约束于圆弧的路线平面直线段重构算法
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摘要
提出一种利用一组(x,y)直角坐标点拟合与给定的圆弧相切的直线的算法。该算法遵循总体最小二乘准则来构建约束于圆弧的拟合直线的数学模型,在分析函数单调隔根区间的基础上,利用反拉格朗日插值迭代法计算出函数在定义域内的所有数值根,再经最小二乘准则检验获得数学模型的解,即拟合直线的回归参数。研究结果表明:拟合数学模型简明,基于单调隔根区间的迭代算法稳健,速度快,效果优,可用于重构路线平面的精确几何参数。
On the basis of a set of(x, y) points, this paper proposes a method for fitting a regression line that is tangent to the given arc. The algorithm, based on Holistic Least Square Method, firstly established mathematical model fitting a line constrained by circular arc. On the basis of analysis of the intervals on which a function's null point exists, the Inverse Lagrange's Interpolation Method was used to calculate all of the function's numerical solutions on the domain of definition. Additionally, the least squares criterion decided which of the numerical solutions can satisfy the mathematical model which had been established. Numerical solutions satisfying the mathematical model were exactly regression parameters of the regression line. Theoretical derivation and practical application indicate that this is a fast and stable algorithm with accurate results. This feature allows it to reconstruct precise geometric parameters of the Horizontal Alignment.
On the basis of a set of(x, y) points, this paper proposes a method for fitting a regression line that is tangent to the given arc. The algorithm, based on Holistic Least Square Method, firstly established mathematical model fitting a line constrained by circular arc. On the basis of analysis of the intervals on which a function's null point exists, the Inverse Lagrange's Interpolation Method was used to calculate all of the function's numerical solutions on the domain of definition. Additionally, the least squares criterion decided which of the numerical solutions can satisfy the mathematical model which had been established. Numerical solutions satisfying the mathematical model were exactly regression parameters of the regression line. Theoretical derivation and practical application indicate that this is a fast and stable algorithm with accurate results. This feature allows it to reconstruct precise geometric parameters of the Horizontal Alignment.
引文
[1]Casal G,Santamarina D,Vázquez-Méndez M E.Optimization of horizontal alignment geometry in road design and reconstruction[J].Transportation Research Part C Emerging Technologies,2017,74(1):261-274
[2]刘威,胡光常,麻丁一,等.铁路既有线平面自动重构方法研究[J].高速铁路技术,2015,6(6):79-84.LIU Wei,HU Guangchang,MA Dingyi,et al.Research on existing railway plane line auto-reconstruction[J].High Speed Railway Technology,2015,6(6):79-84.
[3]郭良浩,刘成龙,宋韬,等.铁路既有线平面和竖面线形精确分段方法研究[J].铁道工程学报,2014,31(7):48-52.GUO Lianghao,LIU Chenglong,SONG Tao,et al.Research on the new method for accurate linear segmentation of plane and vertical curve type in existing railway[J].Journal of Railway Engineering Society,2014,31(7):48-52.
[4]张航,黄云,龚良甫.基于三次样条函数拟合公路平面线形方法研究[J].武汉理工大学学报(交通科学与工程版),2007,31(5):925-927.ZHANG Hang,HUANG Yun,GONG Liangpu.Study on fitting highway alignment based on the cubic spline function[J].Journal of Wuhan University of Technology(Transportation Science&Engineering),2007,31(5):925-927.
[5]刘苏,王文强,查旭东,等.基于法线偏差的旧路平面线形拟合精度评估方法[J].中国公路学报,2007,20(5):36-40.LIU Su,WANG Wenqiang,ZHA Xudong,et al.Estimate method for accuracy of fitting plane linear in old highway based on normal error[J].China Journal of Highway and Transport,2007,20(5):36-40.
[6]LI Wei,PU Hao,Paul Schonfeld,et al.Mountain railway alignment optimization with bidirectional distance transform and genetic algorithm[J].Computer-Aided Civil and Infrastructure Engineering,2017,32(8):691-709
[7]LI Wei,PU Hao,Paul Schonfeld,et al.Methodology for optimizing constrained 3-dimensional railway alignments in mountainous terrain[J].Transportation Research Part C,2016,7(68):549-565
[8]李伟.铁路数字选线关键技术研究与应用[D].长沙:中南大学,2014.LI Wei.Research and application for key technologies of digitalizing railway alignment design[D].Changsha:Central South University,2014.
[9]Robert T,Gunner L,Sarbaz O,et al.Using naturalistic field operational test data to identify horizontal curves[J].Journal of Transportation Engineering,2012,138(9):1151-1160.
[10]丁克良,沈云中,欧吉坤.整体最小二乘法直线拟合[J].辽宁工程技术大学学报(自然科学版),2010,29(1):44-47.DING Keliang,SHEN Yunzhong,OU Jikun.Methods of line-fitting based on total least-square[J].Journal of Liaoning Technical University(Natural Science),2010,29(1):44-47.
[11]刘元朋,张定华,桂元坤,等.用带约束的最小二乘拟合平面圆曲线[J].计算机辅助设计与图形学学报,2001,10(16):1382-1385.LIU Yuanpeng,ZHANG Dinghua,GUI Yuankun,et al.Fitting planar circles with constrained least squares[J].Journal of Computer-Aided Design&Computer Graphics,2001,10(16):1382-1385.
