带一个形状参数的3次三角多项式曲线曲面  

Cubic Trigonometric Polynomial Curves and Surfaces with a Shape Parameter

作　　者：严兰兰[1,2] 韩旭里[2] 黄涛[1] 

机构地区：[1]东华理工大学理学院,南昌330013 [2]中南大学数学与统计学院,长沙410083

出　　处：《计算机辅助设计与图形学学报》2016年第7期1047-1058,共12页Journal of Computer-Aided Design & Computer Graphics

基　　金：国家自然科学基金(11261003;11271376;60970097);江西省教育厅科技项目(GJJ14493)

摘　　要：为了使扩展的3次均匀B样条曲线在具备形状可调性、高阶连续性、精确表示椭圆等性质的同时还具有变差缩减性,构造了一种基于三角全正基的分段组合曲线.首先用已证明具有最优规范全正性的3次T-Bézier基的线性组合来表示欲构造的扩展基,根据预设的曲线性质反推扩展基的性质,进而求出其表达式;然后将扩展基表示成3次T-Bézier基与一个转换矩阵的乘积,证明转换矩阵的全正性,进而说明了扩展基的全正性;最后由扩展基定义含一个形状参数的分段组合曲线曲面,分析其连续性,分别给出了曲线、曲面表示椭圆、椭球面的条件和方法.实例结果表明,扩展基的全正性保证了曲线的变差缩减性,形状参数的融入为曲线曲面提供了形状调整的工具,曲线曲面具有C3连续性,取特殊参数时可达5C连续.To make the extended cubic uniform B-spline curve possesses not only shape adjustability, high order continuity, exact representation of ellipse, but also variation diminishing, a piecewise combination curve based on trigonometric totally positive basis is constructed. Firstly, we use the linear combination of the T-Bézier basis, which has been proved to be an optimal normalized totally positive basis, to express the extended basis to be constructed. According to the preset properties of the curve, we deduce the properties of the extended basis, and then obtain its expression. We then represent the extended basis as the product of the T-Bézier basis and a conversion matrix. We prove the totally positive property of the matrix and then show the total positivity of the extended basis. Finally, based on the extended basis, a piecewise combination curve and surface with a shape parameter is defined, and the continuity is analyzed. The conditions and methods using the curve and surface to express the ellipse and ellipsoid are given. Experimental results show that the total positivity of the extended basis ensures the variation diminishing of the curve, and the introduction of the shape parameter provides the shape adjustment tool for the curve and surface. The curve and surface have C3 continuity and can reach C5 continuity when taking special parameter.

关 键 词：分段曲线曲面 全正性 三角基 形状参数 插值 

分 类 号：TP391.72[自动化与计算机技术—计算机应用技术]