发布时间: 2018-09-16 摘要点击次数: 全文下载次数: DOI: 10.11834/jig.180005 2018 | Volume 23 | Number 9 计算机图形学

Bézier曲线的同次扩展及其参数选择

 收稿日期: 2018-01-22; 修回日期: 2018-02-27 基金项目: 国家自然科学基金项目（11261003，11761008）；江西省自然科学基金项目（20161BAB211028）；江西省教育厅科技项目（GJJ160558） 第一作者简介: 严兰兰, 1982年生, 女, 副教授, 2016年于中南大学获计算数学专业理学博士学位, 主要研究方向为计算机辅助几何设计。E-mail:yxh821011@aliyun.com;饶智勇, 男, 讲师, 主要研究方向为数值计算。E-mail:zyrao@ecit.cn;黄涛, 男, 讲师, 主要研究方向为数值计算。E-mail:thuang@ecit.cn. 中图法分类号: TP391.72 文献标识码: A 文章编号: 1006-8961(2018)09-1411-13

# 关键词

Extension of Bézier curves of the same degree and parameter selection
Yan Lanlan, Rao Zhiyong, Huang Tao
College of Science, East China University of Technology, Nanchang 330013, China
Supported by: National Natural Science Foundation of China (11261003, 11761008)

# Abstract

Objective The purpose of this paper is to construct a type of Bézier curve with a shape parameter. We require the curves defined in algebraic polynomial space. The degree of the basis functions should be the same as the Bernstein basis functions, which needed the same number of control points. The calculation of the basis functions and corresponding curves should be as simple as possible. The selection scheme under common design requirements of the shape parameter in the curves should be provided. Method With the cubic Bézier curve as the initial research object and in accordance with the idea of defining a shape-adjustable curve by using adjustable control points, we introduce a parameter into the two inner control points. Let the control points with the parameter have a linear combination with the Bernstein basis functions to generate the shape adjustable curves. By rewriting the expression of the curves as the linear combination of the fixed control points and the blending functions with the parameter, we obtain the extended basis with the parameter of the cubic Bernstein basis functions. By using the recursive formula, we obtain the extended basis with the parameter of a high degree. Then, we observe the rule of the basis function expression and provide the uniform explicit expression of all extended basis functions with parameters. The properties of the extended basis functions are analyzed, and the corresponding curves with parameters are defined. The properties of the curves are analyzed. The geometric drawing method and smooth joining conditions of the curves are also provided. The calculation formula of the parameter, which causes the stretch, strain, and jerk energies of the curves to be approximately minimum, is deduced. The difference of the curves determined by different energy targets is compared and analyzed by using the graph of the curves and their curvatures. Result The method provides the Bézier curve shape adjustability without increasing the calculation amount due to the fact that the extended basis functions have the same degree as the Bernstein basis functions and have a uniform explicit expression. Determining the shape parameter that conforms to the design requirements when using this method is easy because the calculation formula of the shape parameter can be used directly. The numerical examples intuitively show the correctness and validity of the proposed curve modeling method and the shape parameter selection scheme in the curve. The illustration also shows the superiority of the method provided in this paper over similar methods presented in the literature. Conclusion The method of constructing an extended basis with the parameter and selection method of the shape parameter are general. This method can be extended to construct a triangular Bézier surface with parameter.

# Key words

curve modeling; Bézier curve; shape parameter; energy minimization; parameter selection

# 0 引言

 $\begin{array}{*{20}{c}} {b_i^n\left( t \right) = {\rm{C}}_n^i{{\left( {1 - t} \right)}^{n - i}}{t^i} + \left( {1 - \lambda } \right){{\left( {1 - t} \right)}^{n - i - 1}}{t^{i - 1}} \times }\\ {\left[ {{\rm{C}}_{n - 3}^{i - 3}{{\left( {1 - t} \right)}^2} - {\rm{C}}_{n - 2}^{i - 1}\left( {1 - t} \right)t + {\rm{C}}_{n - 3}^i{t^2}} \right]} \end{array}$ (1)

 $b_i^n = \left( {1 - t} \right)b_i^{n - 1} + tb_{i - 1}^{n - 1},t \in \left[ {0,1} \right]$ (2)

