We present an exact implementation of an efficient algorithm that computes Minkowski sums of convex polyhedra in R 3 . 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 R 3 . 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

We present a tight bound on the exact maximum complexity of Minkowski sums of polytopes in R 3 . In particular, we prove that the maximum number of facets of the Minkowski sum of k polytopes with m 1 , m 2 , . . . , m k facets, respectively, is bounded from above by 1≤i

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Abstract. We introduce a framework for the construction, maintenance, and manipulation of arrangements of curves embedded on certain two-dimensional orientable parametric surfaces in three-dimensional space. The framework applies to planes, cylinders, spheres, tori, and surfaces homeomorphic to them. We reduce the effort needed to generalize existing algorithms, such as the sweep line and zone traversal algorithms, originally designed for arrangements of bounded curves in the plane, by extensive reuse of code. We have realized our approach as the Cgal package Arrangement on surface 2. We define a compact interface for our framework; only the operations in the interface need to be implemented for a specific application. The companion paper [6] describes concretizations for several types of surfaces and curves embedded on them, and applications. This is the first implementation of a generic algorithm that can handle arrangements on a large class of parametric surfaces.

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