| Voronoi diagrams are incredibly useful geometric entities that have been studied, in one form or another, since the 1600s. When we combine them with their dual, the Delaunay tessellation, we obtain an incredible analytical tool. But Voronoi diagrams are useful in more than just analytics. They can be utilized for creating new and unusual forms that still manifest themselves with regularity and predictability.;Moreover, we see Voronoi diagrams and Delaunay tessellations everywhere. Whether it is in the honeycomb of a beehive, or in the atomic structure of a crystalline lattice, Voronoi diagrams surround us.;Despite the long history of Voronoi diagram research, and despite living in a world that is inundated with Voronoi diagrams, we know very little about them. We do know a lot about point-based Voronoi diagrams, and we do know a little about many other forms of Voronoi diagrams, but even simple shapes such as line segments tend to cause trouble when we try to make a Voronoi diagram out of them.;In this dissertation we attempt to define a generalized theory for Voronoi diagrams. That said, much of the work in this dissertation is still theoretical. The vast majority of the work in the first eleven chapters is either self-evident or has been tested; however, much of the work in chapter twelve still requires implementation and testing. The untested nature of some of the work toward the end of the dissertation aside, we do believe that we have been able to develop a rudimentary foundation for a general approach to computing all varieties of Voronoi diagrams and Delaunay tessellations. |
