Editing Spheroid

{{Short description|Surface formed by rotating an ellipse}}
{{Use dmy dates|date=September 2015}}
{| class=wikitable align=right
|+ Spheroids with vertical rotational axes
|- align-center
|colspan=3|[[File:Spheroids.svg|360px]]
|- style="text-align: center"
!colspan=2 width=100|''oblate''||''prolate''
|}

A '''spheroid''', also known as an '''ellipsoid of revolution''' or '''rotational ellipsoid''', is a [[quadric]] [[surface (mathematics)|surface]] obtained by [[Surface of revolution|rotating]] an [[ellipse]] about one of its principal axes; in other words, an [[ellipsoid]] with two equal [[semi-diameter]]s. A spheroid has [[circular symmetry]].

If the ellipse is rotated about its [[major axis]], the result is a '''''prolate spheroid''''', elongated like a [[rugby ball]]. The [[ball (gridiron football)|American football]] is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its [[minor axis]], the result is an '''''oblate spheroid''''', flattened like a [[lentil]] or a plain [[M&M's|M&M]]. If the generating ellipse is a circle, the result is a [[sphere]].

Due to the combined effects of [[gravity]] and [[rotation of the Earth|rotation]], the [[figure of the Earth]] (and of all [[planet]]s) is not quite a sphere, but instead is slightly [[flattening|flattened]] in the direction of its axis of rotation. For that reason, in [[cartography]] and [[geodesy]] the Earth is often approximated by an oblate spheroid, known as the [[reference ellipsoid]], instead of a sphere. The current [[World Geodetic System]] model uses a spheroid whose radius is {{cvt|6,378.137|km|mi}} at the [[Equator]] and {{cvt|6,356.752|km|mi}} at the [[geographical pole|poles]].

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 [[Earth's gravity]] [[geopotential model]]).<ref>{{cite book |url=https://books.google.com/books?id=pFO6VB_czRYC&q=equipotential+ellipsoid&pg=PA104 |title=Geodesy |last=Torge |first=Wolfgang |year=2001 |edition=3rd |page=104 |publisher=[[Walter de Gruyter]]|isbn=9783110170726 }}</ref>

==Equation==
[[File:ellipsoid-rot-ax.svg|thumb|350px| The assignment of semi-axes on a spheroid. It is oblate if {{math|''c'' < ''a''}} (left) and prolate if {{math|''c'' > ''a''}} (right).]]
The equation of a tri-axial ellipsoid centred at the origin with semi-axes {{mvar|a}}, {{mvar|b}} and {{mvar|c}} aligned along the coordinate axes is
:<math>\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1.</math>

The equation of a spheroid with {{mvar|z}} as the [[symmetry axis]] is given by setting {{math|''a'' {{=}} ''b''}}:
:<math>\frac{x^2+y^2}{a^2}+\frac{z^2}{c^2}=1.</math>

The semi-axis {{mvar|a}} is the equatorial radius of the spheroid, and {{mvar|c}} is the distance from centre to pole along the symmetry axis. There are two possible cases:
* {{math|''c'' < ''a''}}: oblate spheroid
* {{math|''c'' > ''a''}}: prolate spheroid
The case of {{math|''a'' {{=}} ''c''}} reduces to a sphere.

==Properties==
===Circumference===
{{anchor|Equatorial circumference}}The equatorial circumference of a spheroid is measured around its [[equator]] and is given as:
:<math>C_\text{e} = 2\pi a</math>
{{anchor|Meridional circumference}}{{anchor|Polar circumference}}The meridional or polar circumference of a spheroid is measured through its [[Semi-major and semi-minor axes|poles]] and is given as:
<math display="block">C_\text{p} \,=\, 4a\int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta</math>
{{anchor|Volumetric circumference}}The volumetric circumference of a spheroid is the circumference of a [[sphere]] of equal volume as the spheroid and is given as:
:<math>C_\text{v} = 2\sqrt[3]{a^2c}</math>

===Area===
An oblate spheroid with {{math|''c'' < ''a''}} has [[surface area]]
:<math>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}. </math>
The oblate spheroid is generated by rotation about the {{mvar|z}}-axis of an ellipse with semi-major axis {{mvar|a}} and semi-minor axis {{mvar|c}}, therefore {{mvar|e}} may be identified as the [[eccentricity (mathematics)|eccentricity]]. (See [[ellipse]].)<ref>A derivation of this result may be found at {{cite web|url=http://mathworld.wolfram.com/OblateSpheroid.html |title=Oblate Spheroid |publisher=Wolfram MathWorld |access-date=24 June 2014}}</ref>

A prolate spheroid with {{math|''c'' > ''a''}} has surface area
:<math>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}. </math>
The prolate spheroid is generated by rotation about the {{mvar|z}}-axis of an ellipse with semi-major axis {{mvar|c}} and semi-minor axis {{mvar|a}}; therefore, {{mvar|e}} may again be identified as the [[eccentricity (mathematics)|eccentricity]]. (See [[ellipse]].) <ref>A derivation of this result may be found at {{cite web|url=http://mathworld.wolfram.com/ProlateSpheroid.html |title=Prolate Spheroid |publisher=Wolfram MathWorld |date=7 October 2003 |access-date=24 June 2014}}</ref>

These formulas are identical in the sense that the formula for {{math|''S''{{sub|oblate}}}} can be used to calculate the surface area of a prolate spheroid and vice versa. However, {{mvar|e}} then becomes [[Imaginary number|imaginary]] 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.

