Reconsidering algorithms for real parametric curves

  • Cesar Alonso
  • Jaime Gutierrez
  • Tomas Recio


Complex parametric curves have been subject of a symbolic algorithm approach in recent years. In this paper we analyze the theoretical applicability of some of these algorithms to the real parametric curve case. In particular, we show how several results are valid both over the real and the complex numbers, as they hold equivalently over a real curve and its complexification. Therefore, the standard algorithms for the complex case can be applied to obtain real answers in the real case. A second issue in our paper is the study of the very different behaviour of the real parametric mapping and we characterize here the properties of being (almost) injective or surjective.


Real parametric curves Parametrizations Algorithms CAD 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abhyankar, S., Bajaj, C.: Automatic parametrization of rational curves and surfaces III: algebraic plane curves. Computer Aided Geometric Design 5-4, 309–323 (1988)Google Scholar
  2. 2.
    Abhyankar, S., Bajaj, C.: Computation with algebraic curves. Proc. ISSAC 88. Lecture Notes in Computer Science, Vol.358, pp. 279–284, Berlin, Heidelberg, New York: Springer 1989Google Scholar
  3. 3.
    Abhyankar, S.: Algebraic Geometry for scientists and engineers. Math Surveys Monographs35. A.M.S. 1990Google Scholar
  4. 4.
    Becker, E., Neuhaus, R.: Computation of the real radical of an ideal. In: Eyssette, F., Galligo, A. (eds) Proceedings Mega 92, Computational Algebraic Geometry. Progress in Mathematics, pp. 1–20. Boston-Basel-Berlin: Birkläuser 1993Google Scholar
  5. 5.
    Bochnak, J., Coste, M., Roy, M.F.: Géométrie Algébrique Réelle. Berlin, Heidelberg, New York: Springer 1987Google Scholar
  6. 6.
    Chevalley, C.: Introduction to the theory of algebraic functions of one variable. Mathematical Surveys, VI, A.M.S. 1951Google Scholar
  7. 7.
    Coste, M., Roy, M.F.: Thorn's lemma, the coding of real algebraic numbers and the topology of semi-algebraic sets. J. Symbolic Computat5, 121–129 (1988)Google Scholar
  8. 8.
    Cox, D., Little, J., O'Shea, D.: Ideals, varieties and algorithms. Berlin, Heidelberg, New York: UTM-Springer 1992Google Scholar
  9. 9.
    Cucker, F., Pardo, L. M., Raimondo, M., Recio, T., Roy, M. F.: On local and global analytic branches of a real algebraic curve. Lecture Notes in Computer Science, Vol. 356, 161–182, Berlin, Heidelberg, New York: Springer 1989Google Scholar
  10. 10.
    Dubois, D. W., Efroymson, G.: Algebraic theory of real varieties 1. Studies and essays presented to Y. H. Chen for his 60th birthday, pp. 107–135. Taipei: Math. Res. Center Nat. Taiwan Univ. 1970Google Scholar
  11. 11.
    Gao, X.-S., Chou, S. Ch., Li, Z. M.: Computation with rational parametric equations. MM. Researchs Preprints.6, 56–78 (1991)Google Scholar
  12. 12.
    Gao, X.-S.: On the theory of resolvents and its applications. MM. Researchs Preprints.6, 79–93 (1991)Google Scholar
  13. 13.
    Gianni, P., Trager, B., Zacharias, G.: Groebner basis and primary decomposition of polynomial ideals. Computational aspects on Commutative Algebra. Special issue of the Journal of Symbolic Computation6(2–3), 149–168 (1988)Google Scholar
  14. 14.
    Giusti, M.: Combinatorial dimension theory of algebraic varieties. Computational aspects on Commutative Algebra. Special issue of the Journal of Symbolic Computation6(2–3), 249–265 (1988)Google Scholar
  15. 15.
    González, L., Lombardi, H., Recio, T., Roy, M.F.: Sous résultants et spécialisation de la suite de Sturm I." Informatique Theorique et Applications.24, 561–588 (1990)Google Scholar
  16. 16.
    Gutierrez, J., Recio, T.: Rational function decomposition and Groebner basis in the parametrization of a plane curve. Lecture Notes in Computer Science, Vol.583, pp. 231–246, Berlin, Heidelberg, New York: Springer 1992Google Scholar
  17. 17.
    Heintz, J., Recio, T., Roy, M.F.: Algorithms in Real Algebraic Geometry and applications to Computational Geometry. DIMACS, Series in Discrete Mathematics and Theoretical Computer Science,6, 137–163, 1991Google Scholar
  18. 18.
    Kalkbrener, M.: Three contributions to elimination theory. Ph.D. thesis, Institut fur Mathematik, University of Linz, Austria 1991Google Scholar
  19. 19.
    Kredel, H., Weispfenning, V.: Computing dimension and independent sets for polynomial ideals. Computational aspects on Commutative Algebra. Special issue of the Journal of Symbolic Computation6(2–3), 213–248 (1988)Google Scholar
  20. 20.
    Manocha, D., Canny, J.: Rational curves with polynomial parametrization. Computer-Aided Design.23(9), 12–19 (1991)Google Scholar
  21. 21.
    Ollivier, F.: Inversibility of rational mappings and structural identifiability in Automatics. Proc. ISSAC 89. Portland, pp. 43–53 A.C.M. Press, 1989Google Scholar
  22. 22.
    Schinzel, A.: Selected topics on polynomials. Ann Arbor, University of Michigan Press, 1982Google Scholar
  23. 23.
    Sederberg, T. W.: Improperly parametrized rational curves. Computer Aided Geometric Design3, 67–75 (1986)Google Scholar
  24. 24.
    Sendra, J. R., Winkler, F.: Symbolic parametrization of curves. Journal of Symbolic Computation12(6), 607–631 (1991)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Cesar Alonso
    • 1
  • Jaime Gutierrez
    • 1
  • Tomas Recio
    • 1
  1. 1.Departamento de Matemáticas, Estadistica y ComputaciónUniversidad de CantabriaE-SantanderSpain

Personalised recommendations