Computing the Topology of Three-Dimensional Algebraic Curves

  • G. Gatellier
  • A. Labrouzy
  • B. Mourrain
  • J. P. Técourt
Conference paper


In this paper, we present a new method for computing the topology of curves defined as the intersection of two implicit surfaces. The main ingredients are projection tools, based on resultant constructions and 0-dimensional polynomial system solvers. We describe a lifting method for points on the projection of the curve on a plane, even in the case of multiple preimages on the 3D curve. Reducing the problem to the comparison of coordinates of so-called critical points, we propose an approach which combines control and efficiency. An emphasis in this work is put on the experimental validation of this new method. Examples treated with the tools of the library axel1 (Algebraic Software-Components for gEometric modeLing) are showing the potential of such techniques.


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  1. 1.
    K. Abdel-Malek and H.-J. Yeh. On the determination of starting points for parametric surface intersections. Computer-Aided Design, 28:21–35, 1997.CrossRefGoogle Scholar
  2. 2.
    E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling an A. McKenney, S. Ostrouchov, and D. Sorensen. LAPACK Users’ Guide. SIAM, Philadelphia, 1992. Scholar
  3. 3.
    S. Basu, R. Pollack, and M.-F. Roy. Algorithms in Real ALgebraic Geometry. Springer-Verlag, Berlin, 2003. ISBN 3-540-00973-6.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,, pages 321–342. Contemporary Mathematics, 2003.Google Scholar
  5. 5.
    D. Cox, J. Little, and D. O’Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics. Springer Verlag, New York, 1992.Google Scholar
  6. 6.
    G. Dos Reis, B. Mourrain, R. Rouillier, and Ph. Trébuchet. An environment for symbolic and numeric computation. In Proc. of the International Conference on Mathematical Software 2002, World Scientific, pages 239–249, 2002.Google Scholar
  7. 7.
    D. Eisenbud. Commutative Algebra with a view toward Algebraic Geometry, volume 150 of Graduate Texts in Math. Berlin, Springer-Verlag, 1994.Google Scholar
  8. 8.
    M. Elkadi and B. Mourrain. Introduction à la résolution des systèmes d’équations algébriques, 2003. Notes de cours, Univ. de Nice (310 p.). Soumis pour publication dans la srie mathmatiques appliques (SMAI).Google Scholar
  9. 9.
    G. Farin. An ssi bibliography. In Geometry Processing for Design and Manufacturing, pages 205–207. SIAM, Philadelphia, 1992.Google Scholar
  10. 10.
    T. Garrity and J. Warren. Geometric continuity. Comp. Aided Geom. Design, 8:51–65, 1991.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Laureano González-Vega and Ioana Necula. Efficient topology determination of implicitly defined algebraic plane curves. Comput. Aided Geom. Design, 19(9):719–743, 2002.MathSciNetCrossRefGoogle Scholar
  12. 12.
    T. A. Grandine. Applications of contouring. SIAM Review, 42:297–316, 2000.MathSciNetCrossRefGoogle Scholar
  13. 13.
    T. A. Grandine and F. W. Klein. A new approach to the surface intersection problem. Computer Aided Geometric Design, 14:111–134, 1997.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gert-Martin Greuel and Gerhard Pfister. A Singular introduction to commutative algebra. Springer-Verlag, Berlin, 2002. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann, With 1 CD-ROM (Windows, Macintosh, and UNIX).Google Scholar
  15. 15.
    J. Harris. Algebraic Geometry, a first course, volume 133 of Graduate Texts in Math. New-York, Springer-Verlag, 1992.Google Scholar
  16. 16.
    S. Krishnan and D. Manocha. An efficient intersection algorithm based on lower dimensional formulation. ACM Transactions on Computer Graphics, 16:74–106, 1997.CrossRefGoogle Scholar
  17. 17.
    B. Mourrain. A new criterion for normal form algorithms. In M. Fossorier, H. Imai, Shu Lin, and A. Poli, editors, Proc. AAECC, vol. 1719 of LNCS, pages 430–443. Springer, Berlin, 1999.Google Scholar
  18. 18.
    J. Owen and A. Rockwood. Intersection of general implicit surfaces. In Geometric Modeling: Algorithms and New Trends, pages 335–345. SIAM, Philadelphia, 1987.Google Scholar
  19. 19.
    M. P. Patrikalakis and T. Maekawa. Shape Interrogation for Computer Aided Design and Manufacturing. Springer Verlag, 2002.Google Scholar
  20. 20.
    Ph. Trébuchet. Vers une résolution stable et rapide des équations algébriques. PhD thesis, Université Pierre et Marie Curie, 2002.Google Scholar
  21. 21.
    J. von zur Gathen and J. Gerhard. Modern computer algebra. Cambridge University Press, New York, 1999.Google Scholar
  22. 22.
    Hassler Whitney. Complex analytic varieties. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • G. Gatellier
    • 1
  • A. Labrouzy
    • 1
  • B. Mourrain
    • 1
  • J. P. Técourt
    • 1
  1. 1.GALAAD, INRIASophia AntipolisFrance

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