基本型曲线中边桩坐标计算及测设课件讲解

柳州铁道职业技术学院建筑技术学院《铁路工程测量》课程团队铁路工程测量---基本型曲线中边桩坐标计算及测设基本型曲线中边桩坐标计算及测设一、线路中桩坐标计算二、线路边桩坐标计算三、基本型曲线缓和曲线中边桩坐标计算一.简单型线路中桩坐标计算设计给定的要素有：α——转向角；R——圆曲线半径；α1、α2——直线的的方位角主点里程KZY（或JD里程）；JD坐标（XJD，YJD）；（一）简单型曲线中桩坐标计算线路K2+400~K2+700设计资料：

转向角α=-18°25′36″；R=400m；曲线长L=128.642m;切线长T=64.881m;α1=75°30′48″；α2=57°05′12″;JD=K2+560(1256.342，892.143）ZY=K2+495.119(1240.112，829.325)YZ=K2+623.761(1291.597，946.611)1.第一直线段中线桩坐标计算示例(1)K2+400中桩:N=1240.112-(2495.119-2400)COS75°30′48″=1216.318E=829.325-(2495.119-2400)Sin75°30′48″=737.230

已知条件：转向角α=-18°25′36″；R=400m；α1=75°30′48″；α2=57°05′12″；ZY=K2+495.119(1240.112，829.325)；YZ=K2+623.761(1291.597，946.611)(2)K2+420中桩:N=1240.112-(2495.119-2420)COS75°30′48″=1221.320E=829.325-(2495.119-2420)Sin75°30′48″=756.594

已知条件：转向角α=-18°25′36″；R=400m；α1=75°30′48″；α2=57°05′12″；ZY=K2+495.119(1240.112，829.325)；YZ=K2+623.761(1291.597，946.611)练习：计算K2+440，K2+460，K2+480中桩坐标K2+440(1226.323,775.958)K2+460(1231.326,795.322)K2+480(1236.330,814.687)

已知条件：转向角α=-18°25′36″；R=400m；α1=75°30′48″；α2=57°05′12″；ZY=K2+495.119(1240.112，829.325)；YZ=K2+623.761(1291.597，946.611)2.圆曲线段中桩坐标计算如图所示，m为圆曲线上

任一点，里程为Km，则，ZY至m点的曲线长度：Lm=Km-KzY那么，m点的坐标计算过程如下：ZYYZ圆曲线上任一点（1）角度计算ZYYZCα1（2）弦长计算（3）m点坐标计算ZYYZCα1Nm=NZY+C.cos(α1-αm)Ym=YZY+C.sin(α1-αm)如果线路转向角为正,计算过程有什么不一样?圆线段中桩坐标计算示例(1)K2+500中桩:Lm=Km-KzY=2500-2495.119=4.881mαm=90Lm/πR=(90×4.881)/(π×400)=0°20′58.4″C=2Rsinαm=2×400sin(0°20′58.4″)=4.881mNm=NZY+Ccos(α1-αm)=1240.112+4.881cos(75°30′48″-0°20′58.4″)=1241.362Em=EZY+Csin(α1-αm)=829.325+4.881sin(75°30′48″-0°20′58.4″)=834.043

已知条件：转向角α=-18°25′36″；R=400m；α1=75°30′48″；α2=57°05′12″；ZY=K2+495.119(1240.112，829.325)；YZ=K2+623.761(1291.597，946.611)(2)K2+520中桩:Lm=Km-KzY=2520-2495.119=24.881mαm=90Lm/πR=(90×24.881)/(π×400)=1°46′55″C=2Rsinαm=2×400sin(1°46′55″)=24.877mNm=NZY+Ccos(α1-αm)=1240.112+24.877cos(75°30′48″-1°46′55″)=1247.081Em=EZY+Csin(α1-αm)=829.325+24.877sin(75°30′48″-1°46′55″)=853.206

