图学学报16146607-中国图象图形学报.doc

中图法分类号：TP391.72 文献标识码：A 文章编号：1006-8961(200 ) 论文引用格式： 高阶连续的形状可调三角多项式曲线曲面 严兰兰1，2 ，韩旭里2 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 摘 要：目的：目前使用的B样条曲线曲面存在着高连续阶与高局部调整性两者无法兼而有之的不足，且B样条曲线曲面的形状被控制顶点和节点向量唯一确定，这些因素影响着B样条方法的几何设计效果与方便性。本文旨在克服这种局限，以期构造具有高次B样条方法的高连续阶，低次B样条方法的高局部调整性，以及有理B样条方法权因子决定的形状调整性的曲线曲面。方法：在三角函数空间上构造了一组含参数的调配函数，进而定义具有与3次B样条曲线曲面相同结构的新曲线与张量积曲面。结果：新曲线曲面继承了B样条方法的凸包性、对称性、几何不变性等诸多性质。不同的是，同样是基于四点分段，3次均匀B样条曲线连续，而对于等距节点，在一般情况下，新曲线连续，当参数取特殊值时可达连续。新曲线在连续的情况下存在1个形状参数，能较好地调整曲线的形状同时又无须改变控制顶点。另外，将形状参数设为特定值，新曲线可以自动插值给定点列。新曲面具有与新曲线相应的优点。结论：本文给出了一种在强局部性下实现高阶连续性的形状可调分段组合曲线曲面，为高阶光滑曲线曲面的设计提供了可能，并且新曲线实现了逼近与插值的统一表示，能较好地应用于工程实际。文中调配函数的构造方法具有一般性，可用相同方式构造其他具有类似性质的调配函数。 continuous shape adjustable trigonometric polynomial curve and surface Yan Lanlan1,2, Han Xuli2 1. College of Science, East China Institute of Technology, Nanchang, 330013, China; 2. School of Mathematics and Statistics, Central South University, Changsha, 410083, China Abstract: Objective: The currently using B-spline method has the shortcoming that the high-order continuity and high local adjustment can not be both at the same that. And the shape of B-spline curve and surface is uniquely determined by the control points and knot vector. These factors affect the design effects and convenience of B-spline method. The purpose of this paper is to overcome these limitations. We hope to construct a new curve and surface, which possess the high-order continuity of high-order B-spline method, and high local adjustment of low-order B-spline method, and the shape adjustment property of rational B-spline method determined by the weight factor. Method: We first construct a set of blending functions with parameter on the trigonometric function space, and then define the new curve and tensor product surface which have the same structure as the cubic B-spline curve and surface. Result: The new curve and surface inherit many properties of B-spline method, such as convex hull, symmetry, and geometric invariability, et