David a brannan, matthew f esplen, jeremy j gray geometry cambridge university press (2012)

This richly illustrated and clearly written undergraduate textbook captures Brannan, Esplen, Gray
the excitement and beauty of geometry. The approach is that of Klein
in his Erlangen programme: a geometry is a space together with a set of Geometry
transformations of the space. The authors explore various geometries:
affine, projective, inversive, hyperbolic and elliptic. In each case they SECOND EDITION
carefully explain the key results and discuss the relationships between the
geometries. Geometry

New features in this Second Edition include concise end-of-chapter SECOND EDITION
summaries to aid student revision, a list of Further Reading and a list
of Special Symbols. The authors have also revised many of the end-of- DAV I D A . BRA N NA N
chapter exercises to make them more challenging and to include some MATTHEW F. ESPLEN
interesting new results. Full solutions to the 200 problems are included J ER EMY J. GRAY
in the text, while complete solutions to all of the end-of-chapter exercises
are available in a new Instructors’ Manual, which can be downloaded from
www.cambridge.org/9781107647831.

Praise for the First Edition

‘To my mind, this is the best introductory book ever written on
introductory university geometry… Not only are students introduced to a
wide range of algebraic methods, but they will encounter a most pleasing
combination of process and product.’

P. N. RUAN E , MAA Focus

‘… an excellent and precisely written textbook that should be studied in
depth by all would-be mathematicians.’

HANS SAC HS, American Mathematical Society

‘It conveys the beauty and excitement of the subject, avoiding the dryness
of many geometry texts.’

J. I. HA LL, Michigan State University

Geometry

SECOND EDITION


Geometry

SECOND EDITION

DAV I D A . B R A N N A N
M AT T H E W F. E S P L E N
J E R E M Y J . G R AY
The Open University

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town,
Singapore, São Paulo, Delhi, Tokyo, Mexico City

Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

Information on this title: www.cambridge.org/9781107647831

© The Open University 1999, 2012

This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.

First published 1999
Second edition 2012

Printed in the United Kingdom at the University Press, Cambridge

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data

Brannan, D. A.
Geometry / David A. Brannan, Matthew F. Esplen, Jeremy J. Gray. – 2nd ed.
p. cm.
ISBN 978-1-107-64783-1 (Paperback)
1. Geometry. I. Esplen, Matthew F. II. Gray, Jeremy, 1947– III. Title.

QA445.B688 2011
516–dc23

2011030683

ISBN 978-1-107-64783-1 Paperback

Additional resources for this publication at www.cambridge.org/9781107647831

Cambridge University Press has no responsibility for the persistence or
accuracy of URLs for external or third-party internet websites referred to
in this publication, and does not guarantee that any content on such
websites is, or will remain, accurate or appropriate.

