分片代数曲线若干问题的研究

The project studies some problems 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 Bezout number of two piecewise algebraic curves under special partitions or different partition. 2. The real root classification of a given univariate spline function. 3. The topological structure and topological invariant such as Betti numer or Euler number for real piecewise algebrical curves. Based on the above theories, we make the following applied researches: 1. Constructing the piecewise algebraic curves satisfying a certain minimized energy and geometric continuity. 2. Applying the piecewise algebraic curves to curve/surface modelling. The research of this project will continue to enrich and improve spline theory and algebraic geometry, which provides new tool for geometric modelling.

本项目拟对分片代数曲线的若干问题展开研究，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.建立特殊剖分或不同剖分下两条分片代数曲线的Bezout数。2.建立一元样条函数实根显式判定和分类算法。3.分析实分片代数曲线的拓扑结构和Betti数、Euler数等拓扑不变量。基于上述理论，我们将展开如下应用研究：1.构建具有某种意义下能量最小且满足几何连续性的分片代数曲线；2.将分片代数曲线应用到曲线曲面逼近造型上。本项目的研究将不断完善样条函数与代数几何理论体系，为几何造型提供新的工具。

本项目拟对分片代数曲线和样条插值若干问题展开研究，为样条函数和分片代数几何的发展提供理论支撑和核心算法。我们取得的主要研究成果包括：1.建立了积分值低次样条插值或多层样条拟插值的有效算法，同时给出了连续区间上积分值的偶次样条插值的存在唯一性、光滑性质和超收敛性质。2.建立了一元参系数样条函数实根分类算法，即给出了每种分类情形下参数所满足的充要条件。3. 建立了参系数分片代数曲线奇点的上界数及其分布情况,，以及给出参系数分片代数曲线的奇点数达到上界时参数满足的充要条件和它在各个胞腔上具有给定的奇点数时参数所满足的充要条件。本项目的研究将不断完善样条函数与代数几何理论体系，为几何造型提供新的工具。