NURBS曲线曲面的显式矩阵表示及其算法

从 B样条的差商定义出发 ,提出差商展开系数的概念 ,通过差商展开系数显式解析表示式的导出 ,得到任意次 NU RBS曲线曲面系数矩阵的显式解析表示式 ,并给出了求差商展开系数和 NURBS曲线曲面系数矩阵的数值算法 .文中给出的方法适用于一切 NU RBS曲线曲面 ,包括有理和非有理的 Bézier、均匀和非均匀的 B样条曲线曲面 .相应的数值算法计算简单 ,易于实现 .差商展开系数解析表示式为 NU RBS曲线曲面的表示、转换和节点插入、升阶等基本运算以及与差商相关的问题的研究提供了一个统一的构造性工具和应用方法 .

Explicit Matrix Representation for NURBS Curves and Surfaces and Its Algorithm

NURBS offers a common mathematical form for representing and designing both analytic and free-form curves and surfaces. So it is specially important to seek the proper forms of representation for nonuniform B-spline itself, which is the basis for NURBS curves and surfaces. The key to matrix representation for NURBS curves and surfaces is the piecewise power function base representation for B-spline. In the view of algebra, the coefficient matrix of a NURBS curve or surface is just the transformation matrix of a B-spline base to a power function base at each subinterval in a polynomial spline space. Therefore seeking explicit matrix representation for NURBS curves and surfaces is actually seeking the explicit piecewise polynomial representation for B-spline base functions. In the past researches, B-spline base functions were generally represented in the form of recursion or divided difference, which makes it difficult to obtain the explicit analytic representation for coefficient matrices of NURBS curves and surfaces. In this paper, the concept of divided difference expanded coefficient is presented and its explicit analytic representation formula is derived. Based on this formula and the divided difference definition of B-spline, the explicit piecewise polynomial representation for B-spline base functions is obtained and then an explicit matrix representation formula for NURBS curves and surfaces of arbitrary order is presented. Besides, efficient numerical algorithms for the calculations of divided difference expanded coefficient and the coefficient matrix of NURBS curves and surfaces are also given respectively. They can be easily realized. The results can be applied to all NURBS curves and surfaces including rational and nonrational Bézier, uniform and nonuniform B-spline forms. The analytic representation formula of divided difference expanded coefficient provides a uniform constructive tool and application approach for the research of NURBS curves and surfaces such as their representation, transformation, knot insertion, degree elevation, etc.