曲线升阶中一类问题的研究

【作者】 孙景楠；

【导师】 王仁宏；

【作者基本信息】 大连理工大学 ， 计算数学， 2003， 硕士

【摘要】 曲线升阶是曲面设计和计算机辅助几何造型中的一项重要技术，它是CAGD系统中的一个基本工具，经常被用于组合曲线、蒙皮曲面或扫描曲面的几何设计中。本文主要研究了从B样条曲线的升阶到任意次几何连续样条曲线的升阶。 B样条曲线经典的升阶算法共有三种：Prautzsch升阶算法、Cohen升阶算法以及Piegl&&Tiller升阶算法。本文通过提出一个新的端点插值方法，对它们进行改进，使之适用于所有均匀和非均匀B样条曲线。 给定节点向量上的B样条曲线完全由其控制顶点决定。通过多项式和对称多元仿射变换的等价性，即blossoming，可以将对称多元仿射变换在节点序列上的值作为B样条曲线的控制顶点。这个结果被用到节点插入和升阶上，取得很好的效果。 最后，对于任意次几何连续样条曲线，我们以通用样条的概念为基础，得到了样条控制顶点和Bézier点的几何构建，并给出了节点插入和升阶算法。几何构建是通过密切面的交实现的，通用样条的定义保证了交的存在。并且，我们还将blossom推广到了任意次几何连续样条曲线上。更多还原

【Abstract】 Degree elevation of spline curves is an important technique in surface design and computer aided geometric modeling. It is a fundamental tool in CAGD systems and frequently used in geometric design of composite curves, sweeping and skinning surfaces. This paper studies degree elevation from B-spline curves to geometrically continuous spline curves of arbitrary degree.There are three traditional degree elevation algorithms, including Prautzsch algorithm, Cohen algorithm and Piegl&&TiHer algorithm. In this paper, a new method of endpoint interpolating is presented, which improved those algorithms to be useful to all uniform and non-uniform curves.A B-spl ine curve over a given knot vector is completely determined by its control points. By the use of the equivalence between polynomials and symmetric multiaffine mapcs, it is possible to compute B-spJine control points as values of symmetric multiaffine mapes at a sequence of consecutive knots. This result is used to knot insertion and degree elevation, which has been proved very useful.At last, it comes to geometrically continuous spline curves of arbitrary degree. Based on the concept of universal splines , we obtain geometric constructions for the spline control points, for the Bezier points, for knot insertion and degree elevation. The geometric constructions are based on the intersection of osculating flats. The concept of universal splines is defined in such a way that the intersections are guaranteed to exist. As a result of this development , we obtain a generalization of blossom to geometrically continuous spline curves by intersecting osculating flats.更多还原