生成函数运算下细分光滑性变化规律研究

Subdivision method is an important geometric modeling method and is becoming increasingly important in a number of applications. Many recent studies have focused on several unified frameworks for effectively constructing subdivision schemes. The unified frameworks construct new schemes by multiplying or adding Laurent polynomials to generating functions of the initial subdivision. However, these unified frameworks do not suggest a theoretical analysis of the smoothness, and the smoothness analysis is still subject to a case-by-case approach or spline-based schemes. This project will investigate the influence of these two generating function operations on the smoothness of constructed subdivision schemes and propose a modular method which can provide some criterions that can determine the increase or decrease of smoothness. The modular method will turn the analysis of the subdivision scheme smoothness into investigating some simpler Laurent polynomial’s properties. At first, this project will investigate the property of a single Laurent polynomial that can increase or decrease the smoothness of subdivision when the initial generating function multiplies or adds the Laurent polynomial. Then the interactions that one Laurent polynomial multiplies itself several times or several different polynomials multiply together will be studied. This project will enrich the theory of joint spectral radius where subdivision matrices change accordingly, provide new insights into the theory of subdivision smoothness analysis, and can give some guidance on the design of subdivision schemes. In this way, we can choose some suitable construction methods to achieve the desired degree of smoothness.

细分方法是一种重要的几何造型方法，广泛应用于许多领域。近年来，众多学者尝试通过对生成函数乘以一些Laurent多项式或者加上一个特定的Laurent多项式，来建立构造细分的统一框架。目前这种框架下对细分光滑性的分析，多是针对样条细分或某些具体的细分格式，还缺乏系统理论的研究。本项目拟对细分生成函数乘法与加法运算下细分的光滑性进行系统理论研究，给出光滑性分析的一种模块化方法，将光滑性的问题转化为若干Laurent多项式对光滑性影响的叠加问题：将给出单个Laurent多项式相乘或者相加对光滑性影响的判定条件；给出同一个Laurent多项式相乘多次以及不同Laurent多项式相乘对光滑性影响叠加的判定条件。本项目的研究将丰富联合谱半径等工具在生成函数变动下的理论，并为细分光滑性分析理论提供一种新的思路。本项目的研究可以为设计细分格式提供理论指导，使得人们可以采用适当的构造手段，达到预计的光滑度。

本项目研究了以偏移量为工具采用加法构造细分的求和规则. 我们发现，如果不按照常规添加偏移量的方法，而采取最灵活的方式仍然可以达到最高阶的求和规则. 但如果采用的是更高阶的中心差分，则不一定可以达到最高阶的求和规则，需要原细分具有更强的性质. 此外，还提出一种扰动策略，直接将具有最高阶求和规则的生成函数进行扰动，降低求和规则阶数，自然地引入一些参数，可以得到新的细分格式. . 还讨论和探究了三元分次插值的适定性问题，提出了三元分次插值适定结点组的基本定义，给出了构造这类插值适定结点组的“添加竖平面、横平面和纵平面法”． 提出带2 个形状参数的三次DP 曲线，讨论形状参数的几何意义, 给出关于形状参数的曲线G^1,G^2 连续条件. 最后引入位置参数m, 使得曲线在端点处的切点位置可以自由调整, 曲线设计更为灵活. . 本项目还利用有限域上分圆类的笛卡尔积和加法特征，利用四阶分圆类构造出了一族新的渐进最优码本. 同时减弱了几乎差集对八阶分圆类的要求，构造出了另一族新的渐进最优码本..