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Prolate Spheroid

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ProlateSpheroid

A prolate spheroid is aspheroid that is "pointy" instead of "squashed," i.e., one for which the polar radiusc is greater than the equatorial radiusa, soc>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 asurface of revolution obtained by rotating anellipse 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)

Thesurface area of a prolate spheroid can be computedas asurface of revolution about thez-axis,

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

with radius as a function ofz given by

r(z)=asqrt(1-(z/c)^2).
(3)

Theintegrand is then

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

and the integral is given by

S=2piaint_(-c)^csqrt(1+((a-c)(a+c)z^2)/(c^4))dz
(5)
=2pia^2+(2piac^2)/(sqrt(c^2-a^2))sin^(-1)((sqrt(c^2-a^2))/c).
(6)

Using the identity

e^2=(c^2-a^2)/(c^2)
(7)

(where the sign of the numerator is flipped from the definition of theeccentricityof anoblate spheroid) then gives

S=2pia^2+2pi(ac)/esin^(-1)e
(8)

(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 theoblate 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)],
(9)

wheree(x,y) is defined by

e(x,y)=sqrt(1-(x^2)/(y^2)).
(10)

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."FromMathWorld--A Wolfram Web Resource.https://mathworld.wolfram.com/ProlateSpheroid.html

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Created, developed and nurtured by Eric Weisstein at Wolfram Research
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