ńņš. 1 |

Geometry provides a whole range of views on the universe, serving as the inspiration, technical

toolkit and ultimate goal for many branches of mathematics and physics. This book introduces

the ideas of geometry, and includes a generous supply of simple explanations and examples.

The treatment emphasises coordinate systems and the coordinate changes that generate symme-

tries. The discussion moves from Euclidean to non-Euclidean geometries, including spherical

and hyperbolic geometry, and then on to afļ¬ne and projective linear geometries. Group theory

is introduced to treat geometric symmetries, leading to the uniļ¬cation of geometry and group

theory in the Erlangen program. An introduction to basic topology follows, with the MĀØ biuso

strip, the Klein bottle and the surface with g handles exemplifying quotient topologies and

the homeomorphism problem. Topology combines with group theory to yield the geometry

of transformation groups, having applications to relativity theory and quantum mechanics. A

ļ¬nal chapter features historical discussions and indications for further reading. While the book

requires minimal prerequisites, it provides a ļ¬rst glimpse of many research topics in modern

algebra, geometry and theoretical physics.

The book is based on many yearsā™ teaching experience, and is thoroughly class tested.

There are copious illustrations, and each chapter ends with a wide supply of exercises. Further

teaching material is available for teachers via the web, including assignable problem sheets

with solutions.

m i l e s r e i d is a Professor of Mathematics at the Mathematics Institute, University of Warwick

b a l a zs szendro is a Faculty Lecturer in the Mathematical Institute, University of Oxford,

Ā“ Ā“Ā“i

and Martin Powell Fellow in Pure Mathematics at St Peterā™s College, Oxford

Geometry and

Topology

Miles Reid

Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

BalĀ“zs Szendro

a Ā“Ā“i

Mathematical Institute, University of Oxford,

24ā“29 St Giles, Oxford OX1 3LB, UK

cambridge university press

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Ā© Cambridge University Press 2005

