spheroid

 oblate spheroid prolate spheroid

A spheroid, or ellipsoid of revolution is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters.

If the ellipse is rotated about its major axis, the result is a prolate (elongated) spheroid, like a rugby ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, like a lentil. If the generating ellipse is a circle, the result is a sphere.

Because of the combined effects of gravitation and rotation, the Earth's shape is roughly that of a sphere slightly flattened in the direction of its axis. For that reason, in cartography the Earth is often approximated by an oblate spheroid instead of a sphere. The current World Geodetic System model, in particular, uses a spheroid whose radius is approximately 6,378.137 km at the equator and 6,356.752 km at the poles (a difference of over 21 km).

Equation

A spheroid centered at the "y" origin and rotated about the z axis is defined by the implicit equation

where a is the horizontal, transverse radius at the equator, and b is the vertical, conjugate radius.[1]

Surface area

A prolate spheroid has surface area

where is the angular eccentricity of the prolate spheroid, and is its (ordinary) eccentricity.

An oblate spheroid has surface area

where is the angular eccentricity of the oblate spheroid.

Volume

The volume of a spheroid (of any kind) is . If A=2a is the equatorial diameter, and B=2b is the polar diameter, the volume is .

Curvature

If a spheroid is parameterized as

where is the reduced or parametric latitude, is the longitude, and and , then its Gaussian curvature is

and its mean curvature is

Both of these curvatures are always positive, so that every point on a spheroid is elliptic.