Prolate spheroid - Definition and Overview

A spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. If the ellipse is rotated about its major axis, the surface is called a prolate spheroid (similar to the shape of a rugby ball). If the minor axis is chosen, the surface is called an oblate spheroid (similar to the shape of the planet Earth).

Prolate spheroid.

Oblate spheroid.

The sphere is a special case of the spheroid in which the generating ellipse is a circle.

A spheroid is a special case of an ellipsoid where two of the three major axes are equal.

Volume

Prolate spheroid:

• volume is <math>\frac{4}{3}\pi a b^2<math>

Oblate spheroid:

• volume is <math>\frac{4}{3}\pi a^2 b<math>

where

• a is the major axis length
• b is the minor axis length

Surface area

A prolate spheroid has surface area

<math>\pi\left(2 a^2 + \frac{b^2}{e} \ln\left(\frac{1+e}{1-e}\right) \right).<math>

An oblate spheroid has surface area

2πb(b + a·arcsin(e)/e).

Here e is the eccentricity of the ellipse, defined as

<math>\left(1-(b^2/a^2)\right)^{1/2}.<math>