【作者】 王文涛；

【导师】 汪国昭；

【作者基本信息】 浙江大学 ， 计算机辅助几何设计与计算机图形学， 2005， 博士

【摘要】 在CAD中参数曲线曲面造型方法运用得非常广泛，本文对参数曲线曲面造型的一种新方法——带形状参数样条曲线曲面造型方法进行了深入的研究。首先系统地介绍了CAD中带形状参数曲线曲面造型方法的发展历史和背景，然后利用递归和积分方法构造了带形状参数样条基函数，根据初始函数和节点向量的选取产生了不同的带形状参数样条基函数，最后对带形状参数样条基函数的性质，形状参数的取值范围，以及带形状参数样条曲线的VD性质进行了讨论。本文就以下几个方面给出了研究成果： （1）带形状参数均匀样条曲线曲面 利用递归和积分方法选取了三种初始函数，分别生成了k阶（k≥2）带形状参数均匀B样条曲线曲面，带形状参数三角多项式均匀B曲线曲面，带形状参数双曲多项式均匀B曲线曲面，它们都有一个可调形状参数，并且有高于β样条曲线的G²连续性（k＞4），能够达到C^k-2连续。在控制多边形不变的情况下，通过调整形状参数能生成不同位置的曲线，同时具有与均匀B样条曲线相同的结构和几何性质。随着阶数的升高，形状参数可调整范围也越来越大。带形状参数三角多项式均匀B样条曲线可以精确表示圆，椭圆，螺旋线。带形状参数双曲多项式均匀B样条曲线可以精确表示双曲线。 （2）带形状参数Bezier样条曲线曲面 利用递归和积分方法选取了三种初始函数，分别生成了七阶（k≥2）带形状参数Bezier曲线曲面，带形状参数三角多项式Bezier曲线曲面，带形状参数双曲多项式Bezier曲线曲面，它们都有一个可调形状参数，拥有许多和Bezier曲线曲面相同的结构和性质以及一些使用的几何特性。在控制多边形不变的情况下通过调整形状参数，能生成位于Bezier曲线曲面附近的不同位置的曲线曲面。随着形状参数的增大，带形状参数的Bezier样条曲线更加逼近其控制多边形。 （3）带形状参数样条曲线曲面 给定参数t轴的一个分割T：{t_i}_-∞^∞，t_i≤t_i+1，i=0，±1，…，利用递归和积分方法选取了三种初始函数，分别生成了七阶（k≥2）带形状参数B样条曲线曲面，带形状参数更多还原

【Abstract】 The parameter curves and surfaces modeling are used very widely in CAD. In this paper we profoundly research a new technique in parameter curves and surfacesmodeling-parameter curves and surfaces modeling with shape parameter. Firstly, weintroduce the development history and background of parameter curves and surfaces modeling with shape parameter. Then basis of initial function and node vector, the different basic spline functions with shape parameter are constructed by recursion and integral approach. The properties of basic spline functions with shape parameter, the range of shape parameter and the VD property of spline curves with shape parameter are all discussed lastly. The main contributions are listed as follows:（1） Uniform Spline curves and surfaces with shape parameterChoosing three initial functions, then we can get k -th （k ≥ 2） order uniform B spline curves and surfaces with shape parameter, trigonometric polynomial and hyperbolic polynomial uniform B spline curves and surfaces with shape parameter. They all possess one adjustable shape parameter and can reach C^k-2 that is higher than βspline curves which is G² continuous. The different curves can be adjusted by changing shape parameter in invariable control polygon and at the same time they have the same structure and geometry properties as uniform B spline curves. With the elevation of order, feasible range of the shape parameter value is extended. The circle, ellipse and helices can be represented with trigonometric polynomial uniform B spline curves with shape parameter. The hyperbola can be represented with the hyperbolic polynomial uniform B spline curves with shape parameter.（2） Bezier Spline curves and surfaces with shape parameterChoosing three initial functions, then we can get k -th （k ≥ 2） order Bezier curves and surfaces with shape parameter, trigonometric polynomial and hyperbolic polynomial Bezier curves and surfaces with shape parameter. They all possess one adjustable shape parameter and have the same structure and geometry properties as Bezier curves. The different curves lying on the Bezier curve can be created by changing shape parameter in invariable control polygon and approximate to the control polygon as the increase of the更多还原