基于Gauss-Chebyshev积分的道路平曲线计算
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摘要
道路平曲线坐标计算中可能会采用线元法,其中线元类型有直线、圆曲线和缓和曲线。本文采用改进的线元表对平曲线数据进行预处理,讨论了Gauss-Chebyshev积分公式的应用并通过数值计算实验研究了高斯点数量对待定点计算的影响,在此基础上使用Gauss-Chebyshev积分方法和5点改进型GaussChebyshev积分方法解决平曲线计算中的定积分计算问题。为验证Gauss-Chebyshev积分的计算效果,选取某铁路一段平曲线作为计算数据,指定16个临近点作为数值实验对象,实验结果显示反算所得各点里程和偏距与起始给定的数值一致。
In order to calculate the coordinates of the points on a horizontal curve,building block method is usually used to establish an integrated mathematical model.There are three types of line elements:straight line,circular curved line,and transition curved line.In this article,how to utilize building block method for data pretreatment is introduced,the Gauss-Chebyshev quadrature rule is taken as a universal computational method,and the effect set by the count of Gauss points on the accuracy of unknown points is also discussed by means of numerical experiments.On the basis,the Gauss-Chebyshev quadrature rules and the improved 5-point Gauss-Chebyshev quadrature rules are used to approximate the value of the definite integral so as to compute coordinates of the unknown points on horizontal curves.To test how the Gauss-Chebyshev quadrature rules work in computing coordinates,mileage,and deviations,aportion of plane curve on a railway is selected as calculation data,and 16 adjacent points are selected as objects of a numerical experiment.The result shows that all the mileage and deviations acquired by the inverse computation are consistent with the initial given values.
In order to calculate the coordinates of the points on a horizontal curve,building block method is usually used to establish an integrated mathematical model.There are three types of line elements:straight line,circular curved line,and transition curved line.In this article,how to utilize building block method for data pretreatment is introduced,the Gauss-Chebyshev quadrature rule is taken as a universal computational method,and the effect set by the count of Gauss points on the accuracy of unknown points is also discussed by means of numerical experiments.On the basis,the Gauss-Chebyshev quadrature rules and the improved 5-point Gauss-Chebyshev quadrature rules are used to approximate the value of the definite integral so as to compute coordinates of the unknown points on horizontal curves.To test how the Gauss-Chebyshev quadrature rules work in computing coordinates,mileage,and deviations,aportion of plane curve on a railway is selected as calculation data,and 16 adjacent points are selected as objects of a numerical experiment.The result shows that all the mileage and deviations acquired by the inverse computation are consistent with the initial given values.
引文
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[10]李文科,敖亭芝.曲线坐标模型在道路计算中的应用[J].北京测绘,2013(4):59-62.Li Wenke,Ao Tingzhi.Application of Curvilinear Integral Model in Coordinate Calculation of Center Line[J].Beijing Surveying and Mapping,2013(4):59-62.
[11]李自康.公路曲线线元坐标计算通用模型[J].勘察科学技术,2017(5):20-23.Li Zikang.General Model for Coordinate Calculation of Highway Curve Elements[J].Site Investigation Science and Technology,2017(5):20-23.
[12]黄金满,林从谋,李军心.高速公(铁)路平竖曲线正反算的统一解法[J].测绘科学,2008,40(10):3-10.Huang Jinman,Lin Congmou,Li Junxin.A General Method of Direct and Inverse Coordinate Computation for Korizontal and Vertical Curves of Expressway and Highspeed Railway[J].Science of Surveying and Mapping,2008,40(10):3-10.
[13]Koc W.Identification of Transition Curves in Vehicular Roads and Railways[J].Logistics Infrastructure,2015,28(4):31-40.
[14]KobryńA.Universal Solutions of Transition Curves[J].Journal of Surveying Engineering,2016 142(4):04016010.
[15]Eslahchi M R,Dehghan M,Masjed-Jamei M.On Numerical Improvement of the First Kind GaussChebyshev Quadrature Rules[J].Applied Mathematics and Computation,2005,165:5-21.
[16]李德光.铁路线路中线空间坐标与里程换算模型的研究[D].成都:西南交通大学,2012.Li Deguang.The Research(on)Conversion Model Between Spatial Coordinate and Mileage in Railway Centerline[D].Chengdu:Southwest Jiaotong University,2012.
[17]同济大学计算数学教研室.现代数值计算[M].北京:人民邮电出版社,2009:123-132.Teaching and Research Section in School of Mathematical Science,Tongji University.Advanced Numerical Computing[M].Beijing:Posts and Telecom Press,2009:123-132.
[18]李炯城,林惜斌,肖恒辉,等.高阶高斯型积分计算机求解算法[J].计算机工程与设计,2012,33(5):1871-1875.Li Jiongcheng,Lin Xibin,Xiao Henghui,et al.Computer Algorithm for High-Degree Gauss-Type Quadrature[J].Computer Engineering and Design,2012,33(5):1871-1875.
