SherbyNet - /page/wikibook/entry.mu

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<title>Spheroid</title>

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Spheroid

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<caption>Spheroids with vertical rotational axes

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<th colspan="2" width="100">oblate</th>

<th>prolate

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A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a <a href="Quadric" title="Quadric">quadric</a> <a href="Surface_(mathematics)" title="Surface (mathematics)">surface</a> obtained by <a href="Surface_of_revolution" title="Surface of revolution">rotating</a> an <a href="Ellipse" title="Ellipse">ellipse</a> about one of its principal axes; in other words, an <a href="Ellipsoid" title="Ellipsoid">ellipsoid</a> with two equal <a href="Semi-diameter" class="mw-redirect" title="Semi-diameter">semi-diameters</a>. A spheroid has <a href="Circular_symmetry" title="Circular symmetry">circular symmetry</a>.

If the ellipse is rotated about its <a href="Major_axis" class="mw-redirect" title="Major axis">major axis</a>, the result is a prolate spheroid, elongated like a <a href="Rugby_ball" title="Rugby ball">rugby ball</a>. The <a href="Ball_(gridiron_football)" 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="Minor_axis" class="mw-redirect" title="Minor axis">minor axis</a>, the result is an oblate spheroid, flattened like a <a href="Lentil" title="Lentil">lentil</a> or a plain <a href="M%26M's" title="M&M's">M&M</a>. If the generating ellipse is a circle, the result is a <a href="Sphere" title="Sphere">sphere</a>.

Due to the combined effects of <a href="Gravity" title="Gravity">gravity</a> and <a href="Rotation_of_the_Earth" class="mw-redirect" title="Rotation of the Earth">rotation</a>, the <a href="Figure_of_the_Earth" title="Figure of the Earth">figure of the Earth</a> (and of all <a href="Planet" title="Planet">planets</a>) is not quite a sphere, but instead is slightly <a href="Flattening" title="Flattening">flattened</a> in the direction of its axis of rotation. For that reason, in <a href="Cartography" title="Cartography">cartography</a> and <a href="Geodesy" title="Geodesy">geodesy</a> the Earth is often approximated by an oblate spheroid, known as the <a href="Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">reference ellipsoid</a>, instead of a sphere. The current <a href="World_Geodetic_System" 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="Equator" title="Equator">Equator</a> and 6,356.752 km (3,949.903 mi) at the <a href="Geographical_pole" title="Geographical pole">poles</a>.

The word spheroid 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="Earth's_gravity" class="mw-redirect" title="Earth's gravity">Earth's gravity</a> <a href="Geopotential_model" class="mw-redirect" title="Geopotential model">geopotential model</a>).<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Equation">Equation</h2></div>

The equation of a tri-axial ellipsoid centred at the origin with semi-axes a, b and c aligned along the coordinate axes is

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<annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1.}</annotation>

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</math><img src="./45d7de3306b9584a6696d085680460270c1de893.svg" 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.}" loading="lazy"></dd></dl>

The equation of a spheroid with z as the <a href="Symmetry_axis" class="mw-redirect" title="Symmetry axis">symmetry axis</a> is given by setting a = b:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}=1.}</annotation>

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</math><img src="./99d8c624552f8ba8c310df2750f4aed1a0111a22.svg" 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.}" loading="lazy"></dd></dl>

The semi-axis a is the equatorial radius of the spheroid, and c is the distance from centre to pole along the symmetry axis. There are two possible cases:

<li>c > a: prolate spheroid</li></ul>

The case of a = c reduces to a sphere.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Circumference">Circumference</h3></div>

The equatorial circumference of a spheroid is measured around its <a href="Equator" title="Equator">equator</a> and is given as:

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<annotation encoding="application/x-tex">{\displaystyle C_{\text{e}}=2\pi a}</annotation>

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</math><img src="./1483aa7effe5a29680ef8bdf377645e35254e354.svg" 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}" loading="lazy"></dd></dl>

The meridional or polar circumference of a spheroid is measured through its <a href="Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">poles</a> and is given as:

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<annotation encoding="application/x-tex">{\displaystyle C_{\text{p}}\,=\,4a\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta }</annotation>

