A unified approach towards computing Voronoi diagram, medial axis, Delaunay graph and α-hull of planar closed curves using touching discs

Abstract

This paper proposes a unified approach towards computing geometry structures viz. Voronoi diagram, medial axis, Delaunay graph and α-hull of planar closed curves. It initially presents an algorithm for computing the Voronoi diagram of a set of planar freeform closed curves without approximating the curves using points, lines or biarcs. The algorithm starts by computing the minimum antipodal discs (MADs) for all pairs of curves and these MADs are systematically processed to identify all branch points. The key feature of the algorithm is that it computes a branch point without computing any of the bisectors a priori. Local computations of Voronoi segments are then done using the identified pairs of the segments of curves. The theoretical foundation of the algorithm has been first laid for a set of convex curves and then extended to non-convex curves. It has also been shown that the developed algorithm for the Voronoi diagram can also be used to compute related structures such as medial axis, Delaunay graph and α-hull. They have also been addressed without computing Voronoi edges/segments. Results of the implementation have been provided along with a detailed discussion of the algorithm.