Implicitization of Rational Curves

  • Yongli Sun
  • Jianping Yu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4120)


A new technique for finding the implicit equation of a rational curve is investigated. It is based on efficient computation of the Bézout resultant and Lagrange interpolation. One of the main features of our approach is that it considerably reduces the size of intermediate expressions and results in significant speed-up in the algorithm.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alonso, C., Gutiérrez, J., Recio, T.: An implicitization algorithm with fewer variables. Computer Aided Geometric Design 12, 251–258 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Becker, T., Weispfenning, V.: Gröbner Bases: A Computational Approach to Commutative Algebra. Springer, Berlin (1993)zbMATHGoogle Scholar
  3. 3.
    Busé, L.: Residual resultant over the projective plane and the implicitization problem. In: Proceedings ISSAC 2001, pp. 48–55. ACM Press, New York (2001)CrossRefGoogle Scholar
  4. 4.
    Corless, R.M., Giesbrecht, M.W., Kotsireas, I.S., Watt, S.M.: Numerical implicitization of parametric hypersurfaces with linear algebra. In: Campbell, J.A., Roanes-Lozano, E. (eds.) AISC 2000. LNCS (LNAI), vol. 1930, pp. 174–183. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    Cox, D., Little, J., O’shea, D.: Ideals, Varieties and Algorithms. Springer, New York (1996)zbMATHGoogle Scholar
  6. 6.
    Cox, D., Little, J., O’shea, D.: Using Algebraic Geometry. Springer, New York (1998)zbMATHGoogle Scholar
  7. 7.
    Cox, D.: Equations of parametric curves and surfaces via syzygies. Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering. Contemporary Mathematics 286, 1–20 (2001)Google Scholar
  8. 8.
    Cox, D.: Curves, surfaces and syzygies. Topics in Algebraic Geometry and Geometric Modeling. In: Contemporary Mathematics, vol. 334, pp. 131–149 (2003)Google Scholar
  9. 9.
    Farin, G.: Curves and Surfaces for Computer Aided Geometric Design, 4th edn. Academic Press, Boston (1996)Google Scholar
  10. 10.
    Gao, X.S., Chou, S.C.: Implicitization of rational parametric equations. J. Symbolic Computation 14, 459–470 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Gathen, J.v.z., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  12. 12.
    Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, Basel (1994)CrossRefzbMATHGoogle Scholar
  13. 13.
    Hoffmann, C.M.: Algebraic and numerical techniques for offsets and blends. In: Dahmen, W., Gasca, M., Micchelli, C.A. (eds.) Computation of Curves and Surfaces. NATO ASI Series C: Mathematical and Physical Sciences, vol. 307, pp. 499–528. Kluwer Academic, Dordrecht (1990)Google Scholar
  14. 14.
    Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design. A.K. Peters, Wellesley (1993)zbMATHGoogle Scholar
  15. 15.
    Kapur, D., Saxena, T.: Comparison of various multivariate resultant formulations. In: Proceedings ISSAC 1995, pp. 87–194. ACM Press, New York (1995)Google Scholar
  16. 16.
    Li, Z.M.: Automatic implicitization of parametric objects. MM Research Preprints, vol. 4, pp. 54–62. Institute of Systems Science, Academia Sinica (1989)Google Scholar
  17. 17.
    Marco, A., Martínez, J.J.: Using polynomial interpolation for implicitizing algebraic curves. Computer Aided Geometric Design 18, 309–319 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Sederberg, T.W.: Improperly parametrized rational curves. Computer Aided Geometric Design 3, 67–75 (1986)CrossRefzbMATHGoogle Scholar
  19. 19.
    Sederberg, T., Chen, F.: Implicitization using moving curves and surfaces. In: Proceedings SIGGRAPH 1995, pp. 301–308. ACM Press, New York (1995)CrossRefGoogle Scholar
  20. 20.
    Sederberg, T., Goldman, R., Du, H.: Implicitizing rational curves by the method of moving algebraic curves. J. Symbolic Computation 23, 153–175 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Shi, H., Sun, Y.L.: Blending of triangular algebraic surfaces. MM Research Preprints, vol. 21, pp. 200–206. Institute of Systems Science, Academia Sinica (2002)Google Scholar
  22. 22.
    Shi, H., Sun, Y.L.: On blending of cylinders. MM Research Preprints, vol. 21, pp. 207–211. Institute of Systems Science, Academia Sinica (2002)Google Scholar
  23. 23.
    Sturmfels, B.: Introduction to resultants. In: Cox, D., Sturmfels, B. (eds.) Proceedings of Symposium in Applied Mathematics, Applications of Computational Geometry, vol. 53, pp. 25–39. American Mathematical Society, Providence (1998)Google Scholar
  24. 24.
    Wang, D.: Elimination Methods. Springer, Wien (2000)Google Scholar
  25. 25.
    Wang, D.: A simple method for implicitizing rational curves and surfaces. J. Symbolic Computation 38, 899–914 (2004)CrossRefGoogle Scholar
  26. 26.
    Wu, W.T.: Mathematics Mechanization. Science Press and Kluwer Academic, Beijing and Dordrecht (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Yongli Sun
    • 1
    • 2
  • Jianping Yu
    • 3
  1. 1.Department of Mathematics and Computer ScienceBUCTBeijing
  2. 2.KLMM, AMSSChinese Academy of SciencesBeijingChina
  3. 3.Department of Mathematics and MechanicsUniversity of Science and Technology BeijingBeijingChina

Personalised recommendations