On higher order Voronoi diagrams of line segments

Abstract

We analyze structural properties of the order-κ Voronoi diagram of line segments, which surprisingly has not received any attention in the computational geometry literature. We show that order-κ Voronoi regions of line segments may be disconnected; in fact a single order-κ Voronoi region may consist of Ω(n) disjoint faces. Nevertheless, the structural complexity of the order-κ Voronoi diagram of non-intersecting segments remains O(κ(n - κ)) similarly to points. For intersecting line segments the structural complexity remains O(κ(n - κ)) for κ ≥ n/2. © Springer-Verlag 2012.