Prolate Spheroid

A prolate spheroid is a spheroid that is "pointy" instead of "squashed," i.e., one for which the polar radius is greater than the equatorial radius , so (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 equation

(1)

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

(2)

with radius as a function of given by

(3)

The integrand is then

(4)

and the integral is given by

			(5)
			(6)

Using the identity

(7)

(where the sign of the numerator is flipped from the definition of the eccentricity of an oblate spheroid) then gives

(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 the oblate spheroid via the identity

(9)

where is defined by

(10)

An angular surface parameterization is

			(11)
			(12)
			(13)

where and . Here is the azimuth and is a parameter angle, which generally differs from the polar angle of the radius vector. The coefficients of the first fundamental form are , , and , so the area element is

(14)

The surface area of a patch bounded by and is consequently

(15)

These angular formulas also apply to an oblate spheroid with .