几何计算方法及其稳定性研究

项目名称： 几何计算方法及其稳定性研究

项目编号： No.60803076

项目类型： 青年科学基金项目

立项/批准年度： 2009

项目学科： 矿业工程

项目作者： 陈小雕

作者单位： 杭州电子科技大学

项目金额： 20万元

中文摘要： 以距离计算与求交等问题为例，单纯的代数方程组求根技术的效率是很低的。几何计算方法将基于问题本身的几何背景、充分利用代数几何等理论和稳定成熟的剖分技术来获取足够高的计算效率和求解的稳定性，从而更好地满足应用中越来越高的稳定性和实时性要求。本项目研究几何计算方法及其稳定性理论。直接从曲线曲面自身的几何信息出发，以挖掘几何计算问题内在的几何性质的角度来研究高效的几何裁剪方法及其稳定性理论，并研究降维简化的方法，进一步提高计算效率。并以距离计算、求交等问题为例，通过sweeping球、曲线束、曲面束等构造方法和理论的研究来探索病态情形到非病态情形的转化方法，以期实现几何计算方法和数值方法等的完美结合。最后探索更多的应用问题到几何计算问题的转化方法。

中文关键词： 几何计算方法；稳定性；曲线曲面；最近距离；求交

英文摘要： The pure algebraic method for solving non-linear algebraic equation system is not efficient in applications such as distance computation problem and intersection problem between curves and surfaces. Based on the geometric background of these problems, the geometric method is derived, which utilizes both the algebraic geometry theory and the subdivision technique, to obtain higher computation efficiency and robustness, or try to meet with the realtime requirement. This project is to study the geometric computation method and its robustness.　Directly based on the geometric information of given curves and surfaces, new methods are derived from the geometric property of the problem itself, which is trying to obtain high efficiency and robustness. The dimension reduction method is also used for further improvement. Taking distance computation and intersection problem as examples, sweeping sphere method and curve pencil method are tried for finding the possible transformation method from an ill-conditioned case to better one, which is able to improve the stability of numerical method. Finally, we try to find more applications with the geometric method.

英文关键词： Geometric computation method; Robustness; Curves and Surfaces; Minimum distance; Intersection