Voronoi diagram of a circle set from Voronoi diagram of a point set: II. Geometry

Abstract

Presented in this paper are algorithms to compute the positions of vertices and equations of edges of the Voronoi diagram of a circle set on a plane, where the radii of the circles are not necessarily equal and the circles are not necessarily disjoint. The algorithms correctly and efficiently work in conjunction with the first paper of the series dealing with the construction of the correct topology of the Voronoi diagram of a circle set from the topology of the Voronoi diagram of a point set, where the points are centers of the circles. Given three circle generators, the position of the Voronoi vertex is computed by treating the plane as a complex plane, Z-plane, and transforming it into another complex plane, W-plane, via a linear fractional transformation. Then, the problem is formulated as a simple point location problem in regions defined by two lines and two circles in the W-plane. After the correct topology is constructed with the geometry of the vertices, the equations of edge are computed in a rational quadratic Bézier curve from.