Projective Geometry: From Foundations to Applications Albrecht Beutelspacher and Ute Rosenbaum RIll CAMBRIDGE ~ UNIVERSITY PRESS Content PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge CB2 lRP, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge, CB2 2RU, United Kingdom 1 Synthetic geometry 1 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia 1.1 Foundations 1.2 The axioms of projective geometry 5 © Cambridge University Press 1998 1.3 Structure of projective geometry 10 1.4 Quotient geometries 20 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, 1.5 Finite projective spaces 23 no reproduction of any part may take place without 1.6 Affine geometries 27 the written permission of Cambridge University Press. 1.7 Diagrams 32 1.8 Application: efficient communication 40 First published 1998 Exercises 43 Printed in the United Kingdom at the University Press, Cambridge True or false? 50 Project 51 You should know the following notions 53 A catalogue record for this book is available from the British Library 2 Analytic geometry 55 Library of Congress Calaloging in Publication Data 2.1 The projective space P(V) 55 Beutelspacher, A. (Albrecht), 1950- 2.2 The theorems of Desargues and Pappus 59 [Projektive Geometrie. English] 2.3 Coordinates 65 Projective Geometry: from foundations to applications I 2.4 The hyperbolic quadric of PG(3, F) 69 A. Beutelspacher, U. Rosenbaum. 2.5 Normal rational curves 74 p. cm. Includes bibliographical references (p. - ) and indexes. 2.6 The Moulton plane 76 ISBN 0 52148277 1. - ISBN 0 521483646 (pbk.) 2.7 Spatial geometries are Desarguesian 78 1. Geometry, Projective. 1. Rosenbaum, Ute. H. Title. 2.8 Application: a communication problem 81 QA471.B5613 1998 Exercises 89 516'.5-dc21 97-18012 CIP True or false? 93 You should know the following notions 93 ISBN 052148277 1 hardback ISBN 0 521 48364 6 paperback 3 The representation theorems, or good descriptions of projective and affine spaces 95 3.1 Central collineations 95 3.2 The group of translations 104 3.3 The division ring 110 3.4 The representation theorems 116 3.5 The representation theorems for collineations 118 3.6 Projective collineations 126 vi Content Exercises 13 3 Preface True or false? 136 You should know the following notions 136 4 Quadratic sets 137 4.1 Fundamental definitions 137 4.2 The index of a quadratic set 141 It is amusing to browse through the prefaces of randomly selected geometry texts 4.3 Quadratic sets in spaces of small dimension 144 4.4 Quadratic sets in finite projective spaces 147 that have been published over the past century. The authors document the ups and 4.5 Elliptic, parabolic, and hyperbolic quadratic sets 150 downs of their discipline as it moves into and out of fashion. It appears today that 4.6 The Klein quadratic set 157 geometry's status has reached a new low, as educators, who themselves have had 4.7 Quadrics 161 meagre training in the subject, recommend cutting back on a student's exposure to 4.8 Plucker coordinates 165 geometry in order to make room in the curriculum for today's more fashionable 4.9 Application: storage reduction for cryptographic keys 173 topics. Of course, to those of us who have studied geometry it is clear that these Exercises 175 educators are moving in the wrong direction. True or false? 178 So why should a person study projective geometry? You should know the following notions 179 First of all, projective geometry is a jewel of mathematics, one of the out­ 5 Applications of geometry to coding theory 181 standing achievements of the nineteenth century, a century of remarkable mathe­ 5.1 Basic notions of coding theory 181 matical achievements such as non-Euclidean geometry, abstract algebra, and the 5.2 Linear codes 185 foundations of calculus. Projective geometry is as much a part of a general educa­ 5.3 Hamming codes 191 tion in mathematics as differential equations and Galois theory. Moreover, projec­ 5.4 MDS codes 196 tive geometry is a prerequisite for algebraic geometry, one of today's most vigor­ 5.5 Reed-Muller codes 203 ous and exciting branches of mathematics. Exercises 208 Secondly, for more than fifty years projective geometry has been propelled in a True or false? 211 new direction by its combinatorial connections. The challenge of describing a Projects 211 classical geometric structure by its paran:ieters - properties that at first glance You should know the following notions 212 might seem superficial - provided much of the impetus for finite geometry, an­ 6 Applications of geometry in cryptography 213 other of today's flourishing branches of mathematics. 6.1 Basic notions of cryptography 213 Finally, in recent years new and important applications have been discovered. 