已知条件：转向角α=-18°25′36″；R=400m；α1=75°30′48″；α2=57°05′12″；ZY=K2+495.119(1240.112，829.325)；YZ=K2+623.761(1291.597，946.611)练习：计算：K2+540，K2+560，K2+580，K2+600，K2+620中桩坐标K2+540(1253.751,872.059)K2+560(1261.354,890.554)K2+580(1269.873,908.647)K2+600(1279.285,926.292)K2+620(1289.568,943.443)

已知条件：转向角α=-18°25′36″；R=400m；α1=75°30′48″；α2=57°05′12″；ZY=K2+495.119(1240.112，829.325)；YZ=K2+623.761(1291.597，946.611)3.第二直线段中线桩坐标计算示例(1)K2+640中桩:N=1291.597+(2640-2623.761)COS57°05′12″=1300.421E=946.611+(2640-2623.761)Sin57°05′12″=960.244

已知条件：转向角α=-18°25′36″；R=400m；α1=75°30′48″；α2=57°05′12″；ZY=K2+495.119(1240.112，829.325)；YZ=K2+623.761(1291.597，946.611)练习：计算K2+660;K2+680;K2+700中桩坐标

K2+660(1311.288,977.033)K2+680(1322.155,993.823)K2+700(1333.023,1010.613)

已知条件：转向角α=-18°25′36″；R=400m；α1=75°30′48″；α2=57°05′12″；ZY=K2+495.119(1240.112，829.325)；YZ=K2+623.761(1291.597，946.611)二、简单型线路边桩坐标计算线路K5+000~K5+400设计资料：

曲线1：转向角α=-28°25′42″；R=300m；L=148.850m;

T=75.991m;α1=100°18′48″；α2=71°53′06″;JD=K5+120(863.234，945.113）ZY=K5+044.009(876.839,870.35)YZ=K5+192.860(886.861,1017.337)曲线2：转向角α=+33°21′55″；R=200m；L=116.467m;

T=59.937m;α1=71°53′06″；α2=105°15′01″;JD=K5+310(923.283,1128.670）ZY=K5+250.063(904.647,1071.704)YZ=K5+366.530(907.517,1186.496)路基宽度为12m,左右等宽，分别计算左右边桩坐标。（一）第一直线（K5+000~+044.009)

边桩计算示例(1)K5+000左6m宽度桩:N=876.839-44.009COS100°18′48″+6

COS(100°18′48″-90)=890.621E=870.350-44.009Sin100°18′48″+6Sin(100°18′48″-90)=828.125

已知：转向角α=-28°25′42″；R=300m；L=148.850m;

T=75.991m;α1=100°18′48″；α2=71°53′06″;JD=K5+120(863.234，945.113）ZY=K5+044.009(876.839,870.35)YZ=K5+192.860(886.861,1017.337)

(2)K5+000右6m宽度桩:N=876.839-44.009COS100°18′48″+6COS(100°18′48″+90)=878.815E=870.350-44.009Sin100°18′48″+6Sin(100°18′48″+90)=825.977

已知：转向角α=-28°25′42″；R=300m；L=148.850m;

T=75.991m;α1=100°18′48″；α2=71°53′06″;JD=K5+120(863.234，945.113）ZY=K5+044.009(876.839,870.35)YZ=K5+192.860(886.861,1017.337)

练习:(1)计算K5+020左右各6m宽度桩左(887.040,847.802)右(875.234,845.654)(2)计算K5+040左右各6m宽度桩左(883.460,867.480)右(871.653,865.3310

已知：转向角α=-28°25′42″；R=300m；L=148.850m;

T=75.991m;α1=100°18′48″；α2=71°53′06″;JD=K5+120(863.234，945.113）ZY=K5+044.009(876.839,870.35)YZ=K5+192.860(886.861,1017.337)（二）第一圆曲线（K5+044.009~+192.860段)

