Table Of Content

B r This richly illustrated and clearly written undergraduate textbook captures a n 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 n transformations of the space. The authors explore various geometries: , E affi ne, projective, inversive, hyperbolic and elliptic. In each case they s p carefully explain the key results and discuss the relationships between the l e geometries. n , New features in this Second Edition include concise end-of-chapter G r summaries to aid student revision, a list of Further Reading and a list a y 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 SG 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 e O www.cambridge.org/9781107647831. No D m E Praise for the First Edition D I ‘To my mind, this is the best introductory book ever written on Te I introductory university geometry… Not only are students introduced to a O t wide range of algebraic methods, but they will encounter a most pleasing N r combination of process and product.’ y 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 ‘‘IItt ccoonnvveeyyss tthhee bbeeaauuttyy aanndd eexxcciitteemmeenntt ooff tthhee ssuubbjjeecctt,, aavvooiiddiinngg tthhee ddrryynneessss ooff mmaannyy ggeeoommeettrryy tteexxttss..’’ JJ.. II.. HHAALLLL,, MMiicchhiiggaann SSttaattee UUnniivveerrssiittyy SECOND EDITION 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 Cambridge,NewYork,Melbourne,Madrid,CapeTown, Singapore,SãoPaulo,Delhi,Tokyo,MexicoCity CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org 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 CambridgeUniversityPresshasnoresponsibilityforthepersistenceor accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto inthispublication,anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. 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