分片代数曲线曲面的理论与应用研究

项目名称： 分片代数曲线曲面的理论与应用研究

项目编号： No.10801024

项目类型： 青年科学基金项目

立项/批准年度： 2009

项目学科： 轻工业、手工业

项目作者： 朱春钢

作者单位： 大连理工大学

项目金额： 17万元

中文摘要： 本项目基本按照研究计划执行，围绕分片代数曲线曲面的理论和应用开展研究工作，主要研究重点为：1. 分片代数曲线、分片代数簇的理论研究，包括分片代数曲线的Noether型定理与Bezout型定理，分片代数簇与多元样条空间的代数几何关系等。2、多元样条的理论及应用研究，包括二元样条插值适定结点组的构造，样条拟插值在数值分析中的应用，样条理论在组合中的应用等。3、曲线曲面造型的理论与方法研究，包括toric曲面退化的几何意义，分片代数曲面拟合散乱数据点，插值空间Bezier曲线的可展曲面的拼接等。在项目的执行过程中，项目组成员团结协作，取得了一系列的研究成果,在ACM TOG, SIAM J DM, CAD, JCAM, JMAA，《中国科学》，《数学学报(英文版)》等国内外著名期刊上发表（含录用）论文32 篇，会议论文2篇，其中SCI 收录24 篇，EI 收录18 篇。项目组成员参加多次国内国际会议，并做邀请报告与分组报告。项目共计培养硕士4 名，协助培养博士6 名。

中文关键词： 计算几何；样条；分片代数曲线曲面；几何造型

英文摘要： This project has been accomplished according to the plan. Our researches focus on the theories and applications of piecewise algebraic curves and surfaces, which consist of the following aspects: 1. The theories of piecewise algebraic curves and piecewise algebraic varieties, including the Nother-type and Bezout-type theorems of piecewise algebraic curves, and the correspondences between piecewise algebraic varieties and multivariate spline spaces. 2. The theories and applications multivariate splines, including the constructions of well-posed interpolation sets for bivariate spline space, the applications of quasi-interpolants by spines in numerical analysis, and the applications of splines in combinatorics. 3. The theories and applications of curves and surfaces design, including the geometric meaning of degenerations of toric patches, scattered points fitting by piecewise algebraic surfaces, and the blending by developable surface interpolating Bezier curves. We have published (including accepted papers) 32 papers in journals, such as ACM TOG, SIAM J DM, CAD, JCAM, JMAA, Sciences in China Ser A, Acta Mathematica Sinica(English Series), and 2 papers in conference proceedings, including 24 papers indexed by SCI, 18 by EI. Project members participated in several domestic and international conferences to give invited and contributed talks. Some young researchers have taken part in this project. During the project period, 6 of them have been awarded PhD degree and 4 of them have been awarded master degree.

英文关键词： Computational Geometry; Splines; Piecewise algebriac curves and surfaces; Geometric Modeling