一类拟细分插值基函数的构造探究
发布时间：2019-03-18 10:44
【摘要】：在计算机辅助几何设计及其相关领域,插值一直是一个非常基本和重要的研究课题,插值方法就是根据一组有序数据点生成曲线或曲面的方法。目前已有许多比较好的插值方法,但是都存在一些局限性。比如,一些经典的多项式插值方法,需要解一系列方程组来获得插值曲线或曲面,计算量大且不稳定,改变数据点将导致整条插值曲线或曲面的变化,在实现高阶曲线曲面的连续性时需要求解复杂的高阶方程。细分曲线曲面无法用数学表达式来表示,用明确的数学函数模拟细分基函数来构造曲线曲面最近被提出来,受到国际学者的关注与赞扬。本文将对这种基函数加以扩充和改进,构造了具有更好性质的插值基函数用来构造插值曲线与曲面。本文先介绍经典细分方法,通过计算机将其基函数的图形表示出来,并从中总结了所需构造插值细分基函数的性质,然后引入一类具有精确的局部支撑和无穷次可微的函数;将其与Sinc函数结合并优化,构造一类相似于插值细分基函数的新基函数。我们称它为拟细分插值基函数,这类新基函数保持了以往基函数的良好性质,并具有以往基函数所不具有的精确局部支撑性的优点.取特定的插值基函数参数值,可以调节局部支撑性的范围。按照新方法生成的曲线具有如下优点:1、插值性;2、曲线形状局部可调;3、无需解方程组;4、通过改变插值点,可以轻松的改变插值曲面的形状;5、算法简单,易于推广等。在文中我们也通过实例结果表明,文中构造的新基函数有很好的效果;与传统的Akima方法相比,所构造的曲线总体上具有较好的光顺性.在构造实例曲面上,也有很好的效果,并且能实现曲面在连接处的光滑拼接,具有很强的实用性。
[Abstract]:Interpolation is always a very basic and important research topic in computer-aided geometric design and its related fields. Interpolation is a method to generate curves or surfaces from a set of ordered data points. At present, there are many good interpolation methods, but there are some limitations. For example, some classical polynomial interpolation methods need to solve a series of equations to obtain interpolation curves or surfaces. The computation is large and unstable. Changing the data points will lead to the change of the entire interpolation curve or surface. In order to realize the continuity of high-order curves and surfaces, complex higher-order equations need to be solved. Subdivision curves and surfaces can not be represented by mathematical expressions. The construction of curves and surfaces by using explicit mathematical functions to simulate subdivision basis functions has recently been proposed and received international scholars' attention and praise. In this paper, this basis function is extended and improved, and the interpolation basis function with better properties is constructed to construct interpolation curves and surfaces. In this paper, the classical subdivision method is introduced firstly, the graph of its basis function is represented by computer, and the properties of constructing interpolation subdivision basis function are summarized, then a class of functions with exact local support and infinitely differentiable function are introduced. It is combined with Sinc function and optimized to construct a class of new basis functions similar to interpolated subdivision basis functions. We call it quasi-subdivision interpolation basis function. This kind of new basis function maintains the good property of the previous basis function and has the advantage of accurate local support which the former basis function does not have. The range of local support can be adjusted by taking specific parameter values of interpolation basis function. The curves generated by the new method have the following advantages: (1) interpolation; (2) the shape of the curve is locally adjustable; (3) there is no need to solve the equation group; (4) by changing the interpolation point, the shape of the interpolated surface can be easily changed; 5, the algorithm is simple, easy to generalize and so on. In this paper, the results show that the new basis function constructed in this paper has a good effect, and compared with the traditional Akima method, the curves constructed in this paper have better smoothness on the whole. In the construction of example surface, it also has a good effect, and can realize the smooth splicing of the surface at the junction, which has a strong practicability.
【学位授予单位】：浙江工商大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.42
本文编号：2442789
[Abstract]:Interpolation is always a very basic and important research topic in computer-aided geometric design and its related fields. Interpolation is a method to generate curves or surfaces from a set of ordered data points. At present, there are many good interpolation methods, but there are some limitations. For example, some classical polynomial interpolation methods need to solve a series of equations to obtain interpolation curves or surfaces. The computation is large and unstable. Changing the data points will lead to the change of the entire interpolation curve or surface. In order to realize the continuity of high-order curves and surfaces, complex higher-order equations need to be solved. Subdivision curves and surfaces can not be represented by mathematical expressions. The construction of curves and surfaces by using explicit mathematical functions to simulate subdivision basis functions has recently been proposed and received international scholars' attention and praise. In this paper, this basis function is extended and improved, and the interpolation basis function with better properties is constructed to construct interpolation curves and surfaces. In this paper, the classical subdivision method is introduced firstly, the graph of its basis function is represented by computer, and the properties of constructing interpolation subdivision basis function are summarized, then a class of functions with exact local support and infinitely differentiable function are introduced. It is combined with Sinc function and optimized to construct a class of new basis functions similar to interpolated subdivision basis functions. We call it quasi-subdivision interpolation basis function. This kind of new basis function maintains the good property of the previous basis function and has the advantage of accurate local support which the former basis function does not have. The range of local support can be adjusted by taking specific parameter values of interpolation basis function. The curves generated by the new method have the following advantages: (1) interpolation; (2) the shape of the curve is locally adjustable; (3) there is no need to solve the equation group; (4) by changing the interpolation point, the shape of the interpolated surface can be easily changed; 5, the algorithm is simple, easy to generalize and so on. In this paper, the results show that the new basis function constructed in this paper has a good effect, and compared with the traditional Akima method, the curves constructed in this paper have better smoothness on the whole. In the construction of example surface, it also has a good effect, and can realize the smooth splicing of the surface at the junction, which has a strong practicability.
【学位授予单位】：浙江工商大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.42
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