Browse > Article
http://dx.doi.org/10.4134/CKMS.c210333

CIRCLE APPROXIMATION USING PARAMETRIC POLYNOMIAL CURVES OF HIGH DEGREE IN EXPLICIT FORM  

Ahn, Young Joon (Department of Mathematics Education Chosun University)
Publication Information
Communications of the Korean Mathematical Society / v.37, no.4, 2022 , pp. 1259-1267 More about this Journal
Abstract
In this paper we present a full circle approximation method using parametric polynomial curves with algebraic coefficients which are curvature continuous at both endpoints. Our method yields the n-th degree parametric polynomial curves which have a total number of 2n contacts with the full circle at both endpoints and the midpoint. The parametric polynomial approximants have algebraic coefficients involving rational numbers and radicals for degree higher than four. We obtain the exact Hausdorff distances between the circle and the approximation curves.
Keywords
Full circle approximation; parametric polynomial approximant; curvature continuous; Hausdorff distance; algebraic coefficients;
Citations & Related Records
연도 인용수 순위
  • Reference
1 K. Morken, Best approximation of circle segments by quadratic B'ezier curves, in Curves and surfaces (Chamonix-Mont-Blanc, 1990), 331-336, Academic Press, Boston, MA, 1991. https://doi.org/10.1016/B978-0-12-438660-0.50052-2
2 M. Sabin, Subdivision surfaces, in Handbook of computer aided geometric design, 309-325, North-Holland, Amsterdam, 2002. https://doi.org/10.1016/B978-044451104-1/50013-7
3 A. Vavpetic and E. Zagar, A general framework for the optimal approximation of circular arcs by parametric polynomial curves, J. Comput. Appl. Math. 345 (2019), 146-158. https://doi.org/10.1016/j.cam.2018.06.020
4 A. Vavpetic and E. Zagar, On optimal polynomial geometric interpolation of circular arcs according to the Hausdorff distance, J. Comput. Appl. Math. 392 (2021), Paper No. 113491, 14 pp. https://doi.org/10.1016/j.cam.2021.113491
5 Y. J. Ahn, Circle approximation by G2 Bezier curves of degree n with 2n - 1 extreme points, J. Comput. Appl. Math. 358 (2019), 20-28. https://doi.org/10.1016/j.cam.2019.02.037
6 S. Hur and T. Kim, The best G1 cubic and G2 quartic Bezier approximations of circular arcs, J. Comput. Appl. Math. 236 (2011), no. 6, 1183-1192. https://doi.org/10.1016/j.cam.2011.08.002
7 G. Jaklic, Uniform approximation of a circle by a parametric polynomial curve, Comput. Aided Geom. Design 41 (2016), 36-46. https://doi.org/10.1016/j.cagd.2015.10.004   DOI
8 G. Jaklic and J. Kozak, On parametric polynomial circle approximation, Numer. Algorithms 77 (2018), no. 2, 433-450. https://doi.org/10.1007/s11075-017-0322-0   DOI
9 S. H. Kim and Y. J. Ahn, Approximation of circular arcs by quartic Bezier curves, Comput.-Aided Design 39 (2007), no. 6, 490-493.   DOI
10 Y. J. Ahn and H. O. Kim, Approximation of circular arcs by Bezier curves, J. Comput. Appl. Math. 81 (1997), no. 1, 145-163. https://doi.org/10.1016/S0377-0427(97)00037-X
11 L. Fang, Circular arc approximation by quintic polynomial curves, Comput. Aided Geom. Design 15 (1998), no. 8, 843-861. https://doi.org/10.1016/S0167-8396(98)00019-3   DOI
12 C. de Boor, K. Hollig, and M. Sabin, High accuracy geometric Hermite interpolation, Comput. Aided Geom. Design 4 (1987), no. 4, 269-278. https://doi.org/10.1016/0167-8396(87)90002-1   DOI
13 Z. Liu, J. Tan, X. Chen, and L. Zhang, An approximation method to circular arcs, Appl. Math. Comput. 219 (2012), no. 3, 1306-1311. https://doi.org/10.1016/j.amc.2012.07.038   DOI
14 G. Jaklic, J. Kozak, M. Krajnc, V. Vitrih, and E. Zagar, High-order parametric polynomial approximation of conic sections, Constr. Approx. 38 (2013), no. 1, 1-18. https://doi.org/10.1007/s00365-013-9189-z   DOI
15 B. Kovac and E. Zagar, Some new G1 quartic parametric approximants of circular arcs, Appl. Math. Comput. 239 (2014), 254-264. https://doi.org/10.1016/j.amc.2014.04.   DOI
16 B.-G. Lee, Y. Park, and J. Yoo, Application of Legendre-Bernstein basis transformations to degree elevation and degree reduction, Comput. Aided Geom. Design 19 (2002), no. 9, 709-718. https://doi.org/10.1016/S0167-8396(02)00164-4   DOI
17 T. Dokken, M. Daehlen, T. Lyche, and K. Morken, Good approximation of circles by curvature-continuous Bezier curves, Comput. Aided Geom. Design 7 (1990), no. 1-4, 33-41. https://doi.org/10.1016/0167-8396(90)90019-N   DOI
18 M. S. Floater, An O(h2n) Hermite approximation for conic sections, Comput. Aided Geom. Design 14 (1997), no. 2, 135-151. https://doi.org/10.1016/S0167-8396(96)00025-8   DOI
19 M. Goldapp, Approximation of circular arcs by cubic polynomials, Comput. Aided Geom. Design 8 (1991), no. 3, 227-238. https://doi.org/10.1016/0167-8396(91)90007-X   DOI