===Volume===
The volume inside a spheroid (of any kind) is
:<math>\tfrac{4}{3}\pi a^2c\approx4.19a^2c.</math>
If {{math|''A'' {{=}} 2''a''}} is the equatorial diameter, and {{math|''C'' {{=}} 2''c''}} is the polar diameter, the volume is
:<math>\tfrac{\pi}{6}A^2C\approx0.523A^2C.</math>

===Curvature===
{{see also|Radius of the Earth#Radii of curvature}}
Let a spheroid be parameterized as
:<math> \boldsymbol\sigma (\beta,\lambda) = (a \cos \beta \cos \lambda, a \cos \beta \sin \lambda, c \sin \beta),</math>
where {{mvar|β}} is the ''reduced latitude'' or ''[[parametric latitude]]'', {{mvar|λ}} is the [[longitude]], and {{math|−{{sfrac|π|2}} < ''β'' < +{{sfrac|π|2}}}} and {{math|−π < ''λ'' < +π}}. Then, the spheroid's [[Gaussian curvature]] is
:<math> K(\beta,\lambda) = \frac{c^2}{\left(a^2 + \left(c^2 - a^2\right) \cos^2 \beta\right)^2},</math>
and its [[mean curvature]] is
:<math> 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)^\frac32}.</math>

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

===Aspect ratio===
The ''[[aspect ratio]]'' of an oblate spheroid/ellipse, {{math|''c'' : ''a''}}, is the ratio of the polar to equatorial lengths, while the ''[[flattening]]'' (also called ''oblateness'') {{mvar|f}}, is the ratio of the equatorial-polar length difference to the equatorial length:
:<math>f = \frac{a - c}{a} = 1 - \frac{c}{a} .</math>

The first [[eccentricity (mathematics)#Ellipses|''eccentricity'']] (usually simply eccentricity, as above) is often used instead of flattening.<ref>Brial P., Shaalan C.(2009), [http://80calcs.pagesperso-orange.fr/Downloads/IntroGeodesie.pdf Introduction à la Géodésie et au geopositionnement par satellites], p.8</ref> It is defined by:

: <math>e = \sqrt{1 - \frac{c^2}{a^2}}</math>

The relations between eccentricity and flattening are:
: <math>\begin{align} e &= \sqrt{2f - f^2} \\ f &= 1 - \sqrt{1 - e^2} \end{align}</math>

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.

==Occurrence and applications==
The most common shapes for the density distribution of protons and neutrons in an [[atomic nucleus]] are [[spherical]], prolate, and oblate spheroidal, where the polar axis is assumed to be the spin axis (or direction of the spin [[angular momentum]] vector). Deformed nuclear shapes occur as a result of the competition between [[Electromagnetic force|electromagnetic]] repulsion between protons, [[surface tension]] and [[Quantum mechanics|quantum]] [[Nuclear shell model|shell effects]].

Spheroids are common in [[3D cell culture]]s.
Rotating equilibrium spheroids include the [[Maclaurin spheroid]] and the [[Jacobi ellipsoid]].
[[Spheroid (lithic)|Spheroid]] is also a shape of archaeological artifacts.

===Oblate spheroids===
[[File:Jupiter_OPAL_2024.png|thumb|The planet [[Jupiter]] is a slight oblate spheroid with a [[flattening]] of 0.06487]]
The oblate spheroid is the approximate shape of rotating [[planet]]s and other [[astronomical object|celestial bodies]], including Earth, [[Saturn]], [[Jupiter]], and the quickly spinning star [[Altair]]. Saturn is the most oblate planet in the [[Solar System]], with a [[flattening]] of 0.09796.<ref>{{Cite web |title=Spheroid - Explanation, Applications, Shape, Example and FAQs |url=https://www.vedantu.com/maths/spheroid |access-date=2024-11-26 |website=VEDANTU |language=en}}</ref> See [[planetary flattening]] and [[equatorial bulge]] for details.

