The chapter gives a short overview of the concepts from differetial geometry that are used in geometry processing: normal, area, first and second fundamental form, the Gauß and Weingarten map, normal and geodesic curvature, principal curvatures and directions, the Gaußian and mean curvature, the Gauß–Bonnet theorem and the Laplace–Beltrami operator. We end by a brief study of implicitly defined surfaces.
It is not meant as a course in differential geometry, but as a brush up and a handy point of reference. For the reader who wishes to know more there is a vast literature to which we refer.
KeywordsTangent Space Tangent Vector Fundamental Form Principal Curvature Principal Direction
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Graustein, W.C.: Differential Geometry. Dover, New York (1966)
Guggenheimer, H.W.: Differential Geometry. McGraw-Hill, New York (1963)
Lipschutz, M.: Theory and Problems of Differential Geometry. Schaum’s Outline Series. McGraw-Hill, New York (1969)
Struik, D.J.: Lectures on Classical Differential Geometry. Addison-Wesley, London (1961)
do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976)
Oprea, J.: The Mathematics of Soap Films: Explorations with Maple®
. Student Mathematical Library, vol. 10. Amer. Math. Soc., Providence (2000)
Pressley, A.: Elementary Differential Geometry. Springer, London (2001)
Gallier, J.: Geometric Methods and Applications. For Computer Science and Engineering. Springer, New York (2001)
Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press, Boca Raton (1998)
Gravesen, J.: Differential geometry and design of shape and motion. http://www2.mat.dtu.dk/people/J.Gravesen/cagd.pdf
Raussen, M.: Elementary differential geometry: Curves and surfaces (2001). http://www.math.auc.dk/~raussen/MS/01/index.html
Hansen, V.L.: Fundamental Concepts in Modern Analysis. World Scientific, River Edge (1999)
Hartmann, E.: On the curvature of curves and surfaces defined by normalforms. Comput. Aided Geom. Des. 16
(5), 355–376 (1999)
© Springer-Verlag London 2012