Polygonal Minkowski Sums via Convolution : Theory and Practice

This thesis studies theoretical and practical aspects of the computation of planar polygonal Minkowski sums via convolution methods. In particular we prove the “Convolution Theorem”, which is fundamental to convolution based methods, for the case of simple polygons. To the best of our knowledge this is the first complete proof for this case. Moreover, we describe a complete, exact and efficient implementation for the reduced convolution method for simple polygons based on a method proposed by Lien. As part of the “Convolution Theorem” we show that the “convolution cycles” exist and provide an optimal algorithm to extract those. A crucial step of the reduced convolution algorithm is to verify whether a loop of the convolution is on the boundary of the Minkowski sum. For this purpose we study three methods, namely a sweep-line collision detection, a bounding volume hierarchy and a novel ray-shooting method. The ray-shooting provides better theoretical bounds, however in practice, the bounding volume hierarchy proved to be the most efficient method. The thesis contains comprehensive comparisons of our reduced convolution implementation with the full convolution method, which are both implemented in CGAL. Up to pathological cases, our method was more efficient in terms of runtime and memory consumption. Furthermore, the benchmarks indicate that the method is competitive with the inexact implementation of Lien.