Elementary Euclidean geometry. An introduction

191 Pages·2003·0.615 MB·English

by  Gibson C.G.

ElementaryEuclideanGeometry
AnIntroduction
This is a genuine introduction to the geometry of lines and conics in the Euclidean
plane. Lines and circles provide the starting point, with the classical invariants of
general conics introduced at an early stage, yielding a broad subdivision into types,
apreludetothecongruenceclassification.Arecurringthemeisthewayinwhichlines
intersectconics.Fromsinglelinesoneproceedstoparallelpencils,leadingtomidpoint
loci, axes and asymptotic directions. Likewise, intersections with general pencils of
linesleadtothecentralconceptsoftangent,normal,poleandpolar.
Thetreatmentisexample-basedandself-contained,assumingonlyabasicground-
inginlinearalgebra.Withnumerousillustrationsandseveralhundredworkedexamples
andexercises,thisbookisidealforusewithundergraduatecoursesinmathematics,or
forpostgraduatesinengineeringandthephysicalsciences.
C. G. GIBSON is a senior fellow in mathematical sciences at the University of
Liverpool.

Elementary Euclidean Geometry
An Introduction
C. G. GIBSON
cambridge university press
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Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
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Information on this title: www.cambridge.org/9780521834483
© Cambridge University Press 2003
This publication is in copyright. Subject to statutory exception and to the provision of
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without the written permission of Cambridge University Press.
First published in print format 2004
isbn-13 978-0-511-18491-8 eBook (NetLibrary)
isbn-10 0-511-18491-3
eBook (NetLibrary)
isbn-13 978-0-521-83448-3 hardback
isbn-10 0-521-83448-1 hardback
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Contents
ListofFigures pageviii
ListofTables x
Preface xi
Acknowledgements xvi
1 PointsandLines 1
1.1 TheVectorStructure 1
1.2 LinesandZeroSets 2
1.3 UniquenessofEquations 3
1.4 PracticalTechniques 4
1.5 ParametrizedLines 7
1.6 PencilsofLines 9
2 TheEuclideanPlane 12
2.1 TheScalarProduct 12
2.2 LengthandDistance 13
2.3 TheConceptofAngle 15
2.4 DistancefromaPointtoaLine 18
3 Circles 22
3.1 CirclesasConics 22
3.2 GeneralCircles 23
3.3 UniquenessofEquations 24
3.4 IntersectionswithLines 26
3.5 PencilsofCircles 27
4 GeneralConics 32
4.1 StandardConics 33
4.2 ParametrizingConics 35
4.3 MatricesandInvariants 37
v
vi Contents
4.4 IntersectionswithLines 39
4.5 TheComponentLemma 41
5 CentresofGeneralConics 44
5.1 TheConceptofaCentre 44
5.2 FindingCentres 45
5.3 GeometryofCentres 49
5.4 SingularPoints 51
6 DegenerateConics 54
6.1 BinaryQuadratics 54
6.2 ReducibleConics 56
6.3 PencilsofConics 59
6.4 PerpendicularBisectors 61
7 AxesandAsymptotes 65
7.1 MidpointLoci 65
7.2 Axes 68
7.3 BisectorsasAxes 72
7.4 AsymptoticDirections 74
8 FocusandDirectrix 76
8.1 FocalConstructions 76
8.2 PrinciplesforFindingConstructions 79
8.3 ConstructionsforParabolas 79
8.4 GeometricGeneralities 81
8.5 ConstructionsofEllipseandHyperbola 83
9 TangentsandNormals 88
9.1 TangentLines 88
9.2 ExamplesofTangents 89
9.3 NormalLines 94
10 TheParabola 98
10.1 TheAxisofaParabola 98
10.2 PracticalProcedures 99
10.3 ParametrizingParabolas 102
11 TheEllipse 105
11.1 AxesandVertices 105
11.2 RationalParametrization 107
11.3 FocalProperties 110
Contents vii
12 TheHyperbola 114
12.1 Asymptotes 114
12.2 ParametrizingHyperbolas 119
12.3 FocalPropertiesofHyperbolas 121
13 PoleandPolar 125
13.1 ThePolarsofaConic 125
13.2 TheJointTangentEquation 127
13.3 OrthopticLoci 132
14 Congruences 137
14.1 Congruences 138
14.2 CongruentLines 142
14.3 CongruentConics 144
14.4 TheInvarianceTheorem 146
15 ClassifyingConics 149
15.1 RotatingtheAxes 149
15.2 ListingNormalForms 151
15.3 SomeConsequences 154
15.4 EigenvaluesandAxes 155
16 DistinguishingConics 159
16.1 DistinguishingClasses 159
16.2 ConicSections 161
16.3 ConicswithinaClass 162
17 UniquenessandInvariance 167
17.1 ProofofUniqueness 167
17.2 ProofofInvariance 169
Index 171
Figures
1.1 Threewaysinwhichlinescanintersect page5
1.2 Parametrizationofaline 8
1.3 Pencilsoflines 10
2.1 Componentsofavector 14
2.2 Anglesbetweentwolines 17
2.3 Theperpendicularbisector 18
2.4 Projectionofapointontoaline 19
3.1 Howcirclesintersectlines 26
3.2 Threewaysinwhichcirclescanintersect 28
3.3 ThefamilyofcirclesinExample3.8 30
4.1 Astandardparabola 33
4.2 Astandardellipse 34
4.3 Astandardhyperbola 35
5.1 Theconceptofacentre 45
5.2 Atranslateofaconic 46
5.3 Auxiliarycirclesofellipse 48
6.1 Conictypesinapencil 60
6.2 Bisectorsoftwolines 62
6.3 Conesassociatedtoaline-pair 63
7.1 Amidpointlocusforanellipse 66
7.2 Aparallelpencilintersectingahyperbola 74
8.1 Constructionofthestandardparabola 77
8.2 Degenerate‘constructible’conics 78
8.3 Axisofaconstructibleconic 82
8.4 Constructionsofastandardellipse 82
8.5 Constructionsofastandardhyperbola 82
9.1 Ideaofatangent 89
9.2 Latusrectumofthestandardparabola 90
viii
ListofFigures ix
9.3 Anormallinetoanellipse 94
9.4 Evoluteofaparabola 96
10.1 Theconic Q ofExample10.1 99
10.2 Reflectivepropertyforaparabola 103
11.1 Rationalparametrizationofthecircle 108
11.2 Metricpropertyofanellipse 111
11.3 Thestringconstruction 112
11.4 Reflectivepropertyforanellipse 113
12.1 Alineinoneasymptoticcone 116
12.2 Ahyperbolaasagraph 118
12.3 Parametrizingarectangularhyperbola 120
12.4 Wideandnarrowhyperbolas 122
12.5 Metricpropertyofahyperbola 123
12.6 Reflectivepropertyforahyperbola 123
13.1 Circleintersectingpencilsoflines 126
13.2 Poleandpolar 128
13.3 Theideaoftheorthopticlocus 132
13.4 Orthopticlocusofanarrowhyperbola 134
14.1 Superimpositionoftwoellipses 138
14.2 Translationoftheplane 140
14.3 Rotationabouttheorigin 141
14.4 Invarianceofmidpointloci 146
16.1 Sectionsofthecone(cid:2) 161