)}{2a\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{\frac {3}{2}}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Both of these curvatures are always positive, so that every point on a spheroid is elliptic.</p> <div class="mw-heading mw-heading3"> <h3 id="Aspect_ratio">Aspect ratio</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Aspect ratio" 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>edit</span> </a> </span> </div> <p>The <i><a href="https://en-m-wikipedia-org.translate.goog/wiki/Aspect_ratio?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Aspect ratio">aspect ratio</a></i> of an oblate spheroid/ellipse, <span class="texhtml"><i>c</i> : <i>a</i></span>, is the ratio of the polar to equatorial lengths, while the <i><a href="https://en-m-wikipedia-org.translate.goog/wiki/Flattening?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Flattening">flattening</a></i> (also called <i>oblateness</i>) <span class="texhtml mvar" style="font-style:italic;">f</span>, is the ratio of the equatorial-polar length difference to the equatorial length:</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 f={\frac {a-c}{a}}=1-{\frac {c}{a}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> a </mi> <mo> −<!-- − --> </mo> <mi> c </mi> </mrow> <mi> a </mi> </mfrac> </mrow> <mo> = </mo> <mn> 1 </mn> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> c </mi> <mi> a </mi> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f={\frac {a-c}{a}}=1-{\frac {c}{a}}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a2c0836b4f0a1b51c882dc733442118de435d33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.104ex; height:5.009ex;" alt="{\displaystyle f={\frac {a-c}{a}}=1-{\frac {c}{a}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 20.104ex;height: 5.009ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a2c0836b4f0a1b51c882dc733442118de435d33" data-alt="{\displaystyle f={\frac {a-c}{a}}=1-{\frac {c}{a}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>The first <a href="https://en-m-wikipedia-org.translate.goog/wiki/Eccentricity_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Ellipses" title="Eccentricity (mathematics)"><i>eccentricity</i></a> (usually simply eccentricity, as above) is often used instead of flattening.<sup id="cite_ref-4" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It is defined by:</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={\sqrt {1-{\frac {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> <mn> 1 </mn> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> c </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> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e={\sqrt {1-{\frac {c^{2}}{a^{2}}}}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/395f1ed832bb9f70df4721193be39a960b17e0a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.629ex; height:7.509ex;" alt="{\displaystyle e={\sqrt {1-{\frac {c^{2}}{a^{2}}}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 13.629ex;height: 7.509ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/395f1ed832bb9f70df4721193be39a960b17e0a3" data-alt="{\displaystyle e={\sqrt {1-{\frac {c^{2}}{a^{2}}}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>The relations between eccentricity and flattening are:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}e&={\sqrt {2f-f^{2}}}\\f&=1-{\sqrt {1-e^{2}}}\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi> e </mi> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> f </mi> <mo> −<!-- − --> </mo> <msup> <mi> f </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> f </mi> </mtd> <mtd> <mi></mi> <mo> = </mo> <mn> 1 </mn> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 1 </mn> <mo> −<!-- − --> </mo> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}e&={\sqrt {2f-f^{2}}}\\f&=1-{\sqrt {1-e^{2}}}\end{aligned}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93faa2346406b1ecc4ae7ff60f48473a37118178" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:17.596ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}e&={\sqrt {2f-f^{2}}}\\f&=1-{\sqrt {1-e^{2}}}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 17.596ex;height: 8.509ex;vertical-align: -3.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93faa2346406b1ecc4ae7ff60f48473a37118178" data-alt="{\displaystyle {\begin{aligned}e&={\sqrt {2f-f^{2}}}\\f&=1-{\sqrt {1-e^{2}}}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>All modern geodetic ellipsoids are defined by the semi-major axis plus either the semi-minor axis (giving the aspect ratio), the flattening, or the first eccentricity. While these definitions are mathematically interchangeable, real-world calculations must lose some precision. To avoid confusion, an ellipsoidal definition considers its own values to be exact in the form it gives.</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="Occurrence_and_applications">Occurrence and applications</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Occurrence and 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>edit</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <p>The most common shapes for the density distribution of protons and neutrons in an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Atomic_nucleus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Atomic nucleus">atomic nucleus</a> are <a href="https://en-m-wikipedia-org.translate.goog/wiki/Spherical?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Spherical">spherical</a>, prolate, and oblate spheroidal, where the polar axis is assumed to be the spin axis (or direction of the spin <a href="https://en-m-wikipedia-org.translate.goog/wiki/Angular_momentum?