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摘要由于自由曲面镜在性能上大大优于球面镜，近年来得到了广泛的应用，但是对于这个自由曲面的加工和检验的困难性就要比普通的对称非球面镜要大出很多。本毕业论文主要是为了解决对于一个自由曲面的描述，为了进一步的计算全息法检验自由曲面面形解决一个关键问题。我所做的工作是研究用于拟合自由曲面的拟合算法，并且使用这些算法模型对非旋转对称相位函数的拟合，然后通过对于拟合结果的评价来分析这种方法的优缺点。这些方法包括最小二乘法，Gram-Schmidt正交法和Forbes函数，这其中重点选择Forbes函数作为研究的重点。本课题研究结果为最小二乘法适用于精度要求不是很高的并且对于中心误差要求不高的自由曲面。而Gram-Schmidt正交法也同样适用于要求精度不是很高的曲面，而Forbes函数的精度相对要高很多，但是算法相对复杂，而对于精度要求使可控的，随着项数的改变出现误差的极大值的位置会出现漂移。5730
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关键词 自由曲面 拟合相位 最小二乘法 Gram-Schmidt正交法 Forbes函数
毕业设计说明书（论文）外文摘要
Title The research of Fitting method for Non-rotationally unsymmetric Phase function
Abstract
In recent years, the asphere has be a wide range of applications, as it’s unparalleled performance. But the difficulty for this free-form surface machining and inspenction is much larger than usual symmetric non-spherical. This paper mainly to solve the description of a free surface. The work I have done is to research the fitting algorithm to fit the free surface, use these algorithms model to fit the non-rotationally unsymmetric phase function, and then use the result of the to analyze the advantages and disadvantages of these method. There are three method, the least squares method, the Gram-Schmidt orthogonalization and Forbes function, which focus on the Forbes function. This project has a conclusion as a least squares method for precision is not very high and the free surface of the central error of less demanding. Gram-Schmidt orthogonal method is also applicable to the surface accuracy is not high, Forbes function of the relative accuracy is much higher, but the algorithm is relatively complex, precision controlled, with the change of the number of items the location of the maximum error will appear to drift.
Keywords Free-form surface, fitting phase polynomial, least squares method, the Gram-Schmidt orthogonalization method, Forbes function
目 次
1绪论 1
1.1课题的来源与背景 1
1.3本课题研究的内容 3
2自由曲面技术以及曲线拟合方法 4
2.1自由曲面简介 4
2.2自由曲面的定义 4
2.3曲线拟合方法 5
2.4 Zernike多项式[16-18] 5
3自由曲面面形的描述方法 8
3.1最小二乘法 8
3.2 Gram-Schmidt正交法 14
4Forbes函数[19-29] 19
4.1 Forbes函数的提出 19
4.2 Forbes函数的意义 19
4.3 Forbes函数的描述方法 19
5对于自由曲面面形的几种描述方法的比较 33
5.1最小二乘法的拟合结果 33
5.2 Gram-Schmidt正交基的拟合结果 34
5.3 Forbes函数的拟合结果 35
5.4 本章小结 36
结论 38
致谢 39
参考文献 40
1绪论
1.1课题的来源与背景