Toric曲面研究

Multi-sided surface patch plays an important role in shape design of the car and ship, filling holes, construction of barycentric coordinates, finite element analysis, image warping and morphing and is becoming one of the fundamental tools in geometric modeling. As the generalization of the rational Bézier surface patches, the toric surface patch inherits the most of the properties of the rational Bézier surface patches. The mathematical theory of toric surface patch is based upon real toric varieties from algebraic geometry and toric ideals from combinatorics. In this project, we intends to study the theory and applications of the toric surface patch, including: the combinatorial relations among the regular decompositions of convex polygon (or the convex polyhedron), toric varieties, toric ideals, and geometric modeling; the geometric continuity of toric surface patches and its application to fill the holes; the applications of toric surface patch to construct barycentric coordinates; the necessary and sufficient condition to determine the injectivity of toric surface patches geometrically; and the degenerations of toric surface patch and NURBS. Through this project, we give the basic theory and algorithms of the toric surface patches by combining the toric varieties, the toric ideals, and geometric modeling.

随着几何造型理论的不断发展，多边形曲面以其在汽车与船舶外形设计、曲面补洞、重心坐标设计、曲面与实体变形、多边形有限元等方面的重要应用而逐渐成为几何造型研究的一个热点问题。作为有理Bézier曲面的多边形推广，toric曲面继承了Bézier曲面的大多数性质，其数学理论来源于toric几何。本项目拟对toric曲面的理论及其在几何造型中的应用进行研究，主要包括：研究凸多边形（或凸多面体）的正则剖分与toric簇、toric理想之间的组合关系及其在几何造型中的应用；研究toric曲面的几何拼接及其在曲面补洞中的应用；研究toric曲面在多边形(或多面体)重心坐标构造中的应用；研究几何上判断toric曲面单值性的充要条件；研究权因子趋于无穷时toric曲面与NURBS曲面的退化问题。通过本项目的研究，将toric簇、toric理想与几何造型相结合，给出toric曲面造型的基本理论与相关算法。

本项目将代数几何与几何造型相结合，主要对toric曲面的理论与及其在几何造型中的应用进行研究，主要包括toric曲面的数学理论与曲面几何性质以及与其相关的曲线曲面方法研究。通过本项目研究，我们取得了如下成果：给出了整数格点集正则分解个数的计算算法，并应用于toric曲面的正则控制曲面个数计算；给出有理Bézier曲线曲面与NURBS曲线曲面对任何正权因子都没有自交点的几何判定方法；给出toric曲面间G1连续拼接时控制顶点与权因子所需满足的条件；给出toric曲面、有理Bézier曲线曲面与NURBS曲线曲面权因子以幂函数或指数函数形式趋于无穷时其极限曲线曲面的几何结构；给出了插值测地线与曲率线为边界曲线的曲面构造方法等。本项目所得结论不仅是对计算几何与计算机辅助几何设计理论上的补充，而且也对几何造型的应用研究有一定的指导作用。