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Angular momentum">angular momentum</a> vector). Deformed nuclear shapes occur as a result of the competition between <a href="https://en-m-wikipedia-org.translate.goog/wiki/Electromagnetic_force?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Electromagnetic force">electromagnetic</a> repulsion between protons, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Surface_tension?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Surface tension">surface tension</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Quantum_mechanics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Quantum mechanics">quantum</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Nuclear_shell_model?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nuclear shell model">shell effects</a>.</p> <p>Spheroids are common in <a href="https://en-m-wikipedia-org.translate.goog/wiki/3D_cell_culture?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="3D cell culture">3D cell cultures</a>. Rotating equilibrium spheroids include the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Maclaurin_spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Maclaurin spheroid">Maclaurin spheroid</a> and the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Jacobi_ellipsoid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Jacobi ellipsoid">Jacobi ellipsoid</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid_(lithic)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Spheroid (lithic)">Spheroid</a> is also a shape of archaeological artifacts.</p> <div class="mw-heading mw-heading3"> <h3 id="Oblate_spheroids">Oblate spheroids</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Oblate spheroids" 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>edit</span> </a> </span> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Jupiter_oblate_spheroid.png?_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/e/e5/Jupiter_oblate_spheroid.png/220px-Jupiter_oblate_spheroid.png" decoding="async" width="220" height="217" class="mw-file-element" data-file-width="841" data-file-height="829"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 217px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Jupiter_oblate_spheroid.png/220px-Jupiter_oblate_spheroid.png" data-width="220" data-height="217" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Jupiter_oblate_spheroid.png/330px-Jupiter_oblate_spheroid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Jupiter_oblate_spheroid.png/440px-Jupiter_oblate_spheroid.png 2x" data-class="mw-file-element"> </span></a> <figcaption> The planet <a href="https://en-m-wikipedia-org.translate.goog/wiki/Jupiter?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Jupiter">Jupiter</a> is a slight oblate spheroid with a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Flattening?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Flattening">flattening</a> of 0.06487 </figcaption> </figure> <p>The oblate spheroid is the approximate shape of rotating <a href="https://en-m-wikipedia-org.translate.goog/wiki/Planet?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Planet">planets</a> and other <a href="https://en-m-wikipedia-org.translate.goog/wiki/Astronomical_object?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Astronomical object">celestial bodies</a>, including Earth, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Saturn?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Saturn">Saturn</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Jupiter?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Jupiter">Jupiter</a>, and the quickly spinning star <a href="https://en-m-wikipedia-org.translate.goog/wiki/Altair?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Altair">Altair</a>. Saturn is the most oblate planet in the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Solar_System?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Solar System">Solar System</a>, with a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Flattening?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Flattening">flattening</a> of 0.09796.<sup id="cite_ref-5" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> See <a href="https://en-m-wikipedia-org.translate.goog/wiki/Planetary_flattening?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Planetary flattening">planetary flattening</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Equatorial_bulge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Equatorial bulge">equatorial bulge</a> for details.</p> <p><a href="https://en-m-wikipedia-org.translate.goog/wiki/Age_of_Enlightenment?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Age of Enlightenment">Enlightenment</a> scientist <a href="https://en-m-wikipedia-org.translate.goog/wiki/Isaac_Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Isaac Newton">Isaac Newton</a>, working from <a href="https://en-m-wikipedia-org.translate.goog/wiki/Jean_Richer?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Jean Richer">Jean Richer</a>'s pendulum experiments and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Christiaan_Huygens?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Christiaan Huygens">Christiaan Huygens</a>'s theories for their interpretation, reasoned that Jupiter and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Earth?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Earth">Earth</a> are oblate spheroids owing to their <a href="https://en-m-wikipedia-org.translate.goog/wiki/Centrifugal_force?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Centrifugal force">centrifugal force</a>.