边桩坐标计算示例如图所示，m为圆曲线上

任一点，里程为Km，则，ZY至m点的曲线长度：Lm=Km-KzY那么，m点的切线方位角计算过程如下：ZYYZ圆曲线上任一点ZYYZCα1βm=2αm则m点切线方位角=α1-2αm(1)K5+060左6m宽度桩计算:Lm=Km-KzY=5060-5044.009=15.991mαm=90Lm/πR=(90×15.991)/(π×300)=1°31′37″C=2Rsinαm=2×300sin(1°31′37″)=15.988mN=NZY+Ccos(α1-αm)+6cos(α1-2αm-90)=876.839+15.988cos(100°18′48″-1°31′37″)+6cos(100°18′48″-2×1°31′37″-90)=880.349E

=EZY+Csin(α1-αm)+6Sin(α1-2αm-90)=870.350+15.9881sin(100°18′48″-1°31′37″)+6Sin(100°18′48″-2×1°31′37″-90)=886.909

第一圆曲线（K5+044.009～+192.860段)计算示例已知：转向角α=-28°25′42″；R=300m；L=148.850m;

T=75.991m;α1=100°18′48″；α2=71°53′06″;JD=K5+120(863.234，945.113）ZY=K5+044.009(876.839,870.35)YZ=K5+192.860(886.861,1017.337)(2)K5+060右6m宽度桩计算:Lm=Km-KzY=5060-5044.009=15.991mαm=90Lm/πR=(90×15.991)/(π×300)=1°31′37″C=2Rsinαm=2×300sin(1°31′37″)=15.988mNm=NZY+Ccos(α1-αm)+6cos(α1-2αm+90)=876.839+15.988cos(100°18′48″-1°31′37″)+6cos(100°18′48″-2×1°31′37″+90)=868.444Em=EZY+Csin(α1-αm)+6Sin(α1-2αm+90)=870.350+15.9881sin(100°18′48″-1°31′37″)+6Sin(100°18′48″-2×1°31′37″+90)=885.393

已知：转向角α=-28°25′42″；R=300m；L=148.850m;

T=75.991m;α1=100°18′48″；α2=71°53′06″;JD=K5+120(863.234，945.113）ZY=K5+044.009(876.839,870.35)YZ=K5+192.860(886.861,1017.337)练习:(1)计算K5+080左右各6m宽度桩左(878.521,906.420)右(866.543,905.700)(2)计算K5+100左右各6m宽度桩左(877.998,926.010)右(865.998,926.089)

已知：转向角α=-28°25′42″；R=300m；L=148.850m;

T=75.991m;α1=100°18′48″；α2=71°53′06″;JD=K5+120(863.234，945.113）ZY=K5+044.009(876.839,870.35)YZ=K5+192.860(886.861,1017.337)（三）第二直线（K5+192.86~+250.063)计算示例(1)K5+200左6m宽度桩:N=886.861+(200-192.86)COS71°53′06″+6

COS(71°53′06″-90)=894.784E=1017.337+(200-192.86)Sin71°53′06″+6Sin(71°53′06″-90)=1022.258

已知：转向角α=-28°25′42″；R=300m；L=148.850m;

T=75.991m;α1=100°18′48″；α2=71°53′06″;JD=K5+120(863.234，945.113）ZY=K5+044.009(876.839,870.35)YZ=K5+192.860(886.861,1017.337)(1)K5+260左6m宽度桩计算:Lm=Km-KzY=5260-5250.063=9.937mαm=90Lm/πR=(90×9.937)/(π×200)=1°25′24″C=2Rsinαm=2×200sin(1°25′24″)=9.935mNm=NZY+Ccos(α1+αm)+6cos(α1+2αm-90)=904.647+9.935cos(71°53′06″+1°25′24″)+6cos(71°53′06″+2×1°25′24″-90)=913.289Em=EZY+Csin(α1+αm)+6Sin(α1+2αm-90)=1071.704+9.935sin(71°53′06″+1°25′24″)+6Sin(71°53′06″+2×1°25′24″-90)=1079.640

（四）第二圆曲线（K5+250.063～+366.530)计算示例已知：转向角α=+33°21′55″；R=200m；L=116.467m;