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Contents

List of ļ¬gures page x

Preface xiii

1 Euclidean geometry 1

1.1 The metric on Rn 1

1.2 Lines and collinearity in Rn 3

1.3 Euclidean space En 4

1.4 Digression: shortest distance 4

1.5 Angles 5

1.6 Motions 6

1.7 Motions and collinearity 7

1.8 A motion is afļ¬ne linear on lines 7

1.9 Motions are afļ¬ne transformations 8

1.10 Euclidean motions and orthogonal transformations 9

1.11 Normal form of an orthogonal matrix 10

1.11.1 The 2 Ć— 2 rotation and reļ¬‚ection matrixes 10

1.11.2 The general case 12

1.12 Euclidean frames and motions 14

1.13 Frames and motions of E2 14

1.14 Every motion of E2 is a translation, rotation, reļ¬‚ection or glide 15

1.15 Classiļ¬cation of motions of E3 17

1.16 Sample theorems of Euclidean geometry 19

1.16.1 Pons asinorum 19

1.16.2 The angle sum of triangles 19

1.16.3 Parallel lines and similar triangles 20

1.16.4 Four centres of a triangle 21

1.16.5 The Feuerbach 9-point circle 23

Exercises 24

2 Composing maps 26

2.1 Composition is the basic operation 26

2.2 Composition of afļ¬ne linear maps x ā’ Ax + b 27

v

vi CONTENTS

2.3 Composition of two reļ¬‚ections of E2 27

2.4 Composition of maps is associative 28

2.5 Decomposing motions 28

2.6 Reļ¬‚ections generate all motions 29

2.7 An alternative proof of Theorem 1.14 31

2.8 Preview of transformation groups 31

Exercises 32

3 Spherical and hyperbolic non-Euclidean geometry 34

3.1 Basic deļ¬nitions of spherical geometry 35

3.2 Spherical triangles and trig 37

3.3 The spherical triangle inequality 38

3.4 Spherical motions 38

3.5 Properties of S 2 like E2 39

3.6 Properties of S 2 unlike E2 40

3.7 Preview of hyperbolic geometry 41

3.8 Hyperbolic space 42

3.9 Hyperbolic distance 43

3.10 Hyperbolic triangles and trig 44

3.11 Hyperbolic motions 46

3.12 Incidence of two lines in H2 47

3.13 The hyperbolic plane is non-Euclidean 49

3.14 Angular defect 51

3.14.1 The ļ¬rst proof 51

3.14.2 An explicit integral 51

3.14.3 Proof by subdivision 53

3.14.4 An alternative sketch proof 54

Exercises 56

4 Affine geometry 62

4.1 Motivation for afļ¬ne space 62

4.2 Basic properties of afļ¬ne space 63

4.3 The geometry of afļ¬ne linear subspaces 65

4.4 Dimension of intersection 67

4.5 Afļ¬ne transformations 68

4.6 Afļ¬ne frames and afļ¬ne transformations 68

4.7 The centroid 69

Exercises 69

5 Projective geometry 72

5.1 Motivation for projective geometry 72

5.1.1 Inhomogeneous to homogeneous 72

5.1.2 Perspective 73

5.1.3 Asymptotes 73

5.1.4 Compactiļ¬cation 75

CONTENTS vii

5.2 Deļ¬nition of projective space 75

5.3 Projective linear subspaces 76

5.4 Dimension of intersection 77

5.5 Projective linear transformations and projective frames of reference 77

5.6 Projective linear maps of P1 and the cross-ratio 79

5.7 Perspectivities 81

5.8 Afļ¬ne space An as a subset of projective space Pn 81

5.9 Desarguesā™ theorem 82

5.10 Pappusā™ theorem 84

5.11 Principle of duality 85

5.12 Axiomatic projective geometry 86

Exercises 88

6 Geometry and group theory 92

6.1 Transformations form a group 93

6.2 Transformation groups 94

6.3 Kleinā™s Erlangen program 95

6.4 Conjugacy in transformation groups 96

6.5 Applications of conjugacy 98

6.5.1 Normal forms 98

6.5.2 Finding generators 100

6.5.3 The algebraic structure of transformation groups 101

6.6 Discrete reļ¬‚ection groups 103

Exercises 104

7 Topology 107

7.1 Deļ¬nition of a topological space 108

7.2 Motivation from metric spaces 108

7.3 Continuous maps and homeomorphisms 111

7.3.1 Deļ¬nition of a continuous map 111

7.3.2 Deļ¬nition of a homeomorphism 111

7.3.3 Homeomorphisms and the Erlangen program 112

7.3.4 The homeomorphism problem 113

7.4 Topological properties 113

7.4.1 Connected space 113

7.4.2 Compact space 115

7.4.3 Continuous image of a compact space is compact 116

7.4.4 An application of topological properties 117

7.5 Subspace and quotient topology 117

7.6 Standard examples of glueing 118

7.7 Topology of Pn 121

R

7.8 Nonmetric quotient topologies 122

7.9 Basis for a topology 124

viii CONTENTS

7.10 Product topology 126

7.11 The Hausdorff property 127

7.12 Compact versus closed 128

7.13 Closed maps 129

7.14 A criterion for homeomorphism 130

7.15 Loops and the winding number 130

7.15.1 Paths, loops and families 131

7.15.2 The winding number 133

7.15.3 Winding number is constant in a family 135

7.15.4 Applications of the winding number 136

Exercises 137

8 Quaternions, rotations and the geometry of

transformation groups 142

8.1 Topology on groups 143

8.2 Dimension counting 144

8.3 Compact and noncompact groups 146

8.4 Components 148

8.5 Quaternions, rotations and the geometry of SO(n) 149

8.5.1 Quaternions 149

8.5.2 Quaternions and rotations 151

8.5.3 Spheres and special orthogonal groups 152

8.6 The group SU(2) 153

8.7 The electron spin in quantum mechanics 154

8.7.1 The story of the electron spin 154

8.7.2 Measuring spin: the Sternā“Gerlach device 155

8.7.3 The spin operator 156

8.7.4 Rotate the device 157

8.7.5 The solution 158

8.8 Preview of Lie groups 159

Exercises 161

9 Concluding remarks 164

9.1 On the history of geometry 165

9.1.1 Greek geometry and rigour 165

9.1.2 The parallel postulate 165

9.1.3 Coordinates versus axioms 168

9.2 Group theory 169

9.2.1 Abstract groups versus transformation groups 169

9.2.2 Homogeneous and principal homogeneous spaces 169

9.2.3 The Erlangen program revisited 170

9.2.4 Afļ¬ne space as a torsor 171

CONTENTS ix

9.3 Geometry in physics 172

9.3.1 The Galilean group and Newtonian dynamics 172

9.3.2 The PoincarĀ“ group and special relativity

e 173

9.3.3 Wignerā™s classiļ¬cation: elementary particles 175

9.3.4 The Standard Model and beyond 176

9.3.5 Other connections 176

9.4 The famous trichotomy 177

9.4.1 The curvature trichotomy in geometry 177

9.4.2 On the shape and fate of the universe 178

9.4.3 The snack bar at the end of the universe 179

Appendix A Metrics 180

Exercises 181

Appendix B Linear algebra 183

B.1 Bilinear form and quadratic form 183

B.2 Euclid and Lorentz 184

B.3 Complements and bases 185

B.4 Symmetries 186

B.5 Orthogonal and Lorentz matrixes 187

B.6 Hermitian forms and unitary matrixes 188

Exercises 189

References 190

Index 193

Figures

A coordinate model of space page xiv

1.1 Triangle inequality 2

1.5 Angle with direction 6

1.6 Rigid body motion 6

Afļ¬ne linear construction of Ī»x + Āµy

1.9 9

1.11a A rotation in coordinates 11

1.11b The rotation and the reļ¬‚ection 11

The Euclidean frames P0 , P1 , P2 and P0 , P1 , P2

1.13 14

Rot(O, Īø ) and Glide(L, v)

1.14a 15

1.14b Construction of glide 15

1.14c Construction of rotation 16

Twist (L, Īø, v) and Rot-Reļ¬‚ (L, Īø, )

1.15a 17

1.15b A grid of parallel planes and their orthogonal lines 17

1.16a Pons asinorum 19

Sum of angles in a triangle is equal to Ļ

1.16b 20

1.16c Parallel lines fall on lines in the same ratio 20

1.16d Similar triangles 21

1.16e The centroid 21

1.16f The circumcentre 22

1.16g The orthocentre 22

1.16h The Feuerbach 9-point circle 23

2.3 Composite of two reļ¬‚ections 28

2.7 Composite of a rotation and a reļ¬‚ection 31

3.0 Plane-like geometry 35

3.2 Spherical trig 38

Overlapping segments of S 2 41

3.6

The hyperbola t 2 = 1 + x 2 and t > 0

3.7 42

Hyperbolic space H2

3.8 43

x

LIST OF FIGURES xi

3.10 Hyperbolic trig 45

3.12 (a) Projection to the (x, y)-plane of the spherical lines y = cz

(b) Projection to the (x, y)-plane of the hyperbolic lines y = ct 48

3.13 The failure of the parallel postulate in H2 50

3.14a The hyperbolic triangle PQR with one ideal vertex 52

3.14b Area and angle sums are ā˜additiveā™ 52

3.14c The subdivision of PQR. 54

3.14d The angular defect formula 55

3.14e Area is an additive function 56

3.14f Area is a monotonic function 56

3.15 H-lines 60

4.2 Points, vectors and addition 64

4.3a The afļ¬ne construction of the line segment [p, q] 66

4.3b Parallel hyperplanes 66

4.7 The afļ¬ne centroid 70

4.8 A weighted centroid 70

5.1a A cube in perspective 74

5.1b Perspective drawing 74

5.1c Hyperbola and parabola 74

The 3-transitive action of PGL(2) on P1

5.6a 80

The cross-ratio {P, Q; R, S}

5.6b 80

The inclusion An ā‚ Pn

5.8 82

The Desargues conļ¬guration in P2 or P3

5.9a 83

Lifting the Desargues conļ¬guration to P3

5.9b 84

5.10 The Pappus conļ¬guration 85

5.12a Axiomatic projective plane 87

5.12b Geometric construction of addition 88

6.0 The plan of Coventry market 93

The conjugate rotation g Rot(P Īø )g ā’1 = Rot(g(P), g(Īø))

,

6.4a 97

Action of Aff(n) on vectors of An

6.4b 98

6.6a Kaleidoscope 104

6.6b ā˜MusĀ“ e GrĀ“ vinā™

e e 104

7.2a Hausdorff property 110

S 1 = [0, 1] with the ends identiļ¬ed

7.2b 110

(0, 1) R

7.3a 112

7.3b Squaring the circle 112

7.4a Path connected set 114

7.6a The MĀØ bius strip M

o 119

The cylinder S 1 Ć— [0, 1]

7.6b 119

xii LIST OF FIGURES

7.6c The torus 120

7.6d Surface with g handles 120

7.6e Boundary and interior points 121

Topology of P2 : MĀØ bius strip with a disc glued in

7.7 o 122

R

7.8a The mousetrap topology 123

Equivalence classes of quadratic forms ax 2 + 2bx y + cy 2

7.8b 124

7.10 Balls for product metrics 126

7.12 Separating a point from a compact subset 128

7.13a Closed map 129

7.13b Nonclosed map 129

7.15a Continuous family of paths 131

Dā— covered by overlapping open radial sectors

7.15b 134

7.15c Overlapping intervals 134

7.16a Glueing patterns on the square 140

7.16b The surface with two handles and the 12-gon 140

8.0 The geometry of the group of planar rotations 143

8.7a The Sternā“Gerlach experiment 155

8.7b The modiļ¬ed Sternā“Gerlach device 156

8.7c Two identical SG devices 156

8.7d Two different SG devices 157

9.1a The parallel postulate. To meet or not to meet? 166

9.1b The parallel postulate in the Euclidean plane 166

9.1c The ā˜parallel postulateā™ in spherical geometry 168

9.4a The cap, ļ¬‚at plane and Pringleā™s chip 178

The genus trichotomy g = 0, g = 1, g ā„ 2 for oriented surfaces

9.4b 178

A.1 The bear 182

Preface

What is geometry about?

