Classification and approximation of Voronoi nets

In this work, the definition of net is formalized and topological, graph theoretic and geometric properties of a net are proven. A new definition of a Voronoi diagram is provided that admits multiple sets of sites. The thesis provides a complete theoretical solution to the inverse problem of "Given an arbitrary net, determine whether or not it is a Voronoi net. If it is, find all sets of sites that generate the net."; Several notions of a net being "close" to a Voronoi net are presented. A data structure for representing an arbitrary net is given. This data structure allows the graph theoretic and topological properties of the net to be abstracted. Using the data structure, all possible (topologically distinct) Voronoi nets resulting from n sites for n = 2..6 are listed. A set of necessary topological and graph theoretic properties for an arbitrary abstract net to be realizable as a Voronoi net is given.