Modern Differential Geometry ofcurves and Surfaces

Transcription

1 K ALFRED GRAY University of Maryland Modern Differential Geometry ofcurves and Surfaces /, CRC PRESS Boca Raton Ann Arbor London Tokyo

2 CONTENTS 1. Curves in the Plane Euclidean Spaces Curves in R" The Length ofa Curve Vector Fields along Curves Curvature of Curves in the Plane The Turning Angle The Semicubical Parabola Exercises Studying Curves in the Plane with Mathematica Computing Curvature of Curves in the Plane Computing Lengths of Curves Filling Curves Examples of Curves in R Plotting Piecewise Defined Curves Exercises 33 ix

3 3. Famous Plane Curves Cycloids Lemniscates ofbernoulli Cardioids The Cissoid of Diocles The Tractrix Clothoids Exercises Alternate Methods for Plotting Plane Curves Implicitly Defined Curves in R Cassinian Ovals Plane Curves in Polar Coordinates Exercises New Curves from Old Evolutes Iterated Evolutes The Evolute ofa Tractrix is a Catenary Involutes Tangent and Normal Lines to Plane Curves Osculating Circles to Plane Curves Parallel Curves Pedal Curves Exercises 100

4 Determming a Plane Curve from its Curvature Euclidean Motions Curves and Euclidean Motions Intrinsic Equations for Plane Curves Drawing Plane Curves with Assigned Curvature Exercises 119 Curves In Space Preliminaries Curvature and Torsion of Unit-Speed Curves in R Curvature and Torsion of Arbitrary-Speed Curves in R Computing Curvature and Torsion with Mathematica The Helix and its Generalizations Viviani's Curve The Fundamental Theorem of Space Curves Drawing Space Curves with Assigned Curvature Exercises 148 Tubes and Knots Tubes about Curves Torus Knots Exercises 161

5 Calculus on Euclidean Space Tangent Vectors to R n Tangent Vectors as Directional Derivatives Tangent Maps Vector Fields on R n Derivatives of Vector Fields on R n Curves Revisited Exercises 181 Surfaces in Euclidean Space Patches in R n Patches in R The Local Gauss Map The Definition of a Regulär Surface in R n Tangent Vectors to Regulär Surfaces in R n Surface Mappings Level Surfaces in R Exercises 207 Examples of Surfaces The Graph ofa Function of Two Variables The Ellipsoid The Stereographic Ellipsoid Tori The Paraboloid Sea Shells 223

6 xiii 11.7 Patches with Singularities Implicit Plots of Surfaces Exercises Nononentable Surfaces Orientability of Surfaces Nonorientable Surfaces Described by Identifications The Möbius Strip The Klein Bottle Realizations of the Real Projective Plane Coloring Surfaces with Mathematica Exercises Metrics on Surfaces The Intuitive Idea of Distance on a Surface Isometries of Surfaces The Intuitive Idea of Area on a Surface Programs for Computing Metrics and Areas on a Surface Examples of Metrics Exercises Surfaces in 3-Dimensional Space The Shape Operator Normal Curvature Calculation ofthe Shape Operator The Eigenvalues of the Shape Operator

7 xiv 14.5 The Gaussian and Mean Curvatures The Three Fundamental Forms Examples of Curvature Calculations by Hand The Curvature of Nonparametrically Defined Surfaces Exercises Surfaces in 3-Dimensional Space via Mathematica Programs for Computing the Shape Operator and Curvature Examples of Curvature Calculations with Mathematica The Gauss Map via Mathematica Exercises Asymptotic Curves on Surfaces Asymptotic Curves Examples of Asymptotic Curves Using Mathematica to Find Asymptotic Curves Exercises Ruied Surfaces Examples of Ruied Surfaces Fiat Ruied Surfaces Noncylindrical Ruied Surfaces Examples of Striction Curves of Noncylindrical Ruied Surfaces A Program for Ruied Surfaces Developables Exercises 354

8 XV 18. Surfaces of Revolution Principal Curves The Curvature ofa Surface of Revolution Generating a Surface of Revolution with Mathematica The Catenoid The Hyperboloid of Revolution The Surfaces of Revolution of Curves with Specified Curvature Exercises Surfaces of Constant Gaussian Curvature The Elliptic Integral of the Second Kind Surfaces of Revolution of Constant Positive Curvature Surfaces of Revolution of Constant Negative Curvature Kuen's Surface Exercises Intrinsic Surface Geometry Intrinsic Formulas for the Gaussian Curvature Gauss's Theorema Egregium Christoffel Symbols The Mainardi-Codazzi Equations Geodesic Curvature Exercises 408

9 21. Principal Curves and Umbiiic Points The Differential Equation for the Principal Curves Umbiiic Points Triply Orthogonal Systems of Surfaces Elliptic Coordinates Parabolic Coordinates Exercises Minimal Surfaces I Normal Variation Examples of Minimal Surfaces The Gauss Map ofa Minimal Surface Exercises Minimal Surfaces II Isothermal Coordinates Minimal Surfaces and Complex Function Theory Finding Conjugate Minimal Surfaces Enneper's Surface of Degree n The Weierstrass Representation The Weierstrass Patches via Mathematica Examples of Weierstrass Patches Exercises Construction of Surfaces Parallel Surfaces 481

10 1. xvii 24.2 The Shape Operator of a Parallel Surface Pedal Surfaces Generalized Helicoids Twisted Surfaces Exercises Differentiable Manifolds The Definition of Differentiable Manifold Differentiable Functions on Differentiable Manifolds Tangent Vectors on Differentiable Manifolds Induced Maps Vector Fields on Differentiable Manifolds Tensor Fields on Differentiable Manifolds Exercises Riemannian Manifolds Covariant Derivatives Indefinite Riemannian Metrics The Classical Treatment of Metrics Abstract Surfaces Metrics on Abstract Surfaces Examples of Abstract Surfaces Computing Curvature of Metrics on Abstract Surfaces Orientability of an Abstract Surface Geodesic Curvature for Abstract Surfaces Exercises 559

11 xviii 28. Geodesics on Surfaces The Geodesic Equations Clairaut Patches Examples of Clairaut Patches Finding Geodesics Numerically with Mathematica 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