2.1

In memory of Wilson Stothers


Contents

Preface page xi

0 Introduction: Geometry and Geometries 1

1 Conics 5

1.1 Conic Sections and Conics 6

1.2 Properties of Conics 23

1.3 Recognizing Conics 36

1.4 Quadric Surfaces 42

1.5 Exercises 52

Summary of Chapter 1 55

2 Affine Geometry 61

2.1 Geometry and Transformations 62

2.2 Affine Transformations and Parallel Projections 70

2.3 Properties of Affine Transformations 84

2.4 Using the Fundamental Theorem of Affine Geometry 93

2.5 Affine Transformations and Conics 108

2.6 Exercises 117

Summary of Chapter 2 121

3 Projective Geometry: Lines 127

3.1 Perspective 128

3.2 The Projective Plane RP2 137

3.3 Projective Transformations 151

3.4 Using the Fundamental Theorem of Projective Geometry 172

3.5 Cross-Ratio 179

3.6 Exercises 192

Summary of Chapter 3 195

4 Projective Geometry: Conics 201

4.1 Projective Conics 202

4.2 Tangents 216

4.3 Theorems 229

vii

viii Contents

4.4 Applying Linear Algebra to Projective Conics 248

4.5 Duality and Projective Conics 250

4.6 Exercises 252

Summary of Chapter 4 255

5 Inversive Geometry 261

5.1 Inversion 262

5.2 Extending the Plane 276

5.3 Inversive Geometry 295

5.4 Fundamental Theorem of Inversive Geometry 310

5.5 Coaxal Families of Circles 317

5.6 Exercises 331

Summary of Chapter 5 335

6 Hyperbolic Geometry: the Poincare´ Model 343

6.1 Hyperbolic Geometry: the Disc Model 345

6.2 Hyperbolic Transformations 356

6.3 Distance in Hyperbolic Geometry 367

6.4 Geometrical Theorems 383

6.5 Area 401

6.6 Hyperbolic Geometry: the Half-Plane Model 412

6.7 Exercises 413

Summary of Chapter 6 417

7 Elliptic Geometry: the Spherical Model 424

7.1 Spherical Space 425

7.2 Spherical Transformations 429

7.3 Spherical Trigonometry 438

7.4 Spherical Geometry and the Extended Complex Plane 450

7.5 Planar Maps 460

7.6 Exercises 464

Summary of Chapter 7 465

8 The Kleinian View of Geometry 470

8.1 Affine Geometry 470

8.2 Projective Reflections 475

8.3 Hyperbolic Geometry and Projective Geometry 477

8.4 Elliptic Geometry: the Spherical Model 482

8.5 Euclidean Geometry 484

Summary of Chapter 8 486

Special Symbols 488

Further Reading 490

Appendix 1: A Primer of Group Theory 492

Contents ix

Appendix 2: A Primer of Vectors and Vector Spaces 495

Appendix 3: Solutions to the Problems 503

Chapter 1 503

Chapter 2 517

Chapter 3 526

Chapter 4 539

Chapter 5 549

Chapter 6 563

Chapter 7 574

Index 583


Preface

Geometry! For over two thousand years it was one of the criteria for recog- Plato (c. 427–347 BC)
nition as an educated person to be acquainted with the subject of geometry. was an Athenian
Euclidean geometry, of course. philosopher who
established a school of
In the golden era of Greek civilization around 400 BC, geometry was studied theoretical research (with
rigorously and put on a firm theoretical basis – for intellectual satisfaction, the a mathematical bias),
intrinsic beauty of many geometrical results, and the utility of the subject. legislation and
For example, it was written above the door of Plato’s Academy ‘Let no-one government.
ignorant of Geometry enter here!’ Indeed, Archimedes is said to have used the
reflection properties of a parabola to focus sunlight on the sails of the Roman Archimedes (c. 287–212
fleet besieging Syracuse and set them on flame. BC) was a Greek
geometer and physicist
For two millennia the children of those families sufficiently well-off to be who used many of the
educated were compelled to have their minds trained in the noble art of rigor- basic limiting ideas of
ous mathematical thinking by the careful study of translations of the work of differential and integral
Euclid. This involved grasping the notions of axioms and postulates, the draw- calculus.
ing of suitable construction lines, and the careful deduction of the necessary Euclid (c. 325–265 BC)
results from the given facts and the Euclidean axioms – generally in two- was a mathematician in
dimensional or three-dimensional Euclidean space (which we shall denote by Hellenistic Alexandria
R2 and R3, respectively). Indeed, in the 1700s and 1800s popular publications during the reign of
such as The Lady’s and Gentleman’s Diary published geometric problems Ptolemy I (323–283 BC),
for the consideration of gentlefolk at their leisure. And as late as the 1950s famous for his book The
translations of Euclid’s Elements were being used as standard school geometry Elements.
textbooks in many countries. We give a careful
algebraic definition of R2
Just as nowadays, not everyone enjoyed Mathematics! For instance, the and R3 in Appendix 2.
German poet and philosopher Goethe wrote that ‘Mathematicians are like
Frenchmen: whatever you say to them, they translate into their own language, Johann Wolfgang von
and forthwith it is something entirely different!’ Goethe (1749–1832) is
said to have studied all

The Golden Era of geometry came to an end rather abruptly. When the areas of science of his day
USSR launched the Sputnik satellite in 1957, the Western World suddenly except mathematics – for
decided for political and military reasons to give increased priority to its which he had no aptitude.
research and educational efforts in science and mathematics, and redeveloped
the curricula in these subjects. In order to make space for subjects newly
developed or perceived as more ‘relevant in the modern age’, the amount of
geometry taught in schools and universities plummeted. Interest in geometry
languished: it was thought ‘old-fashioned’ by the fashionable majority.

