Next: 2.1 Arc length and Up: Shape Interrogation for Computer Previous: 1.5 Generalization of B-spline   Contents   Index


2. Differential Geometry of Curves

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.



Subsections

Next: 2.1 Arc length and Up: Shape Interrogation for Computer Previous: 1.5 Generalization of B-spline   Contents   Index
December 2009