曲面拟合的研究与应用

曲面拟合的研究与应用

Research and Application of surface fitting

I

摘 要

随着科学的发展，数学对世界的影响和改变能力日益突出。目前，曲面拟合作为数据处理与分析的一种数值方法，已被逐步推广到多个领域，并得到了越来越重要的应用，已经成为数学领域中的一个新的分支。

曲面拟合是一种古老而常用的技术，在工程实验统计和计算机图形等方面有着广泛的应用。在实际问题中，通常我们通过测量或者实验得到一组离散的数据，我们需要从这组离散数据出发去构作曲线曲面或者求解拟合函数的参数。

这里，我们首先研究曲线拟合的常用方法，包括曲线拟合的插值法和解析法。插值法这里主要研究的是牛顿插值法，解析法主要研究的是最小二乘法，通过最小二乘法做曲线拟合函数。然后再从二维的曲线拟合过渡到三维的曲面拟合。在曲面拟合过程中，通过最小二乘法得到一个一个非线性方程组，然后利用牛顿法求解，便得到拟合函数或者拟合参数的参数。最后我们通过一个现实中实例来说明曲面拟合的全部过程及其有点。

通过对有曲面拟合的研究与学习，初步了掌握曲面拟合的最小二乘方法及其应用。在科学技术日新月异的发展过程中，曲线曲面拟合已应用在各个领域中，尤其在数据处理方面，发挥着越来越重要的作用，为科学技术的进步作出了重大的贡献，曲线曲面拟合作为一种方法也得到了巨大的发展。

关键词：曲线拟合；最小二乘法；牛顿法；曲面拟合

II

Abstract

With the development of science, the math’s impact and the ability of change to the world has been prominent day after day. Nowadays, Surface fitting, which as a numerical method of data processing and analysis has been extended to other fields, and it has been a new branch in math.

Surface fitting is an old and common technology which is widely used in engineering experiment statistics and computer graphics. In practical problems, Usually, we get a group of discrete data by measurement and experimental. We need to construct curves and surfaces or to solve the parameters of the fitting function.

Here, we firstly study the common method of curve fitting. It includes the interpolation method and the analytical method. We mainly study the Newton interpolation for interpolation method and the least squares for the analytical method. We construct the fitting function by the least squares and popularize the two-dimensional curve fitting to three-dimensional. During the process we can get a non-linear equations. By the Newton we can get the fitting function or the parameters of it. Finally, we will give a real example to illustrate the whole process of surface fitting and its advantages.

By research and studying of surface fitting, basically, I grasp the least squares method of surface fitting and its application. In the rapid development process of science and technology ,curve and surface fitting has been applied in various fields ,especially in data processing ,and play an increasing important role .It has made a significant contribution to the progress of science and technology ,and ,as a method, has also made a huge development.

Keywords: Curve fitting; Least squares method; Newton method; surface fitting

III

目 录

摘 要 ................................................................................................................................... II Abstract .................................................................................................................................... III 第1章 绪论 ....................................................................................................................... - 1 -

1.1论文研究的目的和意义 ......................................................................................... - 1 - 1.2曲面拟合的最小二乘法 ......................................................................................... - 2 - 1.3设计的思想和采用的方法 ..................................................................................... - 2 - 第2章 曲面拟合的基本知识 ............................................................................................. - 4 -

2.1 曲线拟合 ................................................................................................................ - 4 -

2.1.1 问题的提出 ................................................................................................. - 4 - 2.1.2 曲面拟合的牛顿插值法 ............................................................................. - 4 - 2.1.3曲线拟合的最小二乘法 .............................................................................. - 8 - 2.2 解非线性方程组的牛顿迭代法 .......................................................................... - 11 - 2.3 曲面拟合 .............................................................................................................. - 13 -

2.3.1 方法的概述 ............................................................................................... - 13 - 2.3.2 曲面拟合的最小二乘法 ........................................................................... - 16 -

第3章 实例计算 ............................................................................................................... - 18 -

3.1 问题 ...................................................................................................................... - 18 - 3.2 求解过程 .............................................................................................................. - 19 - 第4章 结论 ....................................................................................................................... - 21 - 参 考 文 献 ....................................................................................................................... - 22 - 致 谢 .................................................................................................. 错误！未定义书签。

I

第1章 绪论

1.1论文研究的目的和意义

曲线曲面的拟合问题在理论研究和实际应用中常常遇到。在模式识别和计算机 …… 此处隐藏：3993字，全部文档内容请下载后查看。喜欢就下载吧 ……