A Symbolic-Numeric Approach for Parametrizing Ruled Surfaces


This paper presents symbolic algorithms to determine whether a given surface (implicitly or parametrically defined) is a rational ruled surface and find a proper parametrization of the ruled surface. However, in practical applications, one has to deal with numerical objects that are given approximately, probably because they proceed from an exact data that has been perturbed under some previous measuring process or manipulation. For these numerical objects, the authors adapt the symbolic algorithms presented by means of the use of numerical techniques. The authors develop numeric algorithms that allow to determine ruled surfaces “close” to an input (not necessarily ruled) surface, and the distance between the input and the output surface is computed.

This is a preview of subscription content, log in to check access.


  1. [1]

    Andradas C, Recio R, Sendra J, et al., Proper real reparametrization of rational ruled surfaces, Computer Aided Geometric Design, 2011, 28: 102–113.

    MathSciNet  Article  Google Scholar 

  2. [2]

    Bo P, Bartoň M, and Pottmann H, Automatic fitting of conical envelopes to free-form surfaces for flank CNC machining, Computer-Aided Design, 2017, 91: 84–94.

    Article  Google Scholar 

  3. [3]

    Busé L, Elkadi M, and Galligo A. A computational study of ruled surfaces, Journal of Symbolic Computation, 2009, 44: 232–241.

    MathSciNet  Article  Google Scholar 

  4. [4]

    Chen F, Reparametrization of a rational ruled surface using the μ-basis, Computer Aided Geometric Design, 2009, 20: 11–17.

    MathSciNet  Article  Google Scholar 

  5. [5]

    Chen F, Zheng J, and Sederberg T W, The μ-basis of a rational ruled surface, Computer Aided Geometric Design, 2001, 18: 61–72.

    MathSciNet  Article  Google Scholar 

  6. [6]

    Chen Y, Shen L Y, and Yuan C, Collision and intersection detection of two ruled surfaces using bracket method, Computer Aided Geometry Design, 2011, 28: 114–126.

    MathSciNet  Article  Google Scholar 

  7. [7]

    Dohm M, Implicitization of rational ruled surfaces with μ-bases, Journal of Symbolic Computation, 2009, 44: 479–489.

    MathSciNet  Article  Google Scholar 

  8. [8]

    Jia X, Chen F, and Deng J, Computing self-intersection curves of rational ruled surfaces, Computer Aided Geometric Design, 2009, 26: 287–299.

    MathSciNet  Article  Google Scholar 

  9. [9]

    Izumiya S and Takeuchi N, Special curves and ruled surfaces, Contributions to Algebra and Geometry, 2003, 44(1): 203–212.

    MathSciNet  MATH  Google Scholar 

  10. [10]

    Li J, Shen L Y, and Gao X S, Proper reparametrization of ruled surface, Journal of Comput. Sci. and Tech., 2008, 5: 290–297.

    MathSciNet  Article  Google Scholar 

  11. [11]

    Liu Y, Pottmann H, Wallner J, et al., Geometric modeling with conical meshes and developable surfaces, ACM Transactions on Graphics, 2006, 25(3): 1–9.

    Article  Google Scholar 

  12. [12]

    Pérez-Díaz S and Sendra J R, A univariate resultant based implicitization algorithm for surfaces, Journal of Symbolic Computation, 2008, 43: 118–139.

    MathSciNet  Article  Google Scholar 

  13. [13]

    Peternell M, Sendra J, and Sendra J, Cissoid constructions of augmented rational ruled surfaces, Computer Aided Geometric Design, 2018, 60: 1–9.

    MathSciNet  Article  Google Scholar 

  14. [14]

    Sendra J, Sevilla D, and Villarino C, Covering rational ruled surfaces, Math. Comp., 2017, 86: 2861–2875.

    MathSciNet  Article  Google Scholar 

  15. [15]

    Shen L Y, Computing μ-bases from algebraic ruled surfaces, Computer Aided Geometric Design, 2016, 46: 125–130.

