WPF 画一个3D矩形并旋转

具体的代码还是线性代数。

主要是旋转和平移。

这个例子的中模型是在世界原点建立。所以旋转会以自身轴心旋转。

如果不在世界原点建立模型，还想以自身为旋转轴旋转。

则是需要以下步骤:

模型的中心点为V1（100，100，0）假设中心为轴（平行于Y轴），旋转A度，也就是说自身中心点的Y轴旋转。

步骤：

（1）v1平移到世界原点后其他八个顶点的坐标。（中心点坐标的三个参数如果是大于0就是（每个）顶点减去相对应XYZ，如果中心点坐标的三个参数如果是小于0，则是（每个）顶点加上相对应XYZ，或者使用平移矩阵）

（2）（每个）顶点先是平移到V1在原点时的所在的位置，再使用旋转矩阵经行旋转

（3） （每个）旋转后的顶点在平移回中心点原先所在位置。

ATP 附加属性类

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Windows;
using System.Windows.Controls;
using System.Windows.Data;
using System.Windows.Media;
using System.Windows.Media.Media3D;
using System.Windows.Shapes;

namespace ATP
{
    public class ATP_Y
    {
        public static readonly DependencyProperty P_YProperty = DependencyProperty.RegisterAttached("P_Y", typeof(double), typeof(ATP.ATP_Y), new PropertyMetadata(0.0, new PropertyChangedCallback(OnP_YChanged)));

        private static void OnP_YChanged(DependencyObject d, DependencyPropertyChangedEventArgs e)
        {
            PY = (double)e.NewValue;
            Draw(d, X, Y, Z);
        }

        public static void SetP_Y(DependencyObject d, double v) => d.SetValue(P_YProperty, v);

        public static double GetP_Y(DependencyObject d) => (double)d.GetValue(P_YProperty);

        public static readonly DependencyProperty P_XProperty = DependencyProperty.RegisterAttached("P_X", typeof(double), typeof(ATP.ATP_Y), new PropertyMetadata(0.0, new PropertyChangedCallback(OnP_XChanged)));

        private static void OnP_XChanged(DependencyObject d, DependencyPropertyChangedEventArgs e)
        {
            PX= (double)e.NewValue;
            Draw(d, X, Y, Z);
        }

     

        public static void SetP_X(DependencyObject d, double v) => d.SetValue(P_XProperty, v);

        public static double GetP_X(DependencyObject d) => (double)d.GetValue(P_XProperty);

        public static readonly DependencyProperty P_ZProperty = DependencyProperty.RegisterAttached("P_Z", typeof(double), typeof(ATP.ATP_Y), new PropertyMetadata(0.0, new PropertyChangedCallback(OnP_ZChanged)));

        private static void OnP_ZChanged(DependencyObject d, DependencyPropertyChangedEventArgs e)
        {
            PZ = (double)e.NewValue;
            Draw(d, X, Y, Z);
        }


        public static void SetP_Z(DependencyObject d, double v) => d.SetValue(P_ZProperty, v);

        public static double GetP_Z(DependencyObject d) => (double)d.GetValue(P_ZProperty);

       
        public static readonly DependencyProperty ModeDataProperty = DependencyProperty.RegisterAttached("ModeData", typeof(Point3D), typeof(ATP_Y), new PropertyMetadata(new Point3D(10, 10, 10), new PropertyChangedCallback(OnModeDataChanged)));

        public static void SetModeData(DependencyObject d, Point3D v) => d.SetValue(ModeDataProperty, v);

        public static Point3D GetModeData(DependencyObject d) => (Point3D)d.GetValue(ModeDataProperty);

        private static void OnModeDataChanged(DependencyObject d, DependencyPropertyChangedEventArgs e)
        {
            var data = (Point3D)e.NewValue;
            ModeWidth = data.X;
            ModeHeight = data.Y;
            ModeZWidth = data.Z;
            Draw(d,X,Y,Z);
        }

        public static readonly DependencyProperty YProperty = DependencyProperty.RegisterAttached("Y", typeof(double), typeof(ATP.ATP_Y), new PropertyMetadata(-1.0, new PropertyChangedCallback(OnYChanged)));

