Detecting Voronoi (area-of-influence) polygons

Abstract

Voronoi, or area-of-influence, polygons are convex, space-filling polygons constructed around a set of points (Voronoi centers) such that each polygon contains all points closer to its Voronoi center than to the center of any other polygon. The relationship of Voronoi centers to edges of Voronoi polygons is used to test whether any convex tessellation consists of Voronoi polygons. This test amounts to finding Voronoi centers that best fit the given tessellation. Voronoi centers are found by solving two systems of linear equations. These equations represent (1) conditions on the slope of polygon edges relative to the slope of lines through Voronoi centers, and (2) conditions on the distance from edges to Voronoi centers. Least squares and constrained least-squares solutions are used to solve the two systems. Different methods of solution can provide insight as to how a tessellation varies from Voronoi polygons. A goodness-of-fit statistic is derived and examined by testing randomly generated convex tessellations. Some polygonal ice cracks provide an example of naturally occurring polygons that are approximated closely by Voronoi polygons. © 1987 International Association for Mathematical Geology.