[[Age of Enlightenment|Enlightenment]] scientist [[Isaac Newton]], working from [[Jean Richer]]'s pendulum experiments and [[Christiaan Huygens]]'s theories for their interpretation, reasoned that Jupiter and [[Earth]] are oblate spheroids owing to their [[centrifugal force]].<ref>{{Cite book |url=https://books.google.com/books?id=RKXeEAAAQBAJ&pg=PA91 |title=Background to Discovery: Pacific Exploration from Dampier to Cook |date=1990 |publisher=[[University of California Press]] |isbn=978-0-520-06208-5 |editor-last=Howse |editor-first=Derek |series= |location= |pages=91}}</ref><ref>{{cite journal|last=Greenburg|first=John L.|date=1995|title=Isaac Newton and the Problem of the Earth's Shape|journal=History of Exact Sciences|volume=49|issue=4|pages=371–391|publisher=Springer|doi=10.1007/BF00374704|jstor=41134011|s2cid=121268606}}</ref><ref>{{Cite web |last=Choi |first=Charles Q. |date=2007-04-12 |title=Strange but True: Earth Is Not Round |url=https://www.scientificamerican.com/article/earth-is-not-round/ |access-date=2025-03-02 |website=[[Scientific American]] |language=en}}</ref> Earth's diverse cartographic and geodetic systems are based on [[reference ellipsoid]]s, all of which are oblate.

===Prolate spheroids===
[[File:Gilbert_rugby_ball_on_grass.jpg|thumb|left|A [[rugby ball]].]]
The prolate spheroid is the approximate shape of the ball in several sports, such as in the [[rugby ball]].

Several [[moons]] of the Solar System approximate prolate spheroids in shape, though they are actually [[triaxial ellipsoid]]s. Examples are [[Saturn]]'s satellites [[Mimas (moon)|Mimas]], [[Enceladus (moon)|Enceladus]], and [[Tethys (moon)|Tethys]] and [[Uranus]]'s satellite [[Miranda (moon)|Miranda]].

In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via [[tide|tidal forces]] when they orbit a massive body in a close orbit. The most extreme example is Jupiter's moon [[Io (moon)|Io]], which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intense [[volcanism]]. 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 [[nebula]]e such as the [[Crab Nebula]].<ref name="Trimble1973">{{ citation | last1 = Trimble | first1 = Virginia Louise | author1-link = Virginia Trimble | title = The Distance to the Crab Nebula and NP 0532 | date = October 1973 | journal = Publications of the Astronomical Society of the Pacific | volume = 85 | issue = 507 | page = 579 | bibcode = 1973PASP...85..579T | doi = 10.1086/129507 | doi-access = free }}</ref> [[Fresnel zone]]s, 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 [[atomic nucleus|atomic nuclei]] of the [[actinide]] and [[lanthanide]] elements are shaped like prolate spheroids.<ref>{{Cite web|url=https://www.britannica.com/science/nuclear-fission/Fission-theory|title=Nuclear fission - Fission theory|website=Encyclopedia Britannica}}</ref> In anatomy, near-spheroid organs such as [[testicle|testis]] may be measured by their [[Anatomical terms of location#Axes|long and short axes]].<ref>[https://books.google.com/books?id=safNmcP3lakC&pg=PA559 Page 559] in: {{cite book|title=Introduction to Vascular Ultrasonography|author=John Pellerito, Joseph F Polak|edition=6|publisher=Elsevier Health Sciences|year=2012|isbn=9781455737666}}</ref>

Many submarines have a shape which can be described as prolate spheroid.<ref name="scientific american">{{cite web |url=http://www.scientificamerican.com/article/football-science-shapes/ |title=What Do a Submarine, a Rocket and a Football Have in Common? |work=[[Scientific American]] |date=8 November 2010 |access-date=13 June 2015 }}</ref>

===Dynamical properties===
{{see also|Ellipsoid#Dynamical properties}}
For a spheroid having uniform density, the [[moment of inertia]] is that of an ellipsoid with an additional axis of symmetry. Given a description of a spheroid as having a [[major axis]] {{mvar|c}}, and minor axes {{mvar|a {{=}} b}}, the moments of inertia along these principal axes are {{mvar|C}}, {{mvar|A}}, and {{mvar|B}}. However, in a spheroid the minor axes are symmetrical. Therefore, our inertial terms along the major axes are:<ref>{{cite web |url=http://mathworld.wolfram.com/Spheroid.html |title=Spheroid. |work=MathWorld--A Wolfram Web Resource. |access-date=16 May 2018 |author=Weisstein, Eric W.}}</ref>

:<math>\begin{align}
A = B &= \tfrac15 M\left(a^2+c^2\right), \\
C &= \tfrac15 M\left(a^2+b^2\right) =\tfrac25 M\left(a^2\right),
\end{align}</math>

where {{mvar|M}} is the mass of the body defined as

:<math> M = \tfrac43 \pi a^2 c\rho.</math>

==See also==
* [[Ellipsoidal dome]]
* [[Equatorial bulge]]
* [[Great ellipse]]
* [[Lentoid]]
* [[Oblate spheroidal coordinates]]
* [[Oval|Ovoid]]
* [[Prolate spheroidal coordinates]]
* [[Rotation of axes]]
* [[Translation of axes]]

==References==
{{reflist}}

== External links ==
* {{commons category inline|Spheroids}}
* {{Cite EB1911|wstitle=Spheroid|short=1}}

[[Category:Ellipsoids]]