Geometry ā˜measuring the worldā™ attempts to describe and understand space around

us and all that is in it. It is the central activity and main driving force in many branches

of math and physics, and offers a whole range of views on the nature and meaning

of the universe. This book treats geometry in a wide context, including a wealth of

relations with surrounding areas of math and other aspects of human experience.

Any discussion of geometry involves tension between the twin ideals of intuition

and precision. Descriptive or synthetic geometry takes as its starting point our ideas

and experience of the observed world, and treats geometric objects such as lines and

shapes as objects in their own right. For example, a line could be the path of a light

ray in space; you can envisage comparing line segments or angles by ā˜movingā™ one

over another, thus giving rise to notions of ā˜congruentā™ ļ¬gures, equal lengths, or equal

angles that are independent of any quantitative measurement. If A, B, C are points

along a line segment, what it means for B to be between A and C is an idea hard-wired

into our consciousness. While descriptive geometry is intuitive and natural, and can

be made mathematically rigorous (and, of course, Euclidean geometry was studied in

these terms for more than two millennia, compare 9.1), this is not my main approach

in this book.

My treatment centres rather on coordinate geometry. This uses Descartesā™ idea

(1637) of measuring distances to view points of space and geometric quantities in

terms of numbers, with respect to a ļ¬xed origin, using intuitive ideas such as ā˜a bit

to the rightā™ or ā˜a long way upā™ and using them quantitatively in a systematic and

precise way. In other words, I set up the (x, y)-plane R2 , the (x, y, z)-space R3 or

whatever I need, and use it as a mathematical model of the plane (space, etc.), for

the purposes of calculations. For example, to plan the layout of a car park, I might

map it onto a sheet of paper or a computer screen, pretending that pairs (x, y) of real

numbers correspond to points of the surface of the earth, at least in the limited region

for which I have planning permission. Geometric constructions, such as drawing an

even rectangular grid or planning the position of the ticket machines to ensure the

maximum aggravation to customers, are easier to make in the model than in real

xiii

xiv PREFACE

z

x

y

A coordinate model of space.

life. We admit possible drawbacks of our model, but its use divides any problem into

calculations within the model, and considerations of how well it reļ¬‚ects the practical

world.

Topology is the youngster of the geometry family. Compared to its venerable

predecessors, it really only got going in the twentieth century. It dispenses with

practically all the familiar quantities central to other branches of geometry, such

as distance, angles, cross-ratios, and so on. If you are tempted to the conclusion

that there is not much left for topology to study, think again. Whether two loops of

string are linked or not does not depend on length or shape or perspective; if that

seems too simple to be a serious object of study, what about the linking or knotting

of strands of DNA, or planning the over- and undercrossings on a microchip? The

higher dimensional analogues of disconnecting or knotting are highly nontrivial and

not at all intuitive to denizens of the lower dimensions such as ourselves, and cannot

be discussed without formal apparatus. My treatment of topology runs brieļ¬‚y through

abstract point-set topology, a fairly harmless generalisation of the notion of continuity

from a ļ¬rst course on analysis and metric spaces. However, my main interest is in

topology as rubber-sheet geometry, dealing with manifestly geometric ideas such as

closed curves, spheres, the torus, the MĀØ bius strip and the Klein bottle.

o

Change of coordinates, motions, group theory

and the Erlangen program

Descartesā™ idea to use numbers to describe points in space involves the choice of

a coordinate system or coordinate frame: an origin, together with axes and units of

length along the axes. A recurring theme of all the different geometries in this book

is the question of what a coordinate frame is, and what I can get out of it. While

coordinates provide a convenient framework to discuss points, lines, and so on, it

is a basic requirement that any meaningful statement in geometry is independent of

the choice of coordinates. That is, coordinate frames are a humble technical aid in

determining the truth, and are not allowed the dignity of having their own meaning.

Changing from one coordinate frame to another can be viewed as a transformation

or motion: I can use a motion of space to align the origin and coordinate axes of two

coordinate systems. A statement that remains true under any such motion is indepen-

dent of the choice of coordinates. Felix Kleinā™s 1872 Erlangen program formalises

PREFACE xv

this relation between geometric properties and changes of coordinates by deļ¬ning

geometry to be the study of properties invariant under allowed coordinate changes,

that is, invariant under a group of transformations. This approach is closely related to

the point of view of special relativity in theoretical physics (Einstein, 1905), which

insists that the laws of physics must be invariant under Lorentz transformations.

This course discusses several different geometries: in some case the spaces them-

selves are different (for example, the sphere and the plane), but in others the differ-

ence is purely in the conventions I make about coordinate changes. Metric geometries

such as Euclidean and hyperbolic non-Euclidean geometry include the notions of dis-

tance between two points and angle between two lines. The allowed transformations

are rigid motions (isometries or congruences) of Euclidean or hyperbolic space. Afļ¬ne

and projective geometries consider properties such as collinearity of points, and the

typical group is the general linear group GL(n), the group of invertible n Ć— n ma-

trixes. Projective geometry presents an interesting paradox: while its mathematical

treatment involves what may seem to be quite arcane calculations, your brain has a

sight driver that carries out projective transformations by the thousand every time

you recognise an object in perspective, and does so unconsciously and practically

instantaneously.

The sets of transformations that appear in topology, for example the set of all

continuous one-to-one maps of the interval [0, 1] to itself, or the same thing for the

circle S 1 or the sphere S 2 , are of course too big for us to study by analogy with trans-

formation groups such as GL(n) or the Euclidean group, whose elements depend on

ļ¬nitely many parameters. In the spirit of the Erlangen program, properties of spaces

that remain invariant under such a huge set of equivalences must be correspondingly

coarse. I treat a few basic topological properties such as compactness, connectedness,

winding number and simple connectedness that appear in many different areas of

analysis and geometry. I use these simple ideas to motivate the central problem of

topology: how to distinguish between topologically different spaces? At a more ad-

vanced level, topology has developed systematic invariants that apply to this problem,

notably the fundamental group and homology groups. These are invariants of spaces

that are the same for topologically equivalent spaces. Thus if you can calculate one

of these invariants for two spaces (for example, a disc and a punctured disc) and

prove that the answers are different, then the spaces are certainly not topologically

equivalent. You may want to take subsequent courses in topology to become a real

expert, and this course should serve as a useful guide in this.

