Conversion of Dupin Cyclide Patches into Rational Biquadratic Bézier Form

  • Sebti Foufou
  • Lionel Garnier
  • Michael J. Pratt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3604)


This paper uses the symmetry properties of circles and Bernstein polynomials to establish a series of interesting barycentric properties of rational biquadratic Bézier patches. A robust algorithm is presented, based on these properties, for the conversion of Dupin cyclide patches into Bézier form. A set of conversion examples illustrates the use of this algorithm.


Control Point Curvature Line Geometric Design Median Plane Negative Weight 
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  1. 1.
    Piegl, L., Tiller, W.: A menagerie of rational B-spline circles. IEEE Computer Graphics and Applications 9, 46–56 (1989)CrossRefGoogle Scholar
  2. 2.
    Forrest, A.R.: Curves and Surfaces for Computer-Aided Design. PhD thesis, University of Cambridge, UK (1968)Google Scholar
  3. 3.
    Demengel, G., Pouget, J.: Mathématiques des Courbes et des Surfaces: Modèles de Bézier, des B-Splines et des NURBS. vol. 1. Ellipse (1998)Google Scholar
  4. 4.
    Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design. A.K.Peters, Wellesley (1993)zbMATHGoogle Scholar
  5. 5.
    Pratt, M.J.: Cyclides in computer aided geometric design. Computer Aided Geometric Design 7, 221–242 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chandru, V., Dutta, D., Hoffmann, C.M.: Variable radius blending using Dupin cyclides. In: Wozny, M., Turner, J.U., Preiss, K. (eds.) Geometric Modelling for Product Engineering, North-Holland Publ. Co, Amsterdam (1990) ; In: Proc. IFIP/NSF Workshop on Geometric Modelling, Rensselaerville, NY (September 1988)Google Scholar
  7. 7.
    Degen, W.L.F.: Generalized cyclides for use in CAGD. In: Bowyer, A. (ed.) Computer-Aided Surface Geometry and Design, pp. 349–363. Oxford University Press, Oxford (1994); In: Proc. 4th IMA Conference on the Mathematics of Surfaces, Bath, UK (September 1990)Google Scholar
  8. 8.
    Pratt, M.J.: Cyclides in computer aided geometric design II. Computer Aided Geometric Design 12, 131–152 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dutta, D., Martin, R.R., Pratt, M.J.: Cyclides in surface and solid modeling. IEEE Computer Graphics and Applications 13, 53–59 (1993)CrossRefGoogle Scholar
  10. 10.
    Paluszny, M., Boehm, W.: General cyclides. Computer Aided Geometric Design 15, 699–710 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Shene, C.: Blending two cones with Dupin cyclides. Computer Aided Geometric Design 15, 643–673 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Farin, G.: Curves And Surfaces, 3rd edn. Academic Press, London (1993)Google Scholar
  13. 13.
    Aumann, G.: Curvature continuous connections of cones and cylinders. Computer-aided Design 27, 293–301 (1995)zbMATHCrossRefGoogle Scholar
  14. 14.
    Shene, C.: Do blending and offsetting commute for Dupin cyclides? Computer Aided Geometric Design 17, 891–910 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Wallner, J., Pottmann, H.: Rational blending surfaces between quadrics. Computer Aided Geometric Design 14, 407–419 (1997)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Sebti Foufou
    • 1
  • Lionel Garnier
    • 1
  • Michael J. Pratt
    • 2
  1. 1.LE2I, UMR CNRS 5158, UFR SciencesUniversité de BourgogneDijon CedexFrance
  2. 2.LMR SystemsCarlton, BedfordUK

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