Differential Geometry of Curves and Surfaces, 2nd Edition

Differential Geometry of Curves and Surfaces, 2nd Edition

by Thomas F. Banchoff
Released April 2016
Publisher(s): Chapman and Hall/CRC
ISBN: 9781482247473

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Book description

This self-contained text takes both an analytical/theoretical approach and a visual/intuitive approach to the local and global properties of curves and surfaces. It develops students' geometric intuition through interactive computer graphics applets supported by sound theory. This edition includes more exercises and project ideas, reorganized material on the Gauss-Bonnet theorem, and a new chapter on curves and surfaces in Rn. New sections cover applications to cartography and problems in spherical and hyperbolic geometry.

Table of contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright Page
  5. Table of Contents
  6. Preface
  7. Acknowledgements
  8. 1 Plane Curves: Local Properties
    1. 1.1 Parametrizations
    2. 1.2 Position, Velocity, and Acceleration
    3. 1.3 Curvature
    4. 1.4 Osculating Circles, Evolutes, and Involutes
    5. 1.5 Natural Equations
  9. 2 Plane Curves: Global Properties
    1. 2.1 Basic Properties
    2. 2.2 Rotation Index
    3. 2.3 Isoperimetric Inequality
    4. 2.4 Curvature, Convexity, and the Four-Vertex Theorem
  10. 3 Curves in Space: Local Properties
    1. 3.1 Definitions, Examples, and Differentiation
    2. 3.2 Curvature, Torsion, and the Frenet Frame
    3. 3.3 Osculating Plane and Osculating Sphere
    4. 3.4 Natural Equations
  11. 4 Curves in Space: Global Properties
    1. 4.1 Basic Properties
    2. 4.2 Indicatrices and Total Curvature
    3. 4.3 Knots and Links
  12. 5 Regular Surfaces
    1. 5.1 Parametrized Surfaces
    2. 5.2 Tangent Planes and Regular Surfaces
    3. 5.3 Change of Coordinates
    4. 5.4 The Tangent Space and the Normal Vector
    5. 5.5 Orientable Surfaces
  13. 6 The First and Second Fundamental Forms
    1. 6.1 The First Fundamental Form
    2. 6.2 Map Projections (Optional)
    3. 6.3 The Gauss Map
    4. 6.4 The Second Fundamental Form
    5. 6.5 Normal and Principal Curvatures
    6. 6.6 Gaussian and Mean Curvatures
    7. 6.7 Developable Surfaces and Minimal Surfaces
  14. 7 The Fundamental Equations of Surfaces
    1. 7.1 Gauss’s Equations and the Christoffel Symbols
    2. 7.2 Codazzi Equations and the Theorema Egregium
    3. 7.3 The Fundamental Theorem of Surface Theory
  15. 8 The Gauss-Bonnet Theorem and Geometry of Geodesics
    1. 8.1 Curvatures and Torsion
    2. 8.2 Gauss-Bonnet Theorem, Local Form
    3. 8.3 Gauss-Bonnet Theorem, Global Form
    4. 8.4 Geodesics
    5. 8.5 Geodesic Coordinates
    6. 8.6 Applications to Plane, Spherical, and Elliptic Ge- ometry
    7. 8.7 Hyperbolic Geometry
  16. 9 Curves and Surfaces in n-dimensional Euclidean Space
    1. 9.1 Curves in n-dimensional Euclidean Space
    2. 9.2 Surfaces in n-dimensional Euclidean Space
  17. A Tensor Notation
    1. A.1 Tensor Notation
  18. Bibliography
  19. Index

Product information

  • Title: Differential Geometry of Curves and Surfaces, 2nd Edition
  • Author(s): Thomas F. Banchoff
  • Release date: April 2016
  • Publisher(s): Chapman and Hall/CRC
  • ISBN: 9781482247473