[2]王科.铁路复曲线计算方法研究[D].兰州:兰州交通大学,2018.Wang Ke.Research on Calculation Method of Complex Curve of Railway Lines[J].Lanzhou:Lanzhou Jiaotong University,2018.
[3]冯晓,李敏,杨佳,等.不同类型缓和曲线的正算与反算的通用算法[J].测绘通报,2008(6):10-13.Feng Xiao,Li Min,Yang Jia,et al.A General Method of Direct and Inverse Coordinate Computation for Different Types of Transition Curve[J].Bulletin of Surveying and Mapping,2008(6):10-13.
[4]高淑照,史东宴.缓和曲线非线性方程的快速解算[J].测绘通报,2006(3):28-30.Gao Shuzhao,Shi Dongyan.A Quick Solution to a Nonlinear Equation of Spiral Curve[J].Bulletin of Surveying and Mapping,2006(3):28-30.
[5]孔晨辉,翁呷.线路曲线的测设[J].测绘地理信息,2017 42(6):69-72.Kong Chenhui,Weng Ga.Surveying of the Line Curve[J].Journal of Geomatics,2017 42(6):69-72.
[6]李全信.线路测量中的正反算问题及应用[J].测绘通报,2006(2):36-38.Li Quanxin.The Direct and Inverse Solution in Route Survey and Its Application[J].Bulletin of Surveying and Mapping,2006(2):36-38.
[7]李全信.线路中边桩坐标计算的通用Gauss-Legendre公式[J].工程勘察,2002(3):61-64.Li Quanxin.A Universal Gauss-Legendre Quadrature Formula for Computing Coordinates of Center and Side Stakes in Route Survey[J].Geotechnical Investigation and Surveying,2002(3):61-64.
[8]孙建国,苗贺.基于Chebyshev走时逼近的三维多次反射射线计算[J].吉林大学学报(地球科学版),2018,48(3):890-899.Sun Jianguo,Miao He.Computation of Three Dimensional Multi-Reflection Rays Based on Traveltimes Numerical Approximation[J].Journal of Jilin University(Earth Science Edition),2018,48(3):890-899.
[9]李孟山,李少元.数值积分法计算线路中线坐标[J].石家庄铁道学院学报,1999(3):51-53.Li Mengshan,Li Shaoyuan.The Coordinate Computation of Central Line Using Numerical Integral[J].Journal of Shijiazhuang Railway University,1999(3):51-53
[10]李文科,敖亭芝.曲线坐标模型在道路计算中的应用[J].北京测绘,2013(4):59-62.Li Wenke,Ao Tingzhi.Application of Curvilinear Integral Model in Coordinate Calculation of Center Line[J].Beijing Surveying and Mapping,2013(4):59-62.
[11]李自康.公路曲线线元坐标计算通用模型[J].勘察科学技术,2017(5):20-23.Li Zikang.General Model for Coordinate Calculation of Highway Curve Elements[J].Site Investigation Science and Technology,2017(5):20-23.
[12]黄金满,林从谋,李军心.高速公(铁)路平竖曲线正反算的统一解法[J].测绘科学,2008,40(10):3-10.Huang Jinman,Lin Congmou,Li Junxin.A General Method of Direct and Inverse Coordinate Computation for Korizontal and Vertical Curves of Expressway and Highspeed Railway[J].Science of Surveying and Mapping,2008,40(10):3-10.
[13]Koc W.Identification of Transition Curves in Vehicular Roads and Railways[J].Logistics Infrastructure,2015,28(4):31-40.
[14]KobryńA.Universal Solutions of Transition Curves[J].Journal of Surveying Engineering,2016 142(4):04016010.
[15]Eslahchi M R,Dehghan M,Masjed-Jamei M.On Numerical Improvement of the First Kind GaussChebyshev Quadrature Rules[J].Applied Mathematics and Computation,2005,165:5-21.
[16]李德光.铁路线路中线空间坐标与里程换算模型的研究[D].成都:西南交通大学,2012.Li Deguang.The Research(on)Conversion Model Between Spatial Coordinate and Mileage in Railway Centerline[D].Chengdu:Southwest Jiaotong University,2012.
[17]同济大学计算数学教研室.现代数值计算[M].北京:人民邮电出版社,2009:123-132.Teaching and Research Section in School of Mathematical Science,Tongji University.Advanced Numerical Computing[M].Beijing:Posts and Telecom Press,2009:123-132.
[18]李炯城,林惜斌,肖恒辉,等.高阶高斯型积分计算机求解算法[J].计算机工程与设计,2012,33(5):1871-1875.Li Jiongcheng,Lin Xibin,Xiao Henghui,et al.Computer Algorithm for High-Degree Gauss-Type Quadrature[J].Computer Engineering and Design,2012,33(5):1871-1875.