不同平面直角坐标系之间的转换公式的推导及C#代码实现

    本篇讨论的主题是在平面坐标系中如何将一个坐标系（目标坐标系，以下简称目标系）中的所有点投射到另一个坐标系（基坐标系）中。平面坐标系之间的转化一般有三步操作：1、平移；2、旋转；3、拉伸。

    在转化的过程中需要的几个已知条件分别是：1、目标系的一个已知点（特征点A）对应于基坐标系中的点（特征点A’）。2、目标系的原点（O）对应于基坐标系中的原点（O'）。3、基座标系的原点（O‘’）。

一、坐标系拉伸

①、计算两坐标系X和Y轴分别对应的拉伸比例：

②、将A点按缩放比例映射到基坐标系中（A''）：

二、坐标系旋转和平移

①：坐标系旋转

②坐标系平移加旋转

③得出公式

根据坐标系旋转和平移的规则，可以得出公式

我们的目标是求出θ。

设m = cos(θ),n=sin(θ)，根据已知量可得出公式：

得出方程组的增广矩阵，根据高斯消除元法即可求得m,n：

再根据反三角函数求得θ，此时θ、a、b、T(坐标系倍数)的值都已经得到。

故任意目标系坐标（x、y）可通过公式映射到基坐标系(f(x)、f(y))：

C#代码：

public class CoorPoint    {        public double x { get; set; }        public double y { get; set; }        public CoorPoint()        { }        public CoorPoint(double x, double y)        {            this.x = x;            this.y = y;        }        public CoorPoint(CoorPoint pt)        {            this.x = pt.x;            this.y = pt.y;        }        public static CoorPoint operator +(CoorPoint src, CoorPoint tar)        {            CoorPoint result = new CoorPoint(src);            result.x += tar.x;            result.y += tar.y;            return result;        }        public static CoorPoint operator -(CoorPoint src, CoorPoint tar)        {            CoorPoint result = new CoorPoint(src);            result.x -= tar.x;            result.y -= tar.y;            return result;        }        public static CoorPoint operator /(CoorPoint src, CoorPoint tar)        {            CoorPoint result = new CoorPoint(src);            result.x /= tar.x;            result.y /= tar.y;            return result;        }        public static CoorPoint operator *(CoorPoint src, CoorPoint tar)        {            CoorPoint result = new CoorPoint(src);            result.x *= tar.x;            result.y *= tar.y;            return result;        }    }    class CoordinateSystemChg    {        /// <summary>        /// 基坐标系:特征点(A')        /// </summary>        public CoorPoint baseCoor_basePoint { get; set; }        /// <summary>        /// 基坐标系:原点(O)        /// </summary>        public CoorPoint baseCoor_originPoint { get; set; }        /// <summary>        /// 现实坐标系:特征点(A)        /// </summary>        public CoorPoint realistic_basePoint { get; set; }        /// <summary>        /// 现实坐标系对应基坐标系拉伸后的点(A'')        /// </summary>        public CoorPoint realistic_basePointAfterZoom { get; set; }        /// <summary>        /// 现实坐标系:原点(O)        /// </summary>        public CoorPoint realistic_originPoint { get; set; }        /// <summary>        /// 现实坐标系对应基坐标系:原点(O')        /// </summary>        public CoorPoint realistic_originAsBaseCoorPoint { get; set; }   //定义原点时 已经统一缩放比例        //第一步的结果        //位移量：a(x轴),b(y轴)        double a = 0;        double b = 0;        //坐标系倍数        CoorPoint coorTimes;        //坐标系夹角        double offsetAngle;        //第一步:得到坐标系之间的角度θ        /// <summary>        /// 计算出两坐标系的X,Y比例        /// </summary>        public void GetSale()        {            coorTimes = (realistic_basePoint - realistic_originPoint) / (baseCoor_basePoint - realistic_originAsBaseCoorPoint);            if (coorTimes.x < 0)                coorTimes.x = -coorTimes.x;            if (coorTimes.y < 0)                coorTimes.y = -coorTimes.y;        }        /// <summary>        /// ①按比例缩放（实际坐标系的基点），使之与基坐标系统一        /// </summary>        public void DoZoom()        {            realistic_basePointAfterZoom = realistic_basePoint / coorTimes;        }        /// <summary>        /// ②得到偏移角，入参实际坐标系原点坐标在基坐标系上的坐标        /// </summary>        public void GetOffsetAngle()        {            double[] x = new double[2];            a = (realistic_originAsBaseCoorPoint.x - baseCoor_originPoint.x);            b = (realistic_originAsBaseCoorPoint.y - baseCoor_originPoint.y);            double[,] g = {                { realistic_basePointAfterZoom.x, realistic_basePointAfterZoom.y, (baseCoor_basePoint.x - a) },                { realistic_basePointAfterZoom.y, -realistic_basePointAfterZoom.x, (baseCoor_basePoint.y - b) }            };            GaussianElimination(g, x);            // x[0]:θ角的余弦值；x[1]:θ角的正弦值            offsetAngle = Math.Acos(x[0]);            Console.WriteLine(x[0] + ":" + Math.Acos(x[0]));            Console.WriteLine(x[1] + ":" + Math.Asin(x[1]));        }        //第二步：根据θ角求出在基坐标系中点的映射        /// <summary>        /// 将实际坐标系中的点转换到基坐标系中的点        /// </summary>        /// <param name="src"></param>        /// <param name="rlt"></param>        public void GetCoordinateBaseSystem(CoorPoint src, out CoorPoint rlt)        {            double db1 = Math.Cos(offsetAngle);            double db2 = Math.Sin(offsetAngle);            rlt = new CoorPoint();            rlt.x = ((src.x / coorTimes.x) * Math.Cos(offsetAngle) + (src.y / coorTimes.y) * Math.Sin(offsetAngle) + a);            rlt.y = ((src.y / coorTimes.y) * Math.Cos(offsetAngle) - (src.x / coorTimes.x) * Math.Sin(offsetAngle) + b);        }        /// <summary>        /// 将基坐标系中的点转换到实际坐标系中的点        /// </summary>        /// <param name="src"></param>        /// <param name="rlt"></param>        public void GetCoordinateRealisticSystem(CoorPoint src, out CoorPoint rlt)        {            double[] x = new double[2];            double[,] g = {                {  Math.Cos(offsetAngle), Math.Sin(offsetAngle),(src.x - a)*coorTimes.x },                { -Math.Sin(offsetAngle), Math.Cos(offsetAngle),(src.y - b)*coorTimes.y }            };            GaussianElimination(g, x);            rlt = new CoorPoint();            rlt.x = x[0];            rlt.y = x[1];        }        #region 数学方法        //简单的高斯消元法；        //输入要求解的扩展矩阵g[m,n]；和存放结果的数组x[n];        //返回值为计算结果数组x[n];        public static double[] GaussianElimination(double[,] g, double[] x)        {            int m = g.GetLength(0);//获得扩展矩阵的行（方程个数）；            int n = g.GetLength(1);//获得扩展矩阵的列（未知数个数+1）；                                   //========================================================                                   //消元过程；            for (int i = 1; i < m; i++)            {                for (int j = i; j < m; j++)                {                    for (int k = n - 1; k > i - 2; k--)                    {                        g[j, k] = g[j, k] - (g[j, i - 1] / g[i - 1, i - 1]) * (g[i - 1, k]);                    }                }            }            //回代过程；            //第一步：翻转；            //换行；            double tem;            for (int i = 0; i < m / 2; i++)                for (int j = 0; j < n; j++)                {                    tem = g[i, j];                    g[i, j] = g[m - i - 1, j];                    g[m - i - 1, j] = tem;                }            //倒序            for (int i = 0; i < m; i++)                for (int j = 0; j < n / 2; j++)                {                    tem = g[i, j];                    g[i, j] = g[i, n - 2 - j];                    g[i, n - 2 - j] = tem;                }            //第二步：消元；            for (int i = 1; i < m; i++)            {                for (int j = i; j < m; j++)                {                    for (int k = n - 1; k > i - 2; k--)                    {                        g[j, k] = g[j, k] - (g[j, i - 1] / g[i - 1, i - 1]) * (g[i - 1, k]);                    }                }            }            //第三步：翻回；            //重新换行；            for (int i = 0; i < m / 2; i++)                for (int j = 0; j < n; j++)                {                    tem = g[i, j];                    g[i, j] = g[m - i - 1, j];                    g[m - i - 1, j] = tem;                }            //重新倒序；            for (int i = 0; i < m; i++)                for (int j = 0; j < n / 2; j++)                {                    tem = g[i, j];                    g[i, j] = g[i, n - 2 - j];                    g[i, n - 2 - j] = tem;                }            //取结果（这里是正序结果哦）；            for (int i = 0; i < m; i++)                x[i] = g[i, n - 1] / g[i, i];            return x;//返回计算结果；        }        #endregion    }