A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves
Applied and Computational Mathematics
Volume 5, Issue 2, April 2016, Pages: 40-45
Received: Feb. 18, 2016;
Accepted: Feb. 26, 2016;
Published: Mar. 9, 2016
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Zhi Liu, School of Mathematics, Hefei University of Technology, Hefei, China
Na Wei, School of Mathematics, Hefei University of Technology, Hefei, China
Jieqing Tan, School of Mathematics, Hefei University of Technology, Hefei, China
Xiaoyan Liu, Department of Mathematics, University of La Verne, La Verne, USA
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A new approximation method for conic section by quartic Bézier curves is proposed. This method is based on the quartic Bézier approximation of circular arcs. We give the upper bound of Hausdorff distance between the conic section and the quartic Bézier curve, and also show that the approximation order is eight. And we prove that our approximation method has a smaller upper bound than previous quartic Bézier approximation methods. A quartic G2-continuous spline approximation of conic sections is obtained by using the subdivision scheme at the shoulder point of the conic section.
Conic Section, Quartic Bézier Curve, Hausdorff Distance, Approximation, G2-Continuous, Subdivision Scheme
To cite this article
A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves, Applied and Computational Mathematics.
Vol. 5, No. 2,
2016, pp. 40-45.
Copyright © 2016 Authors retain the copyright of this article.
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Li-Wen Han, Ying Chu, Zhi-Yu Qiu, Generalized Bézier curves and surfaces based on Lupaş q-analogue of Bernstein operator. Journal of Computational and Applied Mathematics 261(2014) 352-363.
J. Sánchez-Reyes. The conditions for the coincidence or overlapping of two Bézier curves. Applied Mathematics and Computation 248(2014) 625-630.
Przemysław Gospodarczyk, Degree reduction of Bézier curves with restricted control points area. Computer-Aided Design 62 (2015) 143-151.
Çetin Dişibüyük, Ron Goldman, A unifying structure for polar forms and for Bernstein Bézier curves. Journal of Approximation Theory 192(2015) 234-249.
Young Joon Ahn, Hong Oh Kim. Approximation of circular arcs by Bézier curves, Journal of Computational and Applied Mathematics 81(1997): 145-163.
Seon-Hong Kim, Young Joon Ahn. An approximation of circular arcs by quartic Bézier curves, Computer-Aided Design 39(2007) 490-493.
Zhi Liu, Jie-qing Tan, Xiao-yan Chen, Li Zhang. An approximation method to circular arcs, Applied Mathematics and Computation 219(2012) 1306-1311.
Boštjan Kovač, Emil Žagar. Some new G1 quartic parametric approximants of circular arcs. Applied Mathematics and Computation 239(2014) 254-264.
Lian Fang. G3 approximation of conic sections by quintic polynomial curves, Computer Aided Geometric Design 16(1999) 755-766.
Michael S. Floater, An o(h2n) Hermite approximation for conic sections, Computer Aided Geometric Design 14(1997) 135-151.
Young Joon Ahn, Approximation of conic sections by curvature continuous quartic Bézier curves, Computers & Mathematics with Applications 60(2010) 1986-1993.
Qian-qian Hu, Approximating conic sections by constrained Bézier curves of arbitrary degree, Journal of Computational and Applied Mathematics 236(2012) 2813-2821.
Qian-qian Hu, G1 approximation of conic sections by quartic Bézier curves, Computers & Mathematics with Applications 68(2014) 1882-1891.
Lian Fang, A rational quartic Bézier representation for conics, Computer Aided Geometric Design 19(2002): 297-312.
Michael S. Floater. High order approximation of conic sections by quadratic splines, Computer Aided Geometric Design 12(1995) 617-637.
Young Joon Ahn, Conic approximation of planar curves, Computer-Aided Design 33(2001): 867-872.