H.S.M. Coxeter - Projective Geometry, 2nd Edition -Springer (2003)

G12 S Two mutually inscribed pentagons H.S.M Coxeter Projective Geometry SECOND EDITION With 71 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo H.S.M Coxeter Department of Mathematics University of Toronto Toronto M5S I A I Canada TO RIEN AMS Classification: 51 A 05 Library of Congress Cataloging-in-Publication Data Coxeter, H S M (Harold Scott Macdonald) Projective geometry Reprint, slightly revised, of 2nd ed originally published by University of Toronto Press, 1974 Includes index Bibliography: p Geometry, Projective I Title 87-9750 QA471.C67 1987 516.5 The first edition of this book was published by Blaisdell Publishing Company 1964: the second edition was published by the University of Toronto Press, 1974 ©1987 by Springer-Verlag New York Inc All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York New York 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Printed and bound by R R Donnelley and Sons, Harrisonburg Virginia Printed in the United States of America 987654321 ISBN 0-387-96532-7 Springer-Verlag New York Berlin Heidelberg ISBN 3-540-96532-7 Springer-Verlag Berlin Heidelberg New York Preface to the First Edition In Euclidean geometry, constructions are made with the ruler and compass Projective geometry is simpler: its constructions require only the ruler We consider the straight line joining two points, and the point of intersection of two lines, with the further simplification that two lines never fail to meet ! In Euclidean geometry we compare figures by measuring them In projective geometry we never measure anything; instead, we relate one set of points to another by a projectivity Chapter introduces the reader to this important idea Chapter provides a logical foundation for the subject The third and fourth chapters describe the famous theorems of Desargues and Pappus The fifth and sixth make use of projectivities on a line and in a plane, respectively In the next three we develop a self-contained account of von Staudt's approach to the theory of conics, made more "modern" by allowing the field to be general (though not of characteristic 2) instead of real or complex This freedom has been exploited in Chapter 10, which deals with the simplest finite geometry that is rich enough to illustrate all our theorems nontrivially (for instance, Pascal's theorem concerns six points on a conic, and in PG(2, 5) these are the only points on the conic) In Chapters 11 and 12 we return to more familiar ground, showing the connections between projective geometry, Euclidean geometry, and the popular subject of "analytic geometry." The possibility of writing an easy book on projective geometry was foreseen as long ago as 1917, when D N Lehmer [12,* Preface, p v] wrote : The subject of synthetic projective geometry is destined shortly to force its way down into the secondary schools More recently, A N Whitehead [22, p 1331 recommended a revised cur- riculum beginning with Congruence, Similarity, Trigonometry, Analytic * References are given on page 158 Vi PREFACE TO THE FIRST EDITION Geometry, and then: In this ideal course of Geometry, the fifth stage is occupied with the elements of Projective Geometry This "fifth" stage has one notable advantage: its primitive concepts are so simple that a self-contained account can be reasonably entertaining, whereas the foundations of Euclidean geometry are inevitably tedious The present treatment owes much to the famous text-book of Veblen and Young [19], which has the same title To encourage truly geometric habits of thought, we avoid the use of coordinates and all metrical ideas (Whitehead's first four "stages") except in Chapters 1, 11, 12, and a few of the Exercises In particular, the only mention of cross ratio is in three exercises at the end of Section 12.3 I gratefully acknowledge the help of M W Al-Dhahir, W L Edge, P R Halmos, S Schuster and S Trott, who constructively criticized the manuscript, and of H G Forder and C Garner, who read the proofs I wish also to express my thanks for permission to quote from Science: Sense and Nonsense by J L Synge (Jonathan Cape, London) H S M COXETER Toronto, Canada February, 1963 Preface to the Second Edition Why should one study Pappian geometry? To this question, put by enthusiasts for ternary rings, I would reply that the classical projective plane is an easy first step The theory