Geometry in applications

Although this book is primarily intended for use in a math course, and the topics are

oriented towards the theoretical foundations of geometry, I must stress that the math

ideas discussed here are applicable in different ways, basic or sophisticated, as stated

or with extra development, on their own or in combination with other disciplines,

Euclidean or non-Euclidean, metric or topological, to a huge variety of scientiļ¬c and

technological problems in the modern world. I discuss in Chapter 8 the quantum

xvi PREFACE

mechanical description of the electron that illustrates a fundamental application of

the ideas of group theory and topology to the physics of elementary particles. To

move away from basic to more applied science, let me mention a few examples

from technology. The typesetting and page layout software now used throughout the

newspaper and publishing industry, as well as in the computer rooms of most univer-

sity departments, can obviously not exist without a knowledge of basic coordinate

geometry: even a primary instruction such as ā˜place letter A or box B, scaled by such-

and-such a factor, slanted at such-and-such an angle, at such-and-such a point on the

pageā™ involves afļ¬ne transformations. Within the same industry, computer typefaces

themselves are designed using Bezier curves. The geometry used in robotics is more

sophisticated. The technological aim is, say, to get a robot arm holding a spanner into

the right position and orientation, by adjusting some parameters, say, angles at joints

or lengths of rods. This translates in a fairly obvious way into the geometric prob-

lem of parametrising a piece of the Euclidean group; but the solution or approximate

solution of this problem is hard, involving the topology and analysis of manifolds,

algebraic geometry and singularity theory. The computer processing of camera im-

ages, whose applications include missile guidance systems, depends among other

things on projective transformations (I say this for the beneļ¬t of students looking

for a career truly worthy of their talents and education). Although scarcely having

the same nobility of purpose, similar techniques apply in ultrasonic scanning used

in antenatal clinics; here the geometric problem is to map the variations in density

in a 3-dimensional medium onto a 2-dimensional computer screen using ultrasonic

radar, from which the human eye can easily make out salient features. By a curious

coincidence, 3 hours before I, the senior author, gave the ļ¬rst lecture of this course in

January 1989, I was at the maternity clinic of Walsgrave hospital Coventry looking

at just such an image of a 16-week old foetus, now my third daughter Murasaki.

About this book

This book is intended for the early years of study of an undergraduate math course.

Who the

For the most part, it is based on a second year module taught at Warwick over many

book is for

years, a module that is also taken by ļ¬rst and third year math students, and by students

from the math/physics course. You will ļ¬nd the book accessible if you are familiar

with most of the following, which is standard material in ļ¬rst and second year math

courses.

How to express lines and circles in R2 in terms of coordi-

Coordinate geometry

nates, and calculate their points of intersection; some idea of how to do the same in

R3 and maybe Rn may also be helpful.

Vector spaces and linear maps over R and C, bases and matrixes,

Linear algebra

change of bases, eigenvalues and eigenvectors. This is the only major piece of math

that I take for granted. The examples and exercises make occasional reference to

PREFACE xvii

vector spaces over ļ¬elds other than R or C (such as ļ¬nite ļ¬elds), but you can always

omit these bits if they make you uncomfortable.

Bilinear and quadratic forms, and how to express them in ma-

Multilinear algebra

trix terms; also Hermitian forms. I summarise all the necessary background material

in Appendix B.

Some prior familiarity with the ļ¬rst ideas of a metric space course

Metric spaces

would not do any harm, but this is elementary material, and Appendix A contains all

that you need to know.

I have gone to some trouble to develop from ļ¬rst principles all

Group theory

the group theory that I need, with the intention that my book can serve as a ļ¬rst

introduction to transformation groups and the notions of abstract group theory if you

have never seen these. However, if you already have some idea of basic things such

as composition laws, subgroups, cosets and the symmetric group, these will come in

handy as motivation. If you prefer to see a conventional introduction to group theory,

there are any number of textbooks, for example Green [10] or Ledermann [14]. If you

intend to study group theory beyond the introductory stage, I strongly recommend

Artin [1] or Segal [22]. My ideological slant on this issue is discussed in more detail

in 9.2.

Although the thousands queueing impatiently at supermarkets and airport bookshops

How to use

to get their hands on a copy of this book for vacation reading was strong motivation

the book

for me in writing it, experience suggests the harsher view of reality: at least some of

my readers may beneļ¬t from coercion in the form of an organised lecture course.

Experience from teaching at Warwick shows that Chapters 1ā“6 make a reasonably

paced 30 hour second year lecture course. Some more meat could be added to subjects

that the lecturer or students ļ¬nd interesting; reļ¬‚ection groups following Coxeter [5],

Chapter 4 would be one good candidate. Topics from Chapters 7ā“8 or the further

topics of Chapter 9 could then proļ¬tably be assigned to students as essay or project

material. An alternative course oriented towards group theory could start with afļ¬ne

and Euclidean geometry and some elements of topology (maybe as a refresher), and

concentrate on Chapters 3, 6 and 8, possibly concluding with some material from

Segal [22]. This would provide motivation and techniques to study matrix groups

from a geometric point of view, one often ignored in current texts.

I want the book to be as informal as possible in style. To this end, I always refer

The authorā™s

to the student as ā˜youā™, which has the additional advantage that it is independent of

identity

your gender and number. I also refer to myself by the ļ¬rst person singular, despite

crisis

the fact that there are two of me. Each of me has lectured the material many times,

and is used to taking personal responsibility for the truth of my assertions. My model

is van der Waerdenā™s style, who always wrote the crisp ā˜Ich behaupte . . . ā™ (often

when describing results he learned from Emmy Noether or Emil Artinā™s lectures). I

xviii PREFACE

leave you to imagine the speaker as your ideal teacher, be it a bearded patriarch or a

fresh-faced bespectacled Central European intellectual.

Acknowledge- A second year course with the title ā˜Geometryā™ or ā˜Geometry and topologyā™ has

been given at Warwick since the 1960s. It goes without saying that my choice of

ments

material, and sometimes the material itself, is taken in part from the experience of

colleagues, including John Jones, Colin Rourke, Brian Sanderson; David Epstein has

also provided some valuable material, notably in the chapter on hyperbolic geom-

etry. I have also copied material consciously or unconsciously from several of the

textbooks recommended for the course, in particular Coxeter [5], Rees [19], Nikulin

and Shafarevich [18] and Feynman [7]. I owe special thanks to Katrin Wendland, the

most recent lecturer of the Warwick course MA243 Geometry, who has provided a

detailed criticism of my text, thereby saving me from a variety of embarrassments.

Wen solche Lehren nicht erfreun,

Disclaimer

Verdienet nicht ein Mensch zu sein.

From Sarastroā™s aria, The Magic Flute, II.3.

This is an optional course. If you donā™t like my teaching, please deregister before the

deadline.

1 Euclidean geometry

This chapter discusses the geometry of n-dimensional Euclidean space En , together

with its distance function. The distance gives rise to other notions such as angles and

congruent triangles. Choosing a Euclidean coordinate frame, consisting of an origin

O and an orthonormal basis of vectors out of O, leads to a description of En by

coordinates, that is, to an identiļ¬cation En = Rn .

