Prolatespheroid and CymbelloidWritten by Paul Bourke
The points on an ellipsoid (3 dimensions) satisfy the equation
Or in polar coordinates
The special case where a = b is called a prolatespheroid, it is a sphere
with a stretch (scale factor > 1, not compression, c > a)
along one axis. The equation can be written as
Where a is the equatorial radius and c the polar radius.
The example below illustrates a prolatesphereoid where c = 1.5 a
The degree of stretching is often called the eccentricity in 2 dimensions
or the ellipticity in 3 dimensions, it is defined by
e is 0 for a sphere (a = c), and e approaches 1 as the prolatespheroid become increasingly elongated.
The surface area is
The volume is