Elementary Geometry of Differentiable Curves: An Undergraduate Introduction

Front cover......Page 1
Title page......Page 3
Date-line......Page 4
Dedication......Page 5
Contents......Page 7
List of Illustrations......Page 10
1.1 Components of a vector......Page 21
1.2 The two sides of a line......Page 27
1.3 The orthogonal bisector......Page 28
1.6 Projection to Lines......Page 29
1.5 Orthogonal projection onto a line......Page 30
2.1 A parametrized curve......Page 32
2.2 The right strophoid......Page 33
2.3 The catenary......Page 34
2.4 Parametrizing the circle......Page 36
2.5 Some rose curves......Page 37
2.2 Self Crossings......Page 38
2.7 The eight-curve......Page 39
2.8 Cayley\'s sextic......Page 40
2.3 Tangent and Normal Vectors......Page 41
2.10 The semicubical parabola......Page 42
2.11 Construction of the piriform......Page 43
2.12 The piriform......Page 44
2.4 Tangent and Normal Lines......Page 45
2.14 The cross curve......Page 46
2.15 The tractrix......Page 47
3.1 Parabola as a standard conic......Page 51
3.2 Ellipse as a standard conic......Page 53
3.3 Hyperbola as a standard conic......Page 54
3.4 Agnesfs versiera......Page 57
3.5 The cissoid of Diodes......Page 58
3.3 Trochoids......Page 60
3.7 Construction of trochoids......Page 61
3.8 Various epicycloids and hypocycloids......Page 63
3.9 Three different types of limacon......Page 64
3.10 Construction of cycloids......Page 65
3.11 Forms of the cycloid......Page 66
4.1 The astroid......Page 69
4.2 The idea of parametric equivalence......Page 72
4.3 Some Lissajous figures......Page 74
4.4 The involute construction......Page 77
4.5 An involute of a circle......Page 78
5.1 The idea of a moving frame......Page 81
5.4 Inflexions......Page 87
5.3 A biflexional limacon......Page 90
5.4 Forms of the curves near the origin......Page 92
6.1 Isometries......Page 94
6.3 Congruent Curves......Page 100
6.3 An equiangular spiral......Page 101
7.4 Inflexions and Undulations......Page 112
7.2 Picturing the sign of the curvature......Page 114
7.3 Inflexions having odd and even contact......Page 115
7.1 Inflexions on limacons......Page 116
7.5 A cuspidal tangent line......Page 118
7.6 Example of a higher cusp......Page 120
8.1 Circles tangent at the vertex of a parabola......Page 125
8.2 A parabola and its evolute......Page 128
8.3 An ellipse and its evolute......Page 129
8.4 The evolute of the eight-curve......Page 130
8.5 Three cusps and their evolutes......Page 131
8.6 Evolutes of some epicycloids and hypocycloids......Page 133
8.7 A cycloid and its evolute......Page 134
8.8 The tractrix and its evolute......Page 135
8.9 Cayley\'s sextic and its evolute......Page 136
8.10 Parallels of a parabola......Page 138
8.11 Parallels of an ellipse......Page 139
8.12 Two parallels of an astroid......Page 140
9.1 Vertices on the graph of $f(x)=x^2(2x+3)$......Page 146
9.2 A cardioid and its evolute......Page 150
9.3 Convex and non-convex ovals......Page 152
9.4 Proof of Lemma 9.3......Page 153
10.2 A family of circles......Page 156
10.3 Two envelopes of the family of circles......Page 157
10.4 Astroid as an envelope of ellipses......Page 160
10.5 Nephroid as an envelope of circles......Page 161
10.6 Cardioid as an envelope of lines......Page 163
10.7 Parabola parallel as a circle envelope......Page 165
10.8 Envelope of the normals for a parabola......Page 166
10.9 Envelope of the normals for an ellipse......Page 167
11.2 Orthotomics......Page 171
11.2 Three orthotomics of a circle......Page 173
11.3 Orthotomics of a parabola......Page 174
11.4 Orthotomic of an ellipse with source a focus......Page 175
11.5 Bernoulli\'s lemniscate......Page 176
11.1 Orthotomics of epicycloids and hypocycloids......Page 179
12 Caustics by Reflexion......Page 183
12.1 Caustics of a Curve......Page 184
12.3 Constructing the caustics of a circle......Page 185
12.4 Caustics by reflexion for a circle......Page 186
12.5 The reflective property for an ellipse......Page 190
12.6 Orthotomic with respect to a line......Page 192
12.7 Orthotomic of a circle with source at infinity......Page 193
12.8 Orthotomic of a parabola with source at infinity......Page 194
12.4 Orthotomics as Envelopes......Page 195
12.10 Orthotomic as an envelope of circles......Page 196
13.1 The Watt four bar......Page 199
13.2 Four bar model......Page 200
13.4 Collapsible four bar linkages......Page 201
13.2 Planar Motions......Page 202
13.6 The double slider......Page 203
13.3 General Roulettes......Page 204
14.1 Centrodes of the double slider......Page 213
14.2 Crossed parallelogram four bars......Page 214
14.3 Centrode for a crossed parallelogram......Page 215
List of Tables......Page 13
3.1 Special epicycloids and hypocycloids......Page 62
8.1 Evolutes of some epicycloids and hypocycloids......Page 132
Preface......Page 14
1.1 The Vector Structure......Page 19
1.3 Length, Distance and Angle......Page 20
1.4 The Complex Structure......Page 24
1.5 Lines......Page 25
2.1 The General Concept......Page 31
3.1 The Standard Conies......Page 50
3.2 General Algebraic Curves......Page 56
4.1 Arc Length......Page 68
4.2 Parametric Equivalence......Page 71
4.3 Unit Speed Curves......Page 75
4.4 Involutes......Page 76
5.1 The Moving Frame......Page 80
5.2 The Concept of Curvature......Page 82
5.3 Calculating the Curvature......Page 83
5.5 Limiting Behaviour......Page 91
6 Existence and Uniqueness......Page 93
6.2 Fixed Points and Formulas......Page 97
6.4 The Uniqueness Theorem......Page 103
7.1 The Factor Theorem......Page 107
7.2 Multiplicity of a Zero......Page 108
7.3 Contact with Lines......Page 110
7.5 Cusps......Page 117
8.1 Contact Functions......Page 123
8.2 Evolutes......Page 127
8.3 Parallels......Page 137
9.1 The Concept of a Vertex......Page 142
9.2 Appearance of Vertices on the Evolute......Page 149
9.3 The Four Vertex Theorem......Page 151
10.1 Envelopes......Page 155
10.2 The Envelope Theorem......Page 158
10.3 Natural Envelopes in Geometry......Page 164
11.1 Reflexions......Page 169
11.3 Orthotomics of Non-Regular Curves......Page 178
11.5 Antiorthotomics......Page 180
12.2 Caustics as Evolutes......Page 187
12.3 Sources at Infinity......Page 191
13.1 Historical Genesis......Page 198
14.1 Generic Parameters......Page 208
14.2 Generic Parameters for Roulettes......Page 210
14.3 Fixed and Moving Centrodes......Page 212
15 Geometry of Trajectories......Page 217
15.1 Equivalence of Motions......Page 218
15.2 Cusps on Trajectories......Page 221
15.3 Inflexions on Trajectories......Page 223
15.4 Vertices on Trajectories......Page 227
Index......Page 229