<sup id="cite_ref-6" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Earth's diverse cartographic and geodetic systems are based on <a href="https://en-m-wikipedia-org.translate.goog/wiki/Reference_ellipsoid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Reference ellipsoid">reference ellipsoids</a>, all of which are oblate.</p> <div class="mw-heading mw-heading3"> <h3 id="Prolate_spheroids">Prolate spheroids</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Prolate spheroids" 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>edit</span> </a> </span> </div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Gilbert_rugby_ball_on_grass.jpg?_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/1/1a/Gilbert_rugby_ball_on_grass.jpg/220px-Gilbert_rugby_ball_on_grass.jpg" decoding="async" width="220" height="150" class="mw-file-element" data-file-width="1955" data-file-height="1329"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 150px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Gilbert_rugby_ball_on_grass.jpg/220px-Gilbert_rugby_ball_on_grass.jpg" data-width="220" data-height="150" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Gilbert_rugby_ball_on_grass.jpg/330px-Gilbert_rugby_ball_on_grass.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Gilbert_rugby_ball_on_grass.jpg/440px-Gilbert_rugby_ball_on_grass.jpg 2x" data-class="mw-file-element"> </span></a> <figcaption> A <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rugby_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rugby ball">rugby ball</a>. </figcaption> </figure> <p>The prolate spheroid is the approximate shape of the ball in several sports, such as in the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rugby_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rugby ball">rugby ball</a>.</p> <p>Several <a href="https://en-m-wikipedia-org.translate.goog/wiki/Moons?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Moons">moons</a> of the Solar System approximate prolate spheroids in shape, though they are actually <a href="https://en-m-wikipedia-org.translate.goog/wiki/Triaxial_ellipsoid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Triaxial ellipsoid">triaxial ellipsoids</a>. Examples are <a href="https://en-m-wikipedia-org.translate.goog/wiki/Saturn?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Saturn">Saturn</a>'s satellites <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mimas_(moon)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Mimas (moon)">Mimas</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Enceladus_(moon)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Enceladus (moon)">Enceladus</a>, and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tethys_(moon)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tethys (moon)">Tethys</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Uranus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uranus">Uranus</a>' satellite <a href="https://en-m-wikipedia-org.translate.goog/wiki/Miranda_(moon)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Miranda (moon)">Miranda</a>.</p> <p>In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tide?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tide">tidal forces</a> when they orbit a massive body in a close orbit. The most extreme example is Jupiter's moon <a href="https://en-m-wikipedia-org.translate.goog/wiki/Io_(moon)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Io (moon)">Io</a>, which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intense <a href="https://en-m-wikipedia-org.translate.goog/wiki/Volcanism?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Volcanism">volcanism</a>. The major axis of the prolate spheroid does not run through the satellite's poles in this case, but through the two points on its equator directly facing toward and away from the primary. This combines with the smaller oblate distortion from the synchronous rotation to cause the body to become triaxial.</p> <p>The term is also used to describe the shape of some <a href="https://en-m-wikipedia-org.translate.goog/wiki/Nebula?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nebula">nebulae</a> such as the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Crab_Nebula?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Crab Nebula">Crab Nebula</a>.<sup id="cite_ref-Trimble1973_8-0" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Trimble1973-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Fresnel_zone?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fresnel zone">Fresnel zones</a>, used to analyze wave propagation and interference in space, are a series of concentric prolate spheroids with principal axes aligned along the direct line-of-sight between a transmitter and a receiver.</p> <p>The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Atomic_nucleus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Atomic nucleus">atomic nuclei</a> of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Actinide?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Actinide">actinide</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Lanthanide?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lanthanide">lanthanide</a> elements are shaped like prolate spheroids.<sup id="cite_ref-9" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In anatomy, near-spheroid organs such as <a href="https://en-m-wikipedia-org.translate.goog/wiki/Testicle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Testicle">testis</a> may be measured by their <a href="https://en-m-wikipedia-org.translate.goog/wiki/Anatomical_terms_of_location?