高斯正反算—投影坐标转大地坐标、大地坐标转投影坐标(附有完整代码及测试结果)

一、常用椭球参数

二、高斯正反算原理 

1、高斯正算(大地坐标转投影坐标) 

 2、高斯反算(投影坐标转大地坐标)

三、高斯正反算代码实现 

#include<iostream>
#include<cmath>
#include "stdio.h"

#define pi 3.141592653589793238463
#define p0 206264.8062470963551564

//wgs84参考椭球
const double e = 0.00669438002290;
const double e1 = 0.00673949677548;
const double b = 6356752.3141;
const double a = 6378137.0;

using namespace std;


//大地坐标转投影坐标
void DadiPoint2ProjectPoint(double B, double L)
{
    //把度转化为弧度
    B = B * pi / 180;
    L = L * pi / 180;

    double N, t, n, c, V, Xz, m1, m2, m3, m4, m5, m6, a0, a2, a4, a6, a8, M0, M2, M4, M6, M8, x0, y0, l;

    int L_num;
    double L_center;

    //中央子午线经度，6°带
    L_num = (int)(L * 180 / pi / 6.0) + 1;
    L_center = 6 * L_num - 3;

    //中央子午线经度，3°带
    //L_num = (int)(L * 180 / pi / 3.0 + 0.5);
    //L_center = 3 * L_num;				  

    l = (L / pi * 180 - L_center) * 3600; //求带号、中央经线、经差

    M0 = a * (1 - e);
    M2 = 3.0 / 2.0 * e * M0;
    M4 = 5.0 / 4.0 * e * M2;
    M6 = 7.0 / 6.0 * e * M4;
    M8 = 9.0 / 8.0 * e * M6;

    a0 = M0 + M2 / 2.0 + 3.0 / 8.0 * M4 + 5.0 / 16.0 * M6 + 35.0 / 128.0 * M8;
    a2 = M2 / 2.0 + M4 / 2 + 15.0 / 32.0 * M6 + 7.0 / 16.0 * M8;
    a4 = M4 / 8.0 + 3.0 / 16.0 * M6 + 7.0 / 32.0 * M8;
    a6 = M6 / 32.0 + M8 / 16.0;
    a8 = M8 / 128.0;

    Xz = a0 * B - a2 / 2.0 * sin(2 * B) + a4 / 4.0 * sin(4 * B) - a6 / 6.0 * sin(6 * B) + a8 / 8.0 * sin(8 * B);  //计算子午线弧长
    c = a * a / b;
    V = sqrt(1 + e1 * cos(B) * cos(B));
    N = c / V;
    t = tan(B);
    n = e1 * cos(B) * cos(B);

    m1 = N * cos(B);
    m2 = N / 2.0 * sin(B) * cos(B);
    m3 = N / 6.0 * pow(cos(B), 3) * (1 - t * t + n);
    m4 = N / 24.0 * sin(B) * pow(cos(B), 3) * (5 - t * t + 9 * n);
    m5 = N / 120.0 * pow(cos(B), 5) * (5 - 18 * t * t + pow(t, 4) + 14 * n - 58 * n * t * t);
    m6 = N / 720.0 * sin(B) * pow(cos(B), 5) * (61 - 58 * t * t + pow(t, 4));
    x0 = Xz + m2 * l * l / pow(p0, 2) + m4 * pow(l, 4) / pow(p0, 4) + m6 * pow(l, 6) / pow(p0, 6);
    y0 = m1 * l / p0 + m3 * pow(l, 3) / pow(p0, 3) + m5 * pow(l, 5) / pow(p0, 5);   //计算x y坐标

    double x = x0;
    //double y = y0 + 500000 + 1000000 * L_num;    //化为国家统一坐标
    double y = y0 + 500000;     //化为国家统一坐标

    cout << "方法一 x=" << x << endl;
    cout << "方法一 y=" << y << endl;
}


//投影坐标转大地坐标
void ProjectPoint2DadiPoint(double x, double y, double l0)
{
    //l0为中央经度
    double Bf, B0, FBf, M, N, V, t, n, c, y1, n1, n2, n3, n4, n5, n6, a0, a2, a4, a6, M0, M2, M4, M6, M8, l;

    int L_num, L_center;

