Projective Geometry, 2nd Edition

175 Pages·2010·7.77 MB·English

by  H. Coxeter

H. S.M. Coxeter
Projective Geometry
Second Edition
/ \
\
/
'
'
/
'
/
'
/
/
"--/
/"-"
'
/
"
/
/
/
/
/
/
/
/
Springer-Verlag
Two mutually inscribed pentagons
H.S.MC.o xeter
ProjectiGveeo metry
SECOND EDITION
With7 1I llustrations
Springer-Verlag
New YorkB erliHne idelberg
LondonP aris Tokyo
H.S.M. Coxctcr
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.
I. Geometry, Projective. I Title.
QA471.C67 1987 516.5 87-9750
The first edition of this book was publi~hed by Blaisdell Publishing Company. 1964: the second
edition was published by the University of Toronto Pre~s. 1974
©1987 by Springer-Verlag New York Inc.
All rights reserved This work may not be translated or copied in whole or in pan 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 Usc 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 de~criptive names, trade names, trademarks, etc in this publication. even if the
former are not especially identified, b not to be taken as a sign that such name~. as under~tood by
the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone
Printed and bound by R R Donnclley and Son~. Harrbonburg. 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 com
pass. 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 I introduces the reader to this important
idea. Chapter 2 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, re
spectively. 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 pro
jective geometry, Euclidean geometry, and the popular subject of "analytic
geometry."
The possibility of writing an easy book on projective geometry was fore
seen as long ago as 1917, when D. N. Lehmer [ll,* 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. 133] 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. AI-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 enthu
siasts 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 con
figuration 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 7 I have inserted a version of von Staudt's
proof that the Desargues configuration (unlike the general Pappus con
figuration) it not merely self-dual but self-polar. The new Exercise 5 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 5 (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, 5 used on
page 27. (The general Desargues configuration has the same 5! auto
morphisms, 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.
viii 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 vii
CHAPTER 1 Introduction
1.1 What is projective geometry? I
1.2 Historical remarks 2
s
1.3 Definitions
1.4 The simplest geometric objects 6
1.5 Projectivities 8
1.6 Perspectivities 10
CHAPTER 2 Triangles and Quadrangles
2.1 Axioms 14
2.2 Simple consequences of the axioms 16
2.3 Perspective triangles 18
2.4 Quadrangular sets 20
2.5 Harmonic sets 22
CHAPTER 3 The Principle of Duality
3.1 The axiomatic basis of the principle of duality 24
3.2 The Desargues configuration 26
3.3 The invariance of the harmonic relation 28
X CONTBNTS
3.4 Trilinear polarity 29
3.5 Harmonic nets 30
CHAPTER 4 The Fundamental Theorem and Pappus's Theorem
4.1 How three pairs determine a projectivity 33
4.2 Some special projectivities 35
4.3 The axis of a projectivity 36
4.4 Pappus and Desargues 38
CHAPTER 5 One-dimensional Projectivities
5.1 Superposed ranges 41
5.2 Parabolic projectivities 43
5.3 Involutions 45
5.4 Hyperbolic involutions 47
CHAPTER 6 Two-dimensional Projectivities
6.1 Projective collineations 49
6.2 Perspective collineations 52
6.3 Involutory collineations 55
6.4 Projective correlations 57
CHAPTER 7 Polarities
7.1 Conjugate points and conjugate lines 60
7.2 The use of a self-polar triangle 62
7.3 Polar triangles 64
7.4 A construction for the polar of a point 65
7.5 The use of a self-polar pentagon 67
7.6 A self-conjugate quadrilateral 68
7.7 The product of two polarities 68
7.8 The self-polarity of the Desargues configuration 70
CHAPTER 8 The Conic
8.1 How a hyperbolic polarity determines a conic 71
8.2 The polarity induced by a conic 75