Efficient Minimum Distance Computation for Solids of Revolution

Abstract

We present a highly efficient algorithm for computing the minimum distance between two solids of revolution, each of which is defined by a planar cross-section region and a rotation axis. The boundary profile curve for the cross-section is first approximated by a bounding volume hierarchy (BVH) of fat arcs. By rotating the fat arcs around the axis, we generate the BVH of fat tori that bounds the surface of revolution. The minimum distance between two solids of revolution is then computed very efficiently using the distance between fat tori, which can be boiled down to the minimum distance computation for circles in the three-dimensional space. Our circle-based approach to the solids of revolution has distinctive features of geometric simplification. The main advantage is in the effectiveness of our approach in handling the complex cases where the minimum distance is obtained in non-convex regions of the solids under consideration. Though we are dealing with a geometric problem for solids, the algorithm actually works in a computational style similar to that of handling planar curves. Compared with conventional BVH-based methods, our algorithm demonstrates outperformance in computing speed, often 10-100 times faster. Moreover, the minimum distance can be computed very efficiently for the solids of revolution under deformation, where the dynamic reconstruction of fat arcs dominates the overall computation time and takes a few milliseconds.