Constructing Voronoi Diagrams from Hollow Spheres Using Conformal Geometric Algebra

Abstract

This paper proposes a new procedure for the construction of Voronoi diagrams using spheres. The auxiliary spheres that assist in the diagram construction are defined following the concept of a “hollow sphere”, and their geometrical properties, within the Conformal Geometric Algebra model, turn out to be crucial elements for increasing the global efficiency procedure with complexity \(O(n \log n)\). Several examples are presented, using the CLUCalc software, that show the effectiveness and ability of the proposed procedure. Also, we argue that the Conformal Geometric Algebra model provides straightforward and intuitive concepts to the field of Computational Geometry, that prove invaluable for allowing the incremental definition and construction of Voronoi diagrams.