曲线细分算法的构造及连续性分析

发布时间：2018-02-04 21:11

本文关键词： 正弦函数双曲函数联合谱半径连续性　出处：《合肥工业大学》2017年硕士论文　论文类型：学位论文

【摘要】：细分方法是曲线曲面造型技术中的一项重要技术,在计算机辅助几何设计(Computer Aided Geometric Design,简称CAGD)和计算机图形学等相关领域得到了广泛的应用。其基本思想是:给定初始控制网格,定义一个细分规则,在给定的初始网格中不断的加入新的点,直至最后产生一条光滑的曲线或一张光滑的曲面。细分法按照细分规则是否随着细分次数的变化而改变,可将细分法分为动态细分法和静态细分法;按照极限曲线是否经过初始控制顶点,又可将其分为插值细分法和逼近细分法。近年来,在曲线细分方面大多研究集中在关于细分格式的构造、细分格式的收敛性和连续性的证明、基函数的对称性以及计算支集宽度等方面。众所周知,动态细分格式能生成一些特殊曲线,如圆锥曲线等,大大增强了造型能力;逼近型细分可以生成光滑性高的极限曲线,而插值型细分可以使生成的极限曲线通过初始控制网格,近年出现的插值逼近型细分将二者结合,可以使生成的极限曲线插值用户期望通过的点同时逼近其余点,这些算法均具有重要的研究意义。鉴于以上研究,本文做了如下工作:1.利用正弦函数构造了一类新的带有形状参数?的(2p-1)点二重动态逼近细分格式。从理论上分析了随p值变化时这类细分格式的一些重要的性质,如kC连续性、基函数的对称性并且计算了它的支集宽度;2.利用双曲函数构造了一类四点m(其中m为偶数)重动态逼近细分格式。利用静态与动态细分格式之间的渐近等价性,证明了动态细分格式的连续性;3.提出了一种含有多个参数混合的五点三重细分格式,利用联合谱半径的方法证明了该细分格式的连续性。
[Abstract]:Subdivision method is an important technique in curve and surface modeling technology. Computer Aided Geometric Design is used in computer aided geometric design. CAGD), computer graphics and other related fields have been widely used. The basic idea is to define a subdivision rule given the initial control grid. A new point is added to a given initial mesh until a smooth curve or a smooth surface is produced. The subdivision method changes with the number of subdivision rules according to the subdivision rule. The subdivision method can be divided into dynamic subdivision method and static subdivision method. According to whether the limit curve passes through the initial control vertex, it can be divided into interpolation subdivision method and approximate subdivision method. In recent years, most researches on curve subdivision are focused on the construction of subdivision scheme. The proof of convergence and continuity of subdivision schemes, the symmetry of basis functions and the calculation of support width, etc. It is well known that dynamic subdivision schemes can generate some special curves, such as conic curves. Greatly enhanced the modeling ability; Approximation subdivision can generate high smoothness limit curve, and interpolation subdivision can make the generated limit curve pass through the initial control grid. The interpolation approximation subdivision in recent years combines the two. We can make the generated limit curve interpolate the points that users expect to pass and approach the other points at the same time. These algorithms have important research significance, in view of the above research. In this paper, we do the following work: 1. We construct a new class with shape parameter by using sine function. Some important properties of this subdivision scheme, such as KC continuity, are theoretically analyzed when the subdivision scheme varies with p value. The symmetry of the basis function and the width of its support set are calculated. 2. By using hyperbolic functions, a class of four-point m (where m is even number) multiple dynamic approximation subdivision schemes are constructed. By using the asymptotic equivalence between static and dynamic subdivision schemes, the continuity of dynamic subdivision schemes is proved. 3. A 5.3 multiple subdivision scheme with multiple parameters is proposed. The continuity of the scheme is proved by the method of joint spectral radius.
【学位授予单位】：合肥工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TP391.7

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