Browse > Article
http://dx.doi.org/10.4134/CKMS.2005.20.4.861

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

Ahn, Young-Joon (Department of Mathematics Education Chosun University)
Kim, Philsu (Department of Mathematics Kyungpook National University)
Publication Information
Communications of the Korean Mathematical Society / v.20, no.4, 2005 , pp. 861-873 More about this Journal
Abstract
In this paper we approximate a cylindrical helix by bi-conic and bi-quadratic Bezier curves. Each approximation method is $G^1$ 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;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Y. J. Ahn, Conic approximation of planar curves, Computer-Aided Design 33 (2001), no. 12, 867-872   DOI   ScienceOn
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   DOI   ScienceOn
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 C. de Boor, K. Hollig, and M. Sabin, High accuracy geometric Hermite interpolation, Comput. Aided Geom. Design 4 (1987), 169-178
6 W. L. F. Degen, High accurate rational approximation of parametric curves, Comput. Aided Geom. Design 10 (1993), 293-313   DOI   ScienceOn
7 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   DOI   ScienceOn
8 G. Farin, Curvature continuity and offsets for piecewise conics, ACM Trans. Graph. 8 (1989), no. 2, 89-99   DOI   ScienceOn
9 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
10 T. Pavlidis, Curve fitting with conic splines, ACM Trans. Graph. 2 (1983), 1-31   DOI
11 L. Piegl, The sphere as a rational Bezier surfaces, Comput. Aided Geom. Design 3 (1986), 45-52   DOI   ScienceOn
12 L. Piegl and W. Tiller, Curve and surface constructions using rational B-splines, Computer-Aided Design 19 (1987), no. 9, 485-498   DOI   ScienceOn
13 T. Pratt, Techniques for conic splines, in Proceedings of SIGGRAPH 85, pp. 151-159. ACM, 1985
14 R. Schaback, Planar curve interpolation by piecewise conics of arbitrary type, Constr. Approx. 9 (1993), 373-389   DOI   ScienceOn
15 G. Seemann, Approximating a helix segment with a rational Bezier curve, Comput. Aided Geom. Design 14 (1997), 475-490   DOI   ScienceOn
16 P. R. Wilson, Conic representations for sphere description, IEEE Computer Graph. Appl. 7 (1987), no. 4, 1-31   DOI   ScienceOn
17 E. T. Lee, The rational Bezier representation for conics, in geometric modeling: Algorithms and new trends, pp. 3-19, Philadelphia, 1987. SIAM, Academic Press
18 G. Farin, Curves and Surfaces for Computer Aided Geometric Design, Academic Press, San Diego, CA, 1998
19 M. Floater, High order approximation of conic sections by quadratic splines, Comput. Aided Geom. Design 12 (1995), 617-637   DOI   ScienceOn
20 I. Juhasz, Approximating the helix with rational cubic Bezier curves, ComputerAided Design 27 (1995), 587-593   DOI   ScienceOn
21 S. Mick and O. Roschel, Interpolation of helical patches by kinematics rational Bezier patches, Computers and Graphics 14 (1990), no. 2, 275-280   DOI   ScienceOn
22 X. Yang, High accuracy approximation of helices by quintic curves, Comput. Aided Geom. Design 20 (2003), 303-317   DOI   ScienceOn
23 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   DOI   ScienceOn
24 M. Goldapp, Approximation of circular arcs by cubic polynomials, Comput. Aided Geom. Design 8 (1991), 227-238   DOI   ScienceOn
25 M. Floater, An O($h^{2n}$) Hermite approximation for conic sections, Comput. Aided Geom. Design 14 (1997), 135-151   DOI   ScienceOn