[12]吴佐慧,廖军,徐行忠,等.一元三次、四次方程根的行列式解法[J].湖北大学学报(自然科学版),2014,4(36):381-388.WU Zuohui,LIAO Jun,XU Xingzhong,et al.A solution of the root of the cubic and quartic equationusing determinant[J].Journal of Hubei University(Natural Science),2014,4(36):381-388.
[13]李惠.用C程序求一元四次方程的近似解[J].中国科教创新导刊,2012,9(25):98-100.LI Hui.Find the approximate solution of a quartic equation with one unknown in C program[J].China Education Innovation Herald,2012,9(25):98-100.
[14]Francisco J C,Ana M P,JoséM C,et al.Use of heading direction for recreating the horizontal alignment of an existing road[J].Computer-Aided Civil and Infrastructure Engineering,2015,30(4):282-299.
[15]喻文健.数值分析与算法[M].北京:清华大学出版社,2012:41-45.YU Wenjian.Numerical analysis and algorithm[M].Beijing:Tsinghua University Press,2012:41-45.
[2]刘威,胡光常,麻丁一,等.铁路既有线平面自动重构方法研究[J].高速铁路技术,2015,6(6):79-84.LIU Wei,HU Guangchang,MA Dingyi,et al.Research on existing railway plane line auto-reconstruction[J].High Speed Railway Technology,2015,6(6):79-84.
[3]郭良浩,刘成龙,宋韬,等.铁路既有线平面和竖面线形精确分段方法研究[J].铁道工程学报,2014,31(7):48-52.GUO Lianghao,LIU Chenglong,SONG Tao,et al.Research on the new method for accurate linear segmentation of plane and vertical curve type in existing railway[J].Journal of Railway Engineering Society,2014,31(7):48-52.
[4]张航,黄云,龚良甫.基于三次样条函数拟合公路平面线形方法研究[J].武汉理工大学学报(交通科学与工程版),2007,31(5):925-927.ZHANG Hang,HUANG Yun,GONG Liangpu.Study on fitting highway alignment based on the cubic spline function[J].Journal of Wuhan University of Technology(Transportation Science&Engineering),2007,31(5):925-927.
[5]刘苏,王文强,查旭东,等.基于法线偏差的旧路平面线形拟合精度评估方法[J].中国公路学报,2007,20(5):36-40.LIU Su,WANG Wenqiang,ZHA Xudong,et al.Estimate method for accuracy of fitting plane linear in old highway based on normal error[J].China Journal of Highway and Transport,2007,20(5):36-40.
[6]LI Wei,PU Hao,Paul Schonfeld,et al.Mountain railway alignment optimization with bidirectional distance transform and genetic algorithm[J].Computer-Aided Civil and Infrastructure Engineering,2017,32(8):691-709
[7]LI Wei,PU Hao,Paul Schonfeld,et al.Methodology for optimizing constrained 3-dimensional railway alignments in mountainous terrain[J].Transportation Research Part C,2016,7(68):549-565
[8]李伟.铁路数字选线关键技术研究与应用[D].长沙:中南大学,2014.LI Wei.Research and application for key technologies of digitalizing railway alignment design[D].Changsha:Central South University,2014.
[9]Robert T,Gunner L,Sarbaz O,et al.Using naturalistic field operational test data to identify horizontal curves[J].Journal of Transportation Engineering,2012,138(9):1151-1160.
[10]丁克良,沈云中,欧吉坤.整体最小二乘法直线拟合[J].辽宁工程技术大学学报(自然科学版),2010,29(1):44-47.DING Keliang,SHEN Yunzhong,OU Jikun.Methods of line-fitting based on total least-square[J].Journal of Liaoning Technical University(Natural Science),2010,29(1):44-47.
[11]刘元朋,张定华,桂元坤,等.用带约束的最小二乘拟合平面圆曲线[J].计算机辅助设计与图形学学报,2001,10(16):1382-1385.LIU Yuanpeng,ZHANG Dinghua,GUI Yuankun,et al.Fitting planar circles with constrained least squares[J].Journal of Computer-Aided Design&Computer Graphics,2001,10(16):1382-1385.
[12]吴佐慧,廖军,徐行忠,等.一元三次、四次方程根的行列式解法[J].湖北大学学报(自然科学版),2014,4(36):381-388.WU Zuohui,LIAO Jun,XU Xingzhong,et al.A solution of the root of the cubic and quartic equationusing determinant[J].Journal of Hubei University(Natural Science),2014,4(36):381-388.
[13]李惠.用C程序求一元四次方程的近似解[J].中国科教创新导刊,2012,9(25):98-100.LI Hui.Find the approximate solution of a quartic equation with one unknown in C program[J].China Education Innovation Herald,2012,9(25):98-100.
[14]Francisco J C,Ana M P,JoséM C,et al.Use of heading direction for recreating the horizontal alignment of an existing road[J].Computer-Aided Civil and Infrastructure Engineering,2015,30(4):282-299.
[15]喻文健.数值分析与算法[M].北京:清华大学出版社,2012:41-45.YU Wenjian.Numerical analysis and algorithm[M].Beijing:Tsinghua University Press,2012:41-45.