Computing the Topology of Voronoï Diagrams of Parallel Half-Lines

Abstract

In this paper we consider the Voronoi diagram of a finite family of parallel half-lines, with the same orientation, constrained to a compact domain $${\mathscr {D}}_{0} \subset {\mathbb {R}}^3$$ , with respect to the Euclidean distance. We present an efficient approximation algorithm for computing such VD, using a subdivision process, which produces a mesh representing the topology of the VD in $${\mathscr {D}}_{0}$$ . The computed topology may not be correct for degenerate configurations or configurations close to degenerate. In this case, the output is a valid partition, which is close to the exact partition in Voronoi cells if the input data were given with no error. We also present the result of an implementation in Julia language with visualization using Axl software (axl.inria.fr) of the algorithm. Some examples and analysis are shown.