Contributing vertices-based Minkowski sum of a non-convex polyhedron without fold and a convex polyhedron

Abstract

We present an original approach for the computation of the Minkowski sum of a non-convex polyhedron without fold and a convex polyhedron, without decomposition and union steps—that constitute the bottleneck of convex decomposition-based algorithms. A non-convex polyhedron without fold is a polyhedron whose boundary is completely recoverable from three orthographic projections defined by three orthogonal basis vectors in ℝ(su3). First, we generate a superset of the Minkowski sum facets using the concept of contributing vertices we accommodate for a non-convex-convex pair of polyhedra. The generated superset guarantees that its envelope is the boundary of the Minkowski sum polyhedron. Secondly, we extract the Minkowski sum facets and handle the intersections among the superset facets by using 3D envelope computation. Our approach is limited to non-convex polyhedra without fold because of the use of 3D envelope computation to recover the Minkowski sum boundary. Models with holes are not handled by our method. The implementation of our algorithm uses exact number types, produces exact results, and is based on CGAL, the Computational Geometry Algorithms Library.