 $\left\{ \begin{array}{l} b_0^3 = \left( {1 - \lambda t} \right){\left( {1 - t} \right)^2}\\ b_1^3 = \left( {2 + \lambda } \right)t{\left( {1 - t} \right)^2}\\ b_2^3 = \left( {2 + \lambda } \right){t^2}\left( {1 - t} \right)\\ b_3^3 = \left( {1 - \lambda + \lambda t} \right){t^2} \end{array} \right.$ (3)

 $\begin{array}{*{20}{c}} {\sum\limits_{i = 0}^{k + 1} {b_i^{k + 1}} = \left( {1 - t} \right)\sum\limits_{i = 0}^k {b_i^k} + t\sum\limits_{i - 1 = 0}^k {b_{i - 1}^k} = }\\ {\left( {1 - t} \right) + t = 1} \end{array}$

 $\sum\limits_{i = 0}^3 {{k_i}b_i^3} = 0$

 $\begin{array}{*{20}{c}} {{k_0}B_0^3 + \left( {\frac{{1 - \lambda }}{3}{k_0} + \frac{{2 + \lambda }}{3}{k_1}} \right)B_1^3 + }\\ {\left( {\frac{{2 + \lambda }}{3}{k_2} + \frac{{1 - \lambda }}{3}{k_3}} \right)B_2^3 + {k_3}B_3^3 = 0} \end{array}$

 $\left\{ \begin{array}{l} {k_0} = 0\\ \frac{{1 - \lambda }}{3}{k_0} + \frac{{2 + \lambda }}{3}{k_1} = 0\\ \frac{{2 + \lambda }}{3}{k_2} + \frac{{1 - \lambda }}{3}{k_3} = 0\\ {k_3} = 0 \end{array} \right.$

 $\sum\limits_{i = 0}^{k + 1} {{l_i}b_i^{k + 1}} = 0$

 $\left( {1 - t} \right)\sum\limits_{i = 0}^k {{l_i}b_i^k} + t\sum\limits_{i = 1}^{k + 1} {{l_i}b_{i - 1}^k} = 0$

 $\sum\limits_{i = 0}^k {{l_i}b_i^k} = 0$ (5)

 $\sum\limits_{i = 1}^{k + 1} {{l_i}b_{i - 1}^k} = 0$ (6)

 $\left\{ \begin{array}{l} b_i^n\left( 0 \right) = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} 1\\ 0 \end{array}&\begin{array}{l} i = 0\\ i \ne 0 \end{array} \end{array}} \right.\\ b_i^n\left( 1 \right) = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} 1\\ 0 \end{array}&\begin{array}{l} i = n\\ i \ne n \end{array} \end{array}} \right. \end{array} \right.$ (8)

 $\left\{ \begin{array}{l} b{_i^n }'\left( 0 \right) = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} - \left( {n - 1 + \lambda } \right)\\ n - 1 + \lambda \\ 0 \end{array}&\begin{array}{l} i = 0\\ i = 1\\ i \ne 0,1 \end{array} \end{array}} \right.\\ b{_i^n }'\left( 1 \right) = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} - \left( {n - 1 + \lambda } \right)\\ n - 1 + \lambda \\ 0 \end{array}&\begin{array}{l} i = n - 1\\ i = n\\ i \ne n - 1,n \end{array} \end{array}} \right. \end{array} \right.$ (9)

 $\left\{ \begin{array}{l} B_i^n\left( 0 \right) = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} 1\\ 0 \end{array}&\begin{array}{l} i = 0\\ i \ne 0 \end{array} \end{array}} \right.\\ B_i^n\left( 1 \right) = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} 1\\ 0 \end{array}&\begin{array}{l} i = n\\ i \ne n \end{array} \end{array}} \right. \end{array} \right.$ (10)

 $\left\{ \begin{array}{l} B{_i^n }'\left( 0 \right) = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} - n\\ n\\ 0 \end{array}&\begin{array}{l} i = 0\\ i = 1\\ i \ne 0,1 \end{array} \end{array}} \right.\\ B{_i^n}'\left( 1 \right) = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} - n\\ n\\ 0 \end{array}&\begin{array}{l} i = n - 1\\ i = n\\ i \ne n - 1,n \end{array} \end{array}} \right. \end{array} \right.$ (11)

