ERROR ANALYSIS FOR APPROXIMATION OF HELIX BY BI-CONIC AND BI-QUADRATIC BEZIER CURVES

- Journal title : Communications of the Korean Mathematical Society
- Volume 20, Issue 4, 2005, pp.861-873
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2005.20.4.861

Title & Authors

ERROR ANALYSIS FOR APPROXIMATION OF HELIX BY BI-CONIC AND BI-QUADRATIC BEZIER CURVES

Ahn, Young-Joon; Kim, Philsu;

Ahn, Young-Joon; Kim, Philsu;

Abstract

In this paper we approximate a cylindrical helix by bi-conic and bi-quadratic Bezier curves. Each approximation method is end-points interpolation of the helix. We present a sharp upper bound of the Hausdorff distance between the helix and each approximation curve. We also show that the error bound has the approximation order three and monotone increases as the length of the helix increases. As an illustration we give some numerical examples.

Keywords

helix;bi-conic;bi-quadratic;Bezier curve;helicoid sur-face;

Language

English

References

1.

2.

Y. J. Ahn, Helix approximation with conic and qadratic Bezier curves, Comput. Aided Geom. Design, to appear, 2005

3.

Y. J. Ahn and H. O. Kim, Approximation of circular arcs by Bezier curves, J. Comput. Appl. Math. 81 (1997), 145-163

4.

Y. J. Ahn and H. O. Kim, Curvatures of the quadratic rational Bezier curves, Comput. Math. Appl. 36 (1998), no. 9, 71-83

5.

Y. J. Ahn, Y. S. Kim, and Y. S. Shin, Approximation of circular arcs and offset curves by Bezier curves of high degree, J. Comput. Appl. Math. 167 (2004), no. 2,405-416

6.

C. de Boor, K. Hollig, and M. Sabin, High accuracy geometric Hermite interpolation, Comput. Aided Geom. Design 4 (1987), 169-178

7.

W. L. F. Degen, High accurate rational approximation of parametric curves, Comput. Aided Geom. Design 10 (1993), 293-313

8.

T. Dokken, M. Deehlen, T. Lyche, and K. Morken, Good approximation of circles by curvature-continuous Bezier curves, Comput. Aided Geom. Design 7 (1990), 33-41

9.

G. Farin, Curvature continuity and offsets for piecewise conics, ACM Trans. Graph. 8 (1989), no. 2, 89-99

10.

G. Farin, Curves and Surfaces for Computer Aided Geometric Design, Academic Press, San Diego, CA, 1998

11.

M. Floater, High order approximation of conic sections by quadratic splines, Comput. Aided Geom. Design 12 (1995), 617-637

12.

M. Floater, An O($h^{2n}$ ) Hermite approximation for conic sections, Comput. Aided Geom. Design 14 (1997), 135-151

13.

M. Goldapp, Approximation of circular arcs by cubic polynomials, Comput. Aided Geom. Design 8 (1991), 227-238

14.

I. Juhasz, Approximating the helix with rational cubic Bezier curves, ComputerAided Design 27 (1995), 587-593

15.

E. T. Lee, The rational Bezier representation for conics, in geometric modeling: Algorithms and new trends, pp. 3-19, Philadelphia, 1987. SIAM, Academic Press

16.

S. Mick and O. Roschel, Interpolation of helical patches by kinematics rational Bezier patches, Computers and Graphics 14 (1990), no. 2, 275-280

17.

K. Morken, Best approximation of circle segments by quadratic Bezier curves, in P.J. Laurent, A. Le Mehaute, and L.L. Schumaker, editors, Curves and Surfaces, New York, 1990. Academic Press

20.

L. Piegl and W. Tiller, Curve and surface constructions using rational B-splines, Computer-Aided Design 19 (1987), no. 9, 485-498

21.

T. Pratt, Techniques for conic splines, in Proceedings of SIGGRAPH 85, pp. 151-159. ACM, 1985

22.

R. Schaback, Planar curve interpolation by piecewise conics of arbitrary type, Constr. Approx. 9 (1993), 373-389

23.

G. Seemann, Approximating a helix segment with a rational Bezier curve, Comput. Aided Geom. Design 14 (1997), 475-490