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The volumetric circumference of a spheroid is the circumference of a <a href="Sphere" title="Sphere">sphere</a> of equal volume as the spheroid and is given as:

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<annotation encoding="application/x-tex">{\displaystyle C_{\text{v}}=2{\sqrt[{3}]{a^{2}c}}}</annotation>

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</math><img src="./271e21c03f9e1f8e341ed781367399f38cf3ec78.svg" 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}}}" loading="lazy"></dd></dl>

An oblate spheroid with c < a has <a href="Surface_area" title="Surface area">surface area</a>

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<mi>arctanh</mi>

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<annotation encoding="application/x-tex">{\displaystyle S_{\text{o}}=2\pi a^{2}\left(1+{\frac {1-e_{o}^{2}}{e_{o}}}\operatorname {arctanh} e_{o}\right)=2\pi a^{2}+\pi {\frac {c^{2}}{e_{o}}}\ln \left({\frac {1+e_{o}}{1-e_{o}}}\right)}</annotation>

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</math><img src="./dfdd41e3180d0504fc88618060758453914f001f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:63.488ex; height:6.343ex;" alt="{\displaystyle S_{\text{o}}=2\pi a^{2}\left(1+{\frac {1-e_{o}^{2}}{e_{o}}}\operatorname {arctanh} e_{o}\right)=2\pi a^{2}+\pi {\frac {c^{2}}{e_{o}}}\ln \left({\frac {1+e_{o}}{1-e_{o}}}\right)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{o}^{2}=1-{\frac {c^{2}}{a^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{o}^{2}=1-{\frac {c^{2}}{a^{2}}}}</annotation>

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</math><img src="./8b5e873b685740f8c9c33310ab2fe326727c3e8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.359ex; height:6.009ex;" alt="{\displaystyle e_{o}^{2}=1-{\frac {c^{2}}{a^{2}}}}" loading="lazy">. A prolate spheroid with c > a has surface area

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<mi>arcsin</mi>

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<annotation encoding="application/x-tex">{\displaystyle S_{\text{p}}=2\pi a^{2}\left(1+{\frac {c}{ae_{p}}}\arcsin \,e_{p}\right)}</annotation>

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</math><img src="./98a1774df02d57177ce39357d4f4a0f6f5eaaea8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.734ex; height:6.176ex;" alt="{\displaystyle S_{\text{p}}=2\pi a^{2}\left(1+{\frac {c}{ae_{p}}}\arcsin \,e_{p}\right)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{p}^{2}=1-{\frac {a^{2}}{c^{2}}}.}">

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<annotation encoding="application/x-tex">{\displaystyle e_{p}^{2}=1-{\frac {a^{2}}{c^{2}}}.}</annotation>

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</math><img src="./45eafc06d679ec3309b2b1cd760905d7c66759d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.011ex; height:6.009ex;" alt="{\displaystyle e_{p}^{2}=1-{\frac {a^{2}}{c^{2}}}.}" loading="lazy">

In both cases, eo and ep may be identified as the <a href="Eccentricity_(mathematics)" title="Eccentricity (mathematics)">eccentricity</a> (see <a href="Ellipse" title="Ellipse">ellipse</a>).<a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a>

These formulas are identical in the sense that the formula for So can be used to calculate the surface area of a prolate spheroid and vice versa. However, eo then becomes <a href="Imaginary_number" 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.

<div class="mw-heading mw-heading3"><h3 id="Volume">Volume</h3></div>

The volume inside a spheroid (of any kind) is

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<annotation encoding="application/x-tex">{\displaystyle V={\tfrac {4}{3}}\pi a^{2}c\approx 4.19a^{2}c.}</annotation>

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</math><img src="./3c3aa4dcb55d0b435fe9fc81b269b7f7b275bd39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.337ex; height:3.676ex;" alt="{\displaystyle V={\tfrac {4}{3}}\pi a^{2}c\approx 4.19a^{2}c.}" loading="lazy"></dd></dl>

If A = 2a is the equatorial diameter, and C = 2c is the polar diameter, the volume is

