From foundations to applications.

Albrecht Beutelspacher and Ute Rosenbaum

x + 258 pages
Cambridge University Press 1998
hardback: ISBN 0-521-48277-1 (£65.00 / $64.95)
paperback: ISBN 0-521-48364-6 (£15.95 / $24.95)

From the preface

Why should a person study projective geometry?
First of all, projective geometry is a jewel of mathematics, one of the outstanding achievements of the nineteenth century, a century of remarkable mathematical achievements such as non-Euclidean geometry, abstract algebra, and the foundations of calculus. Projective geometry is as much a part of a general education in mathematics as differential equations and Galois theory. Moreover, projective geometry is a prerequisite for algebraic geometry, one of today's most vigorous and exciting branches of mathematics.
Secondly, for more than fifty years projective geometry has been propelled in a new direction by its combinatorial connections. The challenge of describing a classical geometric structure by its parameters - properties that at first glance might seem superficial - provided much of the impetus for finite geometry, another of today's flourishing branches of mathematics.
Finally, in recent years new and important applications have been discovered. Surprisingly, the structures of classical projective geometry are ideally suited for modern communications. We mention, in particular, applications of projective geometry to coding theory and to cryptography.

From a didactical point of view, this book is based on three axioms.

1. We do not assume that the reader has had any prior exposure to projective or affine geometry. Therefore we present ever the elementary part in detail. On the other hand, we suppose that the reader has some experience in manipulating mathematical objects as found in a typical first or second year at university. Notions such as 'equivalence relation', 'basis', or 'bijective' should not strike terror in your heart.
   
2. We present those parts of projective geometry that are important for applications.
   
3. Finally, this book contains material that can readily be taught in a one year course.
   

These axioms force us to take shortcuts around many themes of projective geometry that became canonized in the nineteenth and twentieth centuries: there are no cross ratios or harmonic sets, non-Desarguesian planes are barely touched upon, projectivities are missing, and collineation groups do not play a central role. One may regret these losses, but, on the other hand, we note the following gains:

• This is a book that can be read independently by students.
• Most of the many exercises are very easy, in order to reinforce the reader's understanding.
• We are proud to present some topics for the first time in a textbook: for instance, the classification of quadrics (Theorem 4.4.4) in finite spaces, which we get by purely combinatorial considerations. Another example is the geometric-combinatorial description of the Reed-Muller codes. Finally we mention the theorem of Gilbert, MacWilliams, and Sloane (see 6.3.1), whose proof is - in our opinion - very illuminating.
• Last but not least, we describe the geometrical structures in Chapters 1 and 4 by diagrams, an approach that has led over the past twenty years to a fundamental restructuring of geometrical research.

To collaborate on a book is a real adventure, much to our surprise. In our case we had throughout an enjoyable collaboration, which was always intense and exciting - even when our opinions were far apart. Many arguments were resolved when one of us asked a so-called 'silly question', and we were forced to thoroughly reexamine seemingly clear concepts. We hope that all this will be an advantage for the reader.

For more information and ordering details, please contact Cambridge University Press in Cambridge or New York.