A map of Euclidean space preserving Euclidean distance is called a motion or rigid

body motion. Motions are fun to study in their own right. My aims are

(1) to describe motions in terms of linear algebra and matrixes;

(2) to ļ¬nd out how many motions there are;

(3) to describe (or classify) each motion individually.

I do this rather completely for n = 2, 3 and some of it for all n. For example, the

answer to (2) is that all points of En , and all sets of orthonormal coordinate frames at

a point, are equivalent: given any two frames, there is a unique motion taking one to

the other. In other words, any point can serve as the origin, and any set of orthogonal

axes as the coordinate frames. This is the geometric and philosophical principle that

space is homogeneous and isotropic (the same viewed from every point and in every

direction). The answer to (3) in E2 is that there are four types of motions: translations

and rotations, reļ¬‚ections and glides (Theorem 1.14).

The chapter concludes with some elementary sample theorems of plane Euclidean

geometry.

The metric on Rn

1.1

Throughout the book, I write Rn for the vector space of n-tuples (x1 , . . . , xn ) of real

numbers. I start by discussing its metric geometry. The familiar Euclidean distance

function on Rn is deļ¬ned by

ļ£«ļ£¶ ļ£«ļ£¶

x1 y1

ļ£¬.ļ£· ļ£¬.ļ£·

|x ā’ y| = (xi ā’ yi )2 , where x = ļ£ . ļ£ø and y = ļ£ . ļ£ø. (1)

. .

xn yn

1

2 EUCLIDEAN GEOMETRY

z

u v

x y

Figure 1.1 Triangle inequality.

The relationship between this distance function and the Euclidean inner product (or

dot product) x Ā· y = xi yi on Rn is discussed in Appendix B.2. The more important

point is that the Euclidean distance (1) is a metric on Rn . If you have not yet met

the idea of a metric on a set X , see Appendix A; for now recall that it is a distance

function d(x, y) satisfying positivity, symmetry and the triangle inequality. Both the

positivity |x ā’ y| ā„ 0 and symmetry |x ā’ y| = |y ā’ x| are immediate, so the point is

to prove the triangle inequality (Figure 1.1).

Theorem (Triangle inequality)

|x ā’ y| ā¤ |x ā’ z| + |z ā’ y|, for all x, y, z ā Rn , (2)

with equality if and only if z = x + Ī»(y ā’ x) for Ī» a real number between 0 and 1.

Set x ā’ z = u and z ā’ y = v so that x ā’ y = u + v; then (2) is equivalent

Proof

to

u i2 + v i2 ā„ (u i + v i )2 . (3)

Note that both sides are nonnegative, so that squaring, one sees that (3) is equivalent

to

u i2 + v i2 + 2 u i2 Ā· v i2 ā„ (u i + v i )2

= u i2 + v i2 + 2 ui vi . (4)

Cancelling terms, one sees that (4) is equivalent to

u i2 Ā· v i2 ā„ ui vi . (5)

If the right-hand side is negative then (5), hence also (2), is true and strict. If the

right-hand side of (5) is ā„ 0 then it is again permissible to square both sides, giving

u i2 Ā· v2 ā„ u jv j .

ui vi (6)

j

1.2 LINES AND COLLINEARITY IN Rn 3

You will see at once what is going on if you write this out explicitly for n = 2 and

expand both sides. For general n, the trick is to use two different dummy indexes i, j

as in (6): expanding and cancelling gives that (6) is equivalent to

(u i v j ā’ u j v i )2 ā„ 0. (7)

i> j

Now (7) is true, so retracing our steps back through the argument gives that (2) is

true. Finally, equality in (2) holds if and only if u i v j = u j v i for all i, j (from (7))

u i v i ā„ 0 (from the right-hand side of (5)); that is, u and v are proportional,

and

u = Āµv with Āµ ā„ 0. Rewriting this in terms of x, y, z gives the conclusion. QED

Lines and collinearity in Rn

1.2

There are several ways of deļ¬ning a line (already in the usual x, y plane R2 ); I choose

one deļ¬nition for Rn .

Let u ā Rn be a ļ¬xed point and v ā Rn a nonzero direction vector.

Definition

The line L starting at u ā Rn with direction vector v is the set

L := u + Ī»v Ī» ā R ā‚ Rn .

Three distinct points x, y, z ā Rn are collinear if they are on a line.

If I choose the starting point x, and the direction vector v = y ā’ x, then

L = {(1 ā’ Ī»)x + Ī»y}. To say that distinct points x, y, z are collinear means that z =

{(1 ā’ Ī»)x + Ī»y} for some Ī». Writing

[x, y] = x + Ī»(y ā’ x) 0 ā¤ Ī» ā¤ 1

for the line segment between x and y, the possible orderings of x, y, z on the line L

are controlled by

ļ£¼ ļ£±

ļ£“ ļ£“ x ā [z, y]

Ī» ā¤ 0ļ£“ ļ£“

ļ£½ ļ£²

0 ā¤ Ī» ā¤ 1 āā’ z ā [x, y]

ļ£“ ļ£“

ļ£“ ļ£“

1 ā¤ Ī»ļ£¾ ļ£³ y ā [x, z].

Together with the triangle inequality Theorem 1.1, this proves the following result.

Three distinct points x, y, z ā Rn are collinear if and only if (after a

Corollary

permutation of x, y, z if necessary)

|x ā’ y| + |y ā’ z| = |x ā’ z|.

In other words, collinearity is determined by the metric.

4 EUCLIDEAN GEOMETRY

Euclidean space En

1.3

After these preparations, I am ready to introduce the main object of study: Euclidean

n-space (En , d) is a metric space (with metric d) for which there exists a bijective

map En ā’ Rn , such that if P, Q ā En are mapped to x, y ā Rn then

d(P, Q) = |y ā’ x|.

In other words, (En , d) is isometric to the vector space Rn with its usual distance

function, if you like this kind of language.

Since lines and collinearity in Rn are characterised purely in terms of the Euclidean

distance function, these notions carry over to En without any change: three points of

En are collinear if they are collinear for some isometry En ā’ Rn (hence for all

possible isometries); the lines of En are the lines of Rn under any such identiļ¬cation.

For example, for points P, Q ā En , the line segment [P, Q] ā‚ En is the set

[P, Q] = R ā En d(P, R) + d(R, Q) = d(P, Q) ā‚ En .

The main point of the deļ¬nition of En is that the map En ā’ Rn iden-

Remark

tifying the metrics is not ļ¬xed throughout the discussion; I only insist that one such

isometry should exist. A particular choice of identiļ¬cation preserving the metric is

referred to as a choice of (Euclidean) coordinates. Points of En will always be de-

noted by capital letters P, Q; once I choose a bijection, the points acquire coordinates

P = (x1 , . . . , xn ). In particular, any coordinate system distinguishes one point of En

as the origin (0, . . . , 0); however, different identiļ¬cations pick out different points of

En as their origin. If you want to have a Grand Mosque of Mecca or a Greenwich

Observatory, you must either receive it by Divine Grace or make a deliberate extra

choice. The idea of space ought to make sense without a coordinate system, but you

can always ļ¬x one if you like.

