高斯投影坐标正反算

一、基本思想：

高斯投影正算公式就是由大地坐标（L，B）求解高斯平面坐标（x，y），而高斯投影反算公式则是由高斯平面坐标（x，y）求解大地坐标（L，B）。

二、计算模型：

基本椭球参数： 椭球长半轴a 椭球扁率f

椭球短半轴：b?a(1?f)

a2?b2椭球第一偏心率 ：e?

aa2?b2椭球第二偏心率 ：e?? b高斯投影正算公式：此公式换算的精度为0.001m

x?X??NN2??sinBcosB?l?simBcos3B(5?t2?9?2?4?4)l??4242???24???

N5246??sinBcosB(61?58t?t)l720???6y?NN3223??cosB?l???cosB(1?t??)l???6???3N5242225???cosB(5?18t?t?14??58?t)l120???5

其中：角度都为弧度

B为点的纬度，l???L?L0，L为点的经度，L0为中央子午线经度；

N为子午圈曲率半径，N?a(1?esinB)； t?tanB；

22?12?2?e?2cos2B

????180??3600

其中X为子午线弧长：

1616??X?a0B?sinBcosB?(a2?a4?a6)?(2a4?a6)sin2B?a6sin4B?

33??a0,a2,a4,a6,a8为基本常量，按如下公式计算： m23535?a?m??m?m?m8046?02816128??a?m2?m4?15m?7m68?2223216?m437? ?m6?m8?a4?81632?m6m8?a??632?16??a?m88?128?m0,m2,m4,m6,m8为基本常量，按如下公式计算：

m0?a(1?e2);m2?3279em0;m4?5e2m2;m6?e2m4;m8?e2m6； 268

高斯投影反算公式：此公式换算的精度为0.0001’’.

B?Bf??tftf2MfNfy?2tf24MfN3f?5?3t2f224??2f?9?ftf?y720MfN5f46y?61?90t2f?45tf?yyy322l??1?2t???ff?NfcosBf6N3cosBffy5422?5?28t2?2?f?24tf?6?f?8ftf?5120NfcosBfL?l?L0

其中：

L0为中央子午线经度。

Bf为底点纬度，也就是当x?X时的子午线弧长所对应的纬度。按照子午线弧长公式：

X?a0B?aaa2asin2B?4sin4B?6sin6B?8sin8B，迭代进行计算； 24681初始开始时设：Bf?Xa0 以后每次迭代按下式计算：

Bif?1?X(X?F(Bif))a0 a6a8a2a4iiiiF(B)??sin2Bf?sin4Bf?sin6Bf?sin8Bf2468if重复迭代至Bif?1?Bif??为止。

Nf?a(1?esinBf)； Mf?a(1?e)(1?esinBf)

222?3222?12tf?tanBf；

22??2?ecosBf f

海福特椭球(1910) 我国52年以前基准椭球 a=6378388m b=6356911.9461279m α=0.33670033670 球 球

克拉索夫斯基椭球(1940 Krassovsky) 北京54坐标系基准椭球 a=6378245m b=6356863.018773m α=0.33523298692

1975年I.U.G.G推荐椭球(国际大地测量协会1975) 西安80坐标系基准椭a=6378140m b=6356755.2881575m α=0.0033528131778

WGS-84椭球(GPS全球定位系统椭球、17届国际大地测量协会) WGS-84 GPS 基准椭a=6378137m b=6356752.3142451m α=0.00335281006247

三、程序代码函数：

/************高斯投影正算函数***************

输入 : double a ,f 椭球参数，B,L为大地坐标,L0为中央子午线的经度，单位为弧度，x,y为高斯平面坐标,y加上了500000常量 返回：none

******************************************/

void gaosiforward(double a,double f,double B,double L,double L0,double &x,double &y) {

double b, c,e1, e2; //短半轴，极点处的子午线曲率半径，第一偏心率，第二偏心率 double l, W,N, M, daihao;//W为常用辅助函数，N为子午圈曲率半径，M为卯酉圈曲率半径 double X;//子午线弧长，高斯投影的坐标 double ruo, ita, sb, cb,t; double m[5],n[5]; //计算一些基本常量 {

b=a*(1-f);

e1=sqrt(a*a-b*b)/a; e2=sqrt(a*a-b*b)/b; c=a*a/b;

m[0]=a*(1-e1*e1); m[1]=3*(e1*e1*m[0])/2.0;

