It is used in many domains such as: computer-aided design, robotics, spatial planning, mathematical morphology, and image processing. The relevant programs, source code, data sets, and documentation are available at The results of experimentation with a broad family of convex polyhedra are reported. The second is based on Nef polyhedra embedded on the sphere, and the third is an output-sensitive approach based on linear programming. One is a simple approach that computes the convex hull of the pairwise sums of vertices of two convex polyhedra. We compare our Minkowski-sum construction with the only three other methods that produce exact results we are aware of. 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 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. Namely, it can handle degenerate input, and it produces exact results.
Our implementation is complete in the sense that it does not assume general position. We present an exact implementation of an efficient algorithm that computes Minkowski sums of convex polyhedra in R 3. Among our findings are that in general: (i) triangulations are too costly (ii) what constitutes a good decomposition for one of the input polygons depends on the other input polygon - consequently, we develop a procedure for simultaneously decomposing the two polygons such that a 'mixed' objective function is minimized, (iii) there are optimal decomposition algorithms that significantly expedite the Minkowski-sum computation, but the decomposition itself is expensive to compute - in such cases simple heuristics that approximate the optimal decomposition perform very well. We report on our experiments with various decompositions and different input polygons. We study and experiment with various well-known decompositions as well as with several new decomposition schemes. We examine different methods for decomposing polygons by their suitability for efficient construction of Minkowski sums. Several algorithms for computing the Minkowski sum of two polygons in the plane begin by decomposing each polygon into convex subpolygons.
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