xi

xii Preface

Nowadays it is being realized that geometry is still a subject of abiding Topics in computer
beauty that provides tremendous intellectual satisfaction in return for effort put graphics such as ‘hidden’
into its study, and plays a key underlying role in the understanding, develop- surfaces and the shading
ment and applications of many other branches of mathematics. More and more of curved surfaces involve
universities are reintroducing courses in geometry, to give students a ‘feel’ much mathematics.
for the reasons for studying various areas of mathematics (such as Topology),
to service the needs of Computer Graphics courses, and so on. Geometry is Chapter 0
having a revival! Chapter 1
Chapter 2
Since 1971, the Open University in the United Kingdom has taught math- Chapters 3 and 4
ematics to students via specially written correspondence texts, and has tradi- Chapter 5
tionally given geometry a central position in its courses. This book arises from Chapter 6
those correspondence texts.
Chapter 7
We adopt the Klein approach to geometry. That is, we regard the various Chapter 8
geometries as each consisting of an underlying set together with a group of
transformations acting on that set. Those properties of the set that are not

altered by any of the transformations are called the properties of that geometry.

Following a historical review of the development of the various geometries,
we look at conics (and at the related quadric surfaces) in Euclidean geometry.
Then we address a whole series of different geometries in turn. First, affine
geometry (that provides simple proofs of some results in Euclidean geome-
try). Then projective geometry, which can be regarded as the most basic of
all geometries; we divide this material into a chapter on projective lines and
a chapter on projective conics. We then return to study inversive geometry,
which provides beautiful proofs of many results involving lines and circles in
Euclidean geometry. This leads naturally to the study of hyperbolic geometry
in the unit disc, in which there are two lines through any given point that are
parallel to a given line. Via the link of stereographic projection, this leads on
to spherical geometry: a natural enough concept for a human race that lives
on the surface of a sphere! Finally we tie things together, explaining how the
various geometries are inter-related.

Study Guide Appendices 1 and 2.

The book assumes a basic knowledge of Group Theory and of Linear Alge-
bra, as these are used throughout. However, for completeness and students’
convenience we give a very rapid review of both topics in the appendices.

The book follows many of the standard teaching styles of The Open Uni-
versity. Thus, most chapters are divided into five sections (each often further
divided into subsections); sections are numbered using two digits (such as
‘Section 3.2’) and subsections using three digits (such as ‘Subsection 3.2.4’).
Generally a section is considered to be about one evening’s hard work for an
average student.

We number in order the theorems, examples, problems and equations within
each section.

We use wide pages with margins in which we place various historical notes,
cross-references, teaching comments and diagrams; the cross-references need

Preface xiii

not be consulted by students unless they wish to remind themselves of some
point on that topic, but the other margin notes should be read carefully. We use
boxes in the main text to highlight definitions, strategies, and the statements of
theorems and other key results. The end of the proof of a theorem is indicated
by a solid symbol ‘ ’, and the end of the solution of a worked example by a
hollow symbol ‘ ’. Occasionally the text includes a set of ‘Remarks’; these are
comments of the type that an instructor would give orally to a class, to clarify
a definition, result, or whatever, and should be read carefully. There are many
worked examples within the text to explain the concepts being taught, and it is
important that students read these carefully as they contain many key teaching
points; in addition, there is a good stock of in-text problems to reinforce the
teaching, and solutions to these are given in Appendix 3. At the end of each
chapter there are exercises covering the material of that chapter, some of which
are fairly straight-forward and some are more challenging; solutions are not
given to the exercises.

Our philosophy is to provide clear and complete explanations of all geomet-
ric facts, and to teach these in such a way that students can understand them
without much external help. As a result, students should be able to learn (and,
we hope, to enjoy) the key concepts of the subject in an uncluttered way.

Most students will have met many parts of Chapter 1 already, and so can

proceed fairly quickly through it. Thereafter it is possible to tackle Chapters 2
to 4 or Chapters 5 and 6, in either order. It is possible to omit Chapters 7 or 8,
if the time in a course runs short.

