Introduction and Overview

Abstract

This book studies geometry methodically from an analytical, i.e., coordinate-based, viewpoint. In many settings this approach simplifies the computer representation of geometric data. We shall not confine ourselves to linear problems. This is not only appealing from a theoretical viewpoint, it is also practically motivated by advances in computer algebra and the availability of fast computer hardware.

In Chapter 2 we will lay some mathematical foundations. First, we will introduce the language of projective geometry, which is very well suited for many geometric applications. Since this is not usually covered in standard introductory courses in mathematics, we briefly discuss the central concepts of projective spaces and projective transformations. We will also introduce the notion of convexity in this chapter.

Our analytical approach motivates the structure of this book. It is centered around questions about algorithms which solve systems of equations and their increasingly complex variations with regard to the required mathematical tools.

Appendix

In three out of four parts of the appendix we provide foundations for algebraic structures, convex analysis as well as algorithms and complexity. These sections also standardize our notation. The fourth part of the appendix introduces software packages that are used throughout the book: polymake, Maple and Singular. We also mention CGAL and Sage.

The Structure of This Text

This book consists of more material than a standard one semester course can cover. Hence, this text may be used in several different ways as a basis for a series of lectures. The following compilations are meant as a suggestion:

• “Linear Computational Geometry”: Chapters 2 to 7, Chapters 11 and 12. Please note that Chapter 12 uses elimination techniques from Part II of this book. However, the use of Maple or Singular allows us to treat examples without having a detailed knowledge of the theoretical concepts.

• “Non-linear Computational Geometry”: This is complementary to the selection above, hence consisting of Chapters 8 to 10 of the second part of this book and Chapter 13 from the applications part. The amount of material is suitable for a compact course as a follow-up to the course “Linear Computational Geometry”.

• “Cross-section of Polyhedral and Algebraic Methods”: Chapters 2, 3, 5 or 6, 8 until 10, 12, 13. Sections 9.5 and 10.6 may be left out in this.

Every chapter ends with a small section “Remarks” which references further suggested reading and historical remarks. All figures in this book were produced using the mentioned software and using [62].