摘要
本文主要讨论了Said-Ball曲线、曲面的降阶逼近,共分为四章。
第一章绪论介绍了本文所讨论问题的实际背景及研究的意义。
第二章介绍了本文的预备知识。作者首先介绍了Said-Ball奇函数的定义及其基本性质并通过Said-Ball曲线的升阶公式得到了Said-Ball曲线的降一阶的矩阵表示,通过对端点求导,得到了端点任意阶导数的一般表示式。
第三章作者给出了两种Said-Ball曲线降多阶方法,分别利用广义逆理论及两Said-Ball曲线在最小二乘范数下距离函数取最小值,将给定的n次Said-Ball曲线一次降为m次Said-Ball曲线。两种方法均考虑了不带端点插值条件和具有端点高阶插值条件的情形,并给出了降阶Said-Ball曲线控制顶点的显示表示式。
在第四章作者将Said-Ball曲线一次降n-m阶的两种方法推广到张量积Said-Ball曲面,得到了n×m阶张量积Said-Ball曲面一次降为n_1×m_1阶张量积Said-Ball曲面,在降阶过程中作者分别考虑了不带角点插值条件和具有角点插值条件的情形。
文中给出了数值实例,来显示所给出的两种方法的效果。
This paper is concerned about the degree reduction of the Said-Ball curves and surfaces. There are totally four chapters in this paper.
In Chapter one the author surveys the significances and background of the degree reduction techniques for curves and surfaces.
Chapter two is the preparation knowledge of this paper, which firstly covers the current definition of the Said-Ball curves as well as their properties, and secondly finds for the first time the matrix representation of such curves one degree lower based on the degree elevation formula of the Said-Ball curves, and finally offers the general formulae of derivatives of arbitrary orders at endpoints by differentiation.
In Chapter three the author offers two methods of multi-degree reduction of the Said-Ball curves, one of which is proposed by using the theory of generalized inverse matrix theory, another of which is proposed by minimizing L2 norm distance function between two Said-Ball curves, each of which reduces a given degree n Said-Ball curve to a degree m Said-Ball curve. In the process of degree reduction, two cases are considered respectively, one of which is about the case without conditions of endpoint interpolations, the other of which is about the case with conditions of endpoint interpolations of higher orders. The explicit formulas of control points of the reduced Said-Ball curves are obtained.
In the last chapter, the above two methods are generalized to the tensor product of the Said-Ball surfaces, each of which reduces a given degree nxm tensor product Said-Ball surface to an n1 xm1 one. In the process of degree reduction for the tensor product Said-Ball surfaces, we not only discuss the case with constraints of corner interpolations, but also the case without that.
In this paper, some numerical examples are given to demonstrate the effects of our methods.
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