Exact and Efficient Construction of Minkowski Sums of Convex Polyhedra with Applications

Abstract

We present an exact implementation of an efficient algorithm that computes Minkowski sums of convex polyhedra in ℝ3. Our implementation is complete in the sense that it does not assume general position. Namely, it can handle degenerate input, and it produces exact results. We also present applications of the Minkowski-sum computation to answer collision and proximity queries about the relative placement of two convex polyhedra in ℝ3. The algorithms use a dual representation of convex polyhedra, and their implementation is mainly based on the Arrangement package of Cgal, the Computational Geometry Algorithm Library. We compare our Minkowski-sum construction with the only three other methods that produce exact results we are aware of. One is a simple approach that computes the convex hull of the pairwise sums of vertices of two convex polyhedra. The second is based on Nef polyhedra embedded on the sphere, and the third is an output sensitive approach based on linear programming. Our method is significantly faster. The results of experimentation with a broad family of convex polyhedra are reported. The relevant programs, source code, data sets, and documentation are available at http://www.cs.tau.ac.il/∼efif/CD, and a short movie [16] that describes some of the concepts portrayed in this paper can be downloaded from http://www.cs.tau.ac.il/∼efif/CD/Mink3d.avi.