        public static void SetY(DependencyObject d, double v) => d.SetValue(YProperty, v);

        public static double GetY(DependencyObject d) => (double)d.GetValue(YProperty);

        public static readonly DependencyProperty XProperty = DependencyProperty.RegisterAttached("X", typeof(double), typeof(ATP.ATP_Y), new PropertyMetadata(-1.0, new PropertyChangedCallback(OnXChanged)));

        private static void OnXChanged(DependencyObject d, DependencyPropertyChangedEventArgs e)
        {
            var deg = Math.PI / 180 * (double)e.NewValue;
            X = deg;
            Draw(d, deg, Y, Z);
        }

        public static void SetX(DependencyObject d, double v) => d.SetValue(XProperty, v);

        public static double GetX(DependencyObject d) => (double)d.GetValue(XProperty);

        public static readonly DependencyProperty ZProperty = DependencyProperty.RegisterAttached("Z", typeof(double), typeof(ATP.ATP_Y), new PropertyMetadata(-1.0, new PropertyChangedCallback(OnZChanged)));

        private static void OnZChanged(DependencyObject d, DependencyPropertyChangedEventArgs e)
        {
            var deg = Math.PI / 180 * (double)e.NewValue;
            Z = deg;
            Draw(d, X, Y, deg);
        }

        public static void SetZ(DependencyObject d, double v) => d.SetValue(ZProperty, v);

        public static double GetZ(DependencyObject d) => (double)d.GetValue(ZProperty);

        private static void OnYChanged(DependencyObject d, DependencyPropertyChangedEventArgs e)
        {
            var deg = Math.PI / 180 * (double)e.NewValue;
            Y = deg;
            Draw(d, X, deg, Z);
        }

        private static double PX, PY, PZ;
        private static double X, Y, Z;
        private static double ModeHeight, ModeWidth, ModeZWidth;
        private static void Draw(DependencyObject d, double X, double Y, double Z)
        {
            var ui = d as Grid;
            ui.Children.Clear();
            var rect = new Rect(new Size(ModeWidth,ModeHeight));
            Group[0] = new Point3D(rect.Width / 2, rect.Height / 2, -(ModeZWidth/2));
            Group[1] = new Point3D(0 - (rect.Width / 2), rect.Height / 2, -(ModeZWidth / 2));
            Group[2] = new Point3D(0 - (rect.Width / 2), 0 - (rect.Height / 2), -(ModeZWidth / 2));
            Group[3] = new Point3D((rect.Width / 2), 0 - (rect.Height / 2), -(ModeZWidth / 2));
            Group[4] = new Point3D(rect.Width / 2, rect.Height / 2, (ModeZWidth / 2));
            Group[5] = new Point3D(0 - (rect.Width / 2), rect.Height / 2, (ModeZWidth / 2));
            Group[6] = new Point3D(0 - (rect.Width / 2), 0 - (rect.Height / 2), (ModeZWidth / 2));
            Group[7] = new Point3D((rect.Width / 2), 0 - (rect.Height / 2), (ModeZWidth / 2));

            for (var i = 0; i < 8; i++)
                PP[i] = PSP(ReturnP3D(Y轴转置矩阵(X, Y, Z) * GetMatrixMP(Group[i])), new Rect(new Size(Math.Max(rect.Height, rect.Width),ModeZWidth)));


