Differential Geometry of Curves and Surfaces, 2nd Edition

Front Cover......Page 1
Contents......Page 6
Preface......Page 8
Acknowledgements......Page 16
 1.1 Parametrizations......Page 18
 1.2 Position, Velocity, and Acceleration......Page 30
 1.3 Curvature......Page 42
 1.4 Osculating Circles, Evolutes, and Involutes......Page 50
 1.5 Natural Equations......Page 57
 2.1 Basic Properties......Page 64
 2.2 Rotation Index......Page 69
 2.3 Isoperimetric Inequality......Page 78
 2.4 Curvature, Convexity, and the Four-Vertex Theorem......Page 81
 3.1 Definitions, Examples, and Differentiation......Page 88
 3.2 Curvature, Torsion, and the Frenet Frame......Page 97
 3.3 Osculating Plane and Osculating Sphere......Page 107
 3.4 Natural Equations......Page 114
 4.1 Basic Properties......Page 120
 4.2 Indicatrices and Total Curvature......Page 123
  4.3.1 Knots......Page 132
  4.3.2 Links......Page 136
 5.1 Parametrized Surfaces......Page 142
 5.2 Tangent Planes and Regular Surfaces......Page 150
 5.3 Change of Coordinates......Page 167
 5.4 The Tangent Space and the Normal Vector......Page 172
 5.5 Orientable Surfaces......Page 176
 6.1 The First Fundamental Form......Page 182
  6.2.1 Metric Properties of Maps of the Earth......Page 197
  6.2.2 Azimuthal Projections......Page 201
  6.2.3 Cylindrical Projections......Page 203
  6.2.4 Coordinate Changes on the Sphere......Page 206
 6.3 The Gauss Map......Page 209
 6.4 The Second Fundamental Form......Page 215
 6.5 Normal and Principal Curvatures......Page 226
 6.6 Gaussian and Mean Curvatures......Page 238
 6.7 Developable Surfaces and Minimal Surfaces......Page 248
  6.7.1 Developable Surfaces......Page 249
  6.7.2 Minimal Surfaces......Page 256
Chapter 7: The Fundamental Equations of Surfaces......Page 264
 7.1 Gauss\'s Equations and the Christoffel Symbols......Page 265
 7.2 Codazzi Equations and the Theorema Egregium......Page 275
 7.3 The Fundamental Theorem of Surface Theory......Page 285
Chapter 8: The Gauss-Bonnet Theorem and Geometry of Geodesics......Page 290
  8.1.1 Natural Frames......Page 291
  8.1.2 Normal Curvature......Page 292
  8.1.3 Geodesic Curvature......Page 295
  8.1.4 Geodesic Torsion......Page 298
 8.2 Gauss-Bonnet Theorem, Local Form......Page 301
 8.3 Gauss-Bonnet Theorem, Global Form......Page 312
 8.4 Geodesics......Page 322
  8.5.1 General Geodesic Coordinates......Page 340
  8.5.2 Geodesic Polar Coordinates......Page 346
  8.6.1 Plane Geometry......Page 352
  8.6.2 Spherical Geometry......Page 353
  8.6.3 Elliptic Geometry......Page 356
  8.7.1 Synthetic Hyperbolic Geometry......Page 359
  8.7.2 The Poincaré Upper Half-Plane......Page 362
  8.7.3 The Poincaré Disk......Page 365
  8.7.4 The Pseudosphere Revisited......Page 367
 9.1 Curves in n-dimensional Euclidean Space......Page 372
  9.1.1 Curvatures and the Frenet Frame......Page 373
  9.1.2 Osculating Planes, Circles, and Spheres......Page 378
  9.1.3 The Fundamental Theorem of Curves in Rn......Page 379
  9.2.1 Regular Surfaces in Rn......Page 381
  9.2.2 Intrinsic Geometry for Surfaces......Page 382
  9.2.3 Orientability......Page 387
  9.2.4 The Gauss-Bonnet Theorem......Page 390
 A.1 Tensor Notation......Page 394
  A.1.1 Curvilinear Coordinate Systems......Page 395
  A.1.2 Tensors: Definitions and Notation......Page 401
  A.1.3 Operations on Tensors......Page 404
  A.1.4 Examples......Page 405
  A.1.5 Symmetries......Page 408
  A.1.6 Numerical Tensors......Page 411
Bibliography......Page 422
Back Cover......Page 426