Spheroid

A spheroid is an ellipsoid having two axes of equal length, making it a surface of revolution. By convention, the two distinct axis lengths are denoted and , and the spheroid is oriented so that its axis of rotational symmetric is along the -axis, giving it the parametric representation

			(1)
			(2)
			(3)

with , and .

The Cartesian equation of the spheroid is

(4)

If , the spheroid is called oblate (left figure). If , the spheroid is prolate (right figure). If , the spheroid degenerates to a sphere.

In the above parametrization, the coefficients of the first fundamental form are

			(5)
			(6)
			(7)

and of the second fundamental form are

			(8)
			(9)
			(10)

The Gaussian curvature is given by

(11)

the implicit Gaussian curvature by

(12)

and the mean curvature by

(13)

The surface area of a spheroid can be variously written as

			(14)
			(15)
			(16)
			(17)

where

			(18)
			(19)

and is a hypergeometric function.

The volume of a spheroid can be computed from the formula for a general ellipsoid with ,

			(20)
			(21)

(Beyer 1987, p. 131).

The moment of inertia tensor of a spheroid with -axis along the axis of symmetry is given by

(22)