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Axes" title="Anatomical terms of location">long and short axes</a>.<sup id="cite_ref-10" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></p> <p>Many submarines have a shape which can be described as prolate spheroid.<sup id="cite_ref-scientific_american_11-0" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-scientific_american-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></p> <div class="mw-heading mw-heading3"> <h3 id="Dynamical_properties">Dynamical properties</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Dynamical properties" 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>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"> <div role="note" class="hatnote navigation-not-searchable"> See also: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ellipsoid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Dynamical_properties" title="Ellipsoid">Ellipsoid § Dynamical properties</a> </div> <p>For a spheroid having uniform density, the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Moment_of_inertia?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Moment of inertia">moment of inertia</a> is that of an ellipsoid with an additional axis of symmetry. Given a description of a spheroid as having a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Major_axis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Major axis">major axis</a> <span class="texhtml mvar" style="font-style:italic;">c</span>, and minor axes <span class="texhtml mvar" style="font-style:italic;">a = b</span>, the moments of inertia along these principal axes are <span class="texhtml mvar" style="font-style:italic;">C</span>, <span class="texhtml mvar" style="font-style:italic;">A</span>, and <span class="texhtml mvar" style="font-style:italic;">B</span>. However, in a spheroid the minor axes are symmetrical. Therefore, our inertial terms along the major axes are:<sup id="cite_ref-12" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}A=B&={\tfrac {1}{5}}M\left(a^{2}+c^{2}\right),\\C&={\tfrac {1}{5}}M\left(a^{2}+b^{2}\right)={\tfrac {2}{5}}M\left(a^{2}\right),\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi> A </mi> <mo> = </mo> <mi> B </mi> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 5 </mn> </mfrac> </mstyle> </mrow> <mi> M </mi> <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> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <mi> C </mi> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 5 </mn> </mfrac> </mstyle> </mrow> <mi> M </mi> <mrow> <mo> ( </mo> <mrow> <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> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 2 </mn> <mn> 5 </mn> </mfrac> </mstyle> </mrow> <mi> M </mi> <mrow> <mo> ( </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> ) </mo> </mrow> <mo> , </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}A=B&={\tfrac {1}{5}}M\left(a^{2}+c^{2}\right),\\C&={\tfrac {1}{5}}M\left(a^{2}+b^{2}\right)={\tfrac {2}{5}}M\left(a^{2}\right),\end{aligned}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3138678b30e6577355fd8988960a44354dcfc070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.283ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}A=B&={\tfrac {1}{5}}M\left(a^{2}+c^{2}\right),\\C&={\tfrac {1}{5}}M\left(a^{2}+b^{2}\right)={\tfrac {2}{5}}M\left(a^{2}\right),\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 37.283ex;height: 7.509ex;vertical-align: -3.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3138678b30e6577355fd8988960a44354dcfc070" data-alt="{\displaystyle {\begin{aligned}A=B&={\tfrac {1}{5}}M\left(a^{2}+c^{2}\right),\\C&={\tfrac {1}{5}}M\left(a^{2}+b^{2}\right)={\tfrac {2}{5}}M\left(a^{2}\right),\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>where <span class="texhtml mvar" style="font-style:italic;">M</span> is the mass of the body defined as</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 M={\tfrac {4}{3}}\pi a^{2}c\rho .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 4 </mn> <mn> 3 </mn> </mfrac> </mstyle> </mrow> <mi> π<!-- π --> </mi> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> c </mi> <mi> ρ<!-- ρ --> </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M={\tfrac {4}{3}}\pi a^{2}c\rho .} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8249baae73f3bdcb6eeff09f9146711d04a190ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.67ex; height:3.676ex;" alt="{\displaystyle M={\tfrac {4}{3}}\pi a^{2}c\rho .}"> </noscript><span class="lazy-image-placeholder" style="width: 13.67ex;height: 3.676ex;vertical-align: -1.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8249baae73f3bdcb6eeff09f9146711d04a190ed" data-alt="{\displaystyle M={\tfrac {4}{3}}\pi a^{2}c\rho .}" 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(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="See_also">See also</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: See also" 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>edit</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ellipsoidal_dome?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellipsoidal dome">Ellipsoidal dome</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Equatorial_bulge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Equatorial bulge">Equatorial bulge</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Great_ellipse?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Great ellipse">Great ellipse</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Lentoid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lentoid">Lentoid</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Oblate_spheroidal_coordinates?