二次双曲Bézier曲线曲面

J4 ›› 2015, Vol. 37 ›› Issue (01): 162-167.

二次双曲Bézier曲线曲面

严兰兰1,2，韩旭里2，周其华1

（1.东华理工大学理学院,江西南昌 330013;2.中南大学数学与统计学院,湖南长沙 410083）

收稿日期:2013-03-12 修回日期:2013-08-26 出版日期:2015-01-25 发布日期:2015-01-25
基金资助:
国家自然科学基金资助项目（11261003,11271376,60970097）;江西省教育厅科技项目（GJJ14493）

Quadratic hyperbolic Bézier curve and surface

YAN Lanlan1,2，HAN Xuli2,ZHOU Qihua1

(1.College of Science,East China Institute of Technology,Nanchang 330013;
2.School of Mathematics and Statistics,Central South University,Changsha 410083,China)

Received:2013-03-12 Revised:2013-08-26 Online:2015-01-25 Published:2015-01-25

摘要/Abstract

摘要：

为了简化构造组合曲线时，相邻曲线的控制顶点间应满足的光滑拼接条件, 构造了一种结构类似于二次Bézier曲线的含参数的双曲型曲线，称之为H-Bézier曲线。该曲线具有Bézier曲线的许多基本性质, 如凸包性、对称性、几何不变性、端点插值和端边相切性。另外，该曲线具备形状可调性，可以精确表示双曲线。此外, 若取特殊的参数，则当相邻HBézier曲线的控制顶点间满足普通Bézier曲线的G1光滑拼接条件时, 曲线在公共连接点处可以达到G3光滑拼接。另外, 给出了构造与给定多边形相切的H-Bézier曲线的方法, 该方法简单有效, 而且整条曲线对给定的切线多边形是保形的。运用张量积方法，将H-Bézier曲线推广后得到的曲面同样具有很多良好的性质。

关键词: 曲线设计, Bé, zier曲线, 连续性, 形状参数, 切线多边形

Abstract:

When composite curves are constructed, the control points of the adjacent curves must meet certain conditions. In order to simplify the conditions, a kind of hyperbolic polynomial curve with parameters, called H-Bézier curve, which has a structure similar to the quadratic Bézier curve, is constructed. This curve has many basic properties of Bézier curve, such as convex hull property, symmetry, geometric invariance, endpoint interpolation and end edge tangent property. Besides, the curve has shape adjustability and can represent hyperbola. If special parameters are chosen, then when the control points of two adjacent curves satisfy the conditions for Bézier curve, they can achieve continuity. In addition, the method of constructing H-Bézier with given tangent polygon is given. This method is simple and effective, and the entire curve is conformal to the tangent polygon. Using tensor product method, the H-Bézier curve can be extended to surface, which also has many good properties.

Key words: curve design;Bézier curve;continuity;shape parameter;tangent polygon

严兰兰1,2，韩旭里2，周其华1. 二次双曲Bézier曲线曲面[J]. J4, 2015, 37(01): 162-167.

YAN Lanlan1,2，HAN Xuli2,ZHOU Qihua1. Quadratic hyperbolic Bézier curve and surface [J]. J4, 2015, 37(01): 162-167.

参考文献

［1］Piegl L. A generalization of the BernsteinBézier method ［J］. ComputerAided Design, 1984, 16(4):209215.
［2］Wang Wentao, Wang Guozhao. Bézier curves with shape parameter ［J］. Journal of Zhejiang University Science A(Science in Engineering), 2005, 6(6):497501.
［3］Han Xian, Ma Yichen, Huang Xili. A novel generalization of Bézier curve and surface ［J］. Journal of Computational and Applied Mathematics, 2008, 217(1):180193.
［4］Yang Lianqiang,Zeng Xiaoming.Bézier curves and surfaces with shape parameters ［J］. International Journal of Computer Mathematics, 2009, 86(7):12531263.
［5］Qin Xinqiang, Zhang Nianjuan, Hu Gang. A novel Bézier curve with multiple shape parameters ［C］∥Proc of 2010 3rd IEEE International Conference on Computer Science and Information Technology, 2010:494498.
［6］Chen Jie,Wang Guojin.A new type of the generalized Bézier curves ［J］. Applied MathematicsA Journal of Chinese Universities, 2011, 26(1):4756.
［7］Yan Lanlan, Liang Jiongfeng. An extension of the Bézier model ［J］. Applied Mathematics and Computation, 2011, 218(6):28632879.
［8］Chen Qinyu, Wang Guozhao. A class of Bézierlike curves ［J］. Computer Aided Geometric Design, 2003, 20(1):2939.
［9］Han Xian, Ma Yichen, Huang Xili. The cubic trigonometric Bézier curve with two shape parameters ［J］. Applied Mathematics Letters, 2009, 22(2):226231.
［10］Li Juncheng,Chen Guohua,Yang Duqing.Modifible quasi cubic Bézier trgonometric curves ［J］. Computer Engineering & Science, 2010, 32(3):6971.(in Chinese)
［11］Han Xuli,Zhu Yuanpeng.Curve construction based on five trigonometric blending functions ［J］. BIT Numerical Mathematics, 2012, 52(4):953979.
［12］Su Benyue,Sheng Min.Bézierlike curves and surfaces based on hyperbolic functions ［J］. Computer Engineering and Design, 2006, 27(3):370372.(in Chinese)
［13］Xie Jin,Tan Jieqing.Quadratic hyperbolic polynomial curves with multiple shape parameters ［J］. Journal of Image and Graphics,2009,14(6):12061211.(in Chinese)
［14］Zhang Jinxiu, Tan jieqing. Extensions of hyperbolic Bézier curves ［J］. Journal of Engineering Graphics, 2011, 32(1):3138.(in Chinese)
［15］Shi Fazhong. Computer aided geometric design & NURBS ［M］. Beijing:Higher Education Press, 2001.(in Chinese)
［16］Liu Zhi, Tan Jieqing, Jiang Ping, et al. Closed C2continuous quartic generalized ball curves with given tangent polygons ［J］. Higher School Journal of Computational Mathematics, 2012, 34(3):231237.(in Chinese)
附中文参考文献：
［10］李军成, 陈国华, 杨笃庆. 可调的类三次Bézier三角曲线［J］. 计算机工程与科学, 2010, 32(3):6971.
［12］苏本跃, 盛敏. 基于双曲函数的Bézier型曲线曲面［J］. 计算机工程与设计, 2006, 27(3):370372.
［13］谢进, 檀结庆. 多形状参数的二次双曲多项式曲线［J］. 中国图象图形学报, 2009, 14(6):12061211.
［14］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.
［15］施法中. 计算机辅助几何设计与非均匀有理B样条［M］. 北京:高等教育出版社, 2001.
［16］刘植, 檀结庆, 江平, 等. 与给定多边形相切的C2四次广义Ball闭曲线［J］. 高等学校计算数学学报, 2012, 34(3):231237.

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