The
differential geometry of curves and surfaces is fundamental in
Computer Aided Geometric Design (CAGD). The curves and surfaces
treated in differential geometry are defined by functions which can be
differentiated a certain number of times. Books by Hilbert and
Cohn-Vossen [165], Koenderink
[205] provide intuitive introductions to the extensive
mathematical literature on three-dimensional shape analysis. The books by
Struik [412], Willmore [444], Kreyszig
[206], Lipschutz [235], do Carmo
[76] offer firm theoretical basis to the differential geometry
aspects of three-dimensional shape description. A book by Gray
[136] combines the traditional textbook style and a symbolic
manipulation program MATHEMATICA. In a recent textbook, Gallier
[122] provides a thorough introduction to differential
geometry as well as a comprehensive treatment of affine and
projective geometry and their applications to rational curves and
surfaces in addition to basic topics of computational geometry
(eg. convex hulls, Voronoi diagrams and Delaunay triangulations). We
briefly review elementary differential geometry of curves in this
chapter and surfaces in Chap. 3.
