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Spheroid

From Wikipedia, the free encyclopedia
Surface formed by rotating an ellipse

Spheroids with vertical rotational axes
oblateprolate

Aspheroid, also known as anellipsoid of revolution orrotational ellipsoid, is aquadricsurface obtained byrotating anellipse about one of its principal axes; in other words, anellipsoid with two equalsemi-diameters. A spheroid hascircular symmetry.

If the ellipse is rotated about itsmajor axis, the result is aprolate spheroid, elongated like arugby ball. TheAmerican football is similar but has a pointier end than a spheroid could. If the ellipse is rotated about itsminor axis, the result is anoblate spheroid, flattened like alentil or a plainM&M. If the generating ellipse is a circle, the result is asphere.

Due to the combined effects ofgravity androtation, thefigure of the Earth (and of allplanets) is not quite a sphere, but instead is slightlyflattened in the direction of its axis of rotation. For that reason, incartography andgeodesy the Earth is often approximated by an oblate spheroid, known as thereference ellipsoid, instead of a sphere. The currentWorld Geodetic System model uses a spheroid whose radius is 6,378.137 km (3,963.191 mi) at theEquator and 6,356.752 km (3,949.903 mi) at thepoles.

The wordspheroid 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 theEarth's gravitygeopotential model).[1]

Equation

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The assignment of semi-axes on a spheroid. It is oblate ifc <a (left) and prolate ifc >a (right).

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

x2a2+y2b2+z2c2=1.{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1.}

The equation of a spheroid withz as thesymmetry axis is given by settinga =b:

x2+y2a2+z2c2=1.{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}=1.}

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

  • c <a: oblate spheroid
  • c >a: prolate spheroid

The case ofa =c reduces to a sphere.

Properties

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Circumference

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The equatorial circumference of a spheroid is measured around itsequator and is given as:

Ce=2πa{\displaystyle C_{\text{e}}=2\pi a}

The meridional or polar circumference of a spheroid is measured through itspoles and is given as:Cp=4a∫0π/21−e2sin2⁡θ dθ{\displaystyle C_{\text{p}}\,=\,4a\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta }The volumetric circumference of a spheroid is the circumference of asphere of equal volume as the spheroid and is given as:

Cv=2a2c3{\displaystyle C_{\text{v}}=2{\sqrt[{3}]{a^{2}c}}}

Area

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An oblate spheroid withc <a hassurface area

Soblate=2πa2(1+1−e2earctanh⁡e)=2πa2+πc2eln⁡(1+e1−e)wheree2=1−c2a2.{\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}}}.}

The oblate spheroid is generated by rotation about thez-axis of an ellipse with semi-major axisa and semi-minor axisc, thereforee may be identified as theeccentricity. (Seeellipse.)[2]

A prolate spheroid withc >a has surface area

Sprolate=2πa2(1+caearcsine)wheree2=1−a2c2.{\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}}}.}

The prolate spheroid is generated by rotation about thez-axis of an ellipse with semi-major axisc and semi-minor axisa; therefore,e may again be identified as theeccentricity. (Seeellipse.)[3]

These formulas are identical in the sense that the formula forSoblate can be used to calculate the surface area of a prolate spheroid and vice versa. However,e then becomesimaginary 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

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The volume inside a spheroid (of any kind) is

43πa2c≈4.19a2c.{\displaystyle {\tfrac {4}{3}}\pi a^{2}c\approx 4.19a^{2}c.}

IfA = 2a is the equatorial diameter, andC = 2c is the polar diameter, the volume is

π6A2C≈0.523A2C.{\displaystyle {\tfrac {\pi }{6}}A^{2}C\approx 0.523A^{2}C.}

Curvature

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See also:Radius of the Earth § Radii of curvature

Let a spheroid be parameterized as

σ(β,λ)=(acos⁡βcos⁡λ,acos⁡βsin⁡λ,csin⁡β),{\displaystyle {\boldsymbol {\sigma }}(\beta ,\lambda )=(a\cos \beta \cos \lambda ,a\cos \beta \sin \lambda ,c\sin \beta ),}

whereβ is thereduced latitude orparametric latitude,λ is thelongitude, and−⁠π/2⁠ <β < +⁠π/2⁠ and−π <λ < +π. Then, the spheroid'sGaussian curvature is

K(β,λ)=c2(a2+(c2−a2)cos2⁡β)2,{\displaystyle K(\beta ,\lambda )={\frac {c^{2}}{\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{2}}},}

and itsmean curvature is

H(β,λ)=c(2a2+(c2−a2)cos2⁡β)2a(a2+(c2−a2)cos2⁡β)32.{\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}}}}.}

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

Aspect ratio

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Theaspect ratio of an oblate spheroid/ellipse,c :a, is the ratio of the polar to equatorial lengths, while theflattening (also calledoblateness)f, is the ratio of the equatorial-polar length difference to the equatorial length:

f=a−ca=1−ca.{\displaystyle f={\frac {a-c}{a}}=1-{\frac {c}{a}}.}

The firsteccentricity (usually simply eccentricity, as above) is often used instead of flattening.[4] It is defined by:

e=1−c2a2{\displaystyle e={\sqrt {1-{\frac {c^{2}}{a^{2}}}}}}

The relations between eccentricity and flattening are:

e=2f−f2f=1−1−e2{\displaystyle {\begin{aligned}e&={\sqrt {2f-f^{2}}}\\f&=1-{\sqrt {1-e^{2}}}\end{aligned}}}

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

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The most common shapes for the density distribution of protons and neutrons in anatomic nucleus arespherical, prolate, and oblate spheroidal, where the polar axis is assumed to be the spin axis (or direction of the spinangular momentum vector). Deformed nuclear shapes occur as a result of the competition betweenelectromagnetic repulsion between protons,surface tension andquantumshell effects.