}

}

m[2]=5*(e1*e1*m[1])/4.0; m[3]=7*(e1*e1*m[2])/6.0; m[4]=9*(e1*e1*m[3])/8.0;

n[0]=m[0]+m[1]/2+3*m[2]/8+5*m[3]/16+35*m[4]/128; n[1]=m[1]/2+m[2]/2+15*m[3]/32+7*m[4]/16; n[2]=m[2]/8+3*m[3]/16+7*m[4]/32; n[3]=m[3]/32+m[4]/16;

n[4]=m[4]/128; /////by kjh 2014.5.22 把改成了

//由纬度计算子午线弧长 {

X=n[0]*B-sin(B)*cos(B)*((n[1]-n[2]+n[3])+(2*n[2]-(16*n[3]/3.0))*sin(B)*sin(B)+16*n[3]*p}

l=L-L0;//弧度 ita=e2*cos(B); sb=sin(B); cb=cos(B);

W=sqrt(1-e1*e1*sb*sb); N=a/W; t=tan(B);

ruo=(180/Pi)*3600;

x=(X+N*sb*cb*l*l/2+N*sb*cb*cb*cb*(5-t*t+9*ita*ita+4*ita*ita*ita*ita)*l*l*l*l/24+N*sb*cby=(N*cb*l+N*cb*cb*cb*(1-t*t+ita*ita)*l*l*l/6+N*cb*cb*cb*cb*cb*(5-18*t*t+t*t*t*t+14*ita*y=y+500000;

ow(sin(B),4)/3.0);

*cb*cb*cb*cb*(61-58*t*t+t*t*t*t)*l*l*l*l*l*l/720); ita-58*ita*ita*t*t)*l*l*l*l*l/120);

/**************高斯反算函数***************

输入 : double a ,f 椭球参数， x,y为高斯平面坐标，L0为中央子午线的经度； B,L为大地坐标，单位为弧度 *返回：none

*****************************/

void gaosibackward(double a,double f,double x,double y,double L0,double &B,double &L) {

double b, c,e1, e2; //短半轴，极点处的子午线曲率半径，第一偏心率，第二偏心率 double Bf,itaf,tf,Nf,Mf,Wf; double l; double m[5],n[5]; y=y-500000; //计算一些基本常量 {

b=a*(1-f);

}

e1=sqrt(a*a-b*b)/a; e2=sqrt(a*a-b*b)/b; c=a*a/b;

m[0]=a*(1-e1*e1); m[1]=3*(e1*e1*m[0])/2.0; m[2]=5*(e1*e1*m[1])/4.0; m[3]=7*(e1*e1*m[2])/6.0; m[4]=9*(e1*e1*m[3])/8.0;

n[0]=m[0]+m[1]/2+3*m[2]/8+5*m[3]/16+35*m[4]/128; n[1]=m[1]/2+m[2]/2+15*m[3]/32+7*m[4]/16; n[2]=m[2]/8+3*m[3]/16+7*m[4]/32; n[3]=m[3]/32+m[4]/16; n[4]=m[4]/128;

//计算Bf {

Bf1=x/n[0]; Bfi0=Bf1; Bfi1=0; FBfi=0; int num=0; do {

num=0;

FBfi=0.0-n[1]*sin(2*Bfi0)/2.0+n[2]*sin(4*Bfi0)/4.0-n[3]*sin(6*Bfi0)/6.0; Bfi1=(x-FBfi)/n[0];

if (fabs(Bfi1-Bfi0)>(Pi*pow(10.0,-8)/(36*18))) { }

num=1; Bfi0=Bfi1;

double Bf1,Bfi0,Bfi1,FBfi;

} while (num==1); Bf=Bfi1; }

tf=tan(Bf);

Wf=sqrt(1-e1*e1*sin(Bf)*sin(Bf)); Nf=a/Wf;

Mf=a*(1-e1*e1)/(Wf*Wf*Wf); itaf=e2*cos(Bf);

