Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 293746, 25 pages
http://dx.doi.org/10.1155/2012/293746
Research Article

Approximate Implicitization Using Linear Algebra

Applied Mathematics, SINTEF, ICT, P.O. Box 124, Blindern, 0314 Oslo, Norway

Received 12 July 2011; Revised 16 October 2011; Accepted 21 October 2011

Academic Editor: Michela Redivo-Zaglia

Copyright © 2012 Oliver J. D. Barrowclough and Tor Dokken. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. W. Sederberg and F. Chen, “Implicitization using moving curves and surfaces,” in Proceedings of the 22nd Annual ACM SIGGRAPH Conference on Computer Graphics (SIGGRAPH '95), pp. 301–308, ACM, New York, NY, USA, 1995.
  2. C. L. Bajaj, I. Ihm, and J. Warren, “Higher-order interpolation and least-squares approximation using implicit algebraic surfaces,” ACM Transactions on Graphics, vol. 12, no. 4, pp. 327–347, 1993. View at Publisher · View at Google Scholar
  3. J. H. Chuang and C. M. Hoffmann, “On local implicit approximation and its applications,” ACM Transactions on Graphics, vol. 8, no. 4, pp. 298–324, 1989. View at Publisher · View at Google Scholar
  4. R. M. Corless, M. W. Giesbrecht, I. S. Kotsireas, and S. M. Watt, “Numerical implicitization of parametric hypersurfaces with linear algebra,” in Revised Papers from the International Conference on Artificial Intelligence and Symbolic Computation (AISC '00), vol. 1930, pp. 174–183, Springer, London, UK, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. T. Dokken, Aspects of intersection algorithms and approximations, Ph.D. thesis, University of Oslo, 1997.
  6. V. Pratt, “Direct least-squares fitting of algebraic surfaces,” SIGGRAPH Computer Graphics, vol. 21, no. 4, pp. 145–152, 1987. View at Publisher · View at Google Scholar
  7. T. Dokken, “Approximate implicitization,” in Mathematical Methods for Curves and Surfaces, pp. 81–102, Vanderbilt University Press, Nashville, Tenn, USA, 2001. View at Google Scholar · View at Zentralblatt MATH
  8. T. Dokken and J. B. Thomassen, “Weak approximate implicitization,” in Proceedings of the IEEE International Conference on Shape Modeling and Applications (SMI '06), p. 31, IEEE Computer Society, Washington, DC, USA, 2006. View at Publisher · View at Google Scholar
  9. D. Wang, “A simple method for implicitizing rational curves and surfaces,” Journal of Symbolic Computation, vol. 38, no. 1, pp. 899–914, 2004. View at Publisher · View at Google Scholar
  10. T. Dokken, H. K. Kellermann, and C. Tegnander, “An approach to weak approximate implicitization,” in Mathematical Methods for Curves and Surfaces, pp. 103–112, Vanderbilt University Press, Nashville, Tenn, USA, 2001. View at Google Scholar · View at Zentralblatt MATH
  11. I. Z. Emiris and I. S. Kotsireas, “Implicitization exploiting sparseness,” in Geometric and Algorithmic Aspects of Computer-Aided Design and Manufacturing, vol. 67 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 281–297, American Mathematical Society, Providence, RI, USA, 2005. View at Google Scholar
  12. Z. Battles and L. N. Trefethen, “An extension of MATLAB to continuous functions and operators,” SIAM Journal on Scientific Computing, vol. 25, no. 5, pp. 1743–1770, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. B. K. Alpert and V. Rokhlin, “A fast algorithm for the evaluation of Legendre expansions,” SIAM Journal on Scientific and Statistical Computing, vol. 12, pp. 158–179, 1991. View at Google Scholar
  14. A. Gil, J. Segura, and N. M. Temme, Numerical Methods for Special Functions, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1st edition, 2007.
  15. O. J. D. Barrowclough and T. Dokken, “Approximate implicitization of triangular Bézier surfaces,” in Proceedings of the 26th Spring Conference on Computer Graphics (SCCG '10), pp. 133–140, ACM, New York, NY, USA, 2010. View at Publisher · View at Google Scholar
  16. T. D. DeRose, R. N. Goldman, H. Hagen, and S. Mann, “Functional composition algorithms via blossoming,” ACM Transactions on Graphics, vol. 12, no. 2, pp. 113–135, 1993. View at Publisher · View at Google Scholar
  17. M. S. Floater and T. Lyche, “Asymptotic convergence of degree-raising,” Advances in Computational Mathematics, vol. 12, no. 2-3, pp. 175–187, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, USA, 1968.
  19. R. T. Farouki and T. N. T. Goodman, “On the optimal stability of the Bernstein basis,” Mathematics of Computation, vol. 65, no. 216, pp. 1553–1566, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. J. Schicho and I. Szilágyi, “Numerical stability of surface implicitization,” Journal of Symbolic Computation, vol. 40, no. 6, pp. 1291–1301, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. T. W. Sederberg, J. Zheng, K. Klimaszewski, and T. Dokken, “Approximate implicitization using monoid curves and surfaces,” Graphical Models and Image Processing, vol. 61, no. 4, pp. 177–198, 1999. View at Publisher · View at Google Scholar · View at Scopus
  22. B. Jüttler and A. Felis, “Least-squares fitting of algebraic spline surfaces,” Advances in Computational Mathematics, vol. 17, no. 1-2, pp. 135–152, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. J. N. Lyness and R. Cools, “A survey of numerical cubature over triangles,” in Proceedings of Symposia in Applied Mathematics, vol. 48, pp. 127–150, 1994. View at Google Scholar · View at Zentralblatt MATH
  24. R. T. Farouki, “Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains,” Computer Aided Geometric Design, vol. 20, no. 4, pp. 209–230, 2003. View at Google Scholar · View at Zentralblatt MATH
  25. Persistence of Vision Pty. Ltd. Persistence of vision raytracer (version 3.6), 2004, http://www.povray.org/download/.
  26. C. L. Bajaj and I. Ihm, “Algebraic surface design with Hermite interpolation,” ACM Transactions on Graphics, vol. 11, no. 1, pp. 61–91, 1992. View at Publisher · View at Google Scholar