6.2 Enciphering 216 Surprisingly, the structures of classical projective geometry are ideally suited for 6.3 Authentication 224 modem communications. We mention, in particular, applications of projective 6.4 Secret sharing schemes 233 geometry to coding theory and to cryptography. Exercises 240 But what is projective geometry? Our answer might startle the classically Project 242 trained mathematician who would be steeped in the subject's roots in Renaissance You should know the following notions 243 art and would point out that the discipline was first systematised by the seven­ Bibliography 245 teenth century architect Girard Desargues. Here we follow the insight provided by the German mathematician David Hilbert in his influential Foundations of Ge­ Index of notation 253 ometry (1899): a geometry is the collection of the theorems that follow from its General index 255 axiom system. Although this approach frees the geometer from a dependence on viii Preface Preface ix physical space, it exposes him to the real danger of straying too far from nature offer provable security of arbitrarily high level. In Chapter 6 we shall study some and ending up with meaningless abstraction. In this book we avoid that danger by of these systems. dealing with many applications ~ among which are some that even the fertile mind of Hilbert could not have imagined. And so we side-step the question of what From a didactical point of view, this book is based on three axioms. projective geometry is, simply pointing out that it is an extremely good language 1. We do not assume that the reader has had any prior exposure to projective or for describing a multitude of phenomena inside and outside of mathematics. It is affine geometry. Therefore we present ever the elementary part in detail. On the our goal in this book to exploit this point of view. 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. No­ The first four chapters are mainly devoted to pure geometry. In the first chapter tions such as 'equivalence relation', 'basis', or 'bijective' should not strike terror we study geometry from a synthetic point of view; here, notions such as basis, in your heart. dimension, subspace, quotient space, and affine space are introduced. The second 2. We present those parts of projective geometry that are important for applica­ chapter presents what will be for us the most important class of projective spaces, tions. namely those that can be constructed using a vector space; in other words, those 3. Finally, this book contains material that can readily be taught in a one year projective spaces that can be coordinatized using a field. In analytic geometry one course. usually gets the impression that those are all the projective and affine spaces and no other structures are conceivable. In Chapter 3 we deal with precisely that ques­ These axioms force us to take shortcuts around many themes of projective ge­ tion: A masterpiece of classical geometry is the representation theorem for projec­ ometry that became canonized in the nineteenth and twentieth centuries: there are tive and affine spaces. It says that any projective or affine space that satisfies the no cross ratios or harmonic sets, non-Desarguesian planes are barely touched theorem of Desargues is coordinatizable. In particular we shall show that any upon, projectivities are missing, and collineation groups do not play a central role. projective or affine space of dimension 2': 3 can be coordinatized over a vector One may regret these losses, but, on the other hand, we note the following gains: space. Then we shall be able to describe all collineations (that is automorphisms) ~ This is a book that can be read independently by students. of Desarguesian projective spaces. In Chapter 4 we investigate the quadrics, which ~ Most of the many exercises are very easy, in order to reinforce the reader's are probably the most studied objects in classical geometry. We shall look at them understanding. from a modem synthetic point of view and try to proceed as far as possible using ~ We are proud to present some topics for the first time in a textbook: for in­ only the properties of a 'quadratic set'. This has the advantage of a much better stance, the classification of quadrics (Theorem 4.4.4) in finite spaces, which we insight into the geometric properties of these structures. get by purely combinatorial considerations. Another example is the geometric~ We shall consider not only geometries over the reals, but also their finite ana­ combinatorial description of the Reed~Muller codes. Finally we mention the theo­ logues; in particular we shall determine their parameters. Moreover, at the end of rem of Gilbert, MacWilliams, and SIoane (see 6.3.1), whose proof is ~ in our each chapter we shall present an application.