中文摘要：

曲线曲面的NURBS表示适用于主流及普遍的CAD系统.对于给定的一条非均匀B样条曲线,文中设计了以其为公共测地线的曲面束,并给出一种NURBS表示的显式表达式及全套算法.首先利用离散B样条的性质以及B样条的求导公式得到满足已知条件的参数曲面束表示;其次利用度量函数的任选性,推导出3个度量函数与已知曲线的Frenet标架的乘积的B样条表示;进一步,在度量函数的因子式中预先设置好准有理B样条表示的一系列权因子,因势利导地给出了曲面束的一族参数,可方便地实现对所需曲面的修改.通过实例验证了文中算法的有效性和可行性.

英文摘要：

NURBS curves and surfaces can be applied in the lineal and prevalent CAD systems. In this paper, a design of surface pencil through the given non-uniform B-spline curve as geodesic is constructed, and its explicit NURBS’s expression and intact algorithm are got. First, using the character of the discrete B-spline and the derivative formula of B-spline, the expression of surface pencil with constraint is presented. Next, using randomicity of the marching-scale functions, the products of three marching-scale functions and the Frenet frames of thegiven curve have been precisely represented as the B-spline forms. Furthermore, preestablishing a series of weights in the pre-rational B-spline expressions for the marching-scale functions’ factors, we improve the occasion and give a group of family-parameters of the surface pencil. It is convenient for adjusting the shapes of the resulting surfaces. Finally, the examples in the experiment show that the algorithm is efficient and applicable.