Chapters
Pages
1.
Curves in Space 1-124
1.1
Introduction 1
1.2
Parametric Representation of Space Curves 1
1.3
Arc Length of a Curve between Two Points 1
1.4
Tangent Line 3
1.5
Order of Contact 5
1.6
Osculating Plane or Plane of Curvature 14
1.7
Osculating Plane at a Point of the Curve of
Intersection
of Two Surfaces 17
1.8
Normal Plane 18
1.9
Principal Normal and Binormal 19
1.10
Rectifying Plane 20
1.11
Curvature, Torsion and Screw Curvature 21
1.12
Serret-Frenet Formulae 25
1.13
Curvature and Torsion of the Curve r = r (t) 27
1.14
Helices 29
1.15
Fundamental Theorems for Space Curves 73
1.16
Osculating Circle (or Circle of Curvature) 79
1.17
Osculating Sphere 83
1.18
Spherical Indicatrix 106
1.19
Involutes and Evolutes 110
1.20
Bertrand Curves 116
2.
Concept of a Surface and Envelopes and
Developable
125-185
2.1
Introduction 125
2.2
Ordinary Points and Singularities 126
2.3
Curvilinear Equations of the Curve on the
Surface
127
vi
2.4
Parametric Curves 127
2.5
Tangent Plane and Normal 128
2.6
Characteristics and Envelope of One
Parameter
Family of Surfaces 136
2.7
Edge of Regression 138
2.8
Developable Surfaces or Developables 145
2.9
Developables associated with a Space Curve 149
2.10
Envelope of Two Parameter Family of
Surfaces
175
3.
Fundamental Forms and Directions on a Surface 186-226
3.1
First Fundamental Form 186
3.2
Second Fundamental Form 191
3.3
Derivatives of N (Weingarten Equations) 193
3.4
Fundamental Magnitudes for Some Important
Surfaces
195
3.5
Directions on a Surface 206
3.6
Direction Coefficients 208
3.7
Family of Curves 212
3.8
Orthogonal Trajectories 213
3.9
Double Family of Curves 215
4.
Curves on a Surface 227-387
4.1
Curvature of Normal Section or Normal
Curvature
227
4.2
Principal Directions and Principal Curvatures 229
4.3
Lines of Curvature 237
4.4
Elliptic, Hyperbolic and Parabolic Points on a
Surface
250
4.5
Dupin’s Indicatrix 251
4.6
Third Fundamental Form 253
4.7
Conjugate Directions 302
4.8
Conjugate Systems 307
4.9
Asymptotic Lines 310
4.10
Ruled Surfaces 333
vii
4.11
Isometric Lines 357
4.12
Null Lines 360
4.13
The Fundamental Equations of Surface
Theory
363
4.14
Parallel Surfaces 382
5.
Geodesics 388-476
5.1
Definition 388
5.2
Differential Equations of Geodesics 388
5.3
Canonical Geodesic Equations 396
5.4
Normal Property of Geodesics 397
5.5
Differential Equations of Geodesics using
Normal
Property 399
5.6
Geodesics on a Surface of Revolution 406
5.7
Clairaut’s Theorem 408
5.8
Torsion and Curvature of a Geodesic 427
5.9
Curves in relation to Geodesics 436
5.10
Geodesic Curvature 440
5.11
Gauss-Bonnet Theorem 450
5.12
Geodesic Parallels 454
5.13
Geodesic Polar Coordinates 455
5.14
Minding Theorem 459
5.15
Mapping of Surfaces 463
5.16
Tissot’s Theorem 469
5.17
Dini’s Theorem 470
Index
477-479