分片代数曲线的若干理论及其应用

The project studies some basic theories of piecewise algebraic curves and its related areas which provide theoretical support and effective algorithms for the development of piecewise algebraic geometry. The study includes: 1.The establishment of the supremum and distribution of critical points such as singular points and inflection points for the given piecewise algebraic curve and the analysis of its topological structure. 2.The establishment of piecewise algebraic curves with geometric continuity which possess minimum energy in certain sense. 3.Approximate implicitization or approximate parametrization which possess good shape preserving by using piecewise algebraic curves. The research of this project will continue to enrich and improve multivariate spline theory and algebraic geometry. The research results will apply to graphics and engineering design, etc. It injects new vitality to the development of piecewise algebraic geometry。

本项目研究分片代数曲线的若干理论及其在相关领域的应用，为分片代数几何的发展提供理论支撑和核心算法。研究内容包括：1.拟建立特殊剖分下分片代数曲线关于奇点，拐点等临界点的上界数和分布情况，并且分析实分片代数曲线的拓扑结构；2.构建具有某种意义下能量最小且满足几何连续的分片代数曲线；3.利用分片代数曲线来构建具有良好保形特征的近似隐式化方法或近似参数化方法。本项目的研究将不断丰富和完善多元样条与代数几何理论及其应用体系，并且将研究成果应用于图形学和工程设计等领域，为分片代数几何的发展注入活力。

本项目研究内容和研究目标基本按照计划书执行，主要研究分片代数曲线和样条函数若干理论及其在相关领域的应用。研究内容包括：1.建立了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Viro粘合理论及其组合“Patchwork”方法；2.建立了积分型六次样条函数插值方法及其超收敛性；3.基于多层技术和四次B-样条拟插值算子提出了多层拟插值算子并将其应用到二维奇异积分的数值计算上；4.构建平面参数曲线具有良好保形特征的近似隐式化方法；5. 提出了三次B-样条函数计算两点奇异边值问题的改进方法。本项目的研究将不断丰富样条函数与计算几何理论，为样条函数在图形学和工程设计等领域的发展注入活力.