            Set(0, 1, ui,Colors.Black);
            Set(1, 2, ui,Colors.Blue);
            Set(2, 3, ui, Colors.Red);
            Set(3, 0, ui, Colors.Fuchsia);
            Set(4, 5, ui, Colors.DarkSlateBlue);
            Set(5, 6, ui, Colors.Red);
            Set(6, 7, ui, Colors.Red);
            Set(7, 4, ui, Colors.Red);
            Set(0, 4, ui, Colors.Red);
            Set(1, 5, ui, Colors.Red);
            Set(2, 6, ui, Colors.Red);
            Set(3, 7, ui, Colors.Red);
        }
        private static void Set(int g1, int g2, Grid g,Color A)
        {
            var c1 = new Line();
            c1.Stroke = new SolidColorBrush(A);
            c1.X1 = PP[g1].X;
            c1.Y1 = PP[g1].Y;
            c1.X2 = PP[g2].X;
            c1.Y2 = PP[g2].Y;
            g.Children.Add(c1);
        }
        private static Matrix GetMatrixMP(Point3D MP)
        {
            var D = new double[4, 1];
            D[0, 0] = MP.X;
            D[1, 0] = MP.Y;
            D[2, 0] = MP.Z;
            D[3, 0] = 1;
            return new Matrix(D);
        }
        private static Point3D ReturnP3D(Matrix MP) => new Point3D(MP[0, 0], MP[1, 0], MP[2, 0]);

        private static Point PSP(Point3D ModePoint, Rect rect)
        {
            Point3D vp = new Point3D(PX, PY, Math.Max(rect.Height, rect.Width)+PZ);
            Point p;
            int x, y;
            x = (int)(vp.X + (ModePoint.X - vp.X) * vp.Z / (vp.Z - ModePoint.Z + 0.5));
            y = (int)(vp.Y + (ModePoint.Y - vp.Y) * vp.Z / (vp.Z - ModePoint.Z + 0.5));
            p = new Point(x, y);
            return p;
        }
        private static Point3D[] Group = new Point3D[8];
        private static Point[] PP = new Point[8];
        private static Matrix Y轴转置矩阵(double DegX, double DegY, double DegZ)
        {
            var A = new double[4, 4];
            A[0, 0] = 1;
            A[0, 1] = 0;
            A[0, 2] = 0;
            A[1, 0] = 0;
            A[1, 1] = Math.Cos(X);
            A[1, 2] = Math.Sin(X);
            A[2, 0] = 0;
            A[2, 1] = 0 - Math.Sin(X);
            A[2, 2] = Math.Cos(X);
            A[0, 3] = 0;
            A[1, 3] = 0;
            A[2, 3] = 0;
            A[3, 0] = 1;
            A[3, 1] = 1;
            A[3, 2] = 1;
            A[3, 3] = 1;
            var B = new double[4, 4];
            B[0, 0] = Math.Cos(Y);
            B[0, 1] = 0;
            B[0, 2] = -Math.Sin(Y);
            B[1, 0] = 0;
            B[1, 1] = 1;
            B[1, 2] = 0;
            B[2, 0] = Math.Sin(Y);
            B[2, 1] = 0;
            B[2, 2] = Math.Cos(Y);
            B[0, 3] = 1;
            B[1, 3] = 1;
            B[2, 3] = 1;
            B[3, 0] = 1;
            B[3, 1] = 1;
            B[3, 2] = 1;
            B[3, 3] = 1;
            var C = new double[4, 4];
            C[0, 0] = Math.Cos(Z);
            C[0, 1] = Math.Sin(Z);
            C[0, 2] = 0;
            C[1, 0] = 0 - Math.Sin(Z);
            C[1, 1] = Math.Cos(Z);
            C[1, 2] = 0;
            C[2, 0] = 0;
            C[2, 1] = 0;
            C[2, 2] = 1;
            C[0, 3] = 1;
            C[1, 3] = 1;
            C[2, 3] = 1;
            B[3, 0] = 1;
            B[3, 1] = 1;
            B[3, 2] = 1;
            C[3, 3] = 1;
            Matrix MT1 = new Matrix(A);
            Matrix MT2 = new Matrix(B);
            Matrix MT3 = new Matrix(C);
            var MT4 = MT1 * MT2;
            return MT4 * MT3;
        }
    }
}

