The Real Projective Plane: With an Appendix for Mathematica® by George Beck Macintosh Version

235 Pages·1993·9.25 MB·English

by  H. S. M. Coxeter

THE
REAL PROJECTIVE
PLANE
THIRD EDITION
By the same author
NON-EUCLIDEAN GEOMETRY
UNIVERSITY OF TORONTO PRESS
REGULAR POLYTOPES
DOVER, NEW YORK
INTRODUCTION TO GEOMETRY
WILEY, NEW YORK
PROJECTIVE GEOMETRY
SPRINGER-VERLAG, NEW YORK
TWELVE GEOMETRIC ESSAYS
SOUTHERN ILLINOIS UNIVERSITY PRESS
REGULAR COMPLEX POLYTOPES
CAMBRIDGE UNIVERSITY PRESS
H.S.M. COXETER
THE
REAL PROJECTIVE
PLANE
WITH AN APPENDIX FOR MATHEMATICA® BY GEORGE BECK
MACINTOSH VERSION
(Diskette provided)
THIRD EDITION
Springer-Verlag
New York Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
H.8.M. Coxeter
Department of Mathematics
University of Toronto
Toronto, Ontario M5S lAl
CANADA
George Beck
117 Lyndhurst Avenue
Toronto, Ontario M5R 2Z8
CANADA
Library of Congress Cataloging.in.Publication Data
Coxeter, H.S.M. (Harold Scott Macdonald), 1907-
The real projective plane {H.S.M. Coxeter; with an appendix for
Mathematica by George Beck.- 3rd ed.
p. cm.
Includes bibliographical references (p. ) and index.
ISBN-I): 978·1-4612·7647·0 e·ISBN· \3: 978·1·4612·27)4·2
DOl: 10.1007/978·1-4612·2734·2
I. Geometry, Projective. 2. Geometry, Projective-Data
processing. I. Title.
QA47I.C68 1992
516'.5- dc20 92-22637
Printed on acid·free paper.
Fil1lt edition published in 1949 by the McGraw·Hill Book Company. Inc. Second edition
published in 1955 by Cambridge University Press.
<01993 Springer· Verlag New York. Inc.
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9 8 7 6 5 4 321
Preface to the Third Edition
Along with many small improvements, this revised edition contains
van Yzeren's new proof of Pascal's theorem (§1.7) and, in Chapter 2, an
improved treatment of order and sense. The Sylvester-Gallai theorem,
instead of being introduced as a curiosity, is now used as an essential
step in the theory of harmonic separation (§3.34). This makes the logi
cal development self-contained: the footnotes involving the References
(pp. 214-216) are for comparison with earlier treatments, and to give
credit where it is due, not to fill gaps in the argument.
H.S.M.C.
November 1992
v
Preface to the Second Edition
Why should one study the real plane? To this question, put by those
who advocate the complex plane, or geometry over a general field, I
would reply that the real plane is an easy first step. Most of the prop
erties are closely analogous, and the real field has the advantage of
intuitive accessibility. Moreover, real geometry is exactly what is
needed for the projective approach to non· Euclidean geometry. Instead
of introducing the affine and Euclidean metrics as in Chapters 8 and 9,
we could just as well take the locus of 'points at infinity' to be a conic,
or replace the absolute involution by an absolute polarity.
Apart from the correction of many small errors, the changes made
in this revised edition are chiefly as follows: von Staudt's proof that
AA'BB' 7\ A'AB'B (2'71) has been adapted to yield the quadrangular
involution (4'71). The first axiom of order has been weakened (3'11).
More satisfactory proofs have been given for Hesse's theorem (5.55),
for von Staudt's converse of Chasles's theorem (5'71), for Archimedes'
axiom (10'22), and for Enriques's fixed-point theorem (10'62). There is
also an improved treatment of degenerate polarities (5'9), of the inside
and outside of a conic (6'32), of Desargues's involution (6'72), of the
nine-point conic (6'81), of the condition for a quadrangle to be convex
with respect to a line (7'55), and of Klein's classification of geometries
according to the groups of transformations under which their prop
erties are invariant (8'10).
I wish to express my gratitude to many readers of the first edition
who sent useful suggestions, to W.O.J. Moser for helping with the
proofs, and to the Syndics of the Cambridge University Press for un
dertaking the new edition.
H.S.M.e.
