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Prolate Spheroid


ProlateSpheroid

A prolate spheroid is a spheroid that is "pointy" instead of "squashed," i.e., one for which the polar radius c is greater than the equatorial radius a, so c>a (called "spindle-shaped ellipsoid" by Tietze 1965, p. 27). A symmetrical egg (i.e., with the same shape at both ends) would approximate a prolate spheroid. A prolate spheroid is a surface of revolution obtained by rotating an ellipse about its major axis (Hilbert and Cohn-Vossen 1999, p. 10), and has Cartesian equations

 (x^2+y^2)/(a^2)+(z^2)/(c^2)=1.
(1)

The ellipticity of the prolate spheroid is defined by

 e=sqrt((c^2-a^2)/(c^2))=(sqrt(c^2-a^2))/c=sqrt(1-(a^2)/(c^2)).
(2)

The surface area of a prolate spheroid can be computed as a surface of revolution about the z-axis,

 S=2piintr(z)sqrt(1+[r^'(z)]^2)dz
(3)

with radius as a function of z given by

 r(z)=asqrt(1-(z/c)^2).
(4)

The integrand is then

 rsqrt(1+r^('2))=asqrt(1+((a-c)(a+c)z^2)/(c^4)),
(5)

and the integral is given by

S=2piaint_(-c)^csqrt(1+((a-c)(a+c)z^2)/(c^4))dz
(6)
=2pia^2+(2piac^2)/(sqrt(c^2-a^2))sin^(-1)((sqrt(c^2-a^2))/c).
(7)

Using the identity

 sqrt(c^2-a^2)=ce
(8)

gives

 S=2pia^2+2pi(ac)/esin^(-1)e
(9)

(Beyer 1987, p. 131). Note that this is the conventional form in which the surface area of a prolate spheroid is written, although it is formally equivalent to the conventional form for the oblate spheroid via the identity

 (c^2pi)/(e(a,c))ln[(1+e(a,c))/(1-e(a,c))]=(2piac)/(e(c,a))sin^(-1)[e(c,a)],
(10)

where e(x,y) is defined by

 e(x,y)=sqrt(1-(x^2)/(y^2)).
(11)

See also

Capsule, Darwin-de Sitter Spheroid, Ellipsoid, Lemon Surface, Oblate Spheroid, Prolate Spheroidal Coordinates, Sphere, Spheroid, Superegg, Superellipse

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 10, 1999.Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Modern Times. New York: Graylock Press, p. 27, 1965.Wrinch, D. M. "Inverted Prolate Spheroids." Philos. Mag. 280, 1061-1070, 1932.

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Prolate Spheroid

Cite this as:

Weisstein, Eric W. "Prolate Spheroid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProlateSpheroid.html

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