xaml代码

//其中ModeData的三个参数为矩形的长高宽

//X,Y,Z为轴的旋转角度

<Window x:Class="ATP.MainWindow"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
        xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
        xmlns:local="clr-namespace:ATP"
        mc:Ignorable="d"
        Title="MainWindow" Height="450" Width="800">
    <Grid> <Grid.RowDefinitions>
            <RowDefinition Height="*"/>
            <RowDefinition Height="auto"/>
        </Grid.RowDefinitions>
        <Grid x:Name="G"  local:ATP_Y.P_Z="{Binding ElementName=s4,Path=Value}"  local:ATP_Y.ModeData="100,100,100"  HorizontalAlignment="Center" VerticalAlignment="Center" local:ATP_Y.Y="{Binding ElementName=s2,Path=Value}"  local:ATP_Y.X="{Binding ElementName=s1,Path=Value}" local:ATP_Y.Z="{Binding ElementName=s3,Path=Value}"/>
        <StackPanel Grid.Row="1">
            <!--X轴旋转-->
            <Slider Minimum="0"  Maximum="360" x:Name="s1"/>
            <!--Y轴旋转-->
            <Slider Minimum="0"  Maximum="360" x:Name="s2"/>
            <!--Z轴旋转-->
            <Slider Minimum="0"  Maximum="360" x:Name="s3"/>
            <!--视野远近-->
            <Slider Minimum="0"  Maximum="360" x:Name="s4" />
        </StackPanel>
    </Grid>
</Window>

