Appendix C: Exact equations for ellipsoids of revolution

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 $V=\frac{4}{3}\pi abc$. In the case of ellipsoids of revolution, where a = b, this becomes $V=\frac{4}{3} \pi a^2c$.

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

$$S_A = 2\pi a^2\left[1+\frac{\eta^2}{(\eta^2-1)^{-1/2}}\cos^{-1}{(1/\eta)} \right]$$ (41)

while for oblate ellipsoids

$$S_A = 2\pi a^2\left[1+\frac{\eta^2}{(1-\eta^2)^{-1/2}}\cosh^{-1}{(1/\eta)} \right]$$ (42)

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

$$h S_A = 2\pi a\left[\eta+(\eta^2-1)^{-1/2}cosh^{-1}(\eta)\right]$$ (43)

while for the oblate ellipsoid it is

$$h S_A = 2\pi a \left[\eta+(1-\eta^2)^{-1/2}cos^{-1}(\eta)\right]$$ (44)

The moment of inertia tensor, because of the symmetry of an ellipsoid of revolution with uniform density, has only three non-zero elements, I₁₁ = I₂₂ and I₃₃. If semiaxis c is along the z direction, and semiaxes a = b are along the x and y directions, then I₁₁ corresponds to spinning about the x or y-axis, and I₃₃ 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,

$$I_{11}=I_{22}=\frac{M}{5}(a^2+c^2)$$ (45)

and

$$I_{33}=\frac{2M}{5}(a^2).$$ (46)