圆曲线和缓和曲线坐标推算公式(附带例题)

圆曲线和缓和曲线坐标推算公式(附带例题)圆曲线和缓和曲线坐标推算公式一、直线上的坐标推算???++0imi0imisinaLYYcosaLXX＝＝式中：Xm、Ym——直线段起点M坐标Li——直线段上任意点i到线路起点M的距离a0——直线段起点M到JD1的方位角二、圆曲线上任一点的坐标推算①、圆曲线上任一点i相对应的圆心角：iiLR180π??＝式中：Li——圆曲线上任一点i离开ZY或YZ点的弧长②、圆曲线上任一点i的直角坐标：???-）（＝＝iiiicos1RYRsinX??（可不计算）.③、圆曲线ZY或YZ点到任一点i的偏角：iiiLR902π???＝＝④、圆曲线ZY或YZ点到任一点i的弦长：)sin(2)2sin(2CiiiRR?=?＝⑤、圆曲线ZY或YZ点到任一点i的弦长的方位角：ijdyzjdzyiaa?±→→或＝⑥、所以圆曲线上任意点i的坐标为：???++iiYZZYiiiYZZYisinaCYYcosaCXX或或＝＝例题：已知一段圆曲线,R=3500m，Ls＝553.1m，交点里程K50+154.734，ZY点到JD方向方位角为A=129°23′18.3″，右偏9°3′15.8″，ZY点里程K49+877.607，YZ点里程K50+430.707，起点坐标为x＝389823.196，y＝507787.251，求K50+200处中点坐标及左右各偏12.5m的坐标。解：K50+200处的曲线长度为Li＝322.393mK50+200相对应的方位角：'?????52.39165393.3223500180LR180i＝＝＝ππaK50+200相对应的偏角：'??????76.19382393.322350090LR902iii＝＝＝＝ππ?K50+200到zy点的弦长：m279.32276.19382sin35002Rsin2Cii＝＝＝'????zy点到K50+200中桩的方位角：'?'?+'??+→06.38113276.193823.1823129aaijdzyi＝＝＝K50+200左、右偏12.5m的方位角：'??-'??-+82.5739449082.573913490aa＝＝＝左iA'??+'??++82.57391349082.573913490aa＝＝＝右iA所以K50+200处的坐标为：???'??++'??++6484.50802606.381132sin279.322251.507787sinaCYY4354.38960706.381132cos279.322196.389823cosaCXXiiZYiiiZYi＝＝＝＝＝＝K50+200左偏12.5m的坐标为：???'??++'??++4656.50803582.573944sin5.126484.508026sina5.21YY3256.38961682.573944cos5.124354.389607cosa5.21XXii＝＝＝＝＝＝左左左左K50+200右偏12.5m的坐标为：???'??++'??++5386.50803582.5739341sin5.126484.508026sina5.21YY6482.38959882.5739341cos5.124354.389607cosa5.21XXii＝＝＝＝＝＝右右右右三、缓和曲线上任一点的坐标推算切线角：πβ??1802RLLs2ii＝缓和曲线上任意点i的偏角：πβδ??180RL6L3s2iii＝＝缓和曲线ZH或HZ点到任意点i的方位角为：ijdHZjdZHaaiδ±→→或＝缓和曲线上任意点i的坐标为：???????-s3ii2s25iii6RLLyL40RLLx＝＝缓和曲线ZH或HZ点到任意点i的弦长：22iyxC+＝所以缓和曲线上任意点i的坐标为：???++iiHZZHiiiHZZHisinaCYYcosaCXX或或＝＝例题：已知一段缓和曲线，ZH点到JD方向方位角为A=183°17′08.9″，线路左偏43°31′02″，ZH点里程为K52+001.615，ZH点坐标x＝388071.927，y＝508789.089，R＝960m，Ls＝120m，求K52+100处的中点坐标及左右各偏12.5m的坐标。解：根据已知条件可知：K52+100到zh点的曲线长度Li＝98.385m'???????64.252421801209602385.98180RL2L2s2ii＝＝＝ππβK52+100相对应的偏角：'???????55.84801801209606385.98180RL6L32s2iii＝＝＝＝ππβδZH点到K52+100的方位角：'?'?-'?-→35.02918255.848008.917183aaijdZHi＝＝＝δK52+100左、右偏12.5m的方位角：'??-'??--26.43521809026.435218090Aa＝＝＝左iβ'??+'??+-26.43522709026.435218090Aa＝＝＝左iβK52+100的直角坐标为：???????????--m3778.11209606385.986RLLym3676.9812096040385.98385.98L40RLLx3s3ii2252s25iii＝＝＝＝＝＝缓和曲线ZH点到K52+100的弦长：m377.983778.13676.98yxC2222i＝＝＝++所以K52+100的中线坐标为：???'??++'??++8263.50878435.029182sin377.98089.508789sinaCYY6424.38797335.029182cos377.98927.388071cosaCXXiiZHiiiZHi＝＝＝＝＝＝K52+100左偏12.5m的坐标为：???'??++'??++6346.50878426.4352801sin5.128263.508784sina5.21YY1439.38796126.4352801cos5.126424.387973cosa5.21XXii＝＝＝＝＝＝左左左左K50+200右偏12.5m的坐标为：???'??++'??++3278.50877226.4352072sin5.128263.508784sina5.21YY8341.38797326.4352702cos5.126424.387973cosa5.21XXii＝＝＝＝＝＝右右右右