Oblate Spheroid

An oblate spheroid is a "squashed" spheroid for which the equatorial radius is greater than the polar radius , so (called an oblate ellipsoid by Tietze 1965, p. 27). An oblate spheroid is a surface of revolution obtained by rotating an ellipse about its minor axis (Hilbert and Cohn-Vossen 1999, p. 10). To first approximation, the shape assumed by a rotating fluid (including the Earth, which is "fluid" over astronomical time scales) is an oblate spheroid.

For a spheroid with z-axis as the axis of symmetry, the Cartesian equation is

(1)

The eccentricity of an oblate spheroid is defined by

(2)

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

(3)

with radius as a function of given by

(4)

Therefore

			(5)
			(6)
			(7)
			(8)

where the last step makes use of the natural logarithm identity

(9)

valid for . Re-expressing in terms of the eccentricity then gives

(10)

yielding the particular simple form

(11)

(Beyer 1987, p. 131). Another equivalent form is given by

(12)

An angular surface parameterization is

			(13)
			(14)
			(15)

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

(16)

The surface area can therefore also be computed as

			(17)
			(18)

Note that this is the conventional form in which the surface area of an oblate spheroid is written, although it is formally equivalent to the conventional form for the prolate spheroid via the identity

(19)

where is defined by

(20)