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Oblate spheroidal coordinates">Oblate spheroidal coordinates</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Oval?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Oval">Ovoid</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Prolate_spheroidal_coordinates?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Prolate spheroidal coordinates">Prolate spheroidal coordinates</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Rotation_of_axes?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Rotation of axes">Rotation of axes</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Translation_of_axes?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Translation of axes">Translation of axes</a></li> </ul> </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="References">References</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=13&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: References" 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>edit</span> </a> </span> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style> <div class="reflist"> <div class="mw-references-wrap mw-references-columns"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFTorge2001" class="citation book cs1">Torge, Wolfgang (2001). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DpFO6VB_czRYC%26q%3Dequipotential%2Bellipsoid%26pg%3DPA104"><i>Geodesy</i></a> (3rd ed.). <a href="https://en-m-wikipedia-org.translate.goog/wiki/Walter_de_Gruyter?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Walter de Gruyter">Walter de Gruyter</a>. p. 104. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/9783110170726?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/9783110170726"><bdi>9783110170726</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geodesy&rft.pages=104&rft.edition=3rd&rft.pub=Walter+de+Gruyter&rft.date=2001&rft.isbn=9783110170726&rft.aulast=Torge&rft.aufirst=Wolfgang&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpFO6VB_czRYC%26q%3Dequipotential%2Bellipsoid%26pg%3DPA104&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpheroid" class="Z3988"></span></span></li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-2">^</a></b></span> <span class="reference-text">A derivation of this result may be found at <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://mathworld.wolfram.com/OblateSpheroid.html">"Oblate Spheroid"</a>. Wolfram MathWorld<span class="reference-accessdate">. Retrieved <span class="nowrap">24 June</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Oblate+Spheroid&rft.pub=Wolfram+MathWorld&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FOblateSpheroid.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpheroid" class="Z3988"></span></span></li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-3">^</a></b></span> <span class="reference-text">A derivation of this result may be found at <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://mathworld.wolfram.com/ProlateSpheroid.html">"Prolate Spheroid"</a>. Wolfram MathWorld. 7 October 2003<span class="reference-accessdate">. Retrieved <span class="nowrap">24 June</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Prolate+Spheroid&rft.pub=Wolfram+MathWorld&rft.date=2003-10-07&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FProlateSpheroid.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpheroid" class="Z3988"></span></span></li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-4">^</a></b></span> <span class="reference-text">Brial P., Shaalan C.(2009), <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://80calcs.pagesperso-orange.fr/Downloads/IntroGeodesie.pdf">Introduction à la Géodésie et au geopositionnement par satellites</a>, p.8</span></li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-5">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.vedantu.com/maths/spheroid">"Spheroid - Explanation, Applications, Shape, Example and FAQs"</a>. <i>VEDANTU</i><span class="reference-accessdate">. Retrieved <span class="nowrap">26 November</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=VEDANTU&rft.atitle=Spheroid+-+Explanation%2C+Applications%2C+Shape%2C+Example+and+FAQs&rft_id=https%3A%2F%2Fwww.vedantu.com%2Fmaths%2Fspheroid&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpheroid" class="Z3988"></span></span></li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-6">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreenburg1995" class="citation journal cs1">Greenburg, John L. (1995). "Isaac Newton and the Problem of the Earth's Shape". <i>History of Exact Sciences</i>. <b>49</b> (4). Springer: <span class="nowrap">371–</span>391. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007%252FBF00374704">10.1007/BF00374704</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/JSTOR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.jstor.org/stable/41134011">41134011</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/S2CID_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://api.semanticscholar.org/CorpusID:121268606">121268606</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=History+of+Exact+Sciences&rft.atitle=Isaac+Newton+and+the+Problem+of+the+Earth%27s+Shape&rft.volume=49&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E371-%3C%2Fspan%3E391&rft.date=1995&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121268606%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F41134011%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1007%2FBF00374704&rft.aulast=Greenburg&rft.aufirst=John+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpheroid" class="Z3988"></span></span></li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-7">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDurantDurant1997" class="citation book cs1">Durant, Will; Durant, Ariel (28 July 1997). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/storyofcivilizat00dura_3"><i>The Story of Civilization: The Age of Louis XIV</i></a></span>. MJF Books. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/1567310192?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/1567310192"><bdi>1567310192</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Story+of+Civilization%3A+The+Age+of+Louis+XIV&rft.pub=MJF+Books&rft.date=1997-07-28&rft.isbn=1567310192&rft.aulast=Durant&rft.aufirst=Will&rft.au=Durant%2C+Ariel&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstoryofcivilizat00dura_3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpheroid" class="Z3988"></span></span></li> <li id="cite_note-Trimble1973-8"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Trimble1973_8-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrimble1973" class="citation cs2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Virginia_Trimble?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Virginia Trimble">Trimble, Virginia Louise</a> (October 1973), "The Distance to the Crab Nebula and NP 0532", <i>Publications of the Astronomical Society of the Pacific</i>, <b>85</b> (507): 579, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bibcode_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ui.adsabs.harvard.edu/abs/1973PASP...85..579T">1973PASP...85..579T</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1086%252F129507">10.1086/129507</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Publications+of+the+Astronomical+Society+of+the+Pacific&rft.atitle=The+Distance+to+the+Crab+Nebula+and+NP+0532&rft.volume=85&rft.issue=507&rft.pages=579&rft.date=1973-10&rft_id=info%3Adoi%2F10.1086%2F129507&rft_id=info%3Abibcode%2F1973PASP...85..579T&rft.aulast=Trimble&rft.aufirst=Virginia+Louise&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpheroid" class="Z3988"></span></span></li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-9">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.britannica.com/science/nuclear-fission/Fission-theory">"Nuclear fission - Fission theory"</a>. <i>Encyclopedia Britannica</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+Britannica&rft.atitle=Nuclear+fission+-+Fission+theory&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2Fnuclear-fission%2FFission-theory&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpheroid" class="Z3988"></span></span></li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-10">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DsafNmcP3lakC%26pg%3DPA559">Page 559</a> in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Pellerito,_Joseph_F_Polak2012" class="citation book cs1">John Pellerito, Joseph F Polak (2012). <i>Introduction to Vascular Ultrasonography</i> (6 ed.). Elsevier Health Sciences. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/9781455737666?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/9781455737666"><bdi>9781455737666</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Vascular+Ultrasonography&rft.edition=6&rft.pub=Elsevier+Health+Sciences&rft.date=2012&rft.isbn=9781455737666&rft.au=John+Pellerito%2C+Joseph+F+Polak&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpheroid" class="Z3988"></span></span></li> <li id="cite_note-scientific_american-11"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-scientific_american_11-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.scientificamerican.com/article/football-science-shapes/">"What Do a Submarine, a Rocket and a Football Have in Common?"</a>. <i><a href="https://en-m-wikipedia-org.translate.goog/wiki/Scientific_American?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Scientific American">Scientific American</a></i>. 8 November 2010<span class="reference-accessdate">. Retrieved <span class="nowrap">13 June</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Scientific+American&rft.atitle=What+Do+a+Submarine%2C+a+Rocket+and+a+Football+Have+in+Common%3F&rft.date=2010-11-08&rft_id=http%3A%2F%2Fwww.scientificamerican.com%2Farticle%2Ffootball-science-shapes%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpheroid" class="Z3988"></span></span></li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spheroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-12">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein,_Eric_W." class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://mathworld.wolfram.com/Spheroid.html">"Spheroid"</a>. <i>MathWorld--A Wolfram Web Resource</i><span class="reference-accessdate">. Retrieved <span class="nowrap">16 May</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld--A+Wolfram+Web+Resource.&rft.atitle=Spheroid.&rft.au=Weisstein%2C+Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FSpheroid.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpheroid" class="Z3988"></span></span></li> </ol> </div> </div> </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="External_links">External links</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&action=edit&section=14&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: External links" 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>edit</span> </a> </span> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> <ul> <li><span class="noviewer" typeof="mw:File"><a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Commons-logo.