超限插值曲面造型的连分式方法与光滑拼接研究

Based on the generalised inverse and partial generalised inverse of vector-valued function, the rational transfinite interpolation surface mondeling is presented via continued fractions interpolation approach. This will enrich the continued fraction interpolation theory, and traditional polynomial transfinite interpolation surface modeling is extended to the rational case. In this project, smooth joining of the transfinite interpolation surface patch is also discussed via blending approach. For more details, first，the definitions of the generalized inverse and partial generalised inverse are discussed , and by means of the above definitions, the vector-valued functionn continued fraction is defined, recursive formula of the continued fraction calculation are also disscussed; transfinite interpolation surface patch and osculatory transfinite interpolation surface patch in one-direction is discussed via continued fractions interpolation algorithm;The smooth joining of the transfinition interpolation surface patch in one-direction is discussed; The rational transfinite interpolation surface patch on rectangular grid and triangur grid is constructed via bivariate continued fraction and its smooth connection; Then, when profile curve and mesh curve is the B-spline one,how the constructed rational transfinite interpolation surface patch converted to NURBS surface patches is discussed; Finally, we discuss some applications of the continued fraction transfinite interpolation modeling method such as a pipeline surface, medical 3D visualization modeling et.al. The presented approach is the extension of Coons, Gordon transfinite interpolation surface modeling method, and has important significance for surface modeling and engineering applications.

本项目讨论向量函数广义逆及其偏广义逆，提出有理超限插值曲面造型的连分式插值算法，这将丰富连分式插值理论，且把传统的多项式Gordon 超限插值曲面造型推广到有理情形，并讨论基于融合的超限插值曲面片的光滑拼接方法。具体地说，研究向量函数广义逆及其偏广义逆的定义，以及基于此定义下的连分式定义、连分式计算的递推公式；研究单方向一元连分式构造超限插值曲面片以及切触超限插值曲面片的方法；研究单方向超限插值曲面片基于融合的光滑拼接；研究矩形网格与三角网格下有理超限插值曲面片的二元连分式造型算法及其基于融合的光滑拼接；然后研究当轮廓曲线或网格曲线为B-样条曲线时，所构造的有理超限插值曲面片的NURBS表示；最后探讨连分式超限插值造型方法的一些应用如管道曲面、医学三维可视化造型等。本课题研究成果将推广Coons,Gordon等提出的超限插值曲面造型技术，并对曲面造型具有重要的科学意义和工程应用。

本项目给出了一元向量函数广义逆以及二元向量函数的偏广义逆，采用连分式插值算法研究了有理超限插值曲面造型技术，把传统的多项式超限插值造型方法推广到有理超限插值曲面情形。使有理超限插值曲面造型向成为一种实用的CAGD/CAD造型技术方向迈进了一步。为此目的本课题项目提出了一系列造型算法：基于广义逆定义下的连分式定义、建立连分式计算的递推公式；如建立了单方向连分式超限插值曲面片构造算法；建立了矩形网格下二元连分式有理超限插值曲面片以及样条曲面构造算法及样条曲面构造算法。同时探求采用连分式有理超限插值造型方法能解决多项式超限插值造型不能解决的问题，如管道曲面造型方法。实验结果显示这一些工作在CAGD中的有效性，是有理超限插值曲面方法在曲面造型有意义的扩展。另外，本项目借助μ基，讨论了有理曲线曲面的构造算法