ALFRED GRAY University of Maryland

Modern Differential Geometry ofCurves and Surfaces

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CRC PRESS

Boca Raton Ann Arbor London Tokyo

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CONTENTS

1. Curves in the Plane 1

1.1 Euclidean Spaces 2

1.2 Curves in R" 4

1.3 The Length ofa Curve 6

1.4 Vector Fields along Curves 10

1.5 Curvature of Curves in the Plane 11

1.6 The Turning Angle 13

1.7 The Semicubical Parabola 15

1.8 Exercises 16

2. Studying Curves in the Plane with Mathematica 17

2.1 Computing Curvature of Curves in the Plane 20

2.2 Computing Lengths of Curves 23

2.3 Filling Curves 24

2.4 Examples of Curves in R2 25

2.5 Plotting Piecewise Defined Curves 30

2.6 Exercises 33

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3. Famous Plane Curves 37 3.1 Cycloids 37

3.2 Lemniscates ofBernoulli 39

3.3 Cardioids 41

3.4 The Cissoid of Diocles 43

3.5 The Tractrix 46

3.6 Clothoids 50

3.7 Exercises 53

4. Alternate Methods for Plotting Plane Curves 57

4.1 Implicitly Defined Curves in R2 57

4.2 Cassinian Ovals 63

4.3 Plane Curves in Polar Coordinates 66

4.4 Exercises 71

5. New Curves from Old 75

5.1 Evolutes 76

5.2 Iterated Evolutes 79

5.3 The Evolute ofa Tractrix is a Catenary 80

5.4 Involutes 81

5.5 Tangent and Normal Lines to Plane Curves 85

5.6 Osculating Circles to Plane Curves 90

5.7 Parallel Curves 95

5.8 Pedal Curves ...97

5.9 Exercises 100

Determming a Plane Curve from its Curvature 103 6.1 Euclidean Motions 104

6.2 Curves and Euclidean Motions 108

6.3 Intrinsic Equations for Plane Curves 109

6.4 Drawing Plane Curves with Assigned Curvature 113

6.5 Exercises 119

Curves In Space 123

7.1 Preliminaries 124

7.2 Curvature and Torsion of Unit-Speed Curves in R3 125

7.3 Curvature and Torsion of Arbitrary-Speed Curves in R3 129

7.4 Computing Curvature and Torsion with Mathematica 133

7.5 The Helix and its Generalizations 138

7.6 Viviani's Curve 140

7.7 The Fundamental Theorem of Space Curves 142

7.8 Drawing Space Curves with Assigned Curvature 145

7.9 Exercises 148

Tubes and Knots 153

8.1 Tubes about Curves 153

8.2 Torus Knots 155

8.3 Exercises 161

Calculus on Euclidean Space 163

9.1 Tangent Vectors to Rn 164

9.2 Tangent Vectors as Directional Derivatives 165

9.3 Tangent Maps 168

9.4 Vector Fields on Rn 171

9.5 Derivatives of Vector Fields on Rn 175

9.6 Curves Revisited 179

9.7 Exercises 181

Surfaces in Euclidean Space 183

10.1 Patches in Rn 183

10.2 Patches in R3 192

10.3 The Local Gauss Map 193

10.4 The Definition of a Regulär Surface in Rn 195

10.5 Tangent Vectors to Regulär Surfaces in Rn 200

10.6 Surface Mappings 202

10.7 Level Surfaces in R3 204

10.8 Exercises 207

Examples of Surfaces 209

11.1 The Graph ofa Function of Two Variables 210

11.2 The Ellipsoid 215

11.3 The Stereographic Ellipsoid 216

11.4 Tori 218

11.5 The Paraboloid 221

11.6 Sea Shells 223

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11.7 Patches with Singularities 224

11.8 Implicit Plots of Surfaces 226

11.9 Exercises 226

12. Nononentable Surfaces 229

12.1 Orientability of Surfaces 229

12.2 Nonorientable Surfaces Described by Identifications 234

12.3 The Möbius Strip 236

12.4 The Klein Bottle 239

12.5 Realizations of the Real Projective Plane 241

12.6 Coloring Surfaces with Mathematica 245

12.7 Exercises 247

13. Metrics on Surfaces 251

13.1 The Intuitive Idea of Distance on a Surface 251

13.2 Isometries of Surfaces 255

13.3 The Intuitive Idea of Area on a Surface 259

13.4 Programs for Computing Metrics and Areas on a Surface 260

13.5 Examples of Metrics 261

13.6 Exercises 263

14. Surfaces in 3-Dimensional Space 267

14.1 The Shape Operator 268

14.2 Normal Curvature 270

14.3 Calculation ofthe Shape Operator .274

14.4 The Eigenvalues of the Shape Operator 277

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14.5 The Gaussian and Mean Curvatures 279

