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

with , and .

The Cartesian equation of the spheroid is

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

and of the second fundamental form are

The Gaussian curvature is given by

the implicit Gaussian curvature by

and the mean curvature by

The surface area of a spheroid can be variously written as

where

and is a hypergeometric function.

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

(Beyer 1987, p. 131).

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

See also

Darwin-de Sitter Spheroid, Ellipsoid, Latitude, Longitude, North Pole, Oblate Spheroid, Prolate Spheroid, South Pole, Sphere

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References

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

Referenced on Wolfram|Alpha

Spheroid

Cite this as:

Weisstein, Eric W. "Spheroid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Spheroid.html

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