Voronoi Diagrams of Semi-algebraic Sets: Delaunay Graphs of Semi-algebraic Sets

Most of the curves and surfaces encountered in geometric modelling are defined as the set of solutions of a system of algebraic equations and inequalities (i. e., semi-algebraic sets). Many problems from different fields involve proximity queries like finding the (nearest) neighbours. The Voronoi diagram of a set of sites is a decomposition of space into proximal regions (points having a generator as nearest neighbour). The dual graph of the Voronoi diagram is called the Delaunay graph. The book shows the basic algebraic and geometric properties of offsets to algebraic curves and introduces the concept of generalised Voronoi vertex, to reduces the semi-algebraic computation of the Delaunay graph to a linear algebra problem. Then, it presents the certified incremental maintenance of the Delaunay graph of conics and of semi-algebraic sets. The central idea of this book is that symbolic pre-computations can be integrated with interval analysis to accelerate the certified incremental maintenance of the Delaunay graph. The certified computation of the Delaunay graph relies on theorems on the uniqueness of a root in given intervals (Kantorovitch, Moore-Krawczyk) and the ALIAS library.