 $\left\{ \begin{array}{l} b_0^n\left( 0 \right) = \frac{{{\rm{C}}_n^0}}{{{\rm{C}}_n^0}}B_0^n\left( 0 \right) + \frac{{{\rm{C}}_{n - 3}^0\left( {1 - \lambda } \right)}}{{{\rm{C}}_n^1}}B_1^n\left( 0 \right) = \\ B_0^n\left( 0 \right) = 1\\ b_i^n\left( 0 \right) = 0\left( {i \ne 0} \right)\\ b_i^n\left( 1 \right) = 0\left( {i \ne n} \right)\\ b_n^n\left( 1 \right) = \frac{{{\rm{C}}_{n - 3}^{n - 3}\left( {1 - \lambda } \right)}}{{{\rm{C}}_n^{n - 1}}}B_{n - 1}^n\left( 1 \right) + \frac{{{\rm{C}}_n^n}}{{{\rm{C}}_n^n}}B_n^n\left( 1 \right) = \\ B_n^n\left( 1 \right) = 1 \end{array} \right.$

 $\left\{ \begin{array}{l} b{_0^n}'\left( 0 \right) = \frac{{{\rm{C}}_n^0}}{{{\rm{C}}_n^0}}B{_0^n}'\left( 0 \right) + \frac{{{\rm{C}}_{n - 3}^0\left( {1 - \lambda } \right)}}{{{\rm{C}}_n^1}}B{_1^n}'\left( 0 \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\; - \left( {n - 1 + \lambda } \right)\\ b{_1^n}'\left( 0 \right) = \frac{{\left( {{\rm{C}}_n^1 - {\rm{C}}_{n{\rm{ - 2}}}^0} \right) + {\rm{C}}_{n - 2}^0\lambda }}{{{\rm{C}}_{\rm{n}}^1}}B{_1^n}'\left( 0 \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;n - 1 + \lambda \\ b{_i^n}'\left( 0 \right) = 0\left( {i \ne 0,1} \right)\\ b{_i^n}'\left( 1 \right) = 0\left( {i \ne n - 1,n} \right)\\ b{_{n - 1}^n}'\left( 1 \right) = \frac{{\left( {{\rm{C}}_n^{n - 1} - {\rm{C}}_{n - 2}^0} \right) + {\rm{C}}_{n - 2}^{n - 2}\lambda }}{{{\rm{C}}_n^{n - 1}}}B{_{n - 1}^n}'\left( 1 \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\; - \left( {n - 1 + \lambda } \right)\\ b{_n^n}'\left( 1 \right) = \frac{{{\rm{C}}_{n - 3}^{n - 3}\left( {1 - \lambda } \right)}}{{{\rm{C}}_n^{n - 1}}}B{_{n - 1}^n}'\left( 1 \right) + \frac{{{\rm{C}}_n^n}}{{{\rm{C}}_n^i}}B{_n^n}'\left( 1 \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;n - 1 + \lambda \end{array} \right.$

# 2 可调曲线及其性质

 $\mathit{\boldsymbol{p}}\left( t \right) = \sum\limits_{i = 0}^n {b_i^n\left( t \right){\mathit{\boldsymbol{V}}_i}}$ (12)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{p}}\left( t \right) = \sum\limits_{i = 0}^n {b_i^n\mathit{\boldsymbol{V}}_i^0} = \sum\limits_{i = 0}^{n - 1} {b_i^{n - 1}\mathit{\boldsymbol{V}}_i^1} = L = }\\ {\sum\limits_{i = 0}^3 {b_i^3\mathit{\boldsymbol{V}}_i^{n - 3}} = \sum\limits_{i = 0}^3 {B_i^3\mathit{\boldsymbol{Q}}_i^{n - 2}} = L = \mathit{\boldsymbol{Q}}_i^{n + 1}} \end{array}$ (13)