矩阵类

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace ATP
{
   [Serializable]
        public class Matrix
        {
            public double[] element;
            private int rows = 0;
            private int cols = 0;
            /// <summary>
            /// 获取矩阵行数
            /// </summary>
            public int Rows
            {
                get
                {
                    return rows;
                }
            }
            /// <summary>
            /// 获取矩阵列数
            /// </summary>
            public int Cols
            {
                get
                {
                    return cols;
                }
            }
            /// <summary>
            /// 获取或设置第i行第j列的元素值
            /// </summary>
            /// <param name="i">第i行</param>
            /// <param name="j">第j列</param>
            /// <returns>返回第i行第j列的元素值</returns>
            public double this[int i, int j]
            {
                get
                {
                    if (i < Rows && j < Cols)
                    {
                        return element[i * cols + j];
                    }
                    else
                    {
                        throw new Exception("索引越界");
                    }
                }
                set
                {
                    element[i * cols + j] = value;
                }
            }
            /// <summary>
            /// 用二维数组初始化Matrix
            /// </summary>
            /// <param name="m">二维数组</param>
            public Matrix(double[][] m)
            {
                this.rows = m.GetLength(0);
                this.cols = m.GetLength(1);
                int count = 0;
                this.element = new double[Rows * Cols];
                for (int i = 0; i < rows; i++)
                {
                    for (int j = 0; j < cols; j++)
                    {
                        element[count++] = m[i][j];
                    }
                }
            }
            public Matrix(double[,] m)
            {
                this.rows = m.GetLength(0);
                this.cols = m.GetLength(1);
                this.element = new double[this.rows * this.cols];
                int count = 0;
                for (int i = 0; i < rows; i++)
                {
                    for (int j = 0; j < cols; j++)
                    {
                        element[count++] = m[i, j];
                    }
                }
            }
            public Matrix(List<List<double>> m)
            {
                this.rows = m.Count;
                this.cols = m[0].Count;
                this.element = new double[Rows * Cols];
                for (int i = 0; i < rows; i++)
                {
                    for (int j = 0; j < cols; j++)
                    {
                        this[i, j] = m[i][j];
                    }
                }
            }
            #region 矩阵数学运算
            public static Matrix MAbs(Matrix a)
            {
                Matrix _thisCopy = a.DeepCopy();
                for (int i = 0; i < a.Rows; i++)
                {
                    for (int j = 0; j < a.Cols; j++)
                    {
                        _thisCopy[i, j] = Math.Abs(a[i, j]);
                    }
                }
                return _thisCopy;
            }
            /// <summary>
            /// 矩阵相加
            /// </summary>
            /// <param name="a">第一个矩阵,和b矩阵必须同等大小</param>
            /// <param name="b">第二个矩阵</param>
            /// <returns>返回矩阵相加后的结果</returns>
            public static Matrix operator +(Matrix a, Matrix b)
            {
                if (a.cols == b.cols && a.rows == b.rows)
                {
                    double[,] res = new double[a.rows, a.cols];
                    for (int i = 0; i < a.Rows; i++)
                    {
                        for (int j = 0; j < a.Cols; j++)
                        {
                            res[i, j] = a[i, j] + b[i, j];
                        }
                    }
                    return new Matrix(res);
                }
                else
                {
                    throw new Exception("两个矩阵行列不相等");
                }
            }
            /// <summary>
            /// 矩阵相减
            /// </summary>
            /// <param name="a">第一个矩阵,和b矩阵必须同等大小</param>
            /// <param name="b">第二个矩阵</param>
            /// <returns>返回矩阵相减后的结果</returns>
            public static Matrix operator -(Matrix a, Matrix b)
            {
