Minimum distance between subdivision bodies ($d(A,B)={\inf }_{a\in A,b\in B}\Vert a-b\Vert$)

For disjoint compact bodies, the minimum occurs on their boundaries. Patch bounding volumes give certified lower bounds and evaluated limit-point pairs give upper bounds. A branch and bound search refines promising patch pairs until the global bounds differ by the tolerance. Boundary distance alone cannot detect containment; touching, crossing or nested solid bodies have ordinary set distance zero. Negative overlap values require an additional convention such as translational penetration depth.