 $\mathit{\boldsymbol{V}}_i^l = \left( {1 - t} \right)\mathit{\boldsymbol{V}}_i^{l - 1} + t\mathit{\boldsymbol{V}}_{i + 1}^{l - 1};i = 0,1, \cdots ,n - l$ (14)

$\mathit{\boldsymbol{Q}}_i^l\left( {l = n - 1,{\rm{ }}n,{\rm{ }}n + 1} \right)$递推定义为

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{Q}}_i^l = \left( {1 - t} \right)\mathit{\boldsymbol{Q}}_i^{l - 1} + t\mathit{\boldsymbol{Q}}_{i + 1}^{l - 1}}\\ {i = 0,1, \cdots ,n + 1 - l} \end{array}$ (15)

$l=n-2$

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{Q}}_0^{n - 2} = \mathit{\boldsymbol{V}}_0^{n - 3}\\ \mathit{\boldsymbol{Q}}_1^{n - 2} = \frac{{1 - \lambda }}{3}\mathit{\boldsymbol{V}}_0^{n - 3} + \frac{{2 + \lambda }}{3}\mathit{\boldsymbol{V}}_1^{n - 3}\\ \mathit{\boldsymbol{Q}}_2^{n - 2} = \frac{{2 + \lambda }}{3}\mathit{\boldsymbol{V}}_2^{n - 3} + \frac{{1 - \lambda }}{3}\mathit{\boldsymbol{V}}_3^{n - 3}\\ \mathit{\boldsymbol{Q}}_3^{n - 2} = \mathit{\boldsymbol{V}}_3^{n - 3} \end{array} \right.$ (16)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{p}}\left( t \right) = \sum\limits_{i = 0}^n {b_i^n{\mathit{\boldsymbol{V}}_i}} = \left( {1 - t} \right)\sum\limits_{i = 0}^n {b_i^{n - 1}{\mathit{\boldsymbol{V}}_i}} + t\sum\limits_{i = 0}^n {b_{i - 1}^{n - 1}{\mathit{\boldsymbol{V}}_i}} = }\\ {\left( {1 - t} \right)\sum\limits_{i = 0}^{n - 1} {b_i^{n - 1}{\mathit{\boldsymbol{V}}_i}} + t\sum\limits_{i = 0}^{n - 1} {b_i^{n - 1}{\mathit{\boldsymbol{V}}_{i + 1}}} = }\\ {\sum\limits_{i = 0}^{n - 1} {\left[ {\left( {1 - t} \right){\mathit{\boldsymbol{V}}_i} + t{\mathit{\boldsymbol{V}}_{i + 1}}} \right]b_i^{n - 1}} } \end{array}$

 $\begin{array}{*{20}{c}} {\sum\limits_{i = 0}^n {b_i^3{\mathit{\boldsymbol{V}}_i}} = B_0^3{\mathit{\boldsymbol{V}}_0} + B_1^3\left( {\frac{{1 - \lambda }}{3}{\mathit{\boldsymbol{V}}_0} + \frac{{2 + \lambda }}{3}{\mathit{\boldsymbol{V}}_1}} \right) + }\\ {B_2^3\left( {\frac{{2 + \lambda }}{3}{\mathit{\boldsymbol{V}}_2} + \frac{{1 - \lambda }}{3}{\mathit{\boldsymbol{V}}_3}} \right) + B_3^3{\mathit{\boldsymbol{V}}_3} = \sum\limits_{i = 0}^3 {B_i^3{\mathit{\boldsymbol{Q}}_i}} } \end{array}$

 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{p}}_1}\left( t \right) = \sum\limits_{i = 0}^m {b_i^m\left( {t;{\lambda _1}} \right){\mathit{\boldsymbol{V}}_{1i}}} \\ {\mathit{\boldsymbol{p}}_2}\left( t \right) = \sum\limits_{i = 0}^n {b_i^n\left( {t;{\lambda _2}} \right){\mathit{\boldsymbol{V}}_{2i}}} \end{array} \right.$ (17)

