多边参数曲面的表示和逼近是计算几何研究的一个重要内容。Toric曲面是由代数几何中的toric簇生成的一种参数域为任意凸多边形的参数曲面，它是有理Bézier曲面的一种多边推广。本项目拟对toric曲面的几何性质及应用进行研究。首先，使用组合学中的多边形等价理论研究toric曲面的分类，给出toric曲面的基本类型；其次，结合格点集的Minkowski和，推导特殊格点集所定义的toric曲面的金字塔算法与升阶算法；最后，利用toric曲面及toric Bernstein基函数的定义与性质，将定义在特定凸区域上的toric Bernstein基函数所张成的有限维函数空间作为试探函数，结合Galerkin有限元方法求微分方程数值解。本项目将对toric曲面的几何性质及在微分方程数值求解中的应用展开研究，对丰富几何造型的理论和应用具有一定的实际意义。

Multi-sided parametric surfaces play an important role in Computational Geometry. Toric surfaces is a kind of multi-sided parametric surfaces from the theory of toric varieties in Algebraic Geometry and toric ideals in Combinatorics. This project will do some researches on geometric properties and applications of toric surfaces. Firstly, we will discuss the classification of toric surfaces with the lattice equivalence theory in Combinatorics; Secondly, the blossoming and degree elevation algorithms of toric surface will be derived based on the Minkowski sum of lattice points; Lastly, on account of the definition and properties of toric Bernstein polynomials defined on an arbitrary convex domain, we choose the functions space spanned by toric Bernstein polynomials as the trial functions to solve the PDE with the Galerkin method.