| Peer-Reviewed

A Cubic Bézier Model with Shape Parameters

Received: 16 December 2014     Accepted: 28 December 2014     Published: 8 January 2015
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Abstract

A novel extension of the cubic Bézier curve with two shape parameters is presented in this work. The proposed curve is still a cubic polynomial model, which has simpler structure than other similar models. The proposed curve has the same properties with the usual cubic Bézier curve and its shape can be adjusted by altering values of the two shape parameters while the control points are fixed. With the two shape parameters, the proposed curve can approach to its control polygon farther or closer. The corresponding surface with four shape parameters has the similar properties with the proposed curve and enjoys the shape adjustable property.

Published in Applied and Computational Mathematics (Volume 3, Issue 6)
DOI 10.11648/j.acm.20140306.19
Page(s) 343-348
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), 2015. Published by Science Publishing Group

Keywords

Cubic Bézier Curve, Cubic Polynomial, Shape Parameter, Shape Adjustment

References
[1] J. W. Zhang, “C-Bézier curves and surfaces”, Graphical Models and Image Processing, vol. 61, no. 1, pp. 2-15, 1999.
[2] Q. Y. Chen and G. Z. Wang, “A class of Bézier-like curves”, Computer Aided Geometric Design, vol. 20, no. 1, pp. 29–39, 2003.
[3] J. W. Zhang, F. L. Krause and H. Y. Zhang. “Unifying C-curves and H-curves by extending the calculation to complex numbers”, Computer Aided Geometric Design, vol. 22, no. 9, pp. 865-883, 2005.
[4] X.-A. Han, Y. C. Ma and X. L. Huang, “The cubic trigonometric Bézier curve with two shape parameters”, Applied Mathematical Letters, vol. 22, no. 2, pp. 226-231, 2009.
[5] J. C. Li, D. B. Zhao, B. J. Li and G. H. Chen, “A family of quasi-cubic trigonometric curves”, Journal of Information and Computational Science, vol. 7, no. 13, pp. 2847-2854, 2010.
[6] X. L. Han and Y. P. Zhu, “Curve construction based on five trigonometric blending functions”, BIT Numerical Mathematics, vol. 52, no. 4, pp. 953-979, 2012.
[7] J. C. Li, “A class of cubic trigonometric Bézier curve with a shape parameter”, Journal of Information and Computational Science, vol. 10, no.10, pp. 3071-3078, 2013.
[8] U. Bashir, M. Abbsa and J. M.Ali, “The G2 and C2 rational quadratic trigonometric Bézier curve with two shape parameters with applications”, Applied Mathematics and Computation, vol. 219. no. 20, pp. 10183-10197, 2013.
[9] W. T. Wang and G. Z. Wang, “Bézier curves with shape parameters”, Journal of Zhejiang University SCIENCE A, vol. 6, no. 6, pp. 497-501, 2005.
[10] X.-A. Han, Y. C. Ma and X. L.Huang, “A novel generalization of Bézier curve and surface”, Journal of Computational and Applied Mathematics, vol. 217, no. 1, pp. 180-193, 2008.
[11] L. Q. Yang and X M. Zeng, “Bézier curves and surfaces with shape parameters”, International Journal of Computer Mathematics, vol. 86, no. 7, pp. 1253-1263, 2009.
[12] L. L. Yan and Q. F. Liang, “An extension of the Bézier model”, Applied Mathematics and Computation, vol. 218, no. 6, pp. 2863-2879, 2011.
[13] J. Chen and G. J. Wang, “A new type of the generalized Bézier curves”, Applied Mathematics: A Journal of Chinese Universities, vol. 26, no. 1, pp. 47–56, 2011.
[14] Y. P. Zhu and X. L. Han, “A class of αβγ-Bernstein-Bézier basis functions over triangular domain”, Applied Mathematics and Computation, vol. 220, no. 17, pp. 446-454, 2013.
[15] T. N. Xiang, Z. Liu, W. F. Wang and P. Jiang, “A novel extension of Bézier curves and surfaces of the same degree”, Journal of Information and Computational Science, vol. 7, no. 10, pp. 2080-2089, 2010.
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    Juncheng Li. (2015). A Cubic Bézier Model with Shape Parameters. Applied and Computational Mathematics, 3(6), 343-348. https://doi.org/10.11648/j.acm.20140306.19

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    ACS Style

    Juncheng Li. A Cubic Bézier Model with Shape Parameters. Appl. Comput. Math. 2015, 3(6), 343-348. doi: 10.11648/j.acm.20140306.19

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    AMA Style

    Juncheng Li. A Cubic Bézier Model with Shape Parameters. Appl Comput Math. 2015;3(6):343-348. doi: 10.11648/j.acm.20140306.19

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  • @article{10.11648/j.acm.20140306.19,
      author = {Juncheng Li},
      title = {A Cubic Bézier Model with Shape Parameters},
      journal = {Applied and Computational Mathematics},
      volume = {3},
      number = {6},
      pages = {343-348},
      doi = {10.11648/j.acm.20140306.19},
      url = {https://doi.org/10.11648/j.acm.20140306.19},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.19},
      abstract = {A novel extension of the cubic Bézier curve with two shape parameters is presented in this work. The proposed curve is still a cubic polynomial model, which has simpler structure than other similar models. The proposed curve has the same properties with the usual cubic Bézier curve and its shape can be adjusted by altering values of the two shape parameters while the control points are fixed. With the two shape parameters, the proposed curve can approach to its control polygon farther or closer. The corresponding surface with four shape parameters has the similar properties with the proposed curve and enjoys the shape adjustable property.},
     year = {2015}
    }
    

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    AB  - A novel extension of the cubic Bézier curve with two shape parameters is presented in this work. The proposed curve is still a cubic polynomial model, which has simpler structure than other similar models. The proposed curve has the same properties with the usual cubic Bézier curve and its shape can be adjusted by altering values of the two shape parameters while the control points are fixed. With the two shape parameters, the proposed curve can approach to its control polygon farther or closer. The corresponding surface with four shape parameters has the similar properties with the proposed curve and enjoys the shape adjustable property.
    VL  - 3
    IS  - 6
    ER  - 

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Author Information
  • Department of Mathematics, Hunan University of Humanities, Science and Technology, Loudi, China

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