You can also look at this process from the opposite point of view. Going from Rn

to En , I forget the distinguished origin 0 ā Rn , the standard coordinate system, and

the vector space structure of Rn , remembering only the distance and properties that

can be derived from it.

1.4 Digression: shortest distance

As just shown, the metric of Euclidean space En determines the lines. This section

digresses to discuss the idea summarised in the well known clichĀ“ ā˜a straight line is

e

the shortest distance between two pointsā™; while logically not absolutely essential in

this chapter, this idea is important in the philosophy of Euclidean geometry (as well

as spherical and hyperbolic geometry).

The distance d(P, Q) between two points P, Q ā En is the length of

Principle

the shortest curve joining P and Q. The line segment [P, Q] is the unique shortest

curve joining P, Q.

1.5 ANGLES 5

This looks obvious: if a curve C strays off the straight and narrow

Sketch proof

to some point R ā [P, Q], its length is at least

/

d(P, R) + d(R, Q) > d(P, Q).

The statement is, however, more subtle: for instance, it clearly does not make

sense without a deļ¬nition of a curve C and its length. A curve C in En from P to

Q is a family of points Rt ā En , depending on a ā˜time variableā™ t such that R0 = P

and R1 = Q. Clearly Rt should at least be a continuous function of t ā“ if you allow

instantaneous ā˜teleportingā™ between far away points, you can obviously get arbitrarily

short paths.

The proper deļ¬nition of curves and lengths of curves belongs to differential geom-

etry or analysis. Given a ā˜sufļ¬ciently smoothā™ curve, you can deļ¬ne its length as the

n

integral C ds along C of the inļ¬nitesimal arc length ds, given by ds 2 = i=1 dxi2 .

Alternatively, you can mark out successive points P = R0 , R1 , . . . , R N +1 = Q along

N

the curve, view the sum i=0 d(Ri , Ri+1 ) as an approximation to the length of C, and

deļ¬ne the length of C to be the supremum taken over all such piecewise linear ap-

proximations. To avoid the analytic details (which are not at all trivial!), I argue under

the following weak assumption: under any reasonable deļ¬nition of the length of C,

for any Īµ > 0, the curve C can be closely approximated by a piecewise linear path made up of

short intervals [P, R1 ], [R1 , R2 ], etc., such that

length of C ā„ sum of the lengths of the intervals ā’ Īµ.

However, by the triangle inequality d(P, R2 ) ā¤ d(P, R1 ) + d(R1 , R2 ), so that the

piecewise linear path can only get shorter if I omit R1 . Dealing likewise with R2 , R3 ,

etc., it follows that the length of C is ā„ d(P, Q) ā’ Īµ. Since this is true for any Īµ > 0, it

follows that the length of C is ā„ d(P, Q). Thus the line interval [P, Q] joining P, Q

is the shortest path between them, and its length is d(P, Q) by deļ¬nition. QED

1.5 Angles

n

The geometric signiļ¬cance of the Euclidean inner product x Ā· y = i=1 xi yi on Rn

(Section B.2) is that the inner product measures the size of the angle ā xyz based at

y for x, y, z ā Rn :

(x ā’ y) Ā· (z ā’ y)

cos(ā xyz) = . (8)

|x ā’ y||z ā’ y|

By convention, I usually choose the angle to be between 0 and Ļ . In particular, the

vectors x ā’ y, z ā’ y are orthogonal if (x ā’ y) Ā· (z ā’ y) = 0.

The notion of angle is easily transported to Euclidean space En . Namely, the angle

spanned by three points of En is deļ¬ned to be the corresponding angle in Rn under

a choice of coordinates. The angle is independent of this choice, because the inner

product in Rn is determined by the quadratic form (Proposition B.1), and so ultimately

6 EUCLIDEAN GEOMETRY

R

Q P

Figure 1.5 Angle with direction.

T Ī“

Ī“

Figure 1.6 Rigid body motion.

by the metric of En . In other words, the notion of angle is intrinsic to the geometry

of En .

There is one ļ¬nal issue to discuss regarding angles that is speciļ¬c to the Euclidean

plane E2 . Namely, once I ļ¬x a speciļ¬c coordinate system in E2 , angles ā P Q R acquire

a direction as well as a size, once we agree (as we usually do) that an anticlockwise

angle counts as positive, and a clockwise angle as negative. In Figure 1.5,

ā P Q R = ā’ā R Q P = Īø.

Under this convention, angles lie between ā’Ļ and Ļ . Of course formula (8) does not

reveal the sign as cos Īø = cos(ā’Īø). It is important to realise that the direction of the

angle is not intrinsic to E2 , since a different choice of coordinates may reverse the sign.

1.6 Motions

A motion T : En ā’ En is a transformation that preserves distances; that is, T is

bijective, and

d(T (P), T (Q)) = d(P, Q) for all P, Q ā En .

The word motion is short for rigid body motion; it is alternatively called isometry or

congruence. To say that T preserves distances means that there is ā˜no squashing or

bendingā™, hence the term rigid body motion; see Figure 1.6.

I study motions in terms of coordinates. After a choice of coordinates En ā’ Rn , a

motion T gives rise to a map T : Rn ā’ Rn , its coordinate expression, which satisļ¬es

|T (x) ā’ T (y)| = |x ā’ y| for all x, y ā Rn .

1.8 A MOTION IS AFFINE LINEAR ON LINES 7

The ļ¬rst thing I set out to do is to get from the abstract ā˜preserves distanceā™ deļ¬nition of

a motion to the concrete coordinate expression T (x) = Ax + b with A an orthogonal

matrix. In the case of the Euclidean plane E2 , the result is even more concrete; A is

either a rotation matrix or a reļ¬‚ection matrix:

cos Īø ā’ sin Īø cos Īø sin Īø

.

or

sin Īø cos Īø sin Īø ā’ cos Īø

1.7 Motions and collinearity

A motion T : En ā’ En preserves collinearity of points, so it takes

Proposition

lines to lines.

P, Q, R ā E n are collinear if and only if, possibly after a permutation of

Proof

P, Q, R,

d(P, R) + d(R, Q) = d(P, Q).

But T preserves the distance function, so this happens if and only if, possibly after a

permutation,

d(T (P), T (R)) + d(T (R), T (Q)) = d(T (P), T (Q))

which is equivalent to T (P), T (Q), T (R) collinear. QED

The point is of course that, as we saw in 1.3, collinearity can be deļ¬ned

purely in terms of distance; since a motion T preserves distance, it preserves

collinearity.

1.8 A motion is affine linear on lines

If T : Rn ā’ Rn is a motion expressed in coordinates, then

Proposition

T ((1 ā’ Ī»)x + Ī»y) = (1 ā’ Ī»)T (x) + Ī»T (y)

for all x, y ā Rn and all Ī» ā R.