B=Bf-tf*y*y/(2*Mf*Nf)+tf*(5+3*tf*tf+itaf*itaf-9*itaf*itaf*tf*tf)*pow(y,4)/(24*Mf*pow(Nfl=y/(Nf*cos(Bf))-(1+2*tf*tf+itaf*itaf)*pow(y,3)/(6*pow(Nf,3)*cos(Bf))+(5+28*tf*tf+24*po

,3))-tf*(61+90*tf*tf+45*pow(tf,4))*pow(y,6)/(720*Mf*pow(Nf,5));

w(tf,4)+6*itaf*itaf+8*itaf*itaf*tf*tf)*pow(y,5)/(120*pow(Nf,5)*cos(Bf));

}

L=l+L0;

2014-5-22

' 输入: double a ,f 椭球参数，B,L为大地坐标,L0为中央子午线的经度，单位为弧度，x,y为高斯平面坐标,y加上了常量

Private Function gaosiforward(ByVal a As Double, ByVal f As Double, ByVal B As Double, ByVal L As Double, ByVal L0 As Double) As Double() Dim x, y, xy(2) As Double

Dim bb, c, e1, e2 As Double '短半轴，极点处的子午线曲率半径，第一偏心率，第二偏心率 Dim ll, W, N, M, daihao As Double 'W为常用辅助函数，N为子午圈曲率半径，M为卯酉圈曲率半径

Dim xx As Double '子午线弧长，高斯投影的坐标 Dim ruo, ita, sb, cb, t As Double Dim mm(5), nn(5) As Double bb = a * (1 - f)

e1 = Math.Sqrt(a * a - bb * bb) / a e2 = Math.Sqrt(a * a - bb * bb) / bb c = a * a / bb

mm(0) = a * (1 - e1 * e1)

mm(1) = 3 * (e1 * e1 * mm(0)) / 2.0 mm(2) = 5 * (e1 * e1 * mm(1)) / 4.0 mm(3) = 7 * (e1 * e1 * mm(2)) / 6.0 mm(4) = 9 * (e1 * e1 * mm(3)) / 8.0

nn(0) = mm(0) + mm(1) / 2 + 3 * mm(2) / 8 + 5 * mm(3) / 16 + 35 * mm(4) / 128 nn(1) = mm(1) / 2 + mm(2) / 2 + 15 * mm(3) / 32 + 7 * mm(4) / 16 nn(2) = mm(2) / 8 + 3 * mm(3) / 16 + 7 * mm(4) / 32 nn(3) = mm(3) / 32 + mm(4) / 16 nn(4) = mm(4) / 128

xx = nn(0) * B - Sin(B) * Cos(B) * ((nn(1) - nn(2) + nn(3)) + (2 * nn(2) - (16 * nn(3) / 3.0)) * Sin(B) * Sin(B) + 16 * nn(3) * Pow(Sin(B), 4) / 3.0) ll = L - L0 '弧度 ita = e2 * Cos(B) sb = Sin(B) cb = Cos(B)

W = Sqrt(1 - e1 * e1 * sb * sb) N = a / W t = Tan(B)

ruo = (180 / PI) * 3600

x = (xx + N * sb * cb * ll * ll / 2 + N * sb * cb * cb * cb * (5 - t * t + 9 * ita * ita + 4 * ita * ita * ita * ita) * ll * ll * ll * ll / 24 + N * sb * cb * cb * cb * cb * cb * (61 - 58 * t * t + t * t * t * t) * ll * ll * ll * ll * ll * ll / 720)

y = (N * cb * ll + N * cb * cb * cb * (1 - t * t + ita * ita) * ll * ll * ll / 6 + N *

cb * cb * cb * cb * cb * (5 - 18 * t * t + t * t * t * t + 14 * ita * ita - 58 * ita * ita * t * t) * ll * ll * ll * ll * ll / 120) y = y + 500000 xy(0) = x xy(1) = y Return xy End Function

Private Function gaosibackward(ByVal a As Double, ByVal f As Double, ByVal x As Double, ByVal y As Double, ByVal L0 As Double) As Double() Dim b, l, bl(2) As Double

Dim bb, c, e1, e2 As Double '短半轴，极点处的子午线曲率半径，第一偏心率，第二偏心率 Dim Bf, itaf, tf, Nf, Mf, Wf As Double Dim ll As Double

