Prolate Spheroid

A Spheroid which is ``pointy'' instead of ``squashed,'' i.e., one for which the polar radius $c$ is greater than the equatorial radius $a$, so $c>a$. A prolate spheroid has Cartesian equations

$${x^2+y^2\over a^2}+{z^2\over c^2}=1.$$ (1)

The Ellipticity of the prolate spheroid is defined by

$$e \equiv \sqrt{c^2-a^2\over c^2}={\sqrt{c^2-a^2}\over c}=\sqrt{1-{a^2\over c^2}},$$ (2)

so that

$$1-e^2= {a^2\over c^2}.$$ (3)

Then

$$r = a\left({1 + {e^2\over 1-e^2} \sin^2\delta}\right)^{-1/2}.$$ (4)

The Surface Area and Volume are

$$S = 2\pi a^2+2\pi {ac\over e} \sin^{-1} e$$ (5)

$$V = {\textstyle{4\over 3}}\pi a^2c.$$ (6)

See also Darwin-de Sitter Spheroid, Ellipsoid, Oblate Spheroid, Prolate Spheroidal Coordinates, Sphere, Spheroid

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 131, 1987.