A Review of Properties and Variations of Voronoi Diagrams

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS15To close, we will look at another interesting property of Delaunaytriangulations. Let D(p i , p j ) be the shortest path along the Delaunayedges of D(P ) from p i to p j . Then D(p i , p j ) ≤ c ∗ d(p i , p j ), whered(p i , p j ) is the Euclidean distance from p i to p j and c =2π ≈ 2.42.3cos( π 6 )5. The Weighted Voronoi Diagram5.1. Introduction. So far, in our discussion of Voronoi diagrams, wehave assumed that our generator points, besides their location, haveequal value, or weight. The idea of assigning distinct weight to generatorpoints can be more useful than having uniformly weighted points insome scenarios. Weighted generator points are sometimes more applicablewhen looking at, for example, the population size of a settlement,the number of stores in a shopping center, or the size of an atom in acrystal structure [6].Recall from our definition of the Voronoi diagram from Section 3.4.that a Voronoi region V i is the intersection of the dominance regions ofp i over every other generator point in P . While we will have differentformulae for dominance regions in weighted voronoi diagrams, the idearemains the same. The dominance region of a generator point p i overanother, p j , where i ≠ j and d w (p x , p y ) is the weighted distance between