The basic structures of Voronoi and generalized Voronoi polygons

Abstract

For each particle in an aggregate of point particles in the plane, the set of points having it as closest particle is a convex polygon, and the aggregate V of such Voronoi polygons tessellates the plane. The geometric and stochastic structure of a random Voronoi polygon relative to a homogeneous Poisson process is specified.Similarly, those points of the plane possessing the same n nearest particles constitute a convex polygon cell in the generalized Voronoi tessellation 𝒱 (n = 2, 3, ·· ·). In fact, 𝒱 = 𝒱₁, but to ease exposition n always takes the values 2, 3, ···. A key geometrical lemma elucidates the geometric structure of members of 𝒱_n, showing it to be simpler in one important respect than that of members of 𝒱; in that, for each such N-gon of given ‘type', there is a uniquely determined set of N generating particles. The corresponding jacobian is given, and used to derive the basic ergodic structure of 𝒱_n relative to a homogeneous Poisson process.Unlike 𝒱 no 𝒱_n contains any triangles. As n →∞, the vertices of the quadrangles of 𝒱ⁿ tend to circularity, so that the sums of their opposite interior angles tend to π.