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The Non-Degenerate Dupin Cyclides in the Space of Spheres Using Geometric Algebra

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Abstract

Dupin cyclides are algebraic surfaces of degree 4 discovered by the French mathematician Pierre-Charles Dupin early in the 19th century and were introduced in CAD by R. Martin in 1982. A Dupin cyclide can be defined, in two different ways, as the envelope of a oneparameter family of oriented spheres. So, it is very interesting to model the Dupin cyclides in the space of spheres, space wherein each family of spheres can be seen as a conic curve. In this paper, we model the nondegenerate Dupin cyclides and the space of spheres using Conformal Geometric Algebra. This new approach permits us to benefit from the advantages of the use of Geometric Algebra.

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References

  1. M. Berger, M. Cole, and S. Levy, Geometry II. Number vol. 2 in Universitext (1979). Springer-Verlag, 1987.

  2. G. Darboux, Leçons sur la Théorie Générale des Surfaces. Volume 1. Gauthier- Villars, 1887.

  3. Leo Dorst, Daniel Fontijne, and Stephen Mann, Geometric Algebra for Computer Science: An Object Oriented Approach to Geometry. Morgan Kauffmann Publishers, 2007.

  4. Druoton L., Garnier L., Langevin R., Marcellier H., Besnard R.: Les cyclides de Dupin et l’espace des sphères. R.E.F.I.G. 5(1), 41–59 (2011)

    Google Scholar 

  5. C. P. Dupin, Application de Géométrie et de Méchanique la Marine, aux Ponts et Chaussées, etc. Bachelier, Paris, 1822.

  6. Daniel Fontijne and Leo Dorst, GAviewer, interactive visualization software for geometric algebra. Downloadable at http://www.geometricalgebra.net/downloads.html.

  7. A. R. Forsyth, Lecture on Differential Geometry of Curves and Surfaces. Cambridge University Press, 1912.

  8. L. Garnier and L. Druoton, Constructions of principal patches of Dupin cyclides defined by constraints : four vertices on a given circle and two perpendicular tangents at a vertex. Birmingham, September 2013. XIV Mathematics od Surfaces.

  9. Langevin R., Walczak P.G.: Conformal geometry of foliations. Geom Dedicata 132(5), 135–178 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. R. R. Martin, Principal patches for computational geometry. PhD thesis, Engineering Department, Cambridge University, 1982.

  11. Pratt M. J.: Cyclides in computer aided geometric design. Computer Aided Geometric Design 7(1-4), 221–242 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Shene C. K.: Do blending and offsetting commute for Dupin cyclides. Computer Aided Geometric Design 17(9), 891–910 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. C.E.A. DAM, Valduc, F-21120, Is Sur Tille, France

    Lucie Druoton

  2. Institut de Mathématique de Bourgogne, UMR CNRS 5584 University of Burgundy, Faculté Mirande, 21000, Dijon, France

    Lucie Druoton & Rémi Langevin

  3. LE2I, UMR CNRS 6306, University of Burgundy, faculté Mirande, 21000, Dijon, France

    Lucie Druoton & Lionel Garnier

  4. XLIM-SIC, UMR CNRS 7252, University of Poitiers, SP2MI, Bld Marie et Pierre Curie, 86962, Futuroscope Chasseneuil, France

    Laurent Fuchs

Authors
  1. Lucie Druoton
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  2. Laurent Fuchs
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  3. Lionel Garnier
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  4. Rémi Langevin
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Correspondence to Lionel Garnier.

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Druoton, L., Fuchs, L., Garnier, L. et al. The Non-Degenerate Dupin Cyclides in the Space of Spheres Using Geometric Algebra. Adv. Appl. Clifford Algebras 24, 515–532 (2014). https://doi.org/10.1007/s00006-014-0453-5

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  • DOI: https://doi.org/10.1007/s00006-014-0453-5

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