November 1954
Vll
Preface to the First Edition
This introduction to projective geometry can be understood by anyone
familiar with high-school geometry and algebra. The restriction to real
geometry of two dimensions makes it possible for every theorem to be
illustrated by a diagram. The early books of Euclid were concerned
with constructions by means of ruler and compasses; this is even sim
pler, being the geometry of the ruler alone. The subject is used, as
metrical geometry was by Euclid, to reveal the development of a logi
cal system from primitive concepts and axioms. Accordingly, the treat
ment is mainly synthetic; analytic geometry is confined to the last two
of the twelve chapters.
The strict axiomatic treatment is followed far enough to show the
reader how it is done, but is then relaxed to avoid becoming tedious.
Continuity is introduced in Chapter 3 by means of an unusual but
intuitively acceptable axiom. A more thorough treatment is reserved
for Chapter 10, at which stage the reader may be expected to have ac
quired the necessary maturity for appreciating the subtleties involved.
The spirit of the book owes much to the great Projective Geometry of
Veblen and Young. That dealt with geometries of various kinds in any
number of dimensions; but the present book may be found easier be
cause one particular geometry has been extracted for detailed consid
eration. Chapters 5 and 6 constitute what is perhaps the first system
atic account in English of von Staudt's synthetic approach to polari
ties and conics as amplified by Enriques: A polarity is defined as an
involutory point-to-line correspondence preserving incidence, and a
conic as the locus of points that lie on their polars, or the envelope of
lines that pass through their poles. This definition for a conic gives the
whole figure at once and makes it immediately self-dual, a locus and an
envelope, whereas Steiner's definition assigns a special role to two
points on the conic, obscuring its essential symmetry. Moreover, the
restriction to real geometry makes it desirable to consider not only the
hyperbolic polarities which determine conics but also the elliptic po
larities which do not. The latter are important because of their applica-
lX
x PREFACE TO THE FIRST EDITION
tion to elliptic geometry. (In complex geometry this distinction is un
necessary, for an elliptic polarity determines an imaginary conic.) The
linear construction for the polar of a given point (5·64) was adapted
from a question in the Cambridge Mathematical Tripos, 1934, Part II,
Schedule A.
The treatment of conics is followed in Chapter 8 by a description of
affine geometry, where one line of the projective plane is singled out as
a line at infinity, enabling us to define parallel lines. It is interesting to
see how much of the familiar content of metrical geometry depends
only on incidence and parallelism and not on perpendicularity. This
includes the theory of area; the distinction between the ellipse, para
bola and hyperbola; and the theory of diameters, asymptotes, etc. The
further specialization to Euclidean geometry is made in Chapter 9 by
singling out an absolute involution on the line at infinity.
Chapter 10 introduces a revised axiom of continuity for the projec
tive line, so simple that only eight words are needed for its enuncia
tion. (This has not been published elsewhere save as an abstract in the
Bulletin of the American Mathematical Society.) Chapter 11 develops
the formal addition and multiplication of points on a conic and the
synthetic derivation of coordinates. Finally, Chapter 12 contains a
verification that the plane of real homogeneous coordinates has all the
properties of our synethetic geometry. This proves that the chosen
axioms are as consistent as the axioms of arithmetic.
Almost every section of the book ends with a group of problems
involving the latest ideas that have been presented. All the difficult
problems are followed by hints for solving them. The teacher can ren
der them more difficult by taking them out of their context or by omit
ting the hints.
I take this opportunity for expressing my thanks to H.G. Forder and
Alan Robson for reading the manuscript and suggesting improve
ments; also to Leopold Infeld and Alex Rosenberg for helping with the
proofs.
H.S.M.C.
February 1949
Contents
Preface to the Third Edition page v
Preface to the Second Edition Vll
Preface to the First Edition ix
Chapter 1. A COMPARISON OF VARIOUS KINDS OF GEOMETRY
1'1 Introduction, p. 1. 1'2 Parallel projection, p. 1. 1'3 Central projection, p. 2.
1-4 The line at infinity, p. 4. 1'5 Desargues's two-triangle theorem, p. 6. 1'6 The
directed angle, or cross, p. 8. 1'7 Hexagramma mysticum, p. 9. 1'8 An outline of
subsequent work, p. 10.
Chapter 2. INCIDENCE
2'1 Primitive concepts, p. 12. 2'2 The axioms of incidence, p. 14. 2'3 The princi·
pie of duality, p. 15. 2-4 Quadrangle and quadrilateral, p. 17. 2'5 Harmonic con
jugacy, p. 18. 2'6 Ranges and pencils, p. 21. 2'7 Perspectivity, p. 21. 2'8 The
invariance and symmetry of the harmonic relation, p. 23.
