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標題: 以權重及最小平方法對圓、橢圓二次方程式之擬合
Least Square Fitting with Correction Factor of Circle and Ellipse
作者: 張智祥
Chang, Chih-Hsiang
關鍵字: Least Square;最小平方法;Quadratic equation;二次方程式
出版社: 機械工程學系所
引用: [1] Ahn, S. J., Rauh, W., and Warnecke, H.-J., “Least-squares orthogonal distances fitting of circle, sphere ellipse, hyperbola, and parabola”, Pattern Recognition, Vol. 34, 2001, pp. 2283-2303. [2] Bookstein, F. L.,“Fitting conic sections to scattered data”, Computer Graphics, Vol. 21, 1987, pp. 145-152. [3] Butler, B. P.,Forbes, A. B. and Harris, P. M.,“Algorithms for geometric tolerance assessment”, NPL Report DITC 228/94, Teddington, UK, 1994. [4] CHAPRA, S. C.,” Applied Numerical Methods With MATLAB for Engineers and Scientists”,Higher Education, McHraw-Hill, 2005, chapter 12. [5] Forbes, A. B., Mason, J. C., and Cox, M. G., eds, “Least-squares best fit geometric elements, in algorithms for approximation II”, Chapman and Hall, London, 1990, pp. 311-319 [6] Gander, W., Golub, G. H., and Strebel, R., “Least-Squares Fitting of Circles and Ellpses”, BIT, Vol. 34, 1994, pp. 559-578. [7] Kasa, I.,“A circle fitting procedure and its error analysis”, IEEE Trans. Instr. Measurement, Vol. 25, 1976, pp.8-14. [8] Protter, M. H. and Morrey, C. B.,”Calculus With Analytic Geometry”, first course, Addison-Wesley Pub., 1964, chapter 10. [9] Rosin, P. L.,”A note on the least squares fitting of ellipse”, Pattern Recognition Letters, Vol. 14, 1993, pp. 799-808. [10] WATSON, G. A., “ Least Squares Fitting of Circles and Ellipses to measured data”, BIT, Vol. 39, 1999, pp. 176-191. [11] Zwick, D. S., Van Huffel, S., ed. “Applications of orthogonal distance regression in metrology, in Recent Advances in Total Least Squares and Errors-in-Variables Techniques”, SIAM, Philadelphia, 1997, pp. 265-272
圓錐曲線包含了橢圓,雙曲線以及拋物線。工程應用中,從橢圓上廣泛的測量得知數據,然而這些的數據所構成的橢圓,其應用延伸到許多工程應用領域,例如工程繪圖、石油業、統計學、計量學、天文學等等。各個領域中的資料點,這些的數據,不外乎經由實驗取得或是量測實體。量測工件可以使用座標量測機器(CMM)來進行測量工件表面上每一點的座標。CMM是利用探針在工件表面的一個方向來回的移動並建立工件表面上的座標。每次測量得到的資料點並不一定是期待的,也許有更好的數據能得到,測量過程中有誤差或是不夠精準。每一次的實驗,需要許多的時間、精神與金錢。工程繪圖領域,例如Auto CAD對於二次曲線等,繪製過程必須依據相關之參數並利用向量的技術進行繪圖,對於方程式的表示式及細節的資訊(含圓心、半徑、焦點、長短軸、頂點等)卻無法獲得。本論文中,推導二次方程式及特徵參數,然而,擬合的過程本文使用二次方程式所構成圖形的幾何特性,例如資料點形成的圓,由資料點第一次使用Least square產生圓的各項特徵參數,每一資料點對於圓心的距離與半徑做比例,帶入方程式後,決定該點是否進行半徑或圓心的疊代運算,本文認為此法能將圓心與半徑有所改善,接近理論值。最後,曲線的擬合對於每一種二次多項式,可以是最佳的資料點。

Conic includes ellipse, hyperbola and parabola. In engineering application, curve fitting from a set of measured data at discrete points for the ellipse (including circle) are widely desired. Furthermore, since the ellipse gas closed boundary, its application is also much expanded. There application arise in computer graphics, statistics, metrology, astronomy, etc. In practice, the measured data on the manufactured part are normally obtained by using a coordinate measuring machine (CMM), which is a device with a probe moving in a particular direction and identifies the coordinates of the points on the surface. Each measurement data the information is not necessarily expected, it might be better data can be, measurement error or a lack of precision caused. Each one of these experiments, many of the needs of their time, spirit and money. Mapping project areas such as Auto CAD for the second curve, Rendering process must be based on parameters related to the use of vector technology, But the formula is expressed in the details and information (including the center, radius, focus, major axis, minor axis, Vertex, etc.) will not be given. This study , the quadratic equation is derived parameters of all relative information, however, fitting for the process is the use of quadratic equations by each graphic geometric characteristics, such as a circle posed by the data, first use the Least Square method find out every parameters, and then every data points calculate distances form the center to the data and radii ratio by radius, finally bring each data point into the quadratic equation and decide which data point to correct the center or radius by iterative computation or not. This study argues that center and radius can be improved and close to the theoretical value. Then final the curve fitting for each quadratic is derived with these optimal data points.
其他識別: U0005-0708200713473300
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