Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space

We present a novel method for fast retrieval of exact Minkowski sums of pairs of convex polytopes in R3 , where one of the polytopes keeps rotating. The algorithm is based on pre-computing a so-called criticality map, which records the changes in the underlying graph-structure of the Minkowski sum, while one of the polytopes rotates. We give tight combinatorial bounds on the complexity of the criticality map when the rotating polytope rotates about one, two, or three axes. The criticality map can be rather large already for rotations about one axis, even for summand polytopes with a moderate number of vertices each. We therefore focus on the restricted case of rotations about a single, though arbitrary, axis. Our work targets applications that require exact collisiondetection such as motion planning with narrow corridors and assembly maintenance where high accuracy is required. Our implementation handles all degeneracies and produces exact results. It efficiently handles the algebra o...

• Type: Conference
• Year: 2010
• Where: SMA
• Authors: Naama Mayer, Efi Fogel, Dan Halperin
• Keywords: Criticality Map, Minkowski Sum, Polytopes, SMA 2010, Solid Modeling