Construction of Voronoi Diagram using the Hollow Sphere Concept

This paper implements an algorithm for constructing Voronoi regions using the method we call "hollow sphere". This principle uses the circle, sphere or hyper-sphere as a geometric structure considering an Euclidean space of an arbitrary dimension. Boris Delone used the property of the empty circle to build the Delaunay triangulation; in our case, the same property is used to perform the validations of the hollow spheres, but without using triangles as a fundamental structure. The spheres structure is dual to the Voronoi diagram, thanks to the concept of hollow sphere. Furthermore, the properties of the hollow sphere are detailed and an algorithm of incremental construction is used with O(n log n) time using a kd-tree data structure. This data structure provides stability as shown in the results of our work. It is demonstrated with the "walking" algorithm its relation in complexity with the Delaunay triangulation, but with advantage in spheres. A sphere is represented only by its center and radius, its equation and its operations with respect to the point are simple, as opposed to simplexes in any dimension. For convenience sake, the hollow sphere as a circle so as to work in two dimensions will be explained and illustrated.