    MathSciNet  Article  Google Scholar 

  16. [16]

    Shen L Y and Yuan C, Implicitization using univariate resultants, Journal of Systems Science and Complexity, 2010, 23(4): 804–814.

    MathSciNet  Article  Google Scholar 

  17. [17]

    Shen L Y, Cheng J, and Jia X, Homeomorphic approximation of the intersection curve of two rational surfaces, Computer Aided Geometric Design, 2012, 29(8): 613–625.

    MathSciNet  Article  Google Scholar 

  18. [18]

    Shen L Y and Piérez-Díaz S, Characterization of rational ruled surfaces, Journal of Symbolic Computation, 2014, 63: 21–45.

    MathSciNet  Article  Google Scholar 

  19. [19]

    Piérez-Díaz S, Rueda S, Sendra J, et al., Approximate parametrization of plane algebraic curves by linear systems of curves, Computer Aided Geometric Design, 2010, 27: 212–231.

    MathSciNet  Article  Google Scholar 

  20. [20]

    Sendra J, Winkler F, and Piérez-Díaz S, Rational algebraic curves: A computer algebra approach, Series: Algorithms and Computation in Mathematics, Springer Verlag, 2007.

  21. [21]

    Corless R, Giesbrecht M, Kotsireas I, et al., Towards factoring bivariate approximate polynomials, Proc. ISSAC 2001, London, Bernard Mourrain, Ed. ACM Press, 2001, 85–92.

  22. [22]

    Corless R, Giesbrecht M, Jeffrey D, et al., Approximate polynomial decomposition, Proc. ISSAC 1999, Vancouver, Ed. by Dooley S S, ACM Press, 1999, 213–220.

  23. [23]

    Galligo A and Rupprech D, Irreducible decomposition of curves, Journal of Symbolic Computation, 2002, 33: 661–677.

    MathSciNet  Article  Google Scholar 

  24. [24]

    Kaltofen E, May J, Yang Z, et al., Approximate factorization of multivariate polynomials using singular value decomposition, Journal of Symbolic Computation, 2008, 43: 359–376.

    MathSciNet  Article  Google Scholar 

  25. [25]

    Sasaki T, Approximate multivariate polynomial factorization based on zero-sum relations, Proc. ISSAC 2001, New York, NY, ACM Press, 2001, 284–291.

  26. [26]

    Beckermann B and Labahn G, A fast and numerically stable Euclidean-like algorithm for detecting relatively prime numerical polynomials, Journal of Symbolic Computation, 1998, 26: 691–714.

    MathSciNet  Article  Google Scholar 

  27. [27]

    Beckermann B and Labahn G, When are two numerical polynomials relatively prime?, Journal of Symbolic Computation, 1998, 26: 677–689.

    MathSciNet  Article  Google Scholar 

  28. [28]

    Karmarkar N and Lakshman Y, Approximate polynomial greatest common divisors and nearest singular polynomials, ISSAC 1996, ACM Press, 1996, 35–39.

Download references

Author information



Corresponding author

Correspondence to Li-Yong Shen.

Additional information

This work has been partially supported by FEDER/Ministerio de Ciencia, Innovación y Universidades — Agencia Estatal de Investigacin/MTM2017-88796-P (Symbolic Computation: New challenges in Algebra and Geometry together with its applications), and the National Natural Science Foundation of China under Grant No. 61872332 and the University of Chinese Academy of Sciences. The first author belongs to the Research Group ASYNACS (Ref. CCEE2011/R34).

This paper was recommended for publication by Editor LI Hongbo.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pérez-Díaz, S., Shen, L. A Symbolic-Numeric Approach for Parametrizing Ruled Surfaces. J Syst Sci Complex 33, 799–820 (2020). https://doi.org/10.1007/s11424-019-8188-y

Download citation


  • Implicit representation
  • numeric algorithm
  • ruled surface
  • standard parametrization