 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{V}}_{20}} = {\mathit{\boldsymbol{V}}_{1m}}\\ {\mathit{\boldsymbol{V}}_{21}} = {\mathit{\boldsymbol{V}}_{20}} + C\left( {{\mathit{\boldsymbol{V}}_{1m}} - {\mathit{\boldsymbol{V}}_{1,m - 1}}} \right) \end{array} \right.$ (18)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{p}}^{\left( k \right)}}\left( t \right) = \frac{{n!}}{{\left( {n - k} \right)!}} \times }\\ {\sum\limits_{i = 0}^{n - k} {{\Delta ^k}\left( {{\mathit{\boldsymbol{Q}}_i} - \lambda {\mathit{\boldsymbol{W}}_i}} \right)B_i^{n - k} \buildrel \Delta \over = \mathit{\boldsymbol{f}} - \mathit{\boldsymbol{\lambda g}}} } \end{array}$

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{f}} = \frac{{n!}}{{\left( {n - k} \right)!}}\sum\limits_{i = 0}^{n - k} {{\Delta ^k}{\mathit{\boldsymbol{Q}}_i}B_i^{n - k}} \\ \mathit{\boldsymbol{g}} = \frac{{n!}}{{\left( {n - k} \right)!}}\sum\limits_{i = 0}^{n - k} {{\Delta ^k}{\mathit{\boldsymbol{W}}_i}B_i^{n - k}} \end{array} \right.$ (19)

 ${\mathit{\boldsymbol{p}}^{\left( k \right)}}\left( t \right) \cdot {\mathit{\boldsymbol{p}}^{\left( k \right)}}\left( t \right) = \mathit{\boldsymbol{f}} \cdot \mathit{\boldsymbol{f}} - 2\lambda \mathit{\boldsymbol{f}} \cdot \mathit{\boldsymbol{g}} + {\lambda ^2}\mathit{\boldsymbol{g}} \cdot \mathit{\boldsymbol{g}}$

 ${E_k}\left( \lambda \right) = \int_0^1 {\mathit{\boldsymbol{f}} \cdot \mathit{\boldsymbol{f}}{\rm{d}}t} - 2\lambda \int_0^1 {\mathit{\boldsymbol{f}} \cdot \mathit{\boldsymbol{g}}{\rm{d}}t} + {\lambda ^2}\int_0^1 {\mathit{\boldsymbol{g}} \cdot \mathit{\boldsymbol{g}}{\rm{d}}t}$

 $\frac{{{\rm{d}}{E_k}\left( \lambda \right)}}{{{\rm{d}}\lambda }} = - 2\int_0^1 {\mathit{\boldsymbol{f}} \cdot \mathit{\boldsymbol{g}}{\rm{d}}t} + 2\lambda \int_0^1 {\mathit{\boldsymbol{g}} \cdot \mathit{\boldsymbol{g}}{\rm{d}}t} = 0$

 $\lambda = \frac{{\int_0^1 {\mathit{\boldsymbol{f}} \cdot \mathit{\boldsymbol{g}}{\rm{d}}t} }}{{\int_0^1 {\mathit{\boldsymbol{g}} \cdot \mathit{\boldsymbol{g}}{\rm{d}}t} }}$ (20)

