A Grobner Bases Approach to the Detection of Improperly Parameterized Rational Curve

DOI: 10.4236/ajcm.2013.31007   PDF   HTML   XML   2,409 Downloads   4,490 Views  

Abstract

This paper proposes an algorithm for the detection of improper parameterization of rational curves using the concept of Grobner bases. The advantage of the proposed algorithm lies in the fact that the Grobner bases can operate in both univariate and multivariate fields with specified ordering.

Share and Cite:

A. Kamara and M. Koroma, "A Grobner Bases Approach to the Detection of Improperly Parameterized Rational Curve," American Journal of Computational Mathematics, Vol. 3 No. 1, 2013, pp. 48-52. doi: 10.4236/ajcm.2013.31007.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] D. Manocha and J. F. Canny, “Rational Curves with Polynomial Parameterization,” Computer-Aided Design, Vol. 23, No. 9, 1991, pp. 645-652. doi:10.1016/0010-4485(91)90042-U
[2] T. W. Sederberg, “Improperly Parametrized Rational Curves,” Computer Aided Geometric Design, Vol. 3, No. 1, 1986, pp. 67-75. doi:10.1016/0167-8396(86)90025-7
[3] M. V. Hoeij, “Rational Parametrizations of Algebraic Curves Using a Canonical Divisor,” Journal of Symbolic Computation, Vol. 23, No. 2-3, 1997, pp. 209-227. doi:10.1006/jsco.1996.0084
[4] T. Recio and J. R. Sendra, “Real Parametrizations of Real Curves,” Journal of Symbolic Computation, Vol. 23, No. 2-3, 1997, pp. 241-254. doi:10.1006/jsco.1996.0086
[5] J. R. Sendra and F. Winkler, “Parameterization of Algebraic Curves over Optimal Field Extensions,” Journal of Symbolic Computation, Vol. 11, 2008, p. 1-000.
[6] J. R. Sendra and F. Winkler, “Symbolic Parameterization of Curves,” Journal of Symbolic Computation, Vol. 12, No. 6, 1991, pp. 607-631.
[7] J. R. Sendra and F. Winkler, “Tracing Index of Rational Curve Parameterizations”. www.risc.jku.at/publications/download/risc_264/Nr.17_final-1.pdf
[8] G. Salmon, “Higher Plane Curves,” G.E. Stechert and Co., New York, 1879, 30 p.
[9] R. J. Walker, “Algebraic Curves,” Princeton, 1950, 67 p.
[10] D. A. Cox, J. Little and D. O’Shea, “Ideals, Varieties and Algorithms,” Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd Edition, Springer, Berlin, 1991, pp. 49-168.
[11] H. Hong and J. Schicho, “Algorithms for Trigonometric Curves (Simplification, Implicitization and Parameterization),” Technical Report, 1997. http://www.risc.uni-linz/people/hhong-jschicho
[12] J. Harris, “Graduate Texts in Mathematics,” Algebraic Geometry: A First Course, Springer-Verlag, Berlin, 1992, pp. 3-16.
[13] L. Buse and T. L. Ba, “Matrix-Based Implicit Representation of Rational Algebraic Curves and Applications,” Computer Aided Geometric Design, Vol. 27, No. 9, 2010, pp. 681-699. doi:10.1016/j.cagd.2010.09.006
[14] J. W. Archbold, “Introduction to the Algebraic Geometry of a Plane,” Edward Arnold & Co., London, 1948.
[15] J. Li, “General Explicit Difference Formulas for Numerical Differentiation,” Journal of Computational and Applied Mathematics, Vol. 183, No. 1, 2005, pp. 29-52. doi:10.1016/j.cam.2004.12.026

  
comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.