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<annotation encoding="application/x-tex">{\displaystyle V={\tfrac {\pi }{6}}A^{2}C\approx 0.523A^{2}C.}</annotation>

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</math><img src="./a7c161a38f18b5799d83aab18b99ce0159371ff0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:24.833ex; height:3.676ex;" alt="{\displaystyle V={\tfrac {\pi }{6}}A^{2}C\approx 0.523A^{2}C.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Curvature">Curvature</h3></div>

/* start https://en.wikipedia.org/ */

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Radius_of_the_Earth" class="mw-redirect" title="Radius of the Earth">Radius of the Earth § Radii of curvature</a></div>

Let a spheroid be parameterized as

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}(\beta ,\lambda )=(a\cos \beta \cos \lambda ,a\cos \beta \sin \lambda ,c\sin \beta ),}</annotation>

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</math><img src="./f0280001175902c166d332fb24a1e45f45c9ecc9.svg" 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 ),}" loading="lazy"></dd></dl>

where β is the reduced latitude or <a href="Parametric_latitude" class="mw-redirect" title="Parametric latitude">parametric latitude</a> and λ is the <a href="Longitude" title="Longitude">longitude</a>, with domain −<style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠π/2⁠ < β < +⁠π/2⁠ and −π < λ < +π, respectively. Then, the spheroid's <a href="Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a> is:

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<annotation encoding="application/x-tex">{\displaystyle K(\beta )={\frac {c^{2}}{\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{2}}},}</annotation>

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</math><img src="./4602e2f698a82aba7104cf930c38e763312a440d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.043ex; height:7.009ex;" alt="{\displaystyle K(\beta )={\frac {c^{2}}{\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{2}}},}" loading="lazy"></dd></dl>

and its <a href="Mean_curvature" title="Mean curvature">mean curvature</a> is

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<annotation encoding="application/x-tex">{\displaystyle H(\beta )={\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>

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</math><img src="./644ef4c55eae9cc18a407758a15e8372082f8cab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:36.114ex; height:8.009ex;" alt="{\displaystyle H(\beta )={\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}}}}.}" loading="lazy"></dd></dl>

Both of these curvatures are a function of latitude only and are always positive, so that every point on a spheroid is elliptic.

<div class="mw-heading mw-heading3"><h3 id="Aspect_ratio">Aspect ratio</h3></div>

The <a href="Aspect_ratio" title="Aspect ratio">aspect ratio</a> of an oblate spheroid/ellipse, c : a, is the ratio of the polar to equatorial lengths, while the <a href="Flattening" title="Flattening">flattening</a> (also called oblateness) f, is the ratio of the equatorial-polar length difference to the equatorial length:

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<annotation encoding="application/x-tex">{\displaystyle f={\frac {a-c}{a}}=1-{\frac {c}{a}}.}</annotation>

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</math><img src="./6a2c0836b4f0a1b51c882dc733442118de435d33.svg" 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}}.}" loading="lazy"></dd></dl>

The first <a href="Eccentricity_(mathematics)#Ellipses" title="Eccentricity (mathematics)">eccentricity</a> (usually simply eccentricity, as above) is often used instead of flattening.<a href="#cite_note-4">[4]</a> It is defined by:

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<annotation encoding="application/x-tex">{\displaystyle e={\sqrt {1-{\frac {c^{2}}{a^{2}}}}}}</annotation>

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</math><img src="./395f1ed832bb9f70df4721193be39a960b17e0a3.svg" 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}}}}}}" loading="lazy"></dd></dl>

The relations between eccentricity and flattening are:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e&={\sqrt {2f-f^{2}}}\\f&=1-{\sqrt {1-e^{2}}}\end{aligned}}}</annotation>

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</math><img src="./93faa2346406b1ecc4ae7ff60f48473a37118178.svg" 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}}}" loading="lazy"></dd></dl>

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.