Spheroids are common in3D cell cultures.Rotating equilibrium spheroids include theMaclaurin spheroid and theJacobi ellipsoid.Spheroid is also a shape of archaeological artifacts.

Oblate spheroids

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The planetJupiter is a slight oblate spheroid with aflattening of 0.06487

The oblate spheroid is the approximate shape of rotatingplanets and othercelestial bodies, including Earth,Saturn,Jupiter, and the quickly spinning starAltair. Saturn is the most oblate planet in theSolar System, with aflattening of 0.09796.[5] Seeplanetary flattening andequatorial bulge for details.

Enlightenment scientistIsaac Newton, working fromJean Richer's pendulum experiments andChristiaan Huygens's theories for their interpretation, reasoned that Jupiter andEarth are oblate spheroids owing to theircentrifugal force.[6][7][8] Earth's diverse cartographic and geodetic systems are based onreference ellipsoids, all of which are oblate.

Prolate spheroids

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Arugby ball.

The prolate spheroid is the approximate shape of the ball in several sports, such as in therugby ball.

Severalmoons of the Solar System approximate prolate spheroids in shape, though they are actuallytriaxial ellipsoids. Examples areSaturn's satellitesMimas,Enceladus, andTethys andUranus' satelliteMiranda.

In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids viatidal forces when they orbit a massive body in a close orbit. The most extreme example is Jupiter's moonIo, which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intensevolcanism. 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 somenebulae such as theCrab Nebula.[9]Fresnel zones, 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.

Theatomic nuclei of theactinide andlanthanide elements are shaped like prolate spheroids.[10] In anatomy, near-spheroid organs such astestis may be measured by theirlong and short axes.[11]

Many submarines have a shape which can be described as prolate spheroid.[12]

Dynamical properties

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See also:Ellipsoid § Dynamical properties

For a spheroid having uniform density, themoment of inertia is that of an ellipsoid with an additional axis of symmetry. Given a description of a spheroid as having amajor axisc, and minor axesa = b, the moments of inertia along these principal axes areC,A, andB. However, in a spheroid the minor axes are symmetrical. Therefore, our inertial terms along the major axes are:[13]

A=B=15M(a2+c2),C=15M(a2+b2)=25M(a2),{\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}}}

whereM is the mass of the body defined as

M=43πa2cρ.{\displaystyle M={\tfrac {4}{3}}\pi a^{2}c\rho .}

See also

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References

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  1. ^Torge, Wolfgang (2001).Geodesy (3rd ed.).Walter de Gruyter. p. 104.ISBN 9783110170726.
  2. ^A derivation of this result may be found at"Oblate Spheroid". Wolfram MathWorld. Retrieved24 June 2014.
  3. ^A derivation of this result may be found at"Prolate Spheroid". Wolfram MathWorld. 7 October 2003. Retrieved24 June 2014.
  4. ^Brial P., Shaalan C.(2009),Introduction à la Géodésie et au geopositionnement par satellites, p.8
  5. ^"Spheroid - Explanation, Applications, Shape, Example and FAQs".VEDANTU. Retrieved26 November 2024.
  6. ^Howse, Derek, ed. (1990).Background to Discovery: Pacific Exploration from Dampier to Cook.University of California Press. p. 91.ISBN 978-0-520-06208-5.
  7. ^Greenburg, John L. (1995). "Isaac Newton and the Problem of the Earth's Shape".History of Exact Sciences.49 (4). Springer:371–391.doi:10.1007/BF00374704.JSTOR 41134011.S2CID 121268606.
  8. ^Choi, Charles Q. (12 April 2007)."Strange but True: Earth Is Not Round".Scientific American. Retrieved2 March 2025.
  9. ^Trimble, Virginia Louise (October 1973), "The Distance to the Crab Nebula and NP 0532",Publications of the Astronomical Society of the Pacific,85 (507): 579,Bibcode:1973PASP...85..579T,doi:10.1086/129507
  10. ^"Nuclear fission - Fission theory".Encyclopedia Britannica.
  11. ^Page 559 in:John Pellerito, Joseph F Polak (2012).Introduction to Vascular Ultrasonography (6 ed.). Elsevier Health Sciences.ISBN 9781455737666.
  12. ^"What Do a Submarine, a Rocket and a Football Have in Common?".Scientific American. 8 November 2010. Retrieved13 June 2015.
  13. ^Weisstein, Eric W."Spheroid".MathWorld--A Wolfram Web Resource. Retrieved16 May 2018.

External links

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