带两个参数的三角多项式曲线曲面构造  

Constructing trigonometric polynomial curves and surfaces with two parameters

作　　者：汪凯 张贵仓[1] 龚进慧 Wang Kai;Zhang Guicang;Gong Jinhui(Northwest Normal University,Lanzhou 730070,China)

机构地区：[1]西北师范大学,兰州730070

出　　处：《中国图象图形学报》2018年第12期1910-1924,共15页Journal of Image and Graphics

基　　金：国家自然科学基金项目(61861040);甘肃省科技基金项目(17YF1FA119);甘肃省教育厅科技成果转化基金项目(2017D-09)~~

摘　　要：目的为了使扩展的曲线曲面保留传统Bézier方法以及B样条方法良好性质的同时,具备保形性、形状可调性、高阶连续性以及广泛的应用性,本文在拟扩展切比雪夫空间利用开花的性质构造了一组最优规范全正基,并利用该基进行曲线曲面构造。方法首先构造一组最优规范全正基,并给出该基生成的拟三次TC-Bézier曲线的割角算法;接着利用最优规范全正基的线性组合构造拟三次均匀TC-B样条基,根据曲线的性质假设拟三次均匀B样条基函数具有规范性和C^2连续性,进而得到其表达式;然后证明拟三次均匀TC-B样条基具有全正性和高阶连续性;最后定义拟三次均匀TC-B样条曲线曲面,并证明曲线曲面的性质,给出曲线表示整圆和旋转曲面的表示方法,设计出球面和旋转曲面的直接生成方法。结果实验表明,本文在拟扩展切比雪夫空间构造的具有全正性曲线曲面,不仅能够灵活地进行形状调整,而且具有高阶连续性、保形性。结论本文在三角函数空间利用两个形状参数进行曲线曲面构造,大量的分析以及案例说明本文构造的曲线曲面不仅保留了传统的Bézier方法以及B样条方法的良好性质,而且具备保形性、形状可调性、高阶连续性以及广泛的应用性,适合用于曲线曲面设计。Objective The Bézier and B-spline curves play an important role in traditional geometric design. With the development of the geometric industry over the recent years,the traditional Bézier and B-spline curves cannot meet people ’s needs due to defects. At the same time,many rational forms of Bézier curves are proposed,which solve the problems faced by traditional methods. However,rational methods have not only progressive problems,but also employ the improper use of weight factors,which can be destructived to the curve and surface design. In view of the abovementioned problems,a large number of Bernstein-like and B-spline-like basis functions with shape parameters are proposed. These methods are mainly constructed in trigonometric,hyperbolic and exponential function spaces,a combination of said spaces,and polynomial space. Although many improved methods are available,these methods are rarely applied in solving practical problems. In the final analysis,these methods increase the flexibility of the curve by adding shape parameters,compared with the traditional Bézier and B-spline methods. However,the method itself does not have the ability to replace the traditional method.Several aspects still need improvement. For example,the majority of these methods only discuss basic properties,such as non-negativity,partition of unity,symmetry,and linear independence. Shape preservation,total positivity,and variation diminishing are often overlooked,which are important properties for curve design. However,the basis function,which has total positivity,will ensure that the related curve contains variation diminishing and shape preservation. Therefore,possessing total positivity is highly important for basis function. In addition,constructing cubic curves and surfaces remains the main method among the improved methods. In general,these improved methods have C2 continuity,which largely meets engineering requirements. However,in many practical applications,C^2 continuity cannot meet current needs. In summary,this study aims

关 键 词：拟扩展切比雪夫空间 最优规范全正基 全正性 高阶连续性 保形性 

分 类 号：TP391.72[自动化与计算机技术—计算机应用技术]