Dim m(5), n(5) As Double y = y - 500000

bb = a * (1 - f)

e1 = Sqrt(a * a - bb * bb) / a e2 = Sqrt(a * a - bb * bb) / bb c = a * a / bb

m(0) = a * (1 - e1 * e1)

m(1) = 3 * (e1 * e1 * m(0)) / 2.0 m(2) = 5 * (e1 * e1 * m(1)) / 4.0 m(3) = 7 * (e1 * e1 * m(2)) / 6.0 m(4) = 9 * (e1 * e1 * m(3)) / 8.0

n(0) = m(0) + m(1) / 2 + 3 * m(2) / 8 + 5 * m(3) / 16 + 35 * m(4) / 128 n(1) = m(1) / 2 + m(2) / 2 + 15 * m(3) / 32 + 7 * m(4) / 16 n(2) = m(2) / 8 + 3 * m(3) / 16 + 7 * m(4) / 32 n(3) = m(3) / 32 + m(4) / 16 n(4) = m(4) / 128 '计算BF

Dim Bf1, Bfi0, Bfi1, FBfi As Double Bf1 = x / n(0) Bfi0 = Bf1 Bfi1 = 0 FBfi = 0

Dim num As Integer = 0 Do

num = 0

FBfi = 0.0 - n(1) * Sin(2 * Bfi0) / 2.0 + n(2) * Sin(4 * Bfi0) / 4.0 - n(3) * Sin(6 * Bfi0) / 6.0

Bfi1 = (x - FBfi) / n(0)

If (Abs(Bfi1 - Bfi0) > (PI * Pow(10.0, -8) / (36 * 18))) Then num = 1

Bfi0 = Bfi1 End If

Loop While num = 1 Bf = Bfi1 tf = Tan(Bf)

Wf = Sqrt(1 - e1 * e1 * Sin(Bf) * Sin(Bf)) Nf = a / Wf

Mf = a * (1 - e1 * e1) / (Wf * Wf * Wf) itaf = e2 * Cos(Bf)

b = Bf - tf * y * y / (2 * Mf * Nf) + tf * (5 + 3 * tf * tf + itaf * itaf - 9 * itaf * itaf * tf * tf) * Pow(y, 4) / (24 * Mf * Pow(Nf, 3)) - tf * (61 + 90 * tf * tf + 45 * Pow(tf, 4)) * Pow(y, 6) / (720 * Mf * Pow(Nf, 5))

ll = y / (Nf * Cos(Bf)) - (1 + 2 * tf * tf + itaf * itaf) * Pow(y, 3) / (6 * Pow(Nf, 3) * Cos(Bf)) + (5 + 28 * tf * tf + 24 * Pow(tf, 4) + 6 * itaf * itaf + 8 * itaf * itaf * tf * tf) * Pow(y, 5) / (120 * Pow(Nf, 5) * Cos(Bf)) l = ll + L0 bl(0) = hdtod(b) bl(1) = hdtod(l) Return bl End Function '度转换为弧度

Private Function dtohd(ByVal d As Double) As Double Dim hd As Double hd = d * PI / 180 Return hd End Function '弧度转换为度

Private Function hdtod(ByVal hd As Double) As Double Dim d As Double d = hd * 180 / PI Return d End Function

Private Function dfmtod(ByVal dfm As Double) As Double Dim dfm2, dfm3(), du, fen, miao, miao1, miao2 As String Dim duf As Double dfm2 = dfm.ToString dfm3 = dfm2.Split(\) du = dfm3(0)

fen = dfm3(1).Substring(0, 2) miao1 = dfm3(1).Substring(2, 2) miao2 = dfm3(1).Substring(4) miao = miao1 & \ & miao2

duf = Convert.ToInt32(du) + Convert.ToInt32(fen) / 60.0 + Convert.ToDouble(miao) / 3600 Return duf

End Function

'国家坐标系椭球参数2000,84,54,80 a2 = 6378137

f2 = 1 / 298.257222101 a84 = a2

f84 = 1 / 298.257223563 a54 = 6378245 f54 = 0.33523298692 a80 = 6378140

f80 = 0.0033528131778