Implicitization and Distance Bounds

  • Martin Aigner
  • Ibolya Szilágyi
  • Bert Jüttler
  • Josef Schicho
Part of the Mathematics and Visualization book series (MATHVISUAL)


Convex Hull Condition Number Parametric Representation Implicit Representation Rational Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Aigner, B. Jüttler, and M.-S. Kim. Analyzing and enhancing the robustness of implicit representations. In Geometric modeling and Processing, pages 131-142. IEEE Press, 2004.Google Scholar
  2. 2.
    C. Alonso, J. Gutierrez, and T. Recio. An implicitization algorithm with fewer variables. Comp. Aided Geom. Design, 12:251-258, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    B. Buchberger Applications of Gröbner bases in nonlinear computational geometry. In Trends in computer algebra, pages 52-80. Springer, Berlin, 1988.Google Scholar
  4. 4.
    L. Busé. Residual resultant over the projective plane and the implicitization problem. In Proc. ISSAC, pages 48-55, New York, 2001. ACM.Google Scholar
  5. 5.
    F. Chen. Approximate implicitization of rational surfaces. In Computational geometry (Beijing, 1998), volume 34, pages 57-65. Amer. Math. Soc., Provi-dence, RI, 2003.Google Scholar
  6. 6.
    F. Chen and L. Deng. Interval implicitization of rational curves. Comput. Aided Geom. Design, 21:401-415, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    R.M. Corless, M.W. Giesbrecht, I.S. Kotsireas, and S.M. Watt Numerical im-plicitization of parametric hypersurfaces with linear algebra. In AISC 2000, LNCS, pages 174-183. Springer, Berlin, 2001.Google Scholar
  8. 8.
    T. Dokken. Approximate implicitization. In Mathematical methods for curves and surfaces, pages 81-102. Vanderbilt Univ. Press, Nashville, TN, 2001.Google Scholar
  9. 9.
    T. Dokken and J. Thomassen Overview of approximate implicitization. In Topics in algebraic geometry and geometric modeling, volume 334, pages 169-184. Amer. Math. Soc., Providence, RI, 2003.MathSciNetGoogle Scholar
  10. 10.
    M. Elkadi and B. Mourrain. Residue and implicitization problem for rational surfaces. Appl. Algebra Engrg. Comm. Comp., 14:361-379, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    G. Farin, J. Hoschek, and M.-S. Kim, editors. Handbook of Computer Aided Geometric Design. Elsevier, 2002.Google Scholar
  12. 12.
    R.T. Farouki. On the stability of transformations between power and Bernstein polynomial forms. Comp. Aided Geom. Design, 8:29-36, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    L. Gonzalez-Vega. Implicitization of parametric curves and surfaces by using multidimensional Newton formulae. J. Symb. Comp., 23:137-151, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    J.E. Goodman and J. O’Rourke, editors. Handbook of discrete and computational geometry. Chapman & Hall, Boca Raton, FL, 2004.zbMATHGoogle Scholar
  15. 15.
    ınez. Implicitization of rational surfaces by means of polynomial interpolation. Comput. Aided Geom. Design, 19:327-344, 2002.Google Scholar
  16. 16.
    J. Schicho and I. Szilágyi. Numerical Stability of Surface Implicitization. Journal of Symbolic Computation, page to appear, 2004.Google Scholar
  17. 17.
    T.W. Sederberg, T. Saito, D.X. Qi, and K.S. Klimaszewski. Curve implicitiza-tion using moving lines. Comp. Aided Geom. Des., 11:687-706, 1994.CrossRefMathSciNetGoogle Scholar
  18. 18.
    M. Shalaby et al. Piecewise approximate implicitization: Experiments using industrial data. In Algebraic Geometry and Geometric Modeling (this volume). Springer, 2006, 37-51.Google Scholar
  19. 19.
    M. Shalaby, B. Jüttler, and J. Schicho C 1 spline implicitization of planar curves. In Automated deduction in geometry, volume 2930 of LNCS, pages 161-177. Springer, Berlin, 2004.Google Scholar
  20. 20.
    H. Shou, H. Lin, R.R. Martin, and G. Wang. Modified affine arithmetic is more accurate than centred interval arithmetic or affine arithmetic. In The Mathematics of Surfaces X, volume 2768 of LNCS, pages 355-365. Springer, Heidelberg, 2003.Google Scholar
  21. 21.
    J. Zheng, T.W. Sederberg, E.-W. Chionh, and David A. Cox. Implicitizing rational surfaces with base points using the method of moving surfaces. In Topics in algebraic geometry and geometric modeling, pages 151-168. AMS, Providence, RI, 2003.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Martin Aigner
    • 1
  • Ibolya Szilágyi
    • 2
  • Bert Jüttler
    • 3
  • Josef Schicho
    • 4
  1. 1.Institute of Applied GeometryJohannes Kepler UniversityLinzAustria
  2. 2.RISC–Linz/RICAMJohannes Kepler UniversityLinzAustria
  3. 3.Institute of Applied GeometryJohannes Kepler UniversityLinzAustria
  4. 4.RICAMAustrian Academy of SciencesLinzAustria

Personalised recommendations