This appendix gives the exact analytical formulae for ellipsoids of revolution for volume, surface area, moment of inertia, and integrated mean curvature [40].

For any ellipsoid with semiaxes *a*, *b*, and *c*, the volume is simply
given by
.
In the case of ellipsoids of revolution, where *a* = *b*, this
becomes
.

For the other three quantities, let = *c*/*a*.
For prolate ellipsoids, > 1, while for oblate ellipsoids, < 1.
For prolate ellipsoids, the surface area is

while for oblate ellipsoids

The integrated mean curvature for the prolate ellipsoid, without the surface area normalization, is

while for the oblate ellipsoid it is

The moment of inertia tensor, because of the symmetry of an
ellipsoid of revolution with uniform density,
has only three non-zero elements,
*I*_{11} = *I*_{22} and *I*_{33}. If semiaxis *c*
is along the z direction, and semiaxes *a* = *b* are along the x and y
directions, then *I*_{11} corresponds to spinning about the x or y-axis,
and *I*_{33} corresponds to spinning about the z-axis. Spinning around
the z-axis is the easiest way to spin for the prolate ellipsoid,
and the hardest way to spin for the oblate ellipsoid.
The diagonal elements of the moment of inertia tensor are
given in terms of the semiaxes. For both kinds of ellipsoids,

(45) |

and

(46) |