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Basics of the Differential Geometry of Curves

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Geometric Methods and Applications

Part of the book series: Texts in Applied Mathematics ((TAM,volume 38))

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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

  1. 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.

    Google Scholar 

  2. 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.

    Google Scholar 

  3. 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.

    Google Scholar 

  4. Eugenio Calabi, Peter J. Olver, and Allen Tannenbaum. Affine geometry, curve flows, and invariant numerical approximations. Advances in Mathematics, 124:154–196, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  5. ’ Elie Cartan. Les syst`emes diff´erentiels ext´erieurs et leurs applications g´eom´etriques. Hermann,first edition, 1945.

    Google Scholar 

  6. Gaston Darboux. Lec¸ons sur la th´eorie g´en´erale des surfaces, Premi`ere Partie. Gauthier-Villars, second edition, 1914.

    Google Scholar 

  7. Manfredo P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.

    Google Scholar 

  8. Gerald A. Edgar. Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics. Springer-Verlag, first edition, 1992.

    Google Scholar 

  9. Gerald Farin. NURB Curves and Surfaces, from Projective Geometry to Practical Use. AK Peters, first edition, 1995.

    Google Scholar 

  10. Gerald Farin. Curves and Surfaces for CAGD. Academic Press, fourth edition, 1998.

    Google Scholar 

  11. J.-C. Fiorot and P. Jeannin. Courbes et Surfaces Rationelles. RMA 12. Masson, first edition,1989.

    Google Scholar 

  12. J.-C. Fiorot and P. Jeannin. Courbes Splines Rationelles. RMA 24. Masson, first edition,1992.

    Google Scholar 

  13. Jean H. Gallier. Curves and Surfaces in Geometric Modeling: Theory and Algorithms. Morgan Kaufmann, first edition, 1999.

    Google Scholar 

  14. J. Hoschek and D. Lasser. Computer-Aided Geometric Design. AK Peters, first edition, 1993.

    Google Scholar 

  15. Erwin Kreyszig. Differential Geometry. Dover, first edition, 1991.

    Google Scholar 

  16. Henry P. Moreton. Minimum curvature variation curves, networks, and surfaces for fair free-form shape design. PhD thesis, University of California, Berkeley, 1993.

    Google Scholar 

  17. Les Piegl and Wayne Tiller. The NURBS Book. Monograph in Visual Communications. Springer-Verlag, first edition, 1995.

    MATH  Google Scholar 

  18. WilliamWelch. Serious Putty: Topological Design for Variational Curves and Surfaces. PhD thesis, Carnegie Mellon University, Pittsburgh, Pa., 1995.

    Google Scholar 

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Correspondence to Jean Gallier .

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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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