Differential Geometry of Curves and Surfaces, Second Edition

Thomas F. Banchoff, Stephen T. Lovett

September 10, 2015 by Chapman and Hall/CRC
Textbook - 414 Pages - 89 B/W Illustrations
ISBN 9781482247343 - CAT# K23627


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  • Requires only a background in multivariable calculus and linear algebra
  • Shows how the Gauss–Bonnet theorem relates global properties of curves and surfaces to the topology of a surface and leads to fundamental results in non-Euclidean spherical and hyperbolic geometry
  • Covers the local and global theory of plane and space curves
  • Presents applications to problems in spherical and hyperbolic geometry
  • Provides online computer graphics applets that illustrate many concepts and theorems
  • Includes a collection of problems in each section ranging from routine to challenging


Differential Geometry of Curves and Surfaces, Second Edition takes both an analytical/theoretical approach and a visual/intuitive approach to the local and global properties of curves and surfaces. Requiring only multivariable calculus and linear algebra, it develops students’ geometric intuition through interactive computer graphics applets supported by sound theory.

The book explains the reasons for various definitions while the interactive applets offer motivation for certain definitions, allow students to explore examples further, and give a visual explanation of complicated theorems. The ability to change parametric curves and parametrized surfaces in an applet lets students probe the concepts far beyond what static text permits.

New to the Second Edition

  • Reworked presentation to make it more approachable
  • More exercises, both introductory and advanced
  • New section on the application of differential geometry to cartography
  • Additional investigative project ideas
  • Significantly reorganized material on the Gauss–Bonnet theorem
  • Two new sections dedicated to hyperbolic and spherical geometry as applications of intrinsic geometry
  • A new chapter on curves and surfaces in Rn

Suitable for an undergraduate-level course or self-study, this self-contained textbook and online software applets provide students with a rigorous yet intuitive introduction to the field of differential geometry. The text gives a detailed introduction of definitions, theorems, and proofs and includes many types of exercises appropriate for daily or weekly assignments. The applets can be used for computer labs, in-class illustrations, exploratory exercises, or self-study aids.

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