of conics is beautiful in itself and provides a natural introduction to algebraic geometry Apart from the correction of many small errors, the changes made in this revised edition are chiefly as follows Veblen's notation Q(ABC, DEF) for a quadrangular set of six points has been replaced by the "permutation symbol" (AD) (BE) (CF), which indicates more immediately that there is an involution interchanging the points on each pair of opposite sides of the quadrangle Although most of the work is in the projective plane, it has seemed worth while (in Section 3.2) to show how the Desargues configuration can be derived as a section of the "complete 5-point" in space Section 4.4 emphasizes the analogy between the configurations of Desargues and Pappus At the end of Chapter I have inserted a version of von Staudt's proof that the Desargues configuration (unlike the general Pappus configuration) it not merely self-dual but self-polar The new Exercise on page 124 shows that there is a Desargues configuration whose ten points and ten lines have coordinates involving only 0, 1, and -1 This scheme is of special interest because, when these numbers are interpreted as residues modulo (so that the geometry is PG(2, 5), as in Chapter 10), the ten pairs of perspective triangles are interchanged by harmonic homologies, and therefore the whole configuration is invariant for a group of 5! projective collineations, appearing as permutations of the digits 1, 2, 3, 4, used on page 27 (The general Desargues configuration has the same 5! automorphisms, but these are usually not expressible as collineations In fact, the perspective collineation OPQR - OP'Q'R' considered on page 53 is not, in general, of period two.*) Finally, there is a new Section 12.9 on page * This remark corrects a mistake in my Twelve Geometric Essays (Southern Illinois University Press, 1968), p 129 Vin PREFACE TO THE SECOND EDITION 132, briefly indicating how the theory changes if the diagonal points of a quadrangle are collinear I wish to express my gratitude to many readers of the first edition who have suggested improvements; especially to John Rigby, who noticed some very subtle points H S M COXETER Toronto, Canada May, 1973 Contents Preface to the First Edition V Preface to the Second Edition CHAPTER 1.1 1.2 1.3 1.4 1.5 1.6 CHAPTER Introduction What is projective geometry? Historical remarks Definitions The simplest geometric objects Projectivities Perspectivities Axioms 2.2 Simple consequences of the axioms 2.3 Perspective triangles 2.4 Quadrangular sets 2.5 Harmonic sets 3.1 10 Triangles and Quadrangles 2.1 CHAPTE R 14 16 18 20 22 The Principle of Duality The axiomatic basis of the principle of duality 3.2 The Desargues configuration 3.3 The invariance of the harmonic relation 24 26 28 R CONTENTS 3.4 3.5 CHAPTER 4.1 4.2 4.3 4.4 CHAPTER Trilinear polarity Harmonic nets 29 30 The Fundamental Theorem and Pappus's Theorem How three pairs determine a projectivity Some special projectivities The axis of a projectivity Pappus and Desargues 33 35 36 38 One-dimensional Projectivities Superposed ranges Parabolic projectivities 41 5.2 5.3 Involutions 45 5.4 Hyperbolic involutions 47 5.1 CHAPTER 43 Two-dimensional Projectivities 6.2 Projective collineations Perspective collineations 49 52 6.3 Involutory collineations 55 6.4 Projective correlations 57 6.1 CHAPTER Polarities Conjugate points and conjugate lines The use of a self-polar triangle Polar triangles 7.4 A construction for the polar of a point 60 62 64 65 The use of a self-polar pentagon 7.6 A self-conjugate quadrilateral 7.7 The product of two polarities 67 68 68 7.8 70 7.1 7.2 7.3 7.5 CHAPTER 8.1 8.2 The self-polarity of the Desargues configuration The Conic How a hyperbolic polarity determines a conic The polarity induced by a conic 71 75 ... 987654321 ISBN 0-3 8 7-9 653 2-7 Springer-Verlag New York Berlin Heidelberg ISBN 3-5 4 0-9 653 2-7 Springer-Verlag Berlin Heidelberg New York Preface to the First Edition In Euclidean geometry, constructions... Classification: 51 A 05 Library of Congress Cataloging-in-Publication Data Coxeter, H S M (Harold Scott Macdonald) Projective geometry Reprint, slightly revised, of 2nd ed originally published by University... index Bibliography: p Geometry, Projective I Title 8 7-9 750 QA471.C67 1987 516.5 The first edition of this book was published by Blaisdell Publishing Company 1964: the second edition was published