Volume of a body formed by rolling a sphere inside an ellipsoid

Question

Is there any analytical solution or approximation to the volume of a body that is formed by the center of a sphere of radius $r$ that is rolling inside an ellipsoid?

The answer to the extreme limiting cases are obvious:

1. when the ratio of the smallest semi-axis of the ellipsoid to the sphere's radius is less than 1, $\frac{a_{min}}{r}\leq 1$, then $V=0$.
2. when $\frac{a_{min}}{r}>> 1$, then $V\rightarrow V_{ellipsoid}$.

Is there an analytical expression that quantifies $V$ as a function of the ellipsoid eigenvalues (i.e., semi-axes)? Note that, by definition, all points of this body are at least a distance of $r$ from the interior surface of the ellipsoid.

Answer 1

The surface is called an offset surface. It is unfortunately not an ellipsoid. People who need this surface (or its planar equivalent) usually resort to approximations. See, e.g., the Wikipedia article on "parallel curves."

Farouki, Rida T. "Conic approximation of conic offsets." Journal of Symbolic computation 23.2-3 (1997): 301-313.

Anton, F., Emiris, I., Mourrain, B., & Teillaud, M. (2005, May). The offset to an algebraic curve and an application to conics. In International Conference on Computational Science and its Applications (pp. 683-696). Springer Berlin Heidelberg.