Date: 11 Feb 2003

Polygon Decomposition for Efficient Construction of Minkowski Sums

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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.

P.A. is supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by NSF grants EIA-9870724, EIA-997287, and CCR-9732787 and by a grant from the U.S.-Israeli Binational Science Foundation. D.H. and E.F. have been supported in part by ESPRIT IV LTR Projects No. 21957 (CGAL) and No. 28155 (GALIA), and by a Franco-Israeli research grant (monitored by AFIRST/France and The Israeli Ministry of Science). D.H. has also been supported by a grant from the U.S.-Israeli Binational Science Foundation, by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities (Center for Geometric Computing and its Applications), and by the Hermann Minkowski - Minerva Center for Geometry at Tel Aviv University.