Elementary Euclidean geometry: An undergraduate introduction

✍ Scribed by Gibson C.G.

• Publisher: CUP
• Year: 2003
• Tongue: English
• Leaves: 191
• Category: Library

✦ Synopsis

This introduction to the geometry of lines and conics in the Euclidean plane is example-based and self-contained, assuming only a basic grounding in linear algebra. Including numerous illustrations and several hundred worked examples and exercises, the book is ideal for use as a course text for undergraduates in mathematics, or for postgraduates in the engineering and physical sciences.

✦ Table of Contents

Cover
......Page 1
About......Page 2
Title: Elementary Euclidean Geometry: An Introduction......Page 4
Copyright......Page 5
Contents......Page 6
Figures......Page 9
Tables......Page 11
Preface......Page 12
Acknowledgements......Page 17
1.1 The Vector Structure......Page 18
1.2 Lines and Zero Sets......Page 19
1.4 Practical Techniques......Page 21
1.5 Parametrized Lines......Page 24
1.6 Pencils of Lines......Page 26
2.1 The Scalar Product......Page 29
2.2 Length and Distance......Page 30
2.3 The Concept of Angle......Page 32
2.4 Distance from a Point to a Line......Page 35
3.1 Circles as Conics......Page 39
3.2 General Circles......Page 40
3.3 Uniqueness of Equations......Page 41
3.4 Intersections with Lines......Page 43
3.5 Pencils of Circles......Page 44
4. General Conics......Page 49
4.1 Standard Conics......Page 50
4.2 Parametrizing Conics......Page 52
4.3 Matrices and Invariants......Page 54
4.4 Intersections with Lines......Page 56
4.5 The Component Lemma......Page 58
5.1 The Concept of a Centre......Page 61
5.2 Finding Centres......Page 62
5.3 Geometry of Centres......Page 66
5.4 Singular Points......Page 68
6.1 Binary Quadratics......Page 71
6.2 Reducible Conics......Page 73
6.3 Pencils of Conics......Page 76
6.4 Perpendicular Bisectors......Page 78
7.1 Midpoint Loci......Page 82
7.2 Axes......Page 85
7.3 Bisectors as Axes......Page 89
7.4 Asymptotic Directions......Page 91
8.1 Focal Constructions......Page 93
8.3 Constructions for Parabolas......Page 96
8.4 Geometric Generalities......Page 98
8.5 Constructions of Ellipse and Hyperbola......Page 100
9.1 Tangent Lines......Page 105
9.2 Examples of Tangents......Page 106
9.3 Normal Lines......Page 111
10.1 The Axis of a Parabola......Page 115
10.2 Practical Procedures......Page 116
10.3 Parametrizing Parabolas......Page 119
11.1 Axes and Vertices......Page 122
11.2 Rational Parametrization......Page 124
11.3 Focal Properties......Page 127
12.1 Asymptotes......Page 131
12.2 Parametrizing Hyperbolas......Page 136
12.3 Focal Properties of Hyperbolas......Page 138
13.1 The Polars of a Conic......Page 142
13.2 The Joint Tangent Equation......Page 144
13.3 Orthoptic Loci......Page 149
14. Congruences......Page 154
14.1 Congruences......Page 155
14.2 Congruent Lines......Page 159
14.3 Congruent Conics......Page 161
14.4 The Invariance Theorem......Page 163
15.1 Rotating the Axes......Page 166
15.2 Listing Normal Forms......Page 168
15.3 Some Consequences......Page 171
15.4 Eigenvalues and Axes......Page 172
16.1 Distinguishing Classes......Page 176
16.2 Conic Sections......Page 178
16.3 Conics within a Class......Page 179
17.1 Proof of Uniqueness......Page 184
17.2 Proof of Invariance......Page 186
Index......Page 188