Prolatespheroid and Cymbelloid

Written by Paul Bourke
September 2002

The points on an ellipsoid (3 dimensions) satisfy the equation

(x/a)2 + (y/b)2 + (z/c)2 = 1

Or in polar coordinates

x = a cos(theta) cos(phi)
y = b cos(theta) sin(phi)
z = c sin(phi)
where -pi <= phi <= phi
and 0 <= theta <= 2 pi

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

((x + y)/a)2 + (z/c)2 = 1

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 = sqrt(1 - (a/c)2)

e is 0 for a sphere (a = c), and e approaches 1 as the polatespheroid become increasingly elongated.

The surface area is

2 pi a [ a + (c/e) arcsin(e) ]

The volume is

4 pi c a2 / 3

Cymbelloid

The cymbelloid is a slice of a prolatespheroid.

Given the dimensions of the cymbelloid on the right, the volume is given by

4 c a2 arcsin(D/2a) / 3

or
4 c a d / 6

The surface area is

d [ a + (c/e) arcsin(e) ]

or
2 a arcsin(D/2a) [ a + (c/e) arcsin(e) ]