A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves

Zhi Liu; Na Wei; Jieqing Tan; Xiaoyan Liu

doi:doi:10.11648/j.acm.20160502.11

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A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves

Zhi Liu, Na Wei, Jieqing Tan, Xiaoyan Liu

Published in Applied and Computational Mathematics (Volume 5, Issue 2)

Received: 18 February 2016 Accepted: 26 February 2016 Published: 9 March 2016

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Abstract

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.

Published in	Applied and Computational Mathematics (Volume 5, Issue 2)
DOI	10.11648/j.acm.20160502.11
Page(s)	40-45
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Conic Section, Quartic Bézier Curve, Hausdorff Distance, Approximation, G²-Continuous, Subdivision Scheme

References

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[3]	Przemysław Gospodarczyk, Degree reduction of Bézier curves with restricted control points area. Computer-Aided Design 62 (2015) 143-151.
[4]	Çetin Dişibüyük, Ron Goldman, A unifying structure for polar forms and for Bernstein Bézier curves. Journal of Approximation Theory 192(2015) 234-249.
[5]	Young Joon Ahn, Hong Oh Kim. Approximation of circular arcs by Bézier curves, Journal of Computational and Applied Mathematics 81(1997): 145-163.
[6]	Seon-Hong Kim, Young Joon Ahn. An approximation of circular arcs by quartic Bézier curves, Computer-Aided Design 39(2007) 490-493.
[7]	Zhi Liu, Jie-qing Tan, Xiao-yan Chen, Li Zhang. An approximation method to circular arcs, Applied Mathematics and Computation 219(2012) 1306-1311.
[8]	Boštjan Kovač, Emil Žagar. Some new G¹ quartic parametric approximants of circular arcs. Applied Mathematics and Computation 239(2014) 254-264.
[9]	Lian Fang. G³ approximation of conic sections by quintic polynomial curves, Computer Aided Geometric Design 16(1999) 755-766.
[10]	Michael S. Floater, An o(h²ⁿ) Hermite approximation for conic sections, Computer Aided Geometric Design 14(1997) 135-151.
[11]	Young Joon Ahn, Approximation of conic sections by curvature continuous quartic Bézier curves, Computers & Mathematics with Applications 60(2010) 1986-1993.
[12]	Qian-qian Hu, Approximating conic sections by constrained Bézier curves of arbitrary degree, Journal of Computational and Applied Mathematics 236(2012) 2813-2821.
[13]	Qian-qian Hu, G¹ approximation of conic sections by quartic Bézier curves, Computers & Mathematics with Applications 68(2014) 1882-1891.
[14]	Lian Fang, A rational quartic Bézier representation for conics, Computer Aided Geometric Design 19(2002): 297-312.
[15]	Michael S. Floater. High order approximation of conic sections by quadratic splines, Computer Aided Geometric Design 12(1995) 617-637.
[16]	Young Joon Ahn, Conic approximation of planar curves, Computer-Aided Design 33(2001): 867-872.

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APA Style

Zhi Liu, Na Wei, Jieqing Tan, Xiaoyan Liu. (2016). A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves. Applied and Computational Mathematics, 5(2), 40-45. https://doi.org/10.11648/j.acm.20160502.11

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ACS Style

Zhi Liu; Na Wei; Jieqing Tan; Xiaoyan Liu. A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves. Appl. Comput. Math. 2016, 5(2), 40-45. doi: 10.11648/j.acm.20160502.11

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AMA Style

Zhi Liu, Na Wei, Jieqing Tan, Xiaoyan Liu. A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves. Appl Comput Math. 2016;5(2):40-45. doi: 10.11648/j.acm.20160502.11

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@article{10.11648/j.acm.20160502.11,
  author = {Zhi Liu and Na Wei and Jieqing Tan and Xiaoyan Liu},
  title = {A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves},
  journal = {Applied and Computational Mathematics},
  volume = {5},
  number = {2},
  pages = {40-45},
  doi = {10.11648/j.acm.20160502.11},
  url = {https://doi.org/10.11648/j.acm.20160502.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160502.11},
  abstract = {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.},
 year = {2016}
}

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TY  - JOUR
T1  - A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves
AU  - Zhi Liu
AU  - Na Wei
AU  - Jieqing Tan
AU  - Xiaoyan Liu
Y1  - 2016/03/09
PY  - 2016
N1  - https://doi.org/10.11648/j.acm.20160502.11
DO  - 10.11648/j.acm.20160502.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
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EP  - 45
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UR  - https://doi.org/10.11648/j.acm.20160502.11
AB  - 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.
VL  - 5
IS  - 2
ER  -

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Author Information

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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APA Style

Zhi Liu, Na Wei, Jieqing Tan, Xiaoyan Liu. (2016). A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves. Applied and Computational Mathematics, 5(2), 40-45. https://doi.org/10.11648/j.acm.20160502.11

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ACS Style

Zhi Liu; Na Wei; Jieqing Tan; Xiaoyan Liu. A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves. Appl. Comput. Math. 2016, 5(2), 40-45. doi: 10.11648/j.acm.20160502.11

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AMA Style

Zhi Liu, Na Wei, Jieqing Tan, Xiaoyan Liu. A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves. Appl Comput Math. 2016;5(2):40-45. doi: 10.11648/j.acm.20160502.11

Copy | Download

@article{10.11648/j.acm.20160502.11,
  author = {Zhi Liu and Na Wei and Jieqing Tan and Xiaoyan Liu},
  title = {A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves},
  journal = {Applied and Computational Mathematics},
  volume = {5},
  number = {2},
  pages = {40-45},
  doi = {10.11648/j.acm.20160502.11},
  url = {https://doi.org/10.11648/j.acm.20160502.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160502.11},
  abstract = {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.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves
AU  - Zhi Liu
AU  - Na Wei
AU  - Jieqing Tan
AU  - Xiaoyan Liu
Y1  - 2016/03/09
PY  - 2016
N1  - https://doi.org/10.11648/j.acm.20160502.11
DO  - 10.11648/j.acm.20160502.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 40
EP  - 45
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20160502.11
AB  - 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.
VL  - 5
IS  - 2
ER  -

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