Notation for Functions as Mappings Note that we use two
different arrows here, to
Suppose that a function f maps some set A into some set B, and that it maps distinguish between the
a typical point x of A onto some image point y of B. Then we say that A is the mapping of a set and the
domain (or domain of definition) of f , B the codomain of f , and denote the mapping of an element.
function f as a mapping (or map) as follows:

f :A→B
x→y

We often denote y by the expression f (x) to indicate its dependence on f
and x.

Acknowledgements

This material has been critically read by, or contributed to in some way,
by many colleagues in The Open University and the BBC/OU TV Produc-
tion Centre in Milton Keynes, including Andrew Adamyk, Alison Cadle,
Anne-Marie Gallen, Ian Harrison, John Hodgson, Roy Knight, Alan Pears,
Alan Slomson, Wilson Stothers and Robin Wilson. Its appearance in book
form owes a great deal to the work of Toni Cokayne, Pat Jeal and the OU
Mathematics and Computing Faculty’s Course Materials Production Unit.

xiv Preface

Without the assistance and the forbearance of our families, the writing of

the original OU course and its later rewriting in this form would have been
impossible. It was Michael Brannan’s idea to produce it as a book.

Changes in the Second Edition Solutions to the exercises
appear in an Instructors’
In addition to correcting typos and errors, the authors have changed the term Manual available from the
‘gradient’ to ‘slope’, and avoided the use of ‘reversed square brackets’ — so publisher.
that, for instance, the interval {x : 0 < x ≤ 1} is now written as (0,1] rather
than ]0,1]. Also, they have clarified the difference between a geometry and
models of that geometry; in particular, the term ‘non-Euclidean’ geometry has
now been largely replaced by ‘hyperbolic’ geometry, and the term ‘elliptic’
geometry has been introduced where appropriate. The problems and exercises
have been revised somewhat, and more exercises included. Each chapter now
includes a summary of the material in that chapter, and before the appendices
there are now lists of symbols and suggestions for further reading.

The authors have taken the opportunity to add some new material to enrich
the reader’s diet: a treatment of conics as envelopes of tangent families,
barycentric coordinates, Poncelet’s Porism and Ptolemy’s Theorem, and planar
maps. Also, the treatment of a number of existing topics has been significantly
changed: the geometric interpretation of projective transformations, the anal-
ysis of the formula for hyperbolic distance, and the treatment of asymptotic
d -triangles.

The authors appreciate the warm reception of the first edition, and have tried
to take on board as many as possible of the helpful comments received. Special
thanks are due to John Snygg and Jonathan I. Hall for invaluable comments and
advice.

Instructors’ Manual

Complete solutions to all of the end-of-chapter exercises are available in an
Instructors’ Manual, which can be downloaded from www.cambridge.org/
9781107647831.

0 Introduction: Geometry
and Geometries

Geometry is the study of shape. It takes its name from the Greek belief that The word comes from the
geometry began with Egyptian surveyors of two or three millennia ago mea- Greek words geo (Earth)
suring the Earth, or at least the fertile expanse of it that was annually flooded and metria (measuring).
by the Nile.
Isaac Newton
It rapidly became more ambitious. Classical Greek geometry, called (1643–1727) was an
Euclidean geometry after Euclid, who organized an extensive collection of English astronomer,
theorems into his definitive text The Elements, was regarded by all in the early physicist and
modern world as the true geometry of space. Isaac Newton used it to formu- mathematician. He was
late his Principia Mathematica (1687), the book that first set out the theory Professor of Mathematics
of gravity. Until the mid-19th Century, Euclidean geometry was regarded as at Cambridge, Master of
one of the highest points of rational thought, as a foundation for practical the Royal Mint, and
mathematics as well as advanced science, and as a logical system splendidly successor of Samuel
adapted for the training of the mind. We shall see in this book that by the 1850s Pepys as President of the
geometry had evolved considerably – indeed, whole new geometries had been Royal Society.
discovered.
Apollonius of Perga
The idea of using coordinates in geometry can be traced back to Apollo- (c. 255–170 BC) was a
nius’s treatment of conic sections, written a generation after Euclid. But their Greek geometer, whose
use in a systematic way with a view to simplifying the treatment of geome- only surviving work is a
try is really due to Fermat and Descartes. Fermat showed how to obtain an text on conics.
equation in two variables to describe a conic or a straight line in 1636, but his

