A new field

Many disciplines require a knowledge of how to efficiently deal with and build geometric objects. Among many examples, one could quote robotics, computer vision, computer graphics, medical imaging, virtual reality, or computer aided design. The first geometric results with a constructive flavor date back to Euclid and remarkable developments occurred during the nineteenth century. However, only very recently did the design and analysis of geometric algorithms find a systematic treatment: this is the topic of computational geometry which as a field truly emerged in the mid 1970s. Since then, the field has undergone considerable growth, and is now a full-fledged scientific discipline, of which this text presents the foundations.

Contents and layout of this book

The design of efficient geometric algorithms and their analysis are largely based on geometric structures, algorithmic data structuring techniques, and combinatorial results.

A major contribution of computational geometry is to exemplify the central role played by a small number of fundamental geometric structures and their relation to many geometric problems.

Geometric data structures and their systematic analysis guided the layout of this text. We have dedicated a part to each of the fundamental geometric structures: convex hulls, triangulations, arrangements, and Voronoi diagrams.

In order to control the complexity of an algorithm, one must know the complexity of the objects that it generates. For example, it is essential to have a sharp bound on the number of facets of a polytope as a function of the number of its vertices: this is the celebrated upper-bound theorem proved by McMullen in 1970.