A calculation based on the same idea as the previous proof: let z =

Proof

(1 ā’ Ī»)x + Ī»y. If x = y there is nothing to prove; set d = |x ā’ y|. Assume ļ¬rst that

Ī» ā [0, 1], so that z ā [x, y]. Then, as in the previous proposition, T (z) ā [T (x), T (y)],

so T (z) = (1 ā’ Āµ)T (x) + ĀµT (y) for some Āµ. But |z ā’ x| = Ī»d, so T (z) is the point

at distance (1 ā’ Ī»)d from T (y) and Ī»d from T (x), that is, Āµ = Ī».

If Ī» < 0, say, then x ā [y, z] with x = (1 ā’ Ī» )y + Ī» z and the same argument gives

T (x) = (1 ā’ Ī» )T (y) + Ī» T (z), and you can derive the statement as an easy exercise.

(The point is to write Ī» as a function of Ī»; see Exercise 1.3.) QED

8 EUCLIDEAN GEOMETRY

1.9 Motions are affine transformations

A map T : En ā’ En is an afļ¬ne transformation if it is given in a co-

Definition

ordinate system by T (x) = Ax + b, where A = (ai j ) is an n Ć— n matrix with nonzero

determinant and b = (bi ) a vector; in more detail,

ļ£«ļ£¶ ļ£«ļ£¶ ļ£«ļ£¶

x1 x1 b1

n

ļ£¬.ļ£· ļ£¬.ļ£· ļ£¬.ļ£·

x = (xi ) ā’ y = ai j x j + bi , ļ£ . ļ£ø ā’ Aļ£ . ļ£ø + ļ£ . ļ£ø.

or (9)

. . .

j=1

xn xn bn

Let T : En ā’ En be any map. Equivalent conditions:

Proposition

T is given in some coordinate system by T (x) = Ax + b for A an n Ć— n matrix.

(1)

For all vectors x, y ā Rn and all Ī», Āµ ā R we have

(2)

T Ī»x + Āµy ā’ T (0) = Ī» T (x) ā’ T (0) + Āµ T (y) ā’ T (0) .

For all x, y ā Rn and all Ī» ā R

(3)

T (1 ā’ Ī»)x + Ī»y = (1 ā’ Ī»)T (x) + Ī»T (y).

that is, T is afļ¬ne linear when restricted to any line.

The point of the proposition is that condition (3) is a priori much

Discussion

weaker than the other two; it only requires that the map T is afļ¬ne when restricted

to lines. Note also that using the origin 0 in (2) seems to go against my expressed

wisdom that there is no distinguished origin in the geometry of En . However, recall

that any point P ā En can serve as origin after a suitable translation.

(1) =ā’ (2) is an easy exercise. (2) means exactly that if after performing

Proof

T we translate by minus the vector b = T (0) to take T (0) back to 0, then T becomes

a linear map of vector spaces. Thus (2) =ā’ (1) comes from the standard result of

linear algebra expressing a linear map as a matrix.

(3) is just the particular case Ī» + Āµ = 1 of (2). Thus the point of the proposition

is to prove (3) =ā’ (2).

Statement (2) concerns only the 2-dimensional vector subspace spanned by x, y ā

V . We use statement (3) on the two lines 0x and 0y (see Figure 1.9), to get

T (2Ī»x) = (1 ā’ 2Ī»)T (0) + 2Ī»T (x)

and

T (2Āµy) = (1 ā’ 2Āµ)T (0) + 2ĀµT (y).

Now apply (3) again to the line spanned by 2Ī»x and 2Āµy:

1.10 EUCLIDEAN MOTIONS AND ORTHOGONAL TRANSFORMATIONS 9

2Ī»x

Ī»x

Ī»x + y

x

0 y y 2y

Affine linear construction of Ī»x + Āµy.

Figure 1.9

1 1

T Ī»x + Āµy = T (2Ī»x) + T (2Āµy)

2 2

1 1

= (1 ā’ 2Ī»)T (0) + 2Ī»T (x) + (1 ā’ 2Āµ)T (0) + 2ĀµT (y)

2 2

= T (0) + Ī» T (x) ā’ T (0) + Āµ T (y) ā’ T (0) ,

as required. QED

Dividing by 2 here is just for the sake of an easy life: { 1 , 1 } is a conve-

Remark 22

nient solution of Ī» + Āµ = 1. The point is just that Ī»x + Āµy lies on a line containing

chosen points of 0x and 0y. The argument for (3) =ā’ (2) can be made to work

provided every line has ā„ 3 points, that is, over any ļ¬eld with > 2 elements.

A Euclidean motion T : En ā’ En is an afļ¬ne transformation, given

Corollary

in any choice of coordinates En ā’ Rn by T (x) = Ax + b.

This follows at once from Proposition 1.7, the implication (3) =ā’ (1) in the

previous proposition, and the fact that T is bijective, so the matrix A must be invertible.

1.10 Euclidean motions and orthogonal transformations

This section makes a brief use of the relationship between the standard quadratic

form |x|2 = xi2 on Rn and the associated inner product x Ā· y = xi yi . If this is

not familiar to you, I refer you once again to Appendix B for a general discussion.

Let A be an n Ć— n matrix and T : Rn ā’ Rn the map deļ¬ned by

Proposition

x ā’ Ax. Then the following are equivalent conditions:

T is a motion T : En ā’ En .

(1)

A preserves the quadratic form; that is, |Ax| = |x| for all x ā Rn .

(2)

A is an orthogonal matrix; that is, it satisļ¬es tA A = In .

(3)

10 EUCLIDEAN GEOMETRY

(1) =ā’ (2) is trivial. Conversely,

Proof

|Ax ā’ Ay|2 = |A(x ā’ y)|2 = |x ā’ y|2 ,

where the ļ¬rst equality is linearity, and the second follows from (2). Thus T preserves

length, so it is a motion. (2) āā’ (3) is proved in Proposition B.4, where you can

also read more about orthogonal matrixes if you wish to. QED

Together with Corollary 1.7, this proves the following very important statement:

A Euclidean motion T : En ā’ En is expressed in coordinates as

Corollary

T (x) = Ax + b

with A an orthogonal matrix, and b ā Rn a vector.

An immediate check shows that an orthogonal matrix A has determinant det A =

Ā±1 (see Lemma B.4).

Let T : En ā’ En be a motion expressed in coordinates as T (x) =

Definition

Ax + b. I call T direct (or orientation preserving) if det A = 1 and opposite (or

orientation reversing) if det A = ā’1.

The meaning of this notion in E2 and E3 is familiar in terms of leftā“right orientation,

and it may seem pretty intuitive that it does not depend on the choice of coordinates.

However, I leave the proof to Exercise 6.8.

1.11 Normal form of an orthogonal matrix

The point of this section is to express an orthogonal map Ī± : Rn ā’ Rn in a simple form

in a suitable orthonormal basis of Rn . This section may seem an obscure digression

into linear algebra, but the result is central to understanding motions of Euclidean

space.

As a prelude to an attack on the general problem, consider the instructive case n = 2.