work was only published posthumously in 1679. Meanwhile in 1637 Descartes Pierre de Fermat
published his book Discourse on Method, with an extensive appendix enti- (1601–1665) was a French
tled La Géometrie, in which he showed how to introduce coordinates to solve lawyer and amateur
a wide variety of geometrical problems; this idea has become so central a mathematician, who
part of mathematics that whole sections of La Géometrie read like a modern claimed to have a proof of
textbook. the recently proved
Fermat’s Last Theorem in
A contemporary of Descartes, Girard Desargues, was interested in the ideas Number Theory.
of perspective that had been developed over many centuries by artists (anx- René Descartes
ious to portray three-dimensional scenes in a realistic way on two-dimensional (1596–1650) was a French
walls or canvases). For instance, how do you draw a picture of a building, scientist, philosopher and
or a staircase, which your client can understand and commission, and from mathematician. He is also
which artisans can deduce the correct dimensions of each stone? Desargues known for the phrase
also realized that since any two conics can always be obtained as sections of ‘Cogito, ergo sum’ (I
the same cone in R3, it is possible to present the theory of conics in a unified think, therefore I am).

1

2 0: Introduction: Geometry and Geometries

way, using concepts which later mathematicians distilled into the notion of Girard Desargues
the cross-ratio of four points. Desargues’ discoveries came to be known as (1591–1661) was a French
projective geometry. architect.
We deal with these ideas
Blaise Pascal was the son of a mathematician, Étienne, who attended a group in Chapters 4 and 5.
of scholars frequented by Desargues. He heard of Desargues’s work from his
father, and quickly came up with one of the most famous results in the geom- Blaise Pascal
etry of conics, Pascal’s Theorem, which we discuss in Chapter 4. By the late (1623–1662) was a French
19th century projective geometry came to be seen as the most basic geometry, geometer, probabilist,
with Euclidean geometry as a significant but special case. physicist and philosopher.

At the start of the 19th century the world of mathematics began to change. Gaspard Monge
The French Revolution saw the creation of the École Polytechnique in Paris (1746–1818) was a French
in 1794, an entirely new kind of institution for the training of military engi- analyst and geometer. A
neers. It was staffed by mathematicians of the highest calibre, and run for strong republican and
many years by Gaspard Monge, an enthusiastic geometer who had invented a supporter of the
simple system of descriptive geometry for the design of forts and other mili- Revolution, he was French
tary sites. Monge was one of those rare teachers who get students to see what is Minister of the Navy in
going on, and he inspired a generation of French geometers. The École Poly- 1792–93, but deprived of
technique, moreover, was the sole entry-point for any one seeking a career all his honours on the
in engineering in France, and the stranglehold of the mathematicians ensured restoration of the French
that all students received a good, rigorous education in mathematics before monarchy.
entering the specialist engineering schools. Thus prepared they then assisted
Napoleon’s armies everywhere across Europe and into Egypt. Jean Victor Poncelet
(1788–1867) followed a
One of the École’s former students, Jean Victor Poncelet, was taken prisoner career as a military
in 1812 in Napoleon’s retreat from Moscow. He kept his spirits up during a engineer by becoming
terrible winter by reviewing what his old teacher, Monge, had taught him about Professor of Mechanics at
descriptive geometry. This is a system of projections of a solid onto a plane – Metz, where he worked on
or rather two projections, one vertically and one horizontally (giving what are the efficiency of turbines.
called to this day the plan and elevation of the solid). Poncelet realized that
instead of projecting ‘from infinity’ so to speak, one could adapt Monge’s ideas elevation vertical
to the study of projection from a point. In this way he re-discovered Desargues’ projection
ideas of projective geometry. During his imprisonment he wrote his famous
book Traité des propriétés projectives des figures outlining the foundations of horizontal
projective geometry, which he extensively rewrote after his release in 1814 and projection
published in 1822.
plan
Around the same time that projective geometry was emerging, mathemati-
cians began to realize that there was more to be said about circles than they 1 2

had previously thought. For instance, in the study of electrostatics let 1 and
2 be two infinitely long parallel cylinders of opposite charge. Then the inter-
section of the surfaces of equipotential with a vertical plane is two families of
circles (and a single line), and a point charge placed in the electrostatic field
moves along a circular path through a specific point inside each cylinder, at
right angles to circles in the families. The study of properties of such fami-
lies of circles gave rise to a new geometry, called inversive geometry, which
was able to provide particularly striking proofs of previously known results in
Euclidean geometry as well as new results.

