Hunting Voronoi vertices

Abstract

Given three sites in the plane, a Voronoi vertex is a point that is equidistant from each. In this paper, we consider the problem of computing Voronoi vertices for three disjoint planar sites of fixed but possibly unknown shape; we only require the ability to query the closest point on an object from a given point. Each site is assumed to be either convex or a finite union of convex sets. Our technique is simple and iterative in nature: beginning from some initial seed point, it computes a sequence of points based on intermediate closest-point queries. This technique is observed to either converge to a Voronoi vertex or oscillate with some finite period. We study geometric conditions on shape/placement of the objects and choice of the initial point that guarantee convergence or oscillation. We show that our technique is probabilistically complete: selecting seed points at random will guarantee convergence to a finite Voronoi vertex, if one exists. Our motivation for seeking Voronoi vertices comes from robot motion planning: Voronoi vertices are natural havens for mobile robots avoiding obstacles. We conclude by briefly describing the implementation of a retraction-like path planner for a planar robot based on our iterative strategy for seeking Voronoi vertices.