IMPLICITIZATION OF RATIONAL CURVES AND POLYNOMIAL SURFACES

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 44, Issue 1, 2007, pp.13-29
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2007.44.1.013

Title & Authors

IMPLICITIZATION OF RATIONAL CURVES AND POLYNOMIAL SURFACES

Yu, Jian-Ping; Sun, Yong-Li;

Yu, Jian-Ping; Sun, Yong-Li;

Abstract

In this paper, we first present a method for finding the implicit equation of the curve given by rational parametric equations. The method is based on the computation of bases. Then, another method for implicitization of curve and surface is given. In the case of rational curves, the method proceeds via giving the implicit polynomial f with indeterminate coefficients, substituting the rational expressions for the given curve and surface into the implicit polynomial to yield a rational expression in the parameters. Equating coefficients of g in terms of parameters to 0 to get a system of linear equations in the indeterminate coefficients of polynomial f, and finally solving the linear system, we get all the coefficients of f, and thus we obtain the corresponding implicit equation. In the case of polynomial surfaces, we can similarly as in the case of rational curves obtain its implicit equation. This method is based on characteristic set theory. Some examples will show that our methods are efficient.

Keywords

implicitization;rational curve;polynomial surface;implicit curve(surface); bases;characteristic set;

Language

English

References

1.

W. Adams and P. Loustaunau, An Introduction to Grabner bases, Graduate Studies in Mathematics 3, AMS, Providence, 1994

2.

T. Becker and V. Weispfenning, Grabner bases: A computational approach to commutative algebra, Springer-Verlag, 1993

3.

L. Buse, Residual resultant over the projective plane and the implicitization problem, In: Proc. 2001 Int. Symp, Symb. Algebr. Comput., ISSAC'01. ACM Press, New York, 48-55

4.

L. Buse, D. Cox, and C. D'Andrea, Implicitization of surfaces in $P^3$ in the presence of base points. J. Algebra Appl. 2 (2003), no. 2, 189-214

5.

J. H. Chung and C. M. Hoffman, On local implicit approximation and its applications, ACM Tran. in Graphics 8 (1989), 298-324

6.

R. M. Corless, M. W. Giesbrecht, I. S. Kotsireas, and S. M. Watt, Numerical implicitization of parametric hypersurfaces with linear algebra, In: Campbell, J. A. , Roanes-Lozano, E.(Eds.), Artificial Intelligence and Symbolic Computation (LANAI1930). Springer, Berlin, Heidelberg, 2001, 174-183

7.

D. Cox, Equations of parametric curves and surfaces via syzygies, In: Symbolic Computation: Solving Equations in Algebra, Geometry, Engineering. Contemporary Mathematics 286 (2001), 1-20

8.

D. Cox, Curves, surfaces, and syzygies, In: Topics in Algebraic Geometry, and Geometric Modeling. Contemporary Mathematics 334 (2003), 131-149

9.

D. Cox, J. Little, and D. O'Shea, Ideals, Varieties and Algorithms, Springer-Verlag, 1992

10.

D, Cox, Ideals, Varieties, and Algorithms, 2nd edition, Springer, New York, 1997

11.

D, Cox, Using Algebraic Geometry, Graduates Texts in Mathematics, 185, Springer, Berlin, 1998

12.

X. S. Gao and S. C. Chou, Implicitization of rational parametric equations, J. Symbolic Comput. 14 (1992), no. 5, 459-470

13.

Z. M. Li, Automatic implicitization of parametric objects, Ins. of Systems Science, Academia Sinica, MM Research Preprints 4 (1989)

14.

A. Marco and J. J. Martinez, Using polynomial interpolation for implicitizing algebraic curves, Comput. Aided Geom. Design 18 (2001), no. 4, 309-319

15.

A. Marco and J. J. Martinez, Implicitization of rational surfaces by means of polynomial interpolation, Computer Aided Geometric Design 19 (2002), no. 5, 327-344

16.

D. Manocha and J. F. Canny, Implicit representation of rational parametric surfaces, J. Symbolic Comput. 13 (1992), no. 5,485-510

17.

T. W. Sederberg, Improperly parameterized rational curves, Computer Aided Geometric Design 3 (1986), no. 1, 67-75

18.

T. W. Sederberg, D. C. Anderson, and R. N. Goldman, Implicit Representation of parametric curves and surfaces, Computer Vision, Graph, Image Proc. 28 (1984), 7284

19.

H. Shi and Y. L. Sun, Blending of triangular algebraic surfaces, MM-Res. Preprints 21 (2002), 200-206

20.

H. Shi and Y. L. Sun, On blending of cylinders, MM-Res. Preprints 21 (2002), 207-211

21.

Y. L. Sun and J. P. Yu, Implicitization of Parametric Curves via Lagrange Interpolation, Computing 77 (2006), no. 4, 379-386

22.

Y. L. Sun and J. P. Yu, Implicitization of Rational Curves, J. Calmet, T. Ida, and D. Wang: AISC 2006, LNAI 4120, 160-169

23.

D. M. Wang, A simple method for implicitizing rational curves and surfaces. J. Symbolic Comput. 38 (2004), no. 1, 899-914

24.

W. T. Wu, Mathematics Mechanization, Science Press and Kluwer Academic, Beijing and Dordrecht, 2000

25.

J. P. Yu, Y. L. Sun, and Y. J. Ma, Implicitization of Rational Curves and Surfaces, Acta Scientiarum Naturalium Universitatis Pekinensis 42 (2006), 578-583