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" data-file-width="1024" data-file-height="1376"> </noscript><span class="lazy-image-placeholder" style="width: 12px;height: 16px;" data-mw-src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" data-alt="" data-width="12" data-height="16" data-srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-class="mw-file-element"> </span></a></span> Media related to <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://commons.wikimedia.org/wiki/Category:Spheroids" class="extiw" title="commons:Category:Spheroids">Spheroids</a> at Wikimedia Commons</li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><span class="cs1-ws-icon" title="s:1911 Encyclopædia Britannica/Spheroid"><a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikisource.org/wiki/1911_Encyclop%25C3%25A6dia_Britannica/Spheroid">"Spheroid" </a></span>. <i><a href="https://en-m-wikipedia-org.translate.goog/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a></i> (11th ed.). 1911.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Spheroid&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.edition=11th&rft.date=1911&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpheroid" class="Z3988"></span></li> </ul><!-- NewPP limit report Parsed by mw‐api‐ext.codfw.main‐786d8bd985‐nfcqh Cached time: 20250214030354 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 0.461 seconds Real time usage: 0.777 seconds Preprocessor visited node count: 2150/1000000 Post‐expand include size: 28281/2097152 bytes Template argument size: 2436/2097152 bytes Highest expansion depth: 12/100 Expensive parser function count: 4/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 43511/5000000 bytes Lua time usage: 0.228/10.000 seconds Lua memory usage: 7475056/52428800 bytes Number of Wikibase entities loaded: 1/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 463.300 1 -total 38.66% 179.121 1 Template:Reflist 23.34% 108.124 3 Template:Cite_book 19.27% 89.278 1 Template:Short_description 12.99% 60.193 2 Template:Pagetype 8.17% 37.863 14 Template:Math 8.09% 37.470 2 Template:Cvt 6.40% 29.669 6 Template:Cite_web 5.81% 26.924 1 Template:Commons_category_inline 5.61% 25.983 2 Template:See_also --> <!-- Saved in parser cache with key enwiki:pcache:82365:|#|:idhash:canonical and timestamp 20250214030355 and revision id 1275626333. Rendering was triggered because: page-edit --> </section> </div><!-- MobileFormatter took 0.018 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=""> Retrieved from "<a dir="ltr" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/w/index.php?title%3DSpheroid%26oldid%3D1275626333">https://en.wikipedia.org/w/index.php?title=Spheroid&oldid=1275626333</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://en-m-wikipedia-org.translate.goog/w/index.php?title=Spheroid&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="Fgnievinski" data-user-gender="unknown" data-timestamp="1739502233"> <span>Last edited on 14 February 2025, at 03:03</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>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://af.wikipedia.org/wiki/Sfero%25C3%25AFde" title="Sferoïde – Afrikaans" lang="af" hreflang="af" data-title="Sferoïde" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li> <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/%25D9%2583%25D8%25B1%25D9%2588%25D8%25A7%25D9%2586%25D9%258A" title="كرواني – Arabic" lang="ar" hreflang="ar" data-title="كرواني" data-language-autonym="العربية" data-language-local-name="Arabic" 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/Esferoide" title="Esferoide – Asturian" lang="ast" hreflang="ast" data-title="Esferoide" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</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_%25D0%25B2%25D1%258F%25D1%2580%25D1%2587%25D1%258D%25D0%25BD%25D0%25BD%25D1%258F" title="Эліпсоід вярчэння – Belarusian" lang="be" hreflang="be" data-title="Эліпсоід вярчэння" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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%25A1%25D1%2584%25D0%25B5%25D1%2580%25D0%25BE%25D0%25B8%25D0%25B4" title="Сфероид – Bulgarian" lang="bg" hreflang="bg" data-title="Сфероид" data-language-autonym="Български" data-language-local-name="Bulgarian" 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/Esferoide" title="Esferoide – Catalan" lang="ca" hreflang="ca" data-title="Esferoide" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</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/%25C3%2587%25D0%25B0%25D0%25B2%25D1%2580%25C4%2583%25D0%25BC_%25D1%258D%25D0%25BB%25D0%25BB%25D0%25B8%25D0%25BF%25D1%2581%25D0%25BE%25D0%25B8%25D1%2587%25C4%2595" title="Çаврăм эллипсоичĕ – Chuvash" lang="cv" hreflang="cv" data-title="Çаврăм эллипсоичĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</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/Sf%25C3%25A9roid" title="Sféroid – Czech" lang="cs" hreflang="cs" data-title="Sféroid" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</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/Rotationsellipsoid" title="Rotationsellipsoid – Danish" lang="da" hreflang="da" data-title="Rotationsellipsoid" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li> <li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ary.wikipedia.org/wiki/%25D9%2585%25D9%2583%25D9%2588%25D8%25B1%25D8%25A7%25D9%2586%25D9%258A" title="مكوراني – Moroccan Arabic" lang="ary" hreflang="ary" data-title="مكوراني" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</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/Rotationsellipsoid" title="Rotationsellipsoid – German" lang="de" hreflang="de" data-title="Rotationsellipsoid" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</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 – Spanish" lang="es" hreflang="es" data-title="Esferoide" data-language-autonym="Español" data-language-local-name="Spanish" 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 – Persian" lang="fa" hreflang="fa" data-title="کرهوار" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li> <li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/Ellipso%25C3%25AFde_de_r%25C3%25A9volution" title="Ellipsoïde de révolution – French" lang="fr" hreflang="fr" data-title="Ellipsoïde de révolution" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</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 – Korean" lang="ko" hreflang="ko" data-title="회전타원면" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</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-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 – Indonesian" lang="id" hreflang="id" data-title="Sferoid" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</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 – Italian" lang="it" hreflang="it" data-title="Sferoide" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</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 – Hebrew" lang="he" hreflang="he" data-title="ספרואיד" data-language-autonym="עברית" data-language-local-name="Hebrew" 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 – Hungarian" lang="hu" hreflang="hu" data-title="Szferoid" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</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 – Macedonian" lang="mk" hreflang="mk" data-title="Сфероид" data-language-autonym="Македонски" data-language-local-name="Macedonian" 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 – Malay" lang="ms" hreflang="ms" data-title="Sferoid" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li> <li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mn.wikipedia.org/wiki/%25D0%25A1%25D1%2584%25D0%25B5%25D1%2580%25D0%25BE%25D0%25B8%25D0%25B4" title="Сфероид – Mongolian" lang="mn" hreflang="mn" data-title="Сфероид" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</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 – Dutch" lang="nl" hreflang="nl" data-title="Sferoïde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</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 – Japanese" lang="ja" hreflang="ja" data-title="回転楕円体" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</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 – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Rotasjonsellipsoide" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</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 – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Sfæroide" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</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 – Polish" lang="pl" hreflang="pl" data-title="Elipsoida obrotowa" data-language-autonym="Polski" data-language-local-name="Polish" 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 – Portuguese" lang="pt" hreflang="pt" data-title="Esferoide" data-language-autonym="Português" data-language-local-name="Portuguese" 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 – Romanian" lang="ro" hreflang="ro" data-title="Sferoid" data-language-autonym="Română" data-language-local-name="Romanian" 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 – Russian" lang="ru" hreflang="ru" data-title="Эллипсоид вращения" data-language-autonym="Русский" data-language-local-name="Russian" 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 – Slovenian" lang="sl" hreflang="sl" data-title="Sferoid" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" 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 – Serbian" lang="sr" hreflang="sr" data-title="Сфероид" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</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 – Finnish" lang="fi" hreflang="fi" data-title="Pyörähdysellipsoidi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</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 – Swedish" lang="sv" hreflang="sv" data-title="Rotationsellipsoid" data-language-autonym="Svenska" data-language-local-name="Swedish" 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 – Tamil" lang="ta" hreflang="ta" data-title="கோளவுரு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" 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 – Thai" lang="th" hreflang="th" data-title="ทรงคล้ายทรงกลม" data-language-autonym="ไทย" data-language-local-name="Thai" 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 – Turkish" lang="tr" hreflang="tr" data-title="Sferoit" data-language-autonym="Türkçe" data-language-local-name="Turkish" 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 – Ukrainian" lang="uk" hreflang="uk" data-title="Сфероїд" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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 – Vietnamese" lang="vi" hreflang="vi" data-title="Phỏng cầu" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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 – Chinese" lang="zh" hreflang="zh" data-title="類球面" data-language-autonym="中文" data-language-local-name="Chinese" 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-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod">This page was last edited on 14 February 2025, at 03:03<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Content is available under <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.en">CC BY-SA 4.0</a> unless otherwise noted.</li> </ul> <ul id="footer-places" class="footer-places hlist hlist-separated"> <li id="footer-places-privacy"><a 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