                if (a.cols == b.cols && a.rows == b.rows)
                {
                    double[,] res = new double[a.rows, a.cols];
                    for (int i = 0; i < a.Rows; i++)
                    {
                        for (int j = 0; j < a.Cols; j++)
                        {
                            res[i, j] = a[i, j] - b[i, j];
                        }
                    }
                    return new Matrix(res);
                }
                else
                {
                    throw new Exception("两个矩阵行列不相等");
                }
            }
            /// <summary>
            /// 对矩阵每个元素取相反数
            /// </summary>
            /// <param name="a">二维矩阵</param>
            /// <returns>得到矩阵的相反数</returns>
            public static Matrix operator -(Matrix a)
            {
                Matrix res = a;
                for (int i = 0; i < a.rows; i++)
                {
                    for (int j = 0; j < a.cols; j++)
                    {
                        res.element[i * a.cols + j] = -res.element[i * a.cols + j];
                    }
                }
                return res;
            }
            /// <summary>
            /// 矩阵相乘
            /// </summary>
            /// <param name="a">第一个矩阵</param>
            /// <param name="b">第二个矩阵,这个矩阵的行要与第一个矩阵的列相等</param>
            /// <returns>返回相乘后的一个新的矩阵</returns>
            public static Matrix operator *(Matrix a, Matrix b)
            {
                if (a.cols == b.rows)
                {
                    double[,] res = new double[a.rows, b.cols];
                    for (int i = 0; i < a.rows; i++)
                    {
                        for (int j = 0; j < b.cols; j++)
                        {
                            for (int k = 0; k < a.cols; k++)
                            {
                                res[i, j] += a[i, k] * b[k, j];
                            }
                        }
                    }
                    return new Matrix(res);
                }
                else
                {
                    throw new Exception("两个矩阵行和列不等");
                }
            }
            /// <summary>
            /// 矩阵与数相乘
            /// </summary>
            /// <param name="a">第一个矩阵</param>
            /// <param name="num">一个实数</param>
            /// <returns>返回相乘后的新的矩阵</returns>
            public static Matrix operator *(Matrix a, double num)
            {
                Matrix res = a;
                for (int i = 0; i < a.rows; i++)
                {
                    for (int j = 0; j < a.cols; j++)
                    {
                        res.element[i * a.cols + j] *= num;
                    }
                }
                return res;
            }
            /// <summary>
            /// 矩阵转置
            /// </summary>
            /// <returns>返回当前矩阵转置后的新矩阵</returns>
            public Matrix Transpose()
            {
                double[,] res = new double[cols, rows];
                {
                    for (int i = 0; i < cols; i++)
                    {
                        for (int j = 0; j < rows; j++)
                        {
                            res[i, j] = this[j, i];
                        }
                    }
                }
                return new Matrix(res);
            }
            /// <summary>
            /// 矩阵求逆
            /// </summary>
            /// <returns>返回求逆后的新的矩阵</returns>
            public Matrix Inverse()
            {
                //最后原始矩阵并不变，所以需要深拷贝一份
                Matrix _thisCopy = this.DeepCopy();
                if (cols == rows && this.Determinant() != 0)
                {
                    //初始化一个同等大小的单位阵
                    Matrix res = _thisCopy.EMatrix();
                    for (int i = 0; i < rows; i++)
                    {
                        //首先找到第i列的绝对值最大的数，并将该行和第i行互换
                        int rowMax = i;
                        double max = Math.Abs(_thisCopy[i, i]);
                        for (int j = i; j < rows; j++)
                        {
                            if (Math.Abs(_thisCopy[j, i]) > max)
                            {
                                rowMax = j;
                                max = Math.Abs(_thisCopy[j, i]);
                            }
                        }
                        //将第i行和找到最大数那一行rowMax交换
                        if (rowMax != i)
                        {
                            _thisCopy.Exchange(i, rowMax);
                            res.Exchange(i, rowMax);

