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

Most of the curves and surfaces encountered in geometricmodelling are defined as the set of solutions of a system ofalgebraic equations and inequalities (i.e., semi-algebraic sets).Many problems from different fields involve proximity queries likefinding the (nearest) neighbours. The Voronoi diagram of a set ofsites is a decomposition of space into proximal regions (pointshaving a generator as nearest neighbour). The dual graph of theVoronoi diagram is called the Delaunay graph. The book shows thebasic algebraic and geometric properties of offsets to algebraiccurves and introduces the concept of generalised Voronoi vertex, toreduces the semi-algebraic computation of the Delaunay graph to alinear algebra problem. Then, it presents the certified incrementalmaintenance of the Delaunay graph of conics and of semi-algebraicsets. The central idea of this book is that symbolicpre-computations can be integrated with interval analysis toaccelerate the certified incremental maintenance of the Delaunaygraph. The certified computation of the Delaunay graph relies ontheorems on the uniqueness of a root in given intervals(Kantorovitch, Moore-Krawczyk) and the ALIAS library.