<div class="mw-heading mw-heading2"><h2 id="Occurrence_and_applications">Occurrence and applications</h2></div>

The most common shapes for the density distribution of protons and neutrons in an <a href="Atomic_nucleus" title="Atomic nucleus">atomic nucleus</a> are <a href="Spherical" 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="Angular_momentum" title="Angular momentum">angular momentum</a> vector). Deformed nuclear shapes occur as a result of the competition between <a href="Electromagnetic_force" class="mw-redirect" title="Electromagnetic force">electromagnetic</a> repulsion between protons, <a href="Surface_tension" title="Surface tension">surface tension</a> and <a href="Quantum_mechanics" title="Quantum mechanics">quantum</a> <a href="Nuclear_shell_model" title="Nuclear shell model">shell effects</a>.

Spheroids are common in <a href="3D_cell_culture" title="3D cell culture">3D cell cultures</a>.

Rotating equilibrium spheroids include the <a href="Maclaurin_spheroid" title="Maclaurin spheroid">Maclaurin spheroid</a> and the <a href="Jacobi_ellipsoid" title="Jacobi ellipsoid">Jacobi ellipsoid</a>.

<a href="Spheroid_(lithic)" title="Spheroid (lithic)">Spheroid</a> is also a shape of archaeological artifacts.

<div class="mw-heading mw-heading3"><h3 id="Oblate_spheroids">Oblate spheroids</h3></div>

The oblate spheroid is the approximate shape of rotating <a href="Planet" title="Planet">planets</a> and other <a href="Astronomical_object" title="Astronomical object">celestial bodies</a>, including Earth, <a href="Saturn" title="Saturn">Saturn</a>, <a href="Jupiter" title="Jupiter">Jupiter</a>, and the quickly spinning star <a href="Altair" title="Altair">Altair</a>. Saturn is the most oblate planet in the <a href="Solar_System" title="Solar System">Solar System</a>, with a <a href="Flattening" title="Flattening">flattening</a> of 0.09796.<a href="#cite_note-5">[5]</a> See <a href="Planetary_flattening" class="mw-redirect" title="Planetary flattening">planetary flattening</a> and <a href="Equatorial_bulge" title="Equatorial bulge">equatorial bulge</a> for details.

<a href="Age_of_Enlightenment" title="Age of Enlightenment">Enlightenment</a> scientist <a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a>, working from <a href="Jean_Richer" title="Jean Richer">Jean Richer</a>'s pendulum experiments and <a href="Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a>'s theories for their interpretation, reasoned that Jupiter and <a href="Earth" title="Earth">Earth</a> are oblate spheroids owing to their <a href="Centrifugal_force" title="Centrifugal force">centrifugal force</a>.<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a> Earth's diverse cartographic and geodetic systems are based on <a href="Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">reference ellipsoids</a>, all of which are oblate.

<div class="mw-heading mw-heading3"><h3 id="Prolate_spheroids">Prolate spheroids</h3></div>

The prolate spheroid is the approximate shape of the ball used in <a href="Ball_(gridiron_football)" title="Ball (gridiron football)">American football</a> and in <a href="Rugby_ball" title="Rugby ball">rugby</a>.

Several <a href="Moons" class="mw-redirect" title="Moons">moons</a> of the Solar System approximate prolate spheroids in shape, though they are closer to <a href="Triaxial_ellipsoid" class="mw-redirect" title="Triaxial ellipsoid">triaxial ellipsoids</a>. Examples are <a href="Saturn" title="Saturn">Saturn</a>'s satellites <a href="Mimas_(moon)" class="mw-redirect" title="Mimas (moon)">Mimas</a>, <a href="Enceladus_(moon)" class="mw-redirect" title="Enceladus (moon)">Enceladus</a>, and <a href="Tethys_(moon)" title="Tethys (moon)">Tethys</a> and <a href="Uranus" title="Uranus">Uranus</a>'s satellite <a href="Miranda_(moon)" title="Miranda (moon)">Miranda</a>.

In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via <a href="Tide" 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="Io_(moon)" title="Io (moon)">Io</a>, which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intense <a href="Volcanism" 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.

The term is also used to describe the shape of some <a href="Nebula" title="Nebula">nebulae</a> such as the <a href="Crab_Nebula" title="Crab Nebula">Crab Nebula</a>.<a href="#cite_note-Trimble1973-9">[9]</a> <a href="Fresnel_zone" 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.