    L_num = (int)(x / 1000000.0);
    y1 = y - 500000;
    //y1 = y - 500000 - L_num * 1000000;

    //L_center = ((L_num + 1) * 6 - 3)*pi*180;		//中央子午线经度，6°带
    //cout<<"L_center="<<L_center<<endl;
    //L_center = L_num * 3;			//中央子午线经度，3°带

    M0 = a * (1 - e);
    M2 = 3.0 / 2.0 * e * M0;
    M4 = 5.0 / 4.0 * e * M2;
    M6 = 7.0 / 6.0 * e * M4;
    M8 = 9.0 / 8.0 * e * M6;

    a0 = M0 + M2 / 2.0 + 3.0 / 8.0 * M4 + 5.0 / 16.0 * M6 + 35.0 / 128.0 * M8;
    a2 = M2 / 2.0 + M4 / 2 + 15.0 / 32.0 * M6 + 7.0 / 16.0 * M8;
    a4 = M4 / 8.0 + 3.0 / 16.0 * M6 + 7.0 / 32.0 * M8;
    a6 = M6 / 32.0 + M8 / 16.0;

    cout << "a0=" << a0 << endl;
    cout << "a2=" << a2 << endl;
    cout << "a4=" << a4 << endl;
    cout << "a6=" << a6 << endl;

    Bf = x / a0;
    B0 = Bf;
    cout<<"B0="<<B0<<endl;
    
     cout<<"sin(2 * B0)="<<sin(2 * B0)/2<<endl;

    while ((fabs(Bf - B0) > 0.0000001) || (B0 == Bf))
    {
        B0 = Bf;
        FBf = -a2 / 2.0 * sin(2 * B0) + a4 / 4.0 * sin(4 * B0) - a6 / 6.0 * sin(6 * B0);
        Bf = (x - FBf) / a0;
    }    //迭代求数值为x坐标的子午线弧长对应的底点纬度

    cout<<"Bf="<<Bf<<endl;

    t = tan(Bf);                            //一样
    c = a * a / b;
    V = sqrt(1 + e1 * cos(Bf) * cos(Bf));   //一样
    N = c / V;                              //一样
    M = c / pow(V, 3);                      //一样
    n = e1 * cos(Bf) * cos(Bf);             //一样(为n的平方)

    n1 = 1 / (N * cos(Bf));
    n2 = -t / (2.0 * M * N);
    n3 = -(1 + 2 * t * t + n) / (6.0 * pow(N, 3) * cos(Bf));
    n4 = t * (5 + 3 * t * t + n - 9 * n * t * t) / (24.0 * M * pow(N, 3));
    n5 = (5 + 28 * t * t + 24 * pow(t, 4) + 6 * n + 8 * n * t * t) / (120.0 * pow(N, 5) * cos(Bf));
    n6 = -t * (61 + 90 * t * t + 45 * pow(t, 4)) / (720.0 * M * pow(N, 5));

    //秒
    double B = (Bf + n2 * y1 * y1 + n4 * pow(y1, 4) + n6 * pow(y1, 6)) / pi * 180;
    
    double L0=l0;

    l = n1 * y1 + n3 * pow(y1, 3) + n5 * pow(y1, 5);
    //double L = L_center + l / pi * 180;    //反算得大地经纬度
    double L = L0 + l / pi * 180;    //反算得大地经纬度

    cout << "方法一 B=" << B << endl;
    cout << "方法一 L=" << L << endl;
}

说明：高斯正算中的输入为度； 

 四、高斯正反算结果

1、已知旧坐标系投影坐标数据

 2、高斯反算(投影坐标转大地坐标)

 3、高斯正算(大地坐标转投影坐标)

 说明：高斯正反算经度由上述转换结果对比可知；

常见坐标系：北京54、西安80、WGS84、CGCS2000等坐标系的高斯转换都做出了实现并使用 QT 进行封装可视化：

坐标转换整套流程包括：像素坐标转投影坐标、投影坐标转大地坐标、大地坐标转空间直角坐标、七参数转换、空间直角坐标转大地坐标、大地坐标转投影坐标、投影坐标转像素坐标； 本人均已实现，且每一个环节都已经过测试、如有需要欢迎在下方留言评论！！！

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