0: Introduction: Geometry and Geometries 3

In inversive geometry mathematicians had to add a ‘point at infinity’ to the August Ferdinand Möbius
plane, and had to regard circles and straight lines as equivalent figures under (1790–1868) was a
the natural mappings, inversions, as these can turn circles into lines, and vice- German geometer,
versa. Analogously, in projective geometry mathematicians had to add a whole topologist, number
‘line at infinity’ in order to simplify the geometry, and found that there were theorist and astronomer;
projective transformations that turned hyperbolas into ellipses, and so on. So he discovered the famous
mathematicians began to move towards thinking of geometry as the study of Möbius Strip (or Band).
shapes and the transformations that preserve (at least specified properties of)
those shapes. For the surface of the
Earth is very nearly
For example, there are very few theorems in Euclidean geometry that depend spherical.
on the size of the figure. The ability to make scale copies without altering
‘anything important’ is basic to mathematical modelling and a familiar fact a c
of everyday life. If we wish to restrict our attention to the transformations
that preserve length, we deal with Euclidean geometry, whereas if we allow b
arbitrary changes of scale we deal with similarity geometry.
p
Another interesting geometry was discovered by Möbius in the 1820s, in

which transformations of the plane map lines to lines, parallel lines to parallel m l
lines, and preserve ratios of lengths along lines. He called this geometry affine
geometry because any two figures related by such a transformation have a like- Janos Bolyai (1802–1860)
ness or affinity to one another. This is the geometry appropriate, in a sense, to was an officer in the
Monge’s descriptive geometry, and the geometry that describes the shadows of Hungarian Army.
figures in sunlight.
Nicolai Ivanovich
Since the days of Greek mathematics, with a stimulus provided by the needs Lobachevskii
of commercial navigation, mathematicians had studied spherical geometry too; (1792–1856) was a
that is, the geometry of figures on the surface of a sphere. Here geometry Russian geometer who
is rather different from plane Euclidean geometry; for instance the area of became Rector of the
a triangle is proportional to the amount by which its angle sum exceeds π , University of Kazan.
and there is a nice generalization of Pythagoras’ Theorem, which says that
in a right-angled triangle with sides a, b and the hypotenuse c, then cos c =
cos a · cos b. It turns out that there is a close connection between spherical
geometry and inversive geometry.

For nearly two millennia mathematicians had accepted as obvious the
Parallel Postulate of Euclid: namely, that given any line and any point P
not on , there is a unique line m in the same plane as P and which passes
through P and does not meet . Indeed much effort had been put into deter-
mining whether this Postulate could be deduced from the other assumptions
of Euclidean geometry. In the 1820s two young and little-known mathemati-
cians, Bolyai in Hungary and Lobachevskii in Russia, showed that there
were perfectly good so-called ‘non-Euclidean geometries’, namely hyper-
bolic geometry and elliptic geometry, that share all the initial assumptions of
Euclidean geometry except the parallel postulate.

In hyperbolic geometry given any line and any point P not on , there are
infinitely many lines in the same plane as P and which pass through P and do

not meet ; in elliptic geometry all lines intersect each other. However, it still
makes sense in both hyperbolic and elliptic geometries to talk about the length

