Chapter 20 - Voronoi Diagram Distances

Abstract

This chapter discusses Voronoi diagram distances. Given a finite set A of objects Ai in a space S, computing Voronoi diagram of A means partitioning the space S into Voronoi regions V (Ai) in such a way that V (Ai) contains all points of S that are “closer” to Ai than to any other object Aj in A. The boundaries of (n-dimensional) Voronoi polygons are called ((n –1)-dimensional) Voronoi facets, the boundaries of Voronoi facets are called (n–2)-dimensional Voronoi faces, ., the boundaries of two‑dimensional Voronoi faces are called Voronoi edges, the boundaries of Voronoi edges are called Voronoi vertices. A generalization of the ordinary Voronoi diagram is possible in following three ways: (1) The generalization with respect to the generator set A = {A1 . . . Ak} which can be a set of lines, a set of areas; (2) the generalization with respect to the space S which can be a sphere (spherical Voronoi diagram), a cylinder (cylindrical Voronoi diagram), a cone (conic Voronoi diagram), a polyhedral surface (polyhedral Voronoi diagram); and (3) the generalization with respect to the function d, where d(x, Ai) measures the “distance” from a point x ∈S to a generator Ai ∈A.