Differential Geometry of Curves and Surfaces

Description

Cover Half Title Title Page Copyright Page Table of Contents Preface Acknowledgments Authors 1 Plane Curves: Local Properties 1.1 Parametrizations 1.2 Position, Velocity, and Acceleration 1.3 Curvature 1.4 Osculating Circles, Evolutes, Involutes 1.5 Natural Equations 2 Plane Curves: Global Properties 2.1 Basic Properties 2.2 Rotation Index 2.3 Isoperimetric Inequality 2.4 Curvature, Convexity, and the Four-Vertex Theorem 3 Curves in Space: Local Properties 3.1 Definitions, Examples, and Differentiation 3.2 Curvature, Torsion, and the Frenet Frame 3.3 Osculating Plane and Osculating Sphere 3.4 Natural Equations 4 Curves in Space: Global Properties 4.1 Basic Properties 4.2 Indicatrices and Total Curvature 4.3 Knots and Links 5 Regular Surfaces 5.1 Parametrized Surfaces 5.2 Tangent Planes; The Differential 5.3 Regular Surfaces 5.4 Change of Coordinates; Orientability 6 First and Second Fundamental Forms 6.1 The First Fundamental Form 6.2 Map Projections (Optional) 6.3 The Gauss Map 6.4 The Second Fundamental Form 6.5 Normal and Principal Curvatures 6.6 Gaussian and Mean Curvatures 6.7 Developable Surfaces; Minimal Surfaces 7 Fundamental Equations of Surfaces 7.1 Gauss’s Equations; Christoffel Symbols 7.2 Codazzi Equations; Theorema Egregium 7.3 Fundamental Theorem of Surface Theory 8 Gauss-Bonnet Theorem; Geodesics 8.1 Curvatures and Torsion 8.2 Gauss-Bonnet Theorem, Local Form 8.3 Gauss-Bonnet Theorem, Global Form 8.4 Geodesics 8.5 Geodesic Coordinates 8.6 Applications to Plane, Spherical, and Elliptic Geometry 8.7 Hyperbolic Geometry 9 Curves and Surfaces in n-Dimensional Space 9.1 Curves in n-Dimensional Euclidean Space 9.2 Surfaces in Euclidean n-Space A Tensor Notation A.1 Tensor Notation Bibliography Index