Voronoi Diagrams and Arrangements

Abstract

We propose a uniform and general framework for defining and dealing with Voronoi Diagrams. In this framework a Voronoi Diagram is a partition of a domain $D$ induced by a finite number of real valued functions on $D$. Valuable insight can be gained when one considers how these real valued functions partition $D \times R$. With this view it turns out that the standard Euclidean Voronoi Diagram of point sets in $R^{d}$ along with its order-$k$ generalizations are intimately related to certain arrangements of hyperplanes. This fact can be used to obtain new Voronoi Diagram algorithms. We also discuss how the formalism of arrangements can be used to solve certain intersection and union problems.