CAD中变次数B样条升阶性质及应用研究

This project studies the degree elevation property and its applications of changeable degree B-splines in CAD. Changeable degree B-spline breaks the limit of the same degree property of nonzero piecewise polynomials in any B-spline and allows different degrees. Thus, when a low-degree segment is connected with a high-degree segment, low-degree polynomials do not need to be represented by high-degree polynomials, so the data size and computation load are both reduced. For B-spline, degree elevation is a global property, since the degrees of all the polynomial segments are raised together. Compared with it, the degree elevation property of changeable degree B-spline is local, because that for this kind of spline, when the degree of one segment is raised, the degrees of the other segments can be kept. Moreover, through degree raising of changeable degree B-spline, the degree elevation process of B-spline can be represented as a geometric corner-cutting form of control points. This shows that the degree elevation property of changeable degree B-spline has many advantages. However, for the complexity of degrees, the theory of degree elevation of changeable degree B-spline only applies to the two-degree cases. This project aims to overcome difficulties, explore new ideas and methods, and establish a generalized theory of the degree elevation of changeable degree B-spline. This theory is needed to be appropriate for the cases of arbitrary degrees and meet the needs of CAD modeling systems well. Firstly, the degree elevation formulas of basis functions of changeable degree B-spline will be studied. Next, the degree elevation formulas of changeable degree B-spline curves. And then, the conversion algorithms between B-spline and changeable degree B-spline are designed, which meet the conversion requirements of geometric models, and play a huge role in CAD.

本项目研究变次数B样条的升阶性质及应用。变次数B样条打破B样条非零段次数相同的限制，允许次数不同。因此，在高低次段拼接时，低次多项式不需要用高次多项式表示，从而能减少数据量和计算量。B样条的升阶，每段次数都升高，具有整体性。相比之下，变次数B样条的升阶，可以仅升某段的次数，而其余段次数保持不变，具有局部性，升阶情况灵活多样。并且，B样条的升阶能通过变次数B样条的升阶表示成控制顶点几何割角的过程。由此可见，变次数B样条的升阶，优点突出。但是，由于次数变化的复杂性，现有的升阶理论，仅适用于双次数的情况。本项目旨在克服困难，发掘新思路，探索新方法，建立变次数B样条在次数任意可变情况下的一般升阶理论，以满足CAD造型系统的要求。项目研究方案首先从基函数的升阶公式入手，接着给出曲线的升阶公式，进而设计变次数B样条曲线与B样条曲线相互转换的算法，以适应几何模型转换的需求，使其能在CAD中发挥巨大作用。

围绕变次数B样条的升阶，本项目主要研究了三部分内容。第一、首次给出了变次数B样条基函数和曲线的一般升阶公式。该公式对次数任意可变的变次数B样条，其中的任意一段，提升任意阶的情况均适用。第二、研究了升阶的几何意义。关于任意变次数B样条（包括传统的B样条），得出结论：曲线上任意一段提升1阶，恰好是其控制多边形的一个割角过程；曲线上任意多段，每段分别提升任意的阶数，必然可以表示为其控制多边形的一系列渐进的割角过程。第三、给出了变次数B样条和B样条模型的相互转换算法。通过这些算法，任意变次数B样条可以精确转换为以其最高次为次数的B样条模型；任意B样条可以精确转换为每段次数均大于或等于原B样条次数的任意变次数B样条模型。转换过程运用升阶实现，其中：先升哪段，后升哪段，可以在需要升阶的段中任意选择；每段先升多少阶，后升多少阶，可以在不超过需要提升的阶数中任意选择，转换过程灵活多样，并且都可以表示为割角过程。因此，本项目圆满的完成了申请书中的研究内容和研究目标。到目前为止，本项目正式发表了期刊论文6篇，其中，SCI检索1篇，EI检索4篇。所有论文均以项目负责人为第一作者和通讯作者，其中1篇发表在《Computer-Aided Design》上。全国或国际会议报告2次，其中，出国（境）报告1次。.