Abstract
In this chapter we consider parametric curves, and we introduce two important invariants, curvature and torsion (in the case of a 3D curve).
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References
Richard H. Bartels, John C. Beatty, and Brian A. Barsky. An Introduction to Splines for Use in Computer Graphics and Geometric Modelling. Morgan Kaufmann, first edition, 1987.
Marcel Berger and Bernard Gostiaux. G´eom´etrie diff´erentielle: vari´et´es, courbes et surfaces. Collection Math’ematiques. Puf, second edition, 1992. English edition: Differential geometry,manifolds, curves, and surfaces, GTM No. 115, Springer-Verlag.
Eugenio Calabi, Peter J. Olver, C. Shakiban, Allen Tannenbaum, and Steven Haker. Differential and numerically invariant signature curves applied to object recognition. International Journal of Computer Vision, 26(2):107–135, 1998.
Eugenio Calabi, Peter J. Olver, and Allen Tannenbaum. Affine geometry, curve flows, and invariant numerical approximations. Advances in Mathematics, 124:154–196, 1996.
’ Elie Cartan. Les syst`emes diff´erentiels ext´erieurs et leurs applications g´eom´etriques. Hermann,first edition, 1945.
Gaston Darboux. Lec¸ons sur la th´eorie g´en´erale des surfaces, Premi`ere Partie. Gauthier-Villars, second edition, 1914.
Manfredo P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.
Gerald A. Edgar. Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics. Springer-Verlag, first edition, 1992.
Gerald Farin. NURB Curves and Surfaces, from Projective Geometry to Practical Use. AK Peters, first edition, 1995.
Gerald Farin. Curves and Surfaces for CAGD. Academic Press, fourth edition, 1998.
J.-C. Fiorot and P. Jeannin. Courbes et Surfaces Rationelles. RMA 12. Masson, first edition,1989.
J.-C. Fiorot and P. Jeannin. Courbes Splines Rationelles. RMA 24. Masson, first edition,1992.
Jean H. Gallier. Curves and Surfaces in Geometric Modeling: Theory and Algorithms. Morgan Kaufmann, first edition, 1999.
J. Hoschek and D. Lasser. Computer-Aided Geometric Design. AK Peters, first edition, 1993.
Erwin Kreyszig. Differential Geometry. Dover, first edition, 1991.
Henry P. Moreton. Minimum curvature variation curves, networks, and surfaces for fair free-form shape design. PhD thesis, University of California, Berkeley, 1993.
Les Piegl and Wayne Tiller. The NURBS Book. Monograph in Visual Communications. Springer-Verlag, first edition, 1995.
WilliamWelch. Serious Putty: Topological Design for Variational Curves and Surfaces. PhD thesis, Carnegie Mellon University, Pittsburgh, Pa., 1995.
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Gallier, J. (2011). Basics of the Differential Geometry of Curves. In: Geometric Methods and Applications. Texts in Applied Mathematics, vol 38. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9961-0_19
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DOI: https://doi.org/10.1007/978-1-4419-9961-0_19
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