Abstract

The classic Voronoi diagram of a configuration of distinct points in the plane associates to each point that part of the plane that is closer to the point than to any other point in the configuration. In this thesis we no longer require all points to be distinct.
... read more
After the introduction in Chapter 1, we recall some well-known facts on Voronoi diagrams in Chapter 2. In Chapter 4 we assume that the position of any of the points is given by a pair of polynomials in one parameter. We call such points polynomial sites. Think of the polynomial sites as points that move in time. This setup allows the sites to coincide at certain moments. From the coefficients of the polynomials one can determine the limit angles of all limit lines passing through two sites at a moment of collision. It turns out that this information is enough to define a Voronoi diagram for the sites at collision.
In Chapter 5 we analyze for small examples what angles occur if we determine all angles passing through two points of a configuration of distinct points. We compactify the configuration space of distinct points by taking the closure of the graph of the map that associates the angles to a configuration. By considering the equations of the lines through the points we define an algebraic alternative for this compactification. In Chapter 6 we prove a theorem that shows that representing a Voronoi diagram by a data set consisting of points and angles between the points is in some sense stable. The theorem essentialy states that two data sets that are Euclidean close have Voronoi diagrams that are Hausdorff close.
In Chapter 7 we define an angle- and hook-compactification. This is again a compactification of the configuration space of distinct points in the plane but now defined by considering geometric properties of both pairs and triples of points. For every pair of points we write down the angle, as before, and for every ordered triple of points we specify a hook. This hook expresses how to construct the third point, given the first and second point. The angle- and hook-compactification is defined as the closure of the image space of all angles and hooks on n distinct points. We show that configurations that are added by taking the closure have a a natural nested structure, easily revealed by analyzing the hooks. The main result of this chapter is an explicit construction establishing the angle- and hook-compactification as the graph of a function. This construction shows that the compactification is in fact a smooth manifold. Kontsevich and Soibelman introduced a manifold with corners by a related construction. We discuss the connection and show how to apply our explicit construction on defining clickable Voronoi diagrams on their manifold with corners. Finally, in Chapter 3, we do not only consider the closest point but the k closest points. That is, we consider all k-th order Voronoi diagrams simultaneously. We introduce a poset that consists of all higher order Voronoi cells that occur for some configuration of distinct points. We prove that there exists a rank function on the poset and, moreover, that the number of elements of odd rank equal the number of elements of even rank of the poset, provided that the number of points is odd.
show less