Projective Geometry

by Coxeter, H. S. M.

• ISBN13: 9780387406237
• ISBN10: 0387406239
• Edition: 2nd
• Format: Paperback
• Copyright: 2003-10-01
• Publisher: Springer Verlag

Summary

In Euclidean geometry, constructions are made with ruler and compass. Projective geometry is simpler: its constructions require only a ruler. In projective geometry one never measures anything, instead, one relates one set of points to another by a projectivity. The first two chapters of this book introduce the important concepts of the subject and provide the logical foundations. The third and fourth chapters introduce the famous theorems of Desargues and Pappus. Chapters 5 and 6 make use of projectivities on a line and plane, repectively. The next three chapters develop a self-contained account of von Staudt's approach to the theory of conics. The modern approach used in that development is exploited in Chapter 10, which deals with the simplest finite geometry that is rich enough to illustrate all the theorems nontrivially. The concluding chapters show the connections among projective, Euclidean, and analytic geometry.

Table of Contents

Preface to the First Edition	v
Preface to the Second Edition	vii
CHAPTER 1 Introduction
1.1 What is projective geometry?	1	(1)
1.2 Historical remarks	2	(3)
1.3 Definitions	5	(1)
1.4 The simplest geometric objects	6	(2)
1.5 Projectivities	8	(2)
1.6 Perspectivities	10	(4)
CHAPTER 2 Triangles and Quadrangles
2.1 Axioms	14	(2)
2.2 Simple consequences of the axioms	16	(2)
2.3 Perspective triangles	18	(2)
2.4 Quadrangular sets	20	(2)
2.5 Harmonic sets	22	(2)
CHAPTER 3 The Principle of Duality
3.1 The axiomatic basis of the principle of duality	24	(2)
3.2 The Desargues configuration	26	(2)
3.3 The invariance of the harmonic relation	28	(1)
3.4 Trilinear polarity	29	(1)
3.5 Harmonic nets	30	(3)
CHAPTER 4 The Fundamental Theorem and Pappus's Theorem
4.1 How three pairs determine a projectivity	33	(2)
4.2 Some special projectivities	35	(1)
4.3 The axis of a projectivity	36	(2)
4.4 Pappus and Desargues	38	(3)
CHAPTER 5 One-dimensional Projectivities
5.1 Superposed ranges	41	(2)
5.2 Parabolic projectivities	43	(2)
5.3 Involutions	45	(2)
5.4 Hyperbolic involutions	47	(2)
CHAPTER 6 Two-dimensional Projectivities
6.1 Projective collineations	49	(3)
6.2 Perspective collineations	52	(3)
6.3 Involutory collineations	55	(2)
6.4 Projective correlations	57	(3)
CHAPTER 7 Polarities
7.1 Conjugate points and conjugate lines	60	(2)
7.2 The use of a self polar triangle	62	(2)
7.3 Polar triangles	64	(1)
7.4 A construction for the polar of a point	65	(2)
7.5 The use of a self polar pentagon	67	(1)
7.6 A self conjugate quadrilateral	68	(1)
7.7 The product of two polarities	68	(2)
7.8 The self-polarity of the Desargues configuration	70	(1)
CHAPTER 8 The Conic
8.1 How a hyperbolic polarity determines a conic	71	(4)
8.2 The polarity induced by a conic	75	(1)
8.3 Projectively related pencils	76	(2)
8.4 Conics touching two lines at given points	78	(2)
8.5 Steiner's definition for a conic	80	(1)
CHAPTER 9 The Conic, Continued
9.1 The conic touching five given lines	81	(4)
9.2 The conic through five given points	85	(2)
9.3 Conics through four given points	87	(1)
9.4 Two self polar triangles	88	(1)
9.5 Degenerate conics	89	(2)
CHAPTER 10 A Finite Projective Plane
10.1 The idea of a finite geometry	91	(1)
10.2 A combinatorial scheme for PG(2,5)	92	(3)
10.3 Verifying the axioms	95	(1)
10.4 Involutions	96	(1)
10.5 Collineations and correlations	97	(1)
10.6 Conics	98	(4)
CHAPTER 11 Parallelism
11.1 Is the circle a conic?	102	(1)
11.2 Affine space	103	(2)
11.3 How two coplanar lines determine a flat pencil and a bundle	105	(1)
11.4 How two planes determine an axial pencil	106	(1)
11.5 The language of pencils and bundles	107	(1)
11.6 The plane at infinity	108	(1)
11.7 Euclidean space	109	(2)
CHAPTER 12 Coordinates
12.1 The idea of analytic geometry	111	(1)
12.2 Definitions	112	(4)
12.3 Verifying the axioms for the projective plane	116	(3)
12.4 Projective collineations	119	(3)
12.5 Polarities	122	(2)
12.6 Conics	124	(2)
12.7 The analytic geometry of PG(2,5)	126	(3)
12.8 Cartesian coordinates	129	(3)
12.9 Planes of characteristic two	132	(1)
Answers to Exercises	133	(24)
References	157	(2)
Index	159