1.11.1

The 2 Ć— 2 The conditions for a 2 Ć— 2 matrix A = a d to be orthogonal are:

b

c

rotation and ļ£±

ļ£“ a 2 + c2 = 1

ļ£²

reflection

ac ab 10

A A = 1 āā’ = āā’ ab + cd = 0

t

matrixes

ļ£“2

bd cd 01 ļ£³

b + d 2 = 1.

Now (a, c) ā R2 is a point of the unit circle, so I can write a = cos Īø, c = sin Īø

for some Īø ā [0, 2Ļ) (Figure 1.11a). Then there are just two possibilities for b, d,

giving

cos Īø ā’ sin Īø cos Īø sin Īø

A= .

or

sin Īø cos Īø sin Īø ā’ cos Īø

1.11 NORMAL FORM OF AN ORTHOGONAL MATRIX 11

(0, 1)

(cos Īø, sin Īø)

(ā’ sin Īø, cos Īø)

Īø

Īø

(1, 0)

Figure 1.11a A rotation in coordinates.

Ī“

Ī“

Refl(L)

L

Rot(O, Īø)

Īø/2

y=0

Īø/2

Īø Ī“

O

Ī“

Ī“

Figure 1.11b The rotation and the reflection.

The ļ¬rst of these corresponds to a direct motion (because det A = 1), and you

recognise it as a rotation around the origin through Īø. In fact it takes

cos Īø ā’ sin Īø

1 0

ā’ ā’ .

and

sin Īø cos Īø

0 1

The second matrix gives an opposite motion (det A = ā’1), and you can understand

it in several ways; for example, write

cos Īø sin Īø cos Īø ā’ sin Īø 1 0

A= = .

sin Īø ā’ cos Īø sin Īø cos Īø ā’1

0

This says: ļ¬rst reļ¬‚ect in the x-axis, then rotate through Īø. It is easy to see geo-

metrically that this is the reļ¬‚ection in the line L through the origin 0 at angle Īø/2

to the x-axis. Indeed, every point on L is ļ¬xed, and the line perpendicular to L is

reversed, as in Figure 1.11b.

In coordinates, this says that f1 = (cos(Īø/2), sin(Īø/2)) is an eigenvector of A with

eigenvalue 1, and f2 = (sin(Īø/2), ā’ cos(Īø/2)) an eigenvector with eigenvalue ā’1.

The pair (f1 , f2 ) gives a vector space basis of R2 , and in this new basis the map

is given by the matrix 1 ā’1 . You can readily check these statements by matrix

0

0

multiplication and the rules of trig, but the geometric argument is simpler and more

convincing.

12 EUCLIDEAN GEOMETRY

1.11.2 The In the general case I control orthogonal matrixes using a slightly more involved

general case argument.

Let Ī± : Rn ā’ Rn be a linear map

Theorem (Normal form of orthogonal matrix)

given by an orthogonal matrix A. Then there exists an orthonormal basis of Rn in

which the matrix of Ī± is

ļ£« ļ£¶

Ik +

ļ£¬ ļ£·

ļ£¬ ļ£·

ā’Ik ā’

ļ£¬ ļ£·

cos Īøi ā’ sin Īøi

ļ£¬ ļ£·

B=ļ£¬ Bi = .

ļ£· where

B1

ļ£¬ ļ£· sin Īøi cos Īøi

ļ£¬ ļ£·

..

ļ£ ļ£ø

.

Bl

Here k + + k ā’ + 2l = n, and Ik Ā± is the k Ā± Ć— k Ā± identity matrix.

cos Īø ā’ sin Īø

has two special cases Īø = 0 (giving

The rotation matrix

Discussion sin Īø cos Īø

the identity) and Īø = Ļ:

ā’1 cos Ļ ā’ sin Ļ

0

= 180ā—¦ rotation.

=

ā’1 sin Ļ cos Ļ

0

These trivial cases introduce a minor ambiguity in the normal form. The most natural

convention seems to be to disallow Īø = 0, thus taking k + as big as possible, but to

use Īø = Ļ wherever possible, so that k ā’ = 0 or 1.

In sketch form, this holds because A is orthogonal, so its eigenvalues

Proof

have absolute value 1. Therefore they are either Ā±1, or come in complex conjugate

pairs {Ī», Ī»} = exp(Ā±iĪø); after this, it is enough simply to build up a basis of Rn

consisting either of real eigenvectors of A, or of real and imaginary parts of complex

eigenvectors.

Now I say the same thing again in more detail in 5 steps; the sketch proof just

given already reveals that complex numbers are closely involved, so I may as well

extend the action of A to the complex vector space Cn , which I can do without any

problems.

If Ī» is a real eigenvalue of A then Ī» = Ā±1, because

Step 1

Ax = Ī»x and A orthogonal =ā’ |x|2 = |Ax|2 = Ī»2 |x|2 .

If Ī» is a complex eigenvalue of A then |Ī»| = 1 and Ī» = Ī»ā’1 is also an

Step 2

eigenvalue (the bar denotes complex conjugate). Indeed, given 0 = z ā Cn such that

Az = Ī»z (recall I write z = t(z 1 , . . . , z n ) a column vector), write z = t(z 1 , . . . , z n ).

1.11 NORMAL FORM OF AN ORTHOGONAL MATRIX 13

Because A is a real matrix,

Az = Az = Ī»z = Ī»z.

Now write z i = xi + iyi , so that t zz = |z i |2 = (xi2 + yi2 ) > 0. Using the fact

that A is orthogonal,

Ī»Ī»t zz = t(Az)Az = t z t A Az = t zz, Ī»Ī» = 1.

and thus

If Ī» = cos Īø + i sin Īø is a complex eigenvalue of A (with Īø = 0, Ļ ) and

Step 3

z = x + iy ā Cn a complex eigenvector then taking real and imaginary parts in the

equality A(x + iy) = Az = Ī»z = (cos Īø + i sin Īø)(x + iy) gives

Ax = cos Īøx ā’ sin Īøy, Ay = sin Īøx + cos Īøy. (10)

Now I claim that |x|2 = |y|2 and x Ā· y = 0, so that scaling makes x, y ā Rn into a pair

of orthonormal vectors. This is an exercise for the reader. [Hint: write out the condition

for (10) (with Īø = 0, Ļ ) to preserve |x|2 , |y|2 and x Ā· y. See Exercises 1.5ā“1.6.]

If Ī± preserves a subspace W of Rn , then it preserves its orthogonal com-

Step 4

plement under the inner product (compare B.3)

W ā„ = x ā Rn x Ā· w = 0 for all w ā W .

In symbols,

Ī±(W ) = W =ā’ Ī±(W ā„ ) = W ā„ .

This is obvious from the deļ¬nition of W ā„ . Look at Figure 1.15b for an example: if a

motion preserves the horizontal plane W and its translates, then it will also preserve

the orthogonal complement W ā„ , the vertical lines.

Eigenvalues of A come from the polynomial equa-

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