Polygon Decomposition for Efficient Construction of Minkowski Sums

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Abstract

Several algorithms for computing the Minkowski sum of two polygons in the plane begin by decomposing each polygon into convex subpolygons. We examine different methods for decomposing polygons by their suitability for efficient construction of Minkowski sums. We study and experiment with various well-known decompositions as well as with several new decomposition schemes. We report on our experiments with the various decompositions and different input polygons. 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.