14.6 The Three Fundamental Forms 286

14.7 Examples of Curvature Calculations by Hand 287

14.8 The Curvature of Nonparametrically Defined Surfaces 291

14.9 Exercises 297

15. Surfaces in 3-Dimensional Space via Mathematica.... 299

15.1 Programs for Computing the Shape Operator and Curvature.... 299

15.2 Examples of Curvature Calculations with Mathematica 303

15.3 The Gauss Map via Mathematica 310

15.4 Exercises 316

16. Asymptotic Curves on Surfaces 319

16.1 Asymptotic Curves 320

16.2 Examples of Asymptotic Curves 323

16.3 Using Mathematica to Find Asymptotic Curves 328

16.4 Exercises 331

17. Ruied Surfaces 333

17.1 Examples of Ruied Surfaces 334

17.2 Fiat Ruied Surfaces 341

17.3 Noncylindrical Ruied Surfaces 345

17.4 Examples of Striction Curves of Noncylindrical Ruied Surfaces ..349

17.5 A Program for Ruied Surfaces 350

17.6 Developables 352

17.7 Exercises 354

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18. Surfaces of Revolution 357

18.1 Principal Curves 359

18.2 The Curvature ofa Surface of Revolution 361

18.3 Generating a Surface of Revolution with Mathematica 365

18.4 The Catenoid 367

18.5 The Hyperboloid of Revolution 369

18.6 The Surfaces of Revolution of Curves with Specified Curvature 370

18.7 Exercises 372

19. Surfaces of Constant Gaussian Curvature 375

19.1 The Elliptic Integral of the Second Kind 376

19.2 Surfaces of Revolution of Constant Positive Curvature 376

19.3 Surfaces of Revolution of Constant Negative Curvature 380

19.4 Kuen's Surface 384

19.5 Exercises 386

20. Intrinsic Surface Geometry 389

20.1 Intrinsic Formulas for the Gaussian Curvature 390

20.2 Gauss's Theorema Egregium 395

20.3 Christoffel Symbols 397

20.4 The Mainardi-Codazzi Equations 401

20.5 Geodesic Curvature 402

20.6 Exercises 408

21. Principal Curves and Umbiiic Points 409 21.1 The Differential Equation for the

Principal Curves 410

21.2 Umbiiic Points 413

21.3 Triply Orthogonal Systems of Surfaces 418

21.4 Elliptic Coordinates 424

21.5 Parabolic Coordinates 429

21.6 Exercises 432

22. Minimal Surfaces I 435

22.1 Normal Variation 435

22.2 Examples of Minimal Surfaces 438

22.3 The Gauss Map ofa Minimal Surface 449

22.4 Exercises 451

23. Minimal Surfaces II 455

23.1 Isothermal Coordinates 455

23.2 Minimal Surfaces and Complex Function Theory 456

23.3 Finding Conjugate Minimal Surfaces 462

23.4 Enneper's Surface of Degree n 469

23.5 The Weierstrass Representation 473

23.6 The Weierstrass Patches via Mathematica 476

23.7 Examples of Weierstrass Patches 477

23.8 Exercises 479

24. Construction of Surfaces 481

24.1 Parallel Surfaces 481

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24.2 The Shape Operator of a Parallel Surface 485

24.3 Pedal Surfaces 488

24.4 Generalized Helicoids 489

24.5 Twisted Surfaces 495

24.6 Exercises 498

25. Differentiable Manifolds 499

25.1 The Definition of Differentiable Manifold 500

25.2 Differentiable Functions on Differentiable Manifolds 504

25.3 Tangent Vectors on Differentiable Manifolds 510

25.4 Induced Maps 518

25.5 Vector Fields on Differentiable Manifolds 524

25.6 Tensor Fields on Differentiable Manifolds 528

25.7 Exercises 532

26. Riemannian Manifolds 533

26.1 Covariant Derivatives 534

26.2 Indefinite Riemannian Metrics 540

26.3 The Classical Treatment of Metrics 544

27. Abstract Surfaces 549

27.1 Metrics on Abstract Surfaces 550

27.2 Examples of Abstract Surfaces 553

27.3 Computing Curvature of Metrics on Abstract Surfaces 555

27.4 Orientability of an Abstract Surface 557

27.5 Geodesic Curvature for Abstract Surfaces 558

27.6 Exercises 559

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28. Geodesics on Surfaces 561

28.1 The Geodesic Equations 561

28.2 Clairaut Patches 564

28.3 Examples of Clairaut Patches 567

28.4 Finding Geodesics Numerically with Mathematica 569

28.5 Exercises 574

Appendix 575

A. 1 General Programs 575

A.2 Plane Curves 607

A.3 Space Curves 620

A.4 Surfaces 622

A.5 Metrics 636

A.6 Mathematica to Acrospin 636

Bibliography 645 Index 658