4 0: Introduction: Geometry and Geometries

of line segments, the distance between points, the angles between lines, and Jules Henri Poincaré
so forth. Around 1900 Poincaré did a great deal to popularise these geometries (1854–1912) was a
by demonstrating their applications in many surprising areas of mathematics, prolific French
such as Analysis. mathematician, physicist,
astronomer and
By 1870, the situation was that there were many geometries: Euclidean, philosopher at the
affine, projective, inversive, hyperbolic and elliptic geometries. One way math- University of Paris.
ematicians have of coping with the growth of their subject is to re-define it so
that different branches of it become branches of the same subject. This was Christian Felix Klein
done for geometry by Klein, who developed a programme (the Erlangen Pro- (1849–1925) was a
gramme) for classifying geometries. His elegant idea was to regard a geometry German algebraist,
as a space together with a group of transformations of that space; the proper- geometer, topologist and
ties of figures that are not altered by any transformation in the group are their physicist; he became a
geometrical properties. professor at the University
of Erlangen at the
For example, in two-dimensional Euclidean geometry the space is the plane remarkable age of 22.
and the group is the group of all length-preserving transformations of the plane
(or isometries). In projective geometry the space is the plane enlarged (in a way For example, you will
we make precise in Chapter 6) by a line of extra points, and the group is the meet two models of
group of all continuous transformations of the space that preserve cross-ratio. hyperbolic geometry.

Klein’s approach to a geometry involves three components: a set of points
(the space), a set of transformations (that specify the invariant properties – for
example, congruence in Euclidean geometry), and a group (that specifies how
the transformations may be composed). The transformations and their group

are the fundamental components of the geometry that may be applied to differ-
ent spaces. A model of a geometry is a set which possesses all the properties of
the geometry; two different models of any geometry will be isomorphic. There
may be several different models of a given geometry, which have different
advantages and disadvantages. Therefore, we shall use the terms ‘geometry’
and ‘model (of a geometry)’ interchangeably whenever we think that there is
no risk of confusion.

In fact as Klein was keen to stress, most geometries are examples of pro-
jective geometry with some extra conditions. For example, affine geometry
emerges as the geometry obtained from projective geometry by selecting a line
and considering only those transformations that map that line to itself; the line
can then be thought of as lying ‘at infinity’ and safely ignored. The result was
that Klein not only had a real insight into the nature of geometry, he could even
show that projective geometry was almost the most basic geometry.

This philosophy of geometry, called the Kleinian view of geometry, is the
one we have adopted in this book. We hope that you will enjoy this introduction
to the various geometries that it contains, and go on to further study of one of
the oldest, and yet most fertile, branches of mathematics.

1 Conics

The study of conics is well over 2000 years old, and has given rise to some of That is, they intersect at
the most beautiful and striking results in the whole of geometry. right angles.

In Section 1.1 we outline the Greek idea of a conic section – that is, a conic We use the notation R2
as defined by the curve in which a double cone is intersected by a plane. We and R3 to denote
then look at some properties of circles, the simplest of the non-degenerate 2-dimensional and
conics, such as the condition for two circles to be orthogonal and the equations 3-dimensional Euclidean

of the family of all circles through two given points. space, respectively.

We explain the focus–directrix definition of the parabola, ellipse and hyper-
bola, and study the focal-distance properties of the ellipse and hyperbola.
Finally, we use the so-called Dandelin spheres to show that the Greek conic
sections are just the same as the conics defined in terms of a focus and a
directrix.

In Section 1.2 we look at tangents to conics, and the reflection properties of
the parabola, ellipse and hyperbola. It turns out that these are useful in prac-
tical situations as diverse as anti-aircraft searchlights and astronomical optical
telescopes! We also see how we can construct each non-degenerate conic as
the ‘envelope’ of lines in a suitably-chosen family of lines.

The equations of conics are all second degree equations in x and y. In
Section 1.3 we show that the converse result holds – that is, that every sec-
ond degree equation in x and y represents a conic. We also find an algorithm
for determining from its equation in x and y which type of non-degenerate
conic a given second degree equation represents, and for finding its principal
features.

The analogue in R3 of a plane conic in R2 is a quadric surface, specified
by a suitable second degree equation in x, y and z. A well-known example
of a quadric surface is the cooling tower of an electricity generating station.
In Section 1.4 we find an algorithm for identifying from its equation which
type of non-degenerate quadric a given second degree equation in x, y and z
represents. We also discover that two of the non-degenerate quadric surfaces
can be generated by two different families of straight lines, and that this feature
is of practical importance.

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