分片多项式系统在几何造型中的应用基础研究

项目名称： 分片多项式系统在几何造型中的应用基础研究

项目编号： No.11271328

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 赖义生

作者单位： 浙江工商大学

项目金额： 56万元

中文摘要： 本项目对分片多项式系统的若干拓扑理论及其在几何造型中应用的基础理论与方法进行研究。研究内容包括建立关于构造具有预先给定拓扑、给定次数及光滑度的实分片代数超曲面的粘合理论及其"Patchwork"方法；确定分片多项式系统的Betti数的界，建立计算分片系统的Betti数与Euler示性数的理论及算法；建立分片多项式的临界点和奇点的粘合理论；确立分片多项式系统的Betti数、指数为i的临界点数与剖分的胞腔数、内顶点数、内公共面数、分片多项式的次数以及光滑度之间的约束关系；确定低次分片多项式的临界点指数分布的分类，以及各类中分片多项式的系数约束条件。基于上述理论与方法，开展半代数集与半代数函数在曲线曲面造型的应用研究。研究目标：建立半代数函数的光滑性以及半代数集之间的光滑拼接理论；构造半代数集与半代数函数表示曲线曲面的方法。本项目为信息与计算科学提供新的理论和方法，为几何造型提供新的工具。

中文关键词： 分片多项式系统；粘合理论；曲线曲面造型；半代数集与半代数函数；Betti 数

英文摘要： The project studies some topological theories for piecewise polynomial systems and some basic theories and methods of its applications in geometric modeling. The research content includes: the establishment of gluing theory and its "Patchwork" method for constructing the real piecewise algebraic hypersurfaces with prescribed topology, degree and smoothness; determination the boundary of Betti number for piecewise polynomial systems and establishment of the theories and the corresponding algorithms for computing the Betti number and Euler characteristic number of piecewise polynomial systems; the establishment of gluing theory for critical points and singular points of piecewise polynomials; the establishment of the restriction relations between Betti number of piecewise polynomial systems, the number of critical point of index i, the number of cells, the inner vertex and the inner public faces and the degree and smoothness of piecewise polynomials; the classification of index distributions for the critical points of piecewise polynomials with low degree, as well as the coefficient constraint conditions of piecewise polynomials. Based on the above theories and methods, we carry out the application research of semi-algebraic sets and semi-algebraic functions in the curve/surface modeling. The research target cont

英文关键词： Piecewise polynomial systems；Gluing theory；Curve/surface modeling；Semi-algebraic sets and semi-algebraic functions；Betti number