Chapter 3. ORDER AND CONTINUITY
3-1 The axioms of order, p. 25. 3'2 Segment and interval, p. 27. 3'3 Sense,
p. 29. 3'4 Ordered correspondence, p. 30. 3'5 Continuity, p. 34. 3'6 Invariant
points, p. 34. 3'7 Order in a pencil, p. 36. 3'8 The four regions determined by a
triangle, p. 37.
Chapter 4. ONE· DIMENSIONAL PROJECTIVITIES
4'1 Projectivity, p. 39. 4'2 The fundamental theorem of projective geometry,
p. 41. 4'3 Pappus's theorem, p. 43. 4'4 Classification of projectivities, p. 45.
4'5 Periodic projectivities, p. 48. 4'6 Involution, p. 48. 4'7 Quadrangular set of
six points, p. 52. 4'8 Projective pencils, p. 54.
Chapter 5. TWO·DIMENSIONAL PROJECTIVITIES
5'1 Collineation, p. 55. 5'2 Perspective collineation, p. 57. 5'3 Involutory
collineation, p. 59. 5'4 Correlation, p. 61. 5'5 Polarity, p. 62. 5'6 Polar and
self-polar triangles, p. 66. 5'7 The self-polarity of the Desargues configuration,
p. 68. 5'8 Pencil and range of polarities, p. 70. 5'9 Degenerate polarities, p. 71.
Xl
xu CONTENTS
Chapter 6, CONICS
6'1 Historial remarks, p, 73. 6'2 Elliptic and hyperbolic polarities, p. 74. 6'3
How a hyperbolic polarity determines a conic, p. 76. 6'4 Conjugate points and
conjugate lines, p. 78. 6'5 Two possible definitions for a conic, p. 80. 6'6 Con
struction for the conic through five given points, p. 83. 6'7 Two triangles inscribed
in a conic, p. 85. 6'8 Pencils of conics, p. 87.
Chapter 7, PROJECTIVITIES ON A CONIC
7'1 Generalized perspectivity, p. 92. 7'2 Pascal and Brianchon, p. 94. 7'3 Con
struction for a projectivity on a conic, p. 96. 7'4 Construction for the invariant
points of a given hyperbolic projectivity, p. 98. 7'5 Involution on a conic,
p. 99. 7'6 A generalization of Steiner's construction, p. 102. 7'7 Trilinear polar
ity, p. 103.
Chapter 8. AFFINE GEOMETRY
8'1 Parallelism, p. 105. 8'2 Intermediacy, p. 106. 8'3 Congruence, p. 107. 8'4
Distance, p. 109. 8'5 Translation and dilatation, p. 113. 8'6 Area, p. 114. 8'7
Classification of conics, p. 117. 8'8 Conjugate diameters, p. 119. 8'9 Asymptotes,
p. 121. 8'10 Affine transformations and the Erlangen programme, p. 124.
Chapter 9, EUCLIDEAN GEOMETRY
9'1 Perpendicularity, p. 126. 9'2 Circles, p. 128. 9'3 Axes of a conic, p. 131.
9'4 Congruent segments, p. 133. 9'5 Congruent angles, p. 134. 9'6 Congruent
transformations, p. 138. 9'7 Foci, p. 142. 9'8 Directrices, p. 144.
Chapter 10. CONTINUITY
10'1 An improved axiom of continuity, p. 147. 10'2 Proving Archimedes' axiom,
p. 148. 10'3 Proving the line to be perfect, p. 149. 10'4 The fundamental theorem
of projective geometry, p. 152. 10'5 Proving Dedekind's axiom, p. 153. 10'6 En
riques's theorem, p. 153.
Chapter 11. THE INTRODUCTION OF COORDINATES
11'1 Addition of points, p. 156. 11'2 Multiplication of points, p. 158. 11'3 Ratio
nal points, p. 161. 11'4 Projectivities, p. 161. 11'5 The one-dimensional con
tinuum,p.163. 11'6 Homogeneouscoordinates,p.165. 11'7 Proof that a line has
a linear equation, p. 165. 11'8 Line coordinates, p. 167.
Chapter 12. THE USE OF COORDINATES
12'1 Consistency and categoricalness, p. 169. 12'2 Analytic geometry, p. 171.
12'3 Verifying the axioms of incidence, p. 173. 12'4 Verifying the axioms of order