T=59.937m;α1=71°53′06″；α2=105°15′01″;JD=K5+310(923.283,1128.670）；ZY=K5+250.063(904.647,1071.704)YZ=K5+366.530(907.517,1186.496)(2)K5+260右6m宽度桩计算:Lm=Km-KzY=5260-5250.063=9.937mαm=90Lm/πR=(90×15.991)/(π×200)=1°25′24″C=2Rsinαm=2×200sin(1°25′24″)=9.935mNm=NZY+Ccos(α1+αm)+6cos(α1+2αm+90)=904.647+9.935cos(71°53′06″+1°25′24″)+6cos(71°53′06″+2×1°25′24″+90)=901.713Em=EZY+Csin(α1+αm)+6Sin(α1+2αm+90)=1071.704+9.935sin(71°53′06″+1°25′24″)+6Sin(71°53′06″+2×1°25′24″+90)=1082.801

已知：转向角α=+33°21′55″；R=200m；L=116.467m;

T=59.937m;α1=71°53′06″；α2=105°15′01″;JD=K5+310(923.283,1128.670）；ZY=K5+250.063(904.647,1071.704)YZ=K5+366.530(907.517,1186.496)练习:(1)计算K5+280左右各6m宽度桩左(917.712,1099.751)右(905.878,1101.740)(2)计算K5+300左右各6m宽度桩左(920.105,1120.204)右(908.132,1121.001)

已知：转向角α=+33°21′55″；R=200m；L=116.467m;

T=59.937m;α1=71°53′06″；α2=105°15′01″;JD=K5+310(923.283,1128.670）；ZY=K5+250.063(904.647,1071.704)YZ=K5+366.530(907.517,1186.496)1、基本型曲线的组成

由直线——缓和曲线——圆曲线——缓和曲线——直线的顺序组成直线JDZHHYYHYH

三、基本型曲线缓和曲线中边桩坐标计算2、基本型曲线计算的已条件

一般地，基本型曲线的已知条件包括：曲线半径R，缓和曲线长度l0，曲线转向角α，第一直线方位角，JD里程及坐标。

基本型曲线直线段上坐标计算同前。计算之前先计算曲线常数及要素，然后计算ZH和HZ的坐标，然后依次计算便可。

3.第一缓和曲线（K5+031.488~+056.488段)

中边桩计算公式

假设线路左转,ZH坐标坐标已知（XZH，YZH）第一直线方位角为α1。α1

⑴角度计算公式（ZH至n点的弧长为ln）

⑵偏距计算公式（ZH至n点的弧长为ln）

XYα1

⑶n点中桩坐标及n点切线方位角计算

Xn=XZH+Xcosα1+Ycos(α1-90)

Yn=YZH+Xsinα1+Ysin(α1-90)

(4)n里程右侧V米及左侧V米边桩坐标计算

X右=Xn+Vcos（αn+90）Y右=Yn+VSin（αn+90）X左=Xn+Vcos（αn-90）Y左=Yn+VSin（αn-90）

4.基本型曲线缓和曲线中边桩坐标计算示例线路K5+000~K5+300设计资料：

曲线1：转向角α=-28°25′42″；缓和曲线长度l0=25m，R=300m；L=173.850m;

T=88.512m;α1=100°18′48″；α2=71°53′06″;JD=K5+120(863.234，945.113）；ZH=K5+031.488(879.080,858.031)；HY=K5+056.488(874.947,882.685)YH=K5+180.339(883.313,1005.374)；HZ=K5+205.339(890.755,1029.239)路基顶宽度为12m,左右等宽，计算中边桩坐标。

K5+050中桩及左6m边桩坐标计算示例

（1）方位角计算

=90×18.5122/(π×300×25)=1°18′32″

已知转向角α=-28°25′42″；缓和曲线长度l0=25m，R=300m；α1=100°18′48″；α2=71°53′06″;JD=K5+120(863.234，945.113）；ZH=K5+031.488(879.080,858.031)；HY=K5+056.488(874.947,882.685)；H=K5+180.339(883.313,1005.374)；HZ=K5+205.339(890.755,1029.239)

（2）偏距计算:由公式

计算得:X=18.511Y=0.139