# 参考文献

• [1] Wang R H, Li C J, Zhu C G. Computational Geometry Tutorial[M]. Beijing: Science Press, 2008. [ 王仁宏, 李崇君, 朱春钢. 计算几何教程[M]. 北京: 科学出版社, 2008.]
• [2] Wang W T, Wang G Z. Bézier curves with shape parameter[J]. Journal of Zhejiang University Science-Science A, 2005, 6(6): 497–501. [DOI:10.1631/jzus.2005.A0497]
• [3] Yan L L, Song L Z. Bézier curves with two shape parameters[J]. Journal of Engineering Graphics, 2008, 29(3): 88–92. [严兰兰, 宋来忠. 带两个形状参数的Bézier曲线[J]. 工程图学学报, 2008, 29(3): 88–92. ] [DOI:10.3969/j.issn.1003-0158.2008.03.017]
• [4] Yang L Q, Zeng X M. Bézier curves and surfaces with shape parameters[J]. International Journal of Computer Mathematics, 2009, 86(7): 1253–1263. [DOI:10.1080/00207160701821715]
• [5] Hang H J, Yu J, Li W G. Two parameters extension of cubic Bézier curve and its applications[J]. Computer Engineering and Applications, 2010, 46(31): 178–180, 205. [杭后俊, 余静, 李汪根. 三次Bézier曲线的一种双参数扩展及应用[J]. 计算机工程与应用, 2010, 46(31): 178–180, 205. ] [DOI:10.3778/j.issn.1002-8331.2010.31.049]
• [6] Yan L L, Liang J F. An extension of the Bézier model[J]. Applied Mathematics and Computation, 2011, 218(6): 2863–2879. [DOI:10.1016/j.amc.2011.08.030]
• [7] Qin X Q, Hu G, Zhang N J, et al. A novel extension to the polynomial basis functions describing Bezier curves and surfaces of degree n with multiple shape parameters[J]. Applied Mathematics and Computation, 2013, 223: 1–16. [DOI:10.1016/j.amc.2013.07.073]
• [8] Yan L L, Wu G G. An new extension of Bézier method[J]. Journal of Hefei University of Technology, 2013, 36(5): 625–631. [严兰兰, 邬国根. Bézier方法的新扩展[J]. 合肥工业大学学报:自然科学版, 2013, 36(5): 625–631. ] [DOI:10.3969/j.issn.1003-5060.2013.05.025]
• [9] Hui H Y, Zhang G C. Similar Bézier curves with two shape parameters[J]. Computer Science, 2014, 41(11A): 100–102, 122. [葸海英, 张贵仓. 带两个参数的拟Bézier曲线[J]. 计算机科学, 2014, 41(11A): 100–102, 122. ]
• [10] Shi L H, Zhang G C. New extension of cubic TC-Bézier curves[J]. Computer Engineering and Applications, 2011, 47(4): 201–204. [师利红, 张贵仓. 三次TC-Bézier曲线的新扩展[J]. 计算机工程与应用, 2011, 47(4): 201–204. ] [DOI:10.3778/j.issn.1002-8331.2011.04.056]
• [11] Dube M, Mishra U. Tension quasi-quintic trigonometric Bézier curve with two shape parameters[J]. International Journal of Recent Scientific Research, 2016, 7(8): 12866–12870.
• [12] Wu B B, Yin J F, Jin M, et al. Rational quadratic trigonometric Bézier curve based on new basis with exponential functions[J]. Journal of Shanghai Normal University:Natural Sciences, 2017, 46(3): 410–416. [吴蓓蓓, 殷俊锋, 金猛, 等. 基于新指数基函数的有理二次三角Bézier曲线[J]. 上海师范大学学报:自然科学版, 2017, 46(3): 410–416. ] [DOI:10.3969/J.ISSN.100-5137.2017.03.009]
• [13] Chen S G, Huang Y D. Hyperbolic Bézier curves with multiple shape parameters[J]. Journal of Engineering Graphics, 2009, 30(1): 75–79. [陈素根, 黄有度. 带多形状参数的双曲Bézier曲线[J]. 工程图学学报, 2009, 30(1): 75–79. ] [DOI:10.3969/j.issn.1003-0158.2009.01.014]
• [14] Zhang J X, Tan J Q. Extensions of hyperbolic Bézier curves[J]. Journal of Engineering Graphics, 2011, 32(1): 31–38. [张锦秀, 檀结庆. 代数双曲Bézier曲线的扩展[J]. 工程图学学报, 2011, 32(1): 31–38. ] [DOI:10.3969/j.issn.1003-0158.2011.01.007]
