基于toric退化理论的形状修改和变形方法的研究

几何造型是计算机辅助设计与制造的核心内容，而形状修改和变形是几何造型的重要组成部分。Toric曲面是一类与有限整数格点集相关的多边形有理参数曲面，其基函数是Bernstein基的推广，经典的有理Bézier曲线曲面是toric曲面的特殊形式。Toric曲面当权因子都趋于无穷时的退化形式为正则控制曲面，此性质称为toric退化理论。本课题提出基于toric退化理论的形状修改和变形的方法，首先拟研究toric曲面的正则控制曲面个数，补充toric曲面权因子的几何性质，扩展toric退化理论；然后结合NURBS曲线曲面的toric退化理论，拟研究NURBS曲线曲面的正则控制曲线曲面个数上界公式，进一步确定上确界，提出正则控制曲线曲面的高效个数算法与快速生成算法；最后拟研究基于toric退化理论，在单值性约束与控制顶点移动的约束下，利用生成算法，对参数曲线曲面进行形状修改和变形。

Geometric modeling is the kernel of Computer Aided Design/Computer Aided Manufacturing. And shape modification and deformation is an important component of geometric modeling. Toric patch is a kind of multisided rational parametric surface, which is associated to a given finite integer lattice points set. Its basis functions are the generalization of Bernstein basis. The classical rational Bézier curves/surfaces are special cases of toric patches. When all weights tend to infinity, the degeneration of toric patch is regular control surface, so this property is called toric degeneration theory. In this project, we provided a method of shape modification and deformation based on toric degeneration theory.Firstly, we will study the number of regular control surfaces of toric surface. This is an important supplement to the geometric properties of weight factors of toric surface, and the toric degeneration theory is extended. Secondly, together with toric degeneration of NURBS curves/surfaces, we will give the upper bound of the number of regular control curves/surfaces of NURBS curves/surfaces, and then the least upper bound will be provided. Moreover, we provided an algorithm of the number for regular control curves /surfaces and an algorithm of generation for regular control curves /surfaces. Finally, based on the toric degeneration theory, the injective constraint and the moving of control points constraint, we will modify and deform the shape of parametric curves/surfaces by the algorithm of generation.