The <a href="Atomic_nucleus" title="Atomic nucleus">atomic nuclei</a> of the <a href="Actinide" title="Actinide">actinide</a> and <a href="Lanthanide" title="Lanthanide">lanthanide</a> elements are shaped like prolate spheroids.<a href="#cite_note-10">[10]</a> In anatomy, near-spheroid organs such as <a href="Testicle" title="Testicle">testis</a> may be measured by their <a href="Anatomical_terms_of_location#Axes" title="Anatomical terms of location">long and short axes</a>.<a href="#cite_note-11">[11]</a>

Many submarines have a shape which can be described as prolate spheroid.<a href="#cite_note-scientific_american-12">[12]</a>

<div class="mw-heading mw-heading3"><h3 id="Dynamical_properties">Dynamical properties</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Ellipsoid#Dynamical_properties" title="Ellipsoid">Ellipsoid § Dynamical properties</a></div>

For a spheroid having uniform density, the <a href="Moment_of_inertia" 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="Major_axis" class="mw-redirect" title="Major axis">major axis</a> c, and minor axes a = b, the moments of inertia along these principal axes are C, A, and B. However, in a spheroid the minor axes are symmetrical. Therefore, our inertial terms along the major axes are:<a href="#cite_note-13">[13]</a>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</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><img src="./3138678b30e6577355fd8988960a44354dcfc070.svg" 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}}}" loading="lazy"></dd></dl>

where M is the mass of the body defined as

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M={\tfrac {4}{3}}\pi a^{2}c\rho .}</annotation>

</semantics>

</math><img src="./8249baae73f3bdcb6eeff09f9146711d04a190ed.svg" 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 .}" loading="lazy"></dd></dl>

<ul><li><a href="Ellipsoidal_dome" title="Ellipsoidal dome">Ellipsoidal dome</a></li>

<li><a href="Equatorial_bulge" title="Equatorial bulge">Equatorial bulge</a></li>

<li><a href="Great_ellipse" title="Great ellipse">Great ellipse</a></li>

<li><a href="Lentoid" title="Lentoid">Lentoid</a></li>

<li><a href="Oblate_spheroidal_coordinates" title="Oblate spheroidal coordinates">Oblate spheroidal coordinates</a></li>

<li><a href="Oval" title="Oval">Ovoid</a></li>

<li><a href="Prolate_spheroidal_coordinates" title="Prolate spheroidal coordinates">Prolate spheroidal coordinates</a></li>

<li><a href="Rotation_of_axes" class="mw-redirect" title="Rotation of axes">Rotation of axes</a></li>

<li><a href="Translation_of_axes" title="Translation of axes">Translation of axes</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFTorge2001" class="citation book cs1">Torge, Wolfgang (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pFO6VB_czRYC&q=equipotential+ellipsoid&pg=PA104">Geodesy</a> (3rd ed.). <a href="Walter_de_Gruyter" class="mw-redirect" title="Walter de Gruyter">Walter de Gruyter</a>. p. 104. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783110170726</bdi>.</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ProlateSpheroid.html">"Prolate Spheroid"</a>. Wolfram MathWorld. 7 October 2003. Retrieved 24 June 2014.</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> Brial P., Shaalan C.(2009), <a rel="nofollow" class="external text" href="http://80calcs.pagesperso-orange.fr/Downloads/IntroGeodesie.pdf">Introduction à la Géodésie et au geopositionnement par satellites</a>, p.8

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFGreenburg1995" class="citation journal cs1">Greenburg, John L. (1995). "Isaac Newton and the Problem of the Earth's Shape". History of Exact Sciences. 49 (4). Springer: 371–391. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00374704">10.1007/BF00374704</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/41134011">41134011</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121268606">121268606</a>.</cite>

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</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Spheroids" class="extiw external" title="commons:Category:Spheroids">Spheroids</a> at Wikimedia Commons</li>

<li><cite class="citation encyclopaedia cs1"><a class="external text external" href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Spheroid">"Spheroid" </a>. <a href="Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a> (11th ed.). 1911.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-30" href="https://en.wikipedia.org/wiki/?title=Spheroid&oldid=1303269680">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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