• [15] Yan L L, Han X L, Zhou Q H. Quadratic hyperbolic Bézier curve and surface[J]. Computer Engineering and Science, 2015, 37(1): 162–167. [严兰兰, 韩旭里, 周其华. 二次双曲Bézier曲线曲面[J]. 计算机工程与科学, 2015, 37(1): 162–167. ] [DOI:10.3969/j.issn.1007-130X.2015.01.025]
• [16] Xu G, Wang G Z, Chen W Y. Geometric construction of energy-minimizing Béezier curves[J]. Science China Information Sciences, 2011, 54(7): 1395–1406. [DOI:10.1007/s11432-011-4294-8]
• [17] Zhang C M, Zhang P F, Cheng F H. Fairing spline curves and surfaces by minimizing energy[J]. Computer-Aided Design, 2001, 33(13): 913–923. [DOI:10.1016/S0010-4485(00)00114-7]
• [18] Long X P. Fairing of curves and surfaces by local energy optimization[J]. Journal of Computer-Aided Design & Computer Graphics, 2002, 14(12): 1109–1113. [龙小平. 局部能量最优法与曲线曲面的光顺[J]. 计算机辅助设计与图形学学报, 2002, 14(12): 1109–1113. ] [DOI:10.3321/j.issn:1003-9775.2002.12.002]
• [19] Wang Y J, Cao Y. Energy optimization fairing algorithm of non-uniform cubic parametric splines[J]. Journal of Computer-Aided Design & Computer Graphics, 2005, 17(9): 1969–1975. [王远军, 曹沅. 非均匀三次参数样条曲线的能量最优光顺算法[J]. 计算机辅助设计与图形学学报, 2005, 17(9): 1969–1975. ] [DOI:10.3321/j.issn:1003-9775.2005.09.013]
• [20] Sun Y H. Quintic G2 interpolating fair curves via curvature variation minmization[D]. Hangzhou: Zhejiang Gongshang University, 2015. [孙义环. 曲率变化最小的五次G2插值光顺曲线[D]. 杭州: 浙江工商大学, 2015.] http://cdmd.cnki.com.cn/Article/CDMD-10353-1016031384.htm
• [21] Wallner J. Existence of set-interpolating and energy-minimizing curves[J]. Computer Aided Geometric Design, 2004, 21(9): 883–892. [DOI:10.1016/j.cagd.2004.07.010]
• [22] Yong J H, Cheng F H. Geometric hermite curves with minimum strain energy[J]. Computer Aided Geometric Design, 2004, 21(3): 281–301. [DOI:10.1016/j.cagd.2003.08.003]
• [23] Zhang A, Zhang C. Shape interpolating geometric hermite curves with minimum strain energy[J]. Journal of Information and Computational Science, 2006, 3(4): 1025–1033.
• [24] Veltkamp R C, Wieger W. Modeling 3D curves of minimal energy[J]. Computer Graphics Forum, 2010, 14(3): 97–110. [DOI:10.1111/j.1467-8659.1995.cgf143_0097.x]
• [25] Jaklič G, Žagara E. Planar cubic G1 interpolatory splines with small strain energy[J]. Journal of Computational and Applied Mathematics, 2011, 235(8): 2758–2765. [DOI:10.1016/j.cam.2010.11.025]
• [26] Ling C C, Abbas M, Ali J M. Minimum energy curve in polynomial interpolation[J]. Matematika (Johor Bahru), 2011, 27(2): 159–167.
• [27] Adriaenssens S, Malek S, Miki M, et al. Generating smooth curves in 3 dimensions by minimizing higher order strain energy measures[J]. International Journal of Space Structures, 2013, 28(3-4): 119–126. [DOI:10.1260/0266-3511.28.3-4.119]
• [28] Li X M, Zhang Y X, Ma L, et al. Discussion on relationship between minimal energy and curve shapes[J]. Applied Mathematics-A Journal of Chinese Universities, 2014, 29(4): 379–390. [DOI:10.1007/s11766-014-3230-2]
• [29] Yan L L, Han X L. Improvement of the modifiable Bézier curves[J]. Journal of Image and Graphics, 2014, 19(9): 1368–1376. [严兰兰, 韩旭里. 对可调控Bézier曲线的改进[J]. 中国图象图形学报, 2014, 19(9): 1368–1376. ] [DOI:10.11834/jig.20140914]
• [30] Wu X Q. Bézier curve with shape parameter[J]. Journal of Image and Graphics, 2006, 11(2): 269–274. [吴晓勤. 带形状参数的Bézier曲线[J]. 中国图象图形学报, 2006, 11(2): 269–274. ] [DOI:10.3969/j.issn.1006-8961.2006.02.019]