生成函数运算下细分光滑性变化规律研究

项目名称： 生成函数运算下细分光滑性变化规律研究

项目编号： No.61502217

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 计算机科学学科

项目作者： 亓万锋

作者单位： 辽宁师范大学

项目金额： 20万元

中文摘要： 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。

中文关键词： 细分曲线曲面；Lane-Riesenfeld算法；光滑性分析；样条曲面；联合谱半径

英文摘要： 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.

英文关键词： Subdivision curves and surfaces;Lane-Riesenfeld agorithm;Smoothness analysis;Spline surfaces;Joint spectral radius