Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in R. 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 polytopes in R in terms of the number of facets of the summand polytopes. The complexity of Minkowski sum structures is directly related to the time consumption of our Minkowski sum constructions, as they are output sensitive. We demonstrate the effectiveness of our Minkowski-sum constructions through simple applications that exploit these operations to detect collision between, and answer proximity queries about, two convex polyhedra in R. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in R, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (Cgal) [5] called Arrangement on surface 2 [WFZH07a]. The code of Cgal in general, the Arrangement on surface 2 package in particular, and all the rest of the code developed as part of this thesis adhere to the Generic Programming paradigm and follow the Exact Geometric Computation paradigm. We also present exact implementations of other applications that exploit arrangements of arcs of great circles, also known as geodesic arcs, embedded on the sphere. For example, we implemented robust polyhedra central-projection and Boolean set-operations applied to point sets embedded on the sphere bounded by geodesic arcs. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in R 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. We have produced three movies that explain some of the concepts portrayed in this thesis [20]. The first movie [FFHL02] explains what Minkowski sums are and demonstrates how they are used in various applications. The second movie [FH05] demonstrates the first method we have developed to construct Minkowski-sums of convex polyhedra. The third movie [FSH08b] illustrates exact construction and maintenance of arrangements induced by geodesic arcs and applications that exploit such arrangements. Additional information is available at the following web sites: http://acg.cs.tau.ac.il/projects/internal-projects/gaussian-map-cubical Throughout the thesis a number in brackets (e.g., [20]) refers to the link list starting on page 118, and an alphanumeric string in brackets (e.g., [FFHL02]) is a standard bibliographic reference.