Intersecting Biquadratic Bézier Surface Patches

  • Stéphane Chau
  • Margot Oberneder
  • André Galligo
  • Bert Jüttler

We present three symbolic—numeric techniques for computing the intersection and self—intersection curve(s) of two Bézier surface patches of bidegree (2,2). In particular, we discuss algorithms, implementation, illustrative examples and provide a comparison of the methods.


Surface Patch Parameter Domain Intersection Curve Parameter Line Implicit Equation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Andersson, J. Peters, and N. Stewart, Self-intersection of composite curves and sur-faces, Computer Aided Geometric Design, 15 (1998), pp. 507-527.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    L. Busé , Etude du résultant sur une variété algébrique, PhD thesis, University of Nice, December 2001.Google Scholar
  3. 3.
    L. Busé and C. D’Andrea, Inversion of parameterized hypersurfaces by means of subresultants, Proceedings ACM of the ISSAC 2004, pp. 65-71.Google Scholar
  4. 4.
    L. Busé , M. Elkadi, and B. Mourrain, Using projection operators in Computer Aided Geometric Design, In Topics in Algebraic Geometry and Geometric Modeling, pp. 321-342, Contemporary Mathematics, AMS, 2003.Google Scholar
  5. 5.
    L. Busé , I.Z. Emiris and B. Mourrain, MULTIRES, http://www-sop.inria. fr/galaad/logiciels/multires.
  6. 6.
    L. Busé and A. Galligo, Using semi-implicit representation of algebraic surfaces, Proceedings of the SMI 2004 conference, IEEE Computer Society, pp. 342-345.Google Scholar
  7. 7.
    E.W. Chionh and R.N. Goldman, Using multivariate resultants to find the implicit equation of a rational surface, The Visual Computer 8 (1992), pp. 171-180.CrossRefGoogle Scholar
  8. 8.
    D. Cox, J. Little and D. O’Shea, Ideals, Varieties and Algorithms, Springer-Verlag, New York, 1992 and 1997.zbMATHGoogle Scholar
  9. 9.
    D. Cox, J. Little and D. O’Shea, Using Algebraic Geometry, Springer-Verlag, New York, 1998.zbMATHGoogle Scholar
  10. 10.
    C. D’Andrea, Macaulay style formulas for sparse resultants, Trans. Amer. Math. Soc., 354(7)(2002), pp. 2595-2629.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    T. Dokken, Aspects of Intersection Algorithms and Approximation, Thesis for the doctor philosophias degree, University of Oslo, Norway 1997.Google Scholar
  12. 12.
    T. Dokken, Approximate implicitization, Mathematical Methods for Curves and Surfaces, T. Lyche and L.L. Schumaker (eds.), Vanderbilt University Press, 2001, pp. 81-102.Google Scholar
  13. 13.
    T. Dokken and J.B. Thomassen, Overview of Approximate Implicitization, Topics in Algebraic Geometry and Geometric modeling, ed. Ron Goldman and Rimvydas Krasauskas, AMS series on Contemporary Mathematics CONM 334, 2003, pp. 169-184.Google Scholar
  14. 14.
    G. Elber and M-S. Kim, Geometric Constraint Solver using Multivariate Rational Spline Functions, The Sixth ACM/IEEE Symposium on Solid Modeling and Applications, 2001, pp. 1-10.Google Scholar
  15. 15.
    M. Elkadi and B. Mourrain, Some applications of Bezoutians in Effective Algebraic Geometry, Rapport de Recherche 3572, INRIA, Sophia Antipolis, 1998.Google Scholar
  16. 16.
    G. Farin, J. Hoschek and M-S. Kim, Handbook of Computer Aided Geometric Design, Elsevier, 2002.Google Scholar
  17. 17.
    L. González-Vega and I. Necula, Efficient topology determination of implicitly defined algebraic plane curves, Comput. Aided Geom. Design, 19(9) (2002), pp. 719-743.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    C. M. Hoffmann, Implicit Curves and Surfaces in CAGD, Comp. Graphics and Appl. (1993), pp. 79-88.Google Scholar
  19. 19.
    M. E. Hohmeyer, A Surface Intersection Algorithm Based on Loop Detection, ACM Symposium on Solid Modeling Foundations and CAD/CAM Applications, 1991, pp. 197-207.Google Scholar
  20. 20.
    J. Hoschek and D. Lasser, Fundamentals of Computer Aided Geometric Design, A.K. Peters, 1993.Google Scholar
  21. 21.
    S. Chau and M. Oberneder,˜margot/biquad
  22. 22.
    A. Khetan, The resultant of an unmixed bivariate system, J. of Symbolic Computation, 36 (2003), pp. 425-442.∼khetan/software.html
  23. 23.
    S. Krishnan and D. Manocha, An Efficient Surface Intersection Algorithm Based on Lower-Dimensional Formulation, ACM Transactions on Graphics, 16(1) (1997), pp. 74-106.CrossRefGoogle Scholar
  24. 24.
    L. Kunwoo, Principles of CAD/CAM/CAE Systems, Addison-Wesley, 1999.Google Scholar
  25. 25.
    B. Mourrain and J.-P. Pavone, Subdivision methods for solving polynomial equations, Technical Report 5658, INRIA Sophia-Antipolis, 2005.Google Scholar
  26. 26.
    T. Nishita, T.W. Sederberg and M. Kakimoto, Ray tracing trimmed rational surface patches, Siggraph, 1990, pp. 337-345.Google Scholar
  27. 27.
    N.M. Patrikalakis, Surface-to-surface intersections, IEEE Computer Graphics and Applications, 13(1) (1993), pp. 89-95.CrossRefGoogle Scholar
  28. 28.
    N. Patrikalakis and T. Maekawa, Chapter 25: Intersection problems, Handbook of Computer Aided Geometric Design (G. Farin and J. Hoschek and M.-S. Kim, eds.), Elsevier, 2002.Google Scholar
  29. 29.
    J.-P. Pavone, Auto-intersection des surfaces paramétrées réelles, Thèse d’informatique de l’Université de Nice Sophia-Antipolis, Décembre 2004.Google Scholar
  30. 30.
    J.P. Técourt, Sur le calcul effectif de la topologie de courbes et surfaces implicites, PhD thesis in Computer Science at INRIA Sophia-Antipolis, Décembre 2005.Google Scholar
  31. 31.
    J.B. Thomassen, Self-Intersection Problems and Approximate Implicitization, Computational Methods for Algebraic Spline Surfaces, Springer, pp. 155-170, 2005.Google Scholar
  32. 32.
    A. Vlachos, J. Peters, C. Boyd and J. L. Mitchell, Curved PN Triangles, Symposium on Interactive 3D Graphics, Bi-Annual Conference Series, ACM Press, 2001, 159-166.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Stéphane Chau
    • 1
  • Margot Oberneder
    • 2
  • André Galligo
    • 1
  • Bert Jüttler
    • 2
  1. 1.Laboratoire J.A. DieudonnéUniversité de Nice - Sophia-AntipolisFrance
  2. 2.Institute of Applied GeometryJohannes Kepler UniversityAustria

Personalised recommendations