At the risk of posting too low-level a question...

Please consider two 3-tori, with tube radii $r_1$ and $r_2$, and center-of-the-hole to center-of-the-tube radii $c_1$ and $c_2$. These tori must have at least a single intersection at some location along the midpoints of their tubes - i.e. at some location they share a common centerpoint for their tubes.

I'd like to find an analytical expression for the overlap volume as a function of the distance between the hole centers of each tori and the angle between some set of cross-sectional two-dimensional planes. Is there an especially nice way of doing this? Is this problem solved elsewhere, perhaps in a computational geometry package?