具有高阶多项式再生性的细分格式

Subdivision Schemes with High Order of Polynomial Reproduction

摘要: 细分方法是计算图形学中一种非常重要的几何造型方法，它可以生成光滑的极限曲线或曲面，并且计算高效、与拓扑无关，在3D打印、动画设计等方面有着广泛应用。到目前为止，细分方法经过多年发展，相关理论研究已经进入较为成熟的阶段，但构造具有较高再生性的细分多项式仍然是非常值得研究的课题，具有重要意义。论文基于细分方法中生成函数的研究，通过乘以扰动的方式来构造具有较高再生性的细分多项式，并讨论扰动中参数的统一表达式和所构造的细分多项式的生成函数表达式。

Abstract: Subdivision method is a very important geometric modeling method in computational graphics. It can generate smooth limit curve or surface, and the calculation is efficient and has nothing to do with the topology of the initial control polygons. It has a wide range of applications in 3D printing, animation design and other aspects. So far, after years of development of subdivision methods, the relevant theoretical research has entered a relatively mature stage, but the construction of higher order polynomial reproduction of subdivision scheme is still a very worthy research topic, and has important significance. Based on the research of generating function in the subdivision method, this paper constructs several subdivision schemes with high order of polynomial reproduction by mul-tiplying by perturbation, and discusses the unified expression of parameters in the perturbation and the generating function expression of the constructed subdivision polynomial.