【摘要】：在計(jì)算機(jī)輔助幾何設(shè)計(jì)及其相關(guān)領(lǐng)域,插值一直是一個(gè)非�；竞椭匾难芯空n題,插值方法就是根據(jù)一組有序數(shù)據(jù)點(diǎn)生成曲線或曲面的方法。目前已有許多比較好的插值方法,但是都存在一些局限性。比如,一些經(jīng)典的多項(xiàng)式插值方法,需要解一系列方程組來(lái)獲得插值曲線或曲面,計(jì)算量大且不穩(wěn)定,改變數(shù)據(jù)點(diǎn)將導(dǎo)致整條插值曲線或曲面的變化,在實(shí)現(xiàn)高階曲線曲面的連續(xù)性時(shí)需要求解復(fù)雜的高階方程。細(xì)分曲線曲面無(wú)法用數(shù)學(xué)表達(dá)式來(lái)表示,用明確的數(shù)學(xué)函數(shù)模擬細(xì)分基函數(shù)來(lái)構(gòu)造曲線曲面最近被提出來(lái),受到國(guó)際學(xué)者的關(guān)注與贊揚(yáng)。本文將對(duì)這種基函數(shù)加以擴(kuò)充和改進(jìn),構(gòu)造了具有更好性質(zhì)的插值基函數(shù)用來(lái)構(gòu)造插值曲線與曲面。本文先介紹經(jīng)典細(xì)分方法,通過(guò)計(jì)算機(jī)將其基函數(shù)的圖形表示出來(lái),并從中總結(jié)了所需構(gòu)造插值細(xì)分基函數(shù)的性質(zhì),然后引入一類具有精確的局部支撐和無(wú)窮次可微的函數(shù);將其與Sinc函數(shù)結(jié)合并優(yōu)化,構(gòu)造一類相似于插值細(xì)分基函數(shù)的新基函數(shù)。我們稱它為擬細(xì)分插值基函數(shù),這類新基函數(shù)保持了以往基函數(shù)的良好性質(zhì),并具有以往基函數(shù)所不具有的精確局部支撐性的優(yōu)點(diǎn).取特定的插值基函數(shù)參數(shù)值,可以調(diào)節(jié)局部支撐性的范圍。按照新方法生成的曲線具有如下優(yōu)點(diǎn):1、插值性;2、曲線形狀局部可調(diào);3、無(wú)需解方程組;4、通過(guò)改變插值點(diǎn),可以輕松的改變插值曲面的形狀;5、算法簡(jiǎn)單,易于推廣等。在文中我們也通過(guò)實(shí)例結(jié)果表明,文中構(gòu)造的新基函數(shù)有很好的效果;與傳統(tǒng)的Akima方法相比,所構(gòu)造的曲線總體上具有較好的光順性.在構(gòu)造實(shí)例曲面上,也有很好的效果,并且能實(shí)現(xiàn)曲面在連接處的光滑拼接,具有很強(qiáng)的實(shí)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.
【學(xué)位授予單位】：浙江工商大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O174.42

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本文編號(hào)：2442789