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A novel extension to the polynomial basis functions describing Bézier curves and surfaces of degree \(n\) with multiple shape parameters. (English) Zbl 1329.65053

Summary: The construction of Bézier curves using shape control parameters is one of the most popular areas of research in computer aided geometric design (CAGD). A class of new polynomial basis functions with \(n-1\) local shape control parameters is presented here to allow the construction of Bézier curves with \( n\) local shape control parameters, which is an extension to the classical Bernstein basis functions of degree \(n\). The properties of the proposed basis functions and the corresponding piecewise polynomial curve with \(n-1\) local shape control parameters are analyzed. This analysis shows that the new class of polynomial functions meets the conditions required for both \(C^0\), \( C^1\) and \(C^2\) continuity as well as \( G^0\), \( G^1\) and \(G^2\) continuity. Some curve design applications are then discussed and an extended application for surface design is also presented. The modeling examples illustrate that the new extension provides not only a better approximation and mathematical description of Bézier curves, but allows the shape parameters to be altered, making it a valuable method for the design of curves and surfaces.

MSC:

65D17 Computer-aided design (modeling of curves and surfaces)
41A10 Approximation by polynomials
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References:

[1] Mainar, E., Shape preserving alternatives to the rational Bézier model, Computer Aided Geometric Design, 18, 1, 37-60 (2001) · Zbl 0972.68157
[2] Qi, Congqian; Wu, Hongyi, A class of modifiable Bézier curves and its approximation, Journal of Hunan University, 23, 6, 15-19 (1996) · Zbl 0870.68151
[3] Liu, Genghong; Liu, Songtao, Application of generalized Bézier curve and surface in smooth joining, Acta Mathematicae Applicatae Sinica, 23, 1, 107-114 (1996) · Zbl 0862.65008
[4] Oruc, H.; Phillips, G. H., \(q\)-Bernstein polynomials and Bézier curves, Journal of Computational and Applied Mathematics, 151, 1, 1-12 (2003) · Zbl 1014.65015
[5] Liang, Xikun, Bernstein-Bézier class curves and a reparametrization method of Bézier curve, Journal of Computer Research and Development, 41, 6, 1016-1021 (2004)
[6] Han, Xuli; Liu, Shengjun, Extension of a quadratic Bézier curve, Journal of Central South University, 34, 2, 214-217 (2003)
[7] Han, Xian; Huang, Xili; Ma, Yichen, Shape analysis of cubic trigonometric Bézier curves with a shape parameter, Applied Mathematics and Computation, 217, 2, 2527-2533 (2010) · Zbl 1202.65021
[8] Wu, Xiaoqin; Han, Xuli, Extension of cubic Bézier curve, Journal of Engineering Graphics, 26, 6, 98-102 (2005)
[9] Wu, Xiaoqin; Han, Xuli; Luo, Shanming, Two different extensions of quartic Bézier curve, Journal of Engineering Graphics, 27, 5, 59-64 (2006)
[10] Wu, Xiaoqin, Bézier curve with shape parameter, Journal of Image and Graphics, 11, 2, 269-274 (2006)
[11] Yan, Lanlan; Liang, Jiongfeng, An extension of the Bézier model, Applied Mathematics and Computation, 218, 6, 2863-2879 (2011) · Zbl 1251.65015
[12] Wang, Wen-tao; Wang, Guo-zhao, Bézier curve with shape parameter, Journal of Zheijang University Science, 6A, 6, 497-501 (2005)
[13] Cheng, Huang-he; Zeng, Xiao-ming, Bézier curves with shape parameter, Journal of Xiamen University (Natural Science), 45, 3, 320-322 (2006) · Zbl 1096.65007
[14] Farin, G.; Class, A., Bézier curves, Computer Aided Geometric Design, 23, 7, 573-581 (2006) · Zbl 1101.65016
[15] Cao, Juan; Wang, Guo-zhao, A note on class A Bézier curves, Computer Aided Geometric Design, 25, 7, 523-528 (2008) · Zbl 1172.65321
[16] Han, Xi’an; Ma, Yichen; Huang, Xili, Shape modification of cubic Quasi-Bézier curve, Journal of Xi’an Jiaotong University, 41, 8, 903-906 (2007) · Zbl 1164.68443
[17] Qin, Xin-qiang; Hu, Gang; Zhang, Su-xia, New extension of cubic Bézier curve and its applications, Computer Engineering and Applications, 44, 2, 112-115 (2008)
[18] Zhu, Xiumei; Guo, Qingwei; Zhu, Gongqin, Extension of the quadric Bézier curve with parameters, Journal of Hefei University of Technology, 31, 4, 671-674 (2008) · Zbl 1199.65059
[19] Zhang, Nian-juan; Qin, Xin-qiang; Hu, Gang; Dang, Fa-ning, New extensions of quadric Bézier curve with multiple shape parameters, Journal of Wuhan University of Technology, 31, 20, 156-160 (2009)
[20] Bashir, Uzma; Abbas, Muhammad; Jamaludin, Md Ali, The \(G^2\) and \(C^2\) rational quadratic trigonometric Bézier curve with two shape parameters with applications, Applied Mathematics and Computation, 219, 20, 10183-10197 (2013) · Zbl 1293.65025
[21] Yan, Lanlan; Song, Laizhong, Bézier curves with two shape parameters, Journal of Engineering Graphics, 29, 5, 88-92 (2008)
[22] Han, Xi-An; Ma, YiChen; Huang, XiLi, A novel generalization of Bézier curve and surface, Journal of Computational and Applied Mathematics, 217, 180-193 (2008) · Zbl 1147.65013
[23] Liu, Zhi; Chen, Xiao-yan; Xie, Jin; Shi, Jun, A class of adjustable Quasi Bézier curve, Journal of Image and Graphics, 14, 11, 269-274 (2009)
[24] Liu, Zhi; Chen, Xiaoyan; Jiang, Ping, A class of generalized Bézier curves and surfaces with multiple shape parameters, Journal of Computer-Aided Design and Computer Graphics, 22, 5, 838-844 (2010)
[25] Wu, Hongyi; Xia, Chenglin, Extensions of Bézier curves and surfaces with multiple shape parameters, Journal of Computer-Aided Design and Computer Graphics, 17, 12, 2607-2612 (2005)
[26] Lianqiang, Yang; Xiaoming, Zeng, Bézier curves and surfaces with shape parameter, International Journal of Computer Mathematics, 86, 7, 1253-1263 (2009) · Zbl 1169.65012
[27] Ri, Kwang-un, Bézier curves with multi-shape-parameters, Journal of Zhejiang University (Science Edition), 37, 4, 401-405 (2010) · Zbl 1240.65061
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