Plotting missing points and branches of real parametric curves



This paper is devoted to the study (from the theoretic and algorithmic point of view) of the existence of points and branches non-reachable by a parametric representation of a rational algebraic curve (in n-dimensional space) either over the field of complex numbers or over the field of real numbers. In particular, we generalize some of the results on missing points in (J. Symbolic Comput. 33, 863–885, 2002) to the case of space curves. Moreover, we introduce for the first time and we solve the case of missing branches. Another novelty is the emphasis on topological conditions over the curve for the existence of missing points and branches. Finally, we would like to point out that, by developing an “ad hoc” and simplified theory of valuations for the case of parametric curves, we approach in a new and unified way the analysis of the missing points and branches, and the proposal of the algorithmic solution to these problems.


Parametric curves Normal parametrization Branches Valuation rings 

Mathematics Subject Classification (2000)

14-Q05 68-W30 


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  1. 1.
    Abhyankar, S.: Algebraic Geometry for Scientists and Engineers, Mathematical Surverys and Monographs vol. 35 American Mathematical Society, Providence. 1990Google Scholar
  2. 2.
    Alonso, C., Gutierrez, J., Recio, T.: Real parametric curves: some symbolic algorithm issues. 14th IMACS (Institute for Mathematics and Computers in Simulation) World Symposium. Atlanta (1994)Google Scholar
  3. 3.
    Alonso C., Gutierrez J. and Recio T. (1995). Reconsidering algorithms for real parametric curves. J. AAECC 6: 345–352 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Alonso C., Gutierrez J. and Recio T. (1995). A rational fuction decomposition algorithm by near separated polynomials. J. Symbolic Comput. 19: 527–544 CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Bajaj C. and Royappa A. (1995). Finite representations of real parametric curves and surfaces.Technical Report, CAPO report CER-92-28, Purdue University. Also in International J. Comput. Geometry Appl. 5: 313–326 CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Canny J. and Manocha D. (1991). Rational curves with polynomial parametrization. Comput-Aided Des. 23: 645–652 CrossRefMATHGoogle Scholar
  7. 7.
    Cox D., Little J. and O’Shea D. (1991). Ideals, Varieties and Algorithms. Undergraduate Texts in Mathematics. Springer, Berlin Heidelberg New York Google Scholar
  8. 8.
    Chou S.C. and Gao X.S. (1991). On the normal parametrization of curves and surfaces. Int. J. Comput. Geometry Appl. 1: 125–136 CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Gonzalez-Lopez M.J., Recio T. and Santos F. (1996). Parametrization of semialgebraic sets. Math. Comput. Simul. 46: 353–362 CrossRefMathSciNetGoogle Scholar
  10. 10.
    Lang S. (2002). Algebra Graduate Texts in Mathematics 3rd edn. Springer, Berlin Heidelberg New York Google Scholar
  11. 11.
    Sederberg T.W. (1986). Improperly parametrized rational curves. Comput-Aided Geometric Des. 3: 67–75 CrossRefMATHGoogle Scholar
  12. 12.
    Sendra J.R. (2002). Normal parametrizations of algebraic plane curves. J. Symbolic Comput. 33: 863–885 CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Walker R. (1950). Algebraic Curves. Dover, New York MATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Departamento de ÁlgebraUniversidad ComplutenseMadridSpain
  2. 2.Departamento de MatemáticasUniversidad de CantabriaSantanderSpain

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