                        }
                        //将第i行做初等行变换，将第一个非0元素化为1
                        double r = 1.0 / _thisCopy[i, i];
                        _thisCopy.Exchange(i, -1, r);
                        res.Exchange(i, -1, r);
                        //消元
                        for (int j = 0; j < rows; j++)
                        {
                            //到本行后跳过
                            if (j == i)
                                continue;
                            else
                            {
                                r = -_thisCopy[j, i];
                                _thisCopy.Exchange(i, j, r);
                                res.Exchange(i, j, r);
                            }
                        }
                    }
                    return res;
                }
                else
                {
                    throw new Exception("矩阵不是方阵无法求逆");
                }
            }
            #region 重载比较运算符
            public static bool operator <(Matrix a, Matrix b)
            {
                bool issmall = true;
                for (int i = 0; i < a.Rows; i++)
                {
                    for (int j = 0; j < a.Cols; j++)
                    {
                        if (a[i, j] >= b[i, j]) issmall = false;
                    }
                }
                return issmall;
            }
            public static bool operator >(Matrix a, Matrix b)
            {
                bool issmall = true;
                for (int i = 0; i < a.Rows; i++)
                {
                    for (int j = 0; j < a.Cols; j++)
                    {
                        if (a[i, j] <= b[i, j]) issmall = false;
                    }
                }
                return issmall;
            }
            public static bool operator <=(Matrix a, Matrix b)
            {
                bool issmall = true;
                for (int i = 0; i < a.Rows; i++)
                {
                    for (int j = 0; j < a.Cols; j++)
                    {
                        if (a[i, j] > b[i, j]) issmall = false;
                    }
                }
                return issmall;
            }
            public static bool operator >=(Matrix a, Matrix b)
            {
                bool issmall = true;
                for (int i = 0; i < a.Rows; i++)
                {
                    for (int j = 0; j < a.Cols; j++)
                    {
                        if (a[i, j] < b[i, j]) issmall = false;
                    }
                }
                return issmall;
            }
            public static bool operator !=(Matrix a, Matrix b)
            {
                bool issmall = true;
                issmall = ReferenceEquals(a, b);
                if (issmall) return issmall;
                for (int i = 0; i < a.Rows; i++)
                {
                    for (int j = 0; j < a.Cols; j++)
                    {
                        if (a[i, j] == b[i, j]) issmall = false;
                    }
                }
                return issmall;
            }
            public static bool operator ==(Matrix a, Matrix b)
            {
                bool issmall = true;
                issmall = ReferenceEquals(a, b);
                if (issmall) return issmall;
                for (int i = 0; i < a.Rows; i++)
                {
                    for (int j = 0; j < a.Cols; j++)
                    {
                        if (a[i, j] != b[i, j]) issmall = false;
                    }
                }
                return issmall;
            }
            public override bool Equals(object obj)
            {
                Matrix b = obj as Matrix;
                return this == b;
            }
            public override int GetHashCode()
            {
                return base.GetHashCode();
            }
            #endregion
            public double Determinant()
            {
                if (cols == rows)
                {
                    Matrix _thisCopy = this.DeepCopy();
                    //行列式每次交换行，都需要乘以-1
                    double res = 1;
                    for (int i = 0; i < rows; i++)
                    {
                        //首先找到第i列的绝对值最大的数
                        int rowMax = i;
                        double max = Math.Abs(_thisCopy[i, i]);
                        for (int j = i; j < rows; j++)
                        {
                            if (Math.Abs(_thisCopy[j, i]) > max)
                            {
                                rowMax = j;
                                max = Math.Abs(_thisCopy[j, i]);
                            }
                        }
                        //将第i行和找到最大数那一行rowMax交换,同时将单位阵做相同初等变换
                        if (rowMax != i)
                        {
                            _thisCopy.Exchange(i, rowMax);
                            res *= -1;
                        }
                        //消元
                        for (int j = i + 1; j < rows; j++)
                        {
                            double r = -_thisCopy[j, i] / _thisCopy[i, i];
                            _thisCopy.Exchange(i, j, r);
                        }
                    }
                    //计算对角线乘积
                    for (int i = 0; i < rows; i++)
                    {
                        res *= _thisCopy[i, i];
                    }
                    return res;
                }
                else
                {
                    throw new Exception("不是行列式");
                }
            }
            #endregion
            #region 初等变换
            /// <summary>
            /// 初等变换：交换第r1和第r2行
            /// </summary>
            /// <param name="r1">第r1行</param>
            /// <param name="r2">第r2行</param>
            /// <returns>返回交换两行后的新的矩阵</returns>
            public Matrix Exchange(int r1, int r2)
            {
                if (Math.Min(r2, r1) >= 0 && Math.Max(r1, r2) < rows)
                {
                    for (int j = 0; j < cols; j++)
                    {
                        double temp = this[r1, j];
                        this[r1, j] = this[r2, j];
                        this[r2, j] = temp;
                    }
                    return this;
                }
                else
                {
                    throw new Exception("超出索引");
                }
            }
            /// <summary>
            /// 初等变换：将r1行乘以某个数加到r2行
            /// </summary>
            /// <param name="r1">第r1行乘以num</param>
            /// <param name="r2">加到第r2行，若第r2行为负，则直接将r1乘以num并返回</param>
            /// <param name="num">某行放大的倍数</param>
            /// <returns></returns>
            public Matrix Exchange(int r1, int r2, double num)
            {
                if (Math.Min(r2, r1) >= 0 && Math.Max(r1, r2) < rows)
                {
                    for (int j = 0; j < cols; j++)
                    {
                        this[r2, j] += this[r1, j] * num;
                    }
                    return this;
                }
                else if (r2 < 0)
                {
                    for (int j = 0; j < cols; j++)
                    {
                        this[r1, j] *= num;
                    }
                    return this;
                }
                else
                {
                    throw new Exception("超出索引");
                }
            }
            /// <summary>
            /// 得到一个同等大小的单位矩阵
            /// </summary>
            /// <returns>返回一个同等大小的单位矩阵</returns>
            public Matrix EMatrix()
            {
                if (rows == cols)
                {
                    double[,] res = new double[rows, cols];
                    for (int i = 0; i < rows; i++)
                    {
                        for (int j = 0; j < cols; j++)
                        {
                            if (i == j)
                                res[i, j] = 1;
                            else
                                res[i, j] = 0;
                        }
                    }
                    return new Matrix(res);
                }
                else
                    throw new Exception("不是方阵，无法得到单位矩阵");
            }
            #endregion
            /// <summary>
            /// 深拷贝，仅仅将值拷贝给一个新的对象
            /// </summary>
            /// <returns>返回深拷贝后的新对象</returns>
            public Matrix DeepCopy()
            {
                double[,] ele = new double[rows, cols];
                for (int i = 0; i < rows; i++)
                {
                    for (int j = 0; j < cols; j++)
                    {
                        ele[i, j] = this[i, j];
                    }
                }
                return new Matrix(ele);
            }

            public override string ToString()
            {
                string str = "";
                for (int i = 0; i < Rows; i++)
                {
                    for (int j = 0; j < Cols; j++)
                    {
                        str += this[i, j].ToString();
                        if (j != Cols - 1)
                            str += " ";
                        else if (i != Rows - 1)
                            str += Environment.NewLine;
                    }
                }
                return str;
            }
        }
}

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