Skip to main content
SpringerLink
Log in

An improved algorithm for constructing moving quadrics from moving planes

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper proposes an improved algorithm to construct moving quadrics from moving planes that follow a tensor product surface with no base points, assuming that there are no moving planes of low degree following the surface. These moving quadrics provide an efficient method to implicitize the tensor product surface which outperforms a previous approach by the present authors.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Sederberg T W, Anderson D, and Goldman R, Implicit representation of parametric curves and surfaces, Computer Vision, Graphics, and Image Processing, 1984, 28: 72–84.

    MATH  Google Scholar 

  2. Buchberger B, Applications of Groebner bases in nonlinear computational geometry, Geometric Reasoning, eds. by Kapur D, Mundy J, Elsevier Science Publisher, MIT Press, Cambridge, 1989, 413–446.

    Google Scholar 

  3. Chionh E, Zhang M, and Goldman R, Implicitization by Dixon A-resultants, Proceedings of Geometric Modeling and Processing 2000, Hong Kong, 2000, 310–318.

    Google Scholar 

  4. Chionh E and Sederberg T W, On the minors of the implicitization bézout matrix for a rational plane curve, Computer Aided Geometric Design, 2001, 18: 21–36.

    Article  MathSciNet  MATH  Google Scholar 

  5. Wu W T, Mechanical Theorem Proving in Geometries: Basic Principle, Springer-Verlag, Berlin, 1994.

    Book  Google Scholar 

  6. Gao X S and Chou S, Implicitization of rational parameteric equations, Journal of Symbolic Computation, 1992, 14(5): 459–470.

    Article  MathSciNet  MATH  Google Scholar 

  7. Li Z M, Automatic implicitization of parametric objects, MM Research Preprints, 1989, 4: 54–62.

    Google Scholar 

  8. Chionh E and Goldman, R, Using multivariate resultants to find the implicit equation of a rational surface, The Visual Computer: International Journal of Computer Graphics, 1992, 8: 171–180.

    Article  Google Scholar 

  9. Chen F L, Recent advances on surface implicitization, Journal of University of Science and Technology of China, 2014, 44(5): 345–361.

    MathSciNet  Google Scholar 

  10. Sederberg T W and Chen F, Implicitization using moving curves and surfaces, Proceedings of SIGGRAPH 95, Computer Graphics Proceedings, Annual Conference Series, ACM, 1995, 301–308.

    Chapter  Google Scholar 

  11. Chen F, Cox D A, and Liu Y, The μ-bases and implicitization of a rational parametric surface, Journal of Symbolic Computation, 2005, 39(6): 689–706.

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen F, Wang W, and Liu Y, Computing singularpoints of rational curve, Journal of Symbolic Computation, 2008, 43(2): 92–117.

    Article  MathSciNet  Google Scholar 

  13. Busé L and Chardin M, Implicitizing rational hypersurfaces using approximation complexes, Journal of Symbolic Computation, 2005, 40(4–5): 1150–1168.

    Article  MathSciNet  MATH  Google Scholar 

  14. Cox D, Goldman R, and Zhang M, On the validity of implicitization by moving quadrics for rational surfaces with no base points, Journal of Symbolic Computation, 2000, 29(3): 419–440.

    Article  MathSciNet  MATH  Google Scholar 

  15. Deng J, Chen F, and Shen L, Computing μ-bases of rational curves and surfaces using polynomial matrix factorization, Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation, ACM, 2005, 132–139.

    Chapter  Google Scholar 

  16. D’andrea C, Resultants and Moving Surfaces, Journal of Symbolic Computation, 2001, 31(5): 585–602.

    Article  MathSciNet  MATH  Google Scholar 

  17. Sederberg T W and Saito T, Rational ruled surfaces: Implicitization and section curves, CVGIP: Graphical Models and Image Processing, 1995, 57: 334–342.

    Google Scholar 

  18. Zhang M, Chionh E W, and Goldman R, On a relationship between the moving line and moving conic coefficient matrices, Computer Aided Geometric Design, 1999, 16: 517–527.

    Article  MathSciNet  MATH  Google Scholar 

  19. Khetan A and D’Andrea C, Implicitization of rational surfaces using toric varieties, Journal of Algebra, 2006, 303(2): 543–565.

    Article  MathSciNet  MATH  Google Scholar 

  20. Shi X and Goldman R, Implicitizing rational surfaces of revolution using μ-bases, Computer Aided Geometric Design, 2012, 29: 348–362.

    Article  MathSciNet  MATH  Google Scholar 

  21. Zheng J and Sederberg T W, A direct approach to computing the-basis of planar rational curves, Journal of Symbolic Computation, 2001, 31(5): 619–629.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Chen F and Wang W, Revisiting the μ-basis of a rational ruled surface, Journal of Symbolic Computation, 2003, 36(5): 699–716.

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang X and Chen F, Implicitization, parameterization and singularity computation of Steiner surfaces using moving surfaces, Journal of Symbolic Computation, 2012, 47(6): 733–750.

    Article  MathSciNet  MATH  Google Scholar 

  25. Shen L, Chen F, and Deng J, Implicitization and parametrization of quadratic and cubic surfaces by μ-bases, Computing, 2006, 5: 131–142.

    MathSciNet  Google Scholar 

  26. Dohm M, Implicitization of rational ruled surfaces with μ-bases, Journal of Symbolic Computation, 2009, 44(5): 479–489.

    Article  MathSciNet  MATH  Google Scholar 

  27. Busé L, Elkadi M, and Galligo A, A computational study of ruled surfaces, Journal of Symbolic Computation, 2009, 44(3): 232–241.

    Article  MathSciNet  MATH  Google Scholar 

  28. Busé L, Cox D A, and D’andrea C, Implicitization of surfaces in P3 in the presense of base points, Journal of Algebra and Its Applications, 2003, 2: 189–214.

    Article  MathSciNet  MATH  Google Scholar 

  29. Adkins W A, Hoffman J W, and Wang H, Equations of parametric surfaces with base points via syzygies, Journal of Symbolic Computation, 2005, 39(1): 73–101.

    Article  MathSciNet  MATH  Google Scholar 

  30. Zheng J, Sederberg T W, Chionh E W, et al., Implicitizing rational surfaces with base points using the method of moving surfaces, Contemporary Mathematics, 2003, 334: 151–168.

    Article  MathSciNet  MATH  Google Scholar 

  31. Lai Y and Chen F, Implicitizing rational surfaces using moving quadrics constructed from moving planes, Journal of Symbolic Computation, 2016, 77: 127–161.

    Article  MathSciNet  MATH  Google Scholar 

  32. Shen L Y and Pérez-Díaz S, Determination and (re)parametrization of rational developable surfaces, Journal of Systems Science and Complexity, 2015, 28(6): 1426–1439.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Gongshang University, Hangzhou, 310018, China

    Yisheng Lai

  2. Department of Mathematics, University of Science and Technology of China, Hefei, 230026, China

    Falai Chen

Authors
  1. Yisheng Lai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Falai Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yisheng Lai.

Additional information

This paper was supported by the National Natural Science Foundation of China under Grant Nos. 11271328 and 11571338, the Zhejiang Provincial Natural Science Foundation under Grant No. Y7080068.

This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lai, Y., Chen, F. An improved algorithm for constructing moving quadrics from moving planes. J Syst Sci Complex 30, 1483–1506 (2017). https://doi.org/10.1007/s11424-017-6049-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-017-6049-0

Keywords

Search

Navigation