Mathematical Sciences: Multivariate Splines

本项目旨在解决多元样条基空间的尺寸和基的突出问题，并将其应用于最有效的曲面插补和通过计算机图形学的三维物体的计算机表示。多元样条族空间的结构比单变量样条族的结构复杂得多，基本问题仍然没有得到回答。这些问题包括维数，基的构造，局部基的存在，以及在插补问题中的应用。这项研究的主要工具将是符号处理语言“归约”，它将使研究人员能够准确地求解由相邻三角形上多项式的光滑性条件产生的大型线性方程组。这位研究人员将把贝塞尔-伯恩斯坦技术与符号操纵相结合，并试图将他的结果应用于工程中出现的表面表示问题。

This project aims at solving the outstanding problem of the dimension and basis for the multivariate spline spaces and applying the results to the most effective surface interpolation and computer representation of three dimensional objects via computer graphics. The structure of multivariate spline spaces is vastly more complex than in the univariate case and fundamental questions are still unanswered. Among these questions are the dimension, the construction of bases, existence of a local basis, and application to interpolation problems. The main tool of this research will be the symbolic manipulation language "Reduce", which will enable the investigator to solve exactly the large linear systems that arise from the smoothness conditions linking polynomials on neighboring triangles. The investigator will blend the Bezier-Bernstein techniques with the symbolic manipulation and will attempt to apply his results to surface representation problems arising in engineering.