Geometry

603 Pages·2012·11.89 MB·English

by  David A. Brannan

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This richly illustrated and clearly written undergraduate textbook captures a
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the excitement and beauty of geometry. The approach is that of Klein n
in his Erlangen programme: a geometry is a space together with a set of a
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transformations of the space. The authors explore various geometries: ,
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affi ne, projective, inversive, hyperbolic and elliptic. In each case they s
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carefully explain the key results and discuss the relationships between the l
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geometries. n
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New features in this Second Edition include concise end-of-chapter G
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summaries to aid student revision, a list of Further Reading and a list a
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of Special Symbols. The authors have also revised many of the end-of-
chapter exercises to make them more challenging and to include some
interesting new results. Full solutions to the 200 problems are included
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in the text, while complete solutions to all of the end-of-chapter exercises E
are available in a new Instructors’ Manual, which can be downloaded from C
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www.cambridge.org/9781107647831.
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Praise for the First Edition
D
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‘To my mind, this is the best introductory book ever written on Te
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introductory university geometry… Not only are students introduced to a O
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wide range of algebraic methods, but they will encounter a most pleasing N
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combination of process and product.’
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P. N. RUANE, MAA Focus
Geometry
‘… an excellent and precisely written textbook that should be studied in
depth by all would-be mathematicians.’
HANS SACHS, American Mathematical Society
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DAVID A. BRANNAN
MATTHEW F. ESPLEN
JEREMY J. GRAY
BRANNAN: GEOMETRY CVR CMYBLK
Geometry
SECOND EDITION

Geometry
SECOND EDITION
DAVID A. BRANNAN
MATTHEW F. ESPLEN
JEREMY J. GRAY
TheOpenUniversity
CAMBRIDGEUNIVERSITYPRESS
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Informationonthistitle:www.cambridge.org/9781107647831
©TheOpenUniversity1999,2012
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished1999
Secondedition2012
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Brannan,D.A.
Geometry/DavidA.Brannan,MatthewF.Esplen,JeremyJ.Gray.–2nded.
p.cm.
ISBN978-1-107-64783-1(Paperback)
1. Geometry. I. Esplen,MatthewF. II. Gray,Jeremy,1947– III. Title.
QA445.B6882011
516–dc23
2011030683
ISBN978-1-107-64783-1Paperback
Additionalresourcesforthispublicationatwww.cambridge.org/9781107647831
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2.1
InmemoryofWilsonStothers

Contents
Preface pagexi
0 Introduction:GeometryandGeometries 1
1 Conics 5
1.1 ConicSectionsandConics 6
1.2 PropertiesofConics 23
1.3 RecognizingConics 36
1.4 QuadricSurfaces 42
1.5 Exercises 52
SummaryofChapter1 55
2 AffineGeometry 61
2.1 GeometryandTransformations 62
2.2 AffineTransformationsandParallelProjections 70
2.3 PropertiesofAffineTransformations 84
2.4 UsingtheFundamentalTheoremofAffineGeometry 93
2.5 AffineTransformationsandConics 108
2.6 Exercises 117
SummaryofChapter2 121
3 ProjectiveGeometry:Lines 127
3.1 Perspective 128
3.2 TheProjectivePlaneRP2 137
3.3 ProjectiveTransformations 151
3.4 UsingtheFundamentalTheoremofProjectiveGeometry 172
3.5 Cross-Ratio 179
3.6 Exercises 192
SummaryofChapter3 195
4 ProjectiveGeometry:Conics 201
4.1 ProjectiveConics 202
4.2 Tangents 216
4.3 Theorems 229
vii
viii Contents
4.4 ApplyingLinearAlgebratoProjectiveConics 248
4.5 DualityandProjectiveConics 250
4.6 Exercises 252
SummaryofChapter4 255
5 InversiveGeometry 261
5.1 Inversion 262
5.2 ExtendingthePlane 276
5.3 InversiveGeometry 295
5.4 FundamentalTheoremofInversiveGeometry 310
5.5 CoaxalFamiliesofCircles 317
5.6 Exercises 331
SummaryofChapter5 335
6 HyperbolicGeometry:thePoincare´ Model 343
6.1 HyperbolicGeometry:theDiscModel 345
6.2 HyperbolicTransformations 356
6.3 DistanceinHyperbolicGeometry 367
6.4 GeometricalTheorems 383
6.5 Area 401
6.6 HyperbolicGeometry:theHalf-PlaneModel 412
6.7 Exercises 413
SummaryofChapter6 417
7 EllipticGeometry:theSphericalModel 424
7.1 SphericalSpace 425
7.2 SphericalTransformations 429
7.3 SphericalTrigonometry 438
7.4 SphericalGeometryandtheExtendedComplexPlane 450
7.5 PlanarMaps 460
7.6 Exercises 464
SummaryofChapter7 465
8 TheKleinianViewofGeometry 470
8.1 AffineGeometry 470
8.2 ProjectiveReflections 475
8.3 HyperbolicGeometryandProjectiveGeometry 477
8.4 EllipticGeometry:theSphericalModel 482
8.5 EuclideanGeometry 484
SummaryofChapter8 486
SpecialSymbols 488
FurtherReading 490
Appendix1:APrimerofGroupTheory 492
Contents ix
Appendix2:APrimerofVectorsandVectorSpaces 495
Appendix3:SolutionstotheProblems 503
Chapter1 503
Chapter2 517
Chapter3 526
Chapter4 539
Chapter5 549
Chapter6 563
Chapter7 574
Index 583