Minkowski Sum Construction and other Applications of Arrangements

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polytopes in three-dimensional space. We do not assume general position. Namely, we handle degenerate input and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of convex polytopes in the space in terms of the number of facets of the summands. The complexity of Minkowski sum structures is directly related to the time consumption of our Minkowski sum constructions, as they are output sensitive. The algorithms employ a data structure that represents arrangements embedded on two-dimensional parametric surfaces in the space and make use of many operations applied to arrangements. We also present an exact implementations an efficient algorithm that partitions an assembly of polytopes in the space with two hands using infinite translations. This application makes extensive use of Minkowski-sum constructions and other operations on arrangements of geodesic arcs embedded on the sphere. It distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.