线性代数：Matrix2x2和Matrix3x3

      今天整理自己的框架代码，将Matrix2x2和Matrix3x3给扩展了一下，发现网上unity数学计算相关挺少的，所以记录一下。
      首先扩展Matrix2x2：

using System.Collections;
using System.Collections.Generic;
using Unity.Mathematics;
using UnityEngine;public class Matrix2x2
{#region ///propertiespublic const int ROW = 2;public const int COLUMN = 2;public float M00{get { return dataArr[0, 0]; }set { dataArr[0, 0] = value; }}public float M01{get { return dataArr[0, 1]; }set { dataArr[0, 1] = value; }}public float M10{get { return dataArr[1, 0]; }set { dataArr[1, 0] = value; }}public float M11{get { return dataArr[1, 1]; }set { dataArr[1, 1] = value; }}private float[,] dataArr = new float[ROW, COLUMN];public Vector2 Row0 { get { return new Vector2(M00, M01); } }public Vector2 Row1 { get { return new Vector2(M10, M11); } }public Vector2 Column0 { get { return new Vector2(M00, M10); } }public Vector2 Column1 { get { return new Vector2(M01, M11); } }#endregionpublic float this[int row, int col]{get { return dataArr[row, col]; }set { dataArr[row, col] = value; }}public Matrix2x2() { }public Matrix2x2(float m00, float m01, float m10, float m11){M00 = m00;M01 = m01;M10 = m10;M11 = m11;}/// <summary>/// xy基向量/// 竖向排列/// </summary>/// <param name="ax"></param>/// <param name="ay"></param>public Matrix2x2(Vector2 ax, Vector2 ay){M00 = ax.x;M10 = ax.y;M10 = ay.x;M11 = ay.y;}/// <summary>/// 2*2数组/// </summary>/// <param name="arr"></param>public Matrix2x2(float[,] arr){M00 = arr[0, 0];M01 = arr[0, 1];M10 = arr[1, 0];M11 = arr[1, 1];}/// <summary>/// 矩阵*vector2/// </summary>/// <param name="m2x2"></param>/// <param name="v2"></param>/// <returns></returns>public static Vector2 operator *(Matrix2x2 m2x2, Vector2 v2){float x = Vector2.Dot(m2x2.Row0, v2);float y = Vector2.Dot(m2x2.Row1, v2);return new Vector2(x, y);}/// <summary>/// 矩阵*矩阵/// </summary>/// <param name="m2x2a"></param>/// <param name="m2x2b"></param>/// <returns></returns>public static Matrix2x2 operator *(Matrix2x2 m2x2a, Matrix2x2 m2x2b){float c00 = Vector2.Dot(m2x2a.Row0, m2x2b.Column0);float c01 = Vector2.Dot(m2x2a.Row0, m2x2b.Column1);float c10 = Vector2.Dot(m2x2a.Row1, m2x2b.Column0);float c11 = Vector2.Dot(m2x2a.Row1, m2x2b.Column1);Matrix2x2 ret = new Matrix2x2(c00, c01, c10, c11);return ret;}/// <summary>/// 矩阵/标量/// </summary>/// <param name="m2x2"></param>/// <param name="f"></param>/// <returns></returns>public static Matrix2x2 operator /(Matrix2x2 m2x2, float f){for (int x = 0; x < ROW; x++){for (int y = 0; y < COLUMN; y++){m2x2[x, y] /= f;}}return m2x2;}/// <summary>/// 行列式/// </summary>/// <returns></returns>public float GetDeterminant(){float det = M00 * M11 - M01 * M10;return det;}/// <summary>/// 求转置矩阵/// </summary>/// <returns></returns>public Matrix2x2 GetTransposeMatrix(){Matrix2x2 m2x2T = new Matrix2x2();for (int x = 0; x < ROW; x++){for (int y = 0; y < COLUMN; y++){m2x2T[x, y] = this[y, x];}}return m2x2T;}/// <summary>/// 求余子式标量/// </summary>/// <param name="r"></param>/// <param name="c"></param>/// <returns></returns>public float GetCofactorScalar(int r, int c){float cof = 0f;for (int x = 0; x < ROW; x++){for (int y = 0; y < COLUMN; y++){if (x != r && y != c){cof = dataArr[x, y];break;}}}return cof;}/// <summary>/// 求余子式标量矩阵/// + -/// - +/// </summary>/// <returns></returns>public Matrix2x2 GetCofactorScalarMatrix(){Matrix2x2 m2x2 = new Matrix2x2();for (int x = 0; x < ROW; x++){for (int y = 0; y < COLUMN; y++){float cof = GetCofactorScalar(x, y);bool ispostive = (x + y) % 2 == 0;m2x2[x, y] = ispostive ? cof : -cof;}}return m2x2;}/// <summary>/// 求伴随矩阵/// 算法：余子式标量矩阵的转置/// </summary>/// <returns></returns>public Matrix2x2 GetAdjointMatrix(){Matrix2x2 m2x2 = GetCofactorScalarMatrix();Matrix2x2 m2x2T = m2x2.GetTransposeMatrix();return m2x2T;}/// <summary>/// 求逆矩阵/// 算法：伴随矩阵/行列式值/// </summary>/// <returns></returns>public Matrix2x2 GetInverseMatrix(){Matrix2x2 m2x2 = GetAdjointMatrix();float det = GetDeterminant();Matrix2x2 m2x2I = m2x2 / det;return m2x2I;}public override string ToString(){string ret = $"换行\nM00:{M00} M01:{M01} \nM10:{M10} M11:{M11}";return ret;}
}

      关于Matrix2x2，我设计了构造、转置、余子式（2x2矩阵的余子式为标量，或称1x1矩阵）、余子式标量矩阵、伴随矩阵和逆矩阵。
      基本上数学运算开发够用了，每个函数的意义只在代码注释上简单说明。
      这里只举一个例子：逆矩阵可以将矩阵变换后向量再变换回来，比如：

Matrix2x2 m2x2 = new Matrix2x2();
m2x2.M00 = 0.3f;
m2x2.M01 = 1.2f;
m2x2.M10 = 5.2f;
m2x2.M11 = -1f;Debug.LogErrorFormat($"m2x2 = {m2x2}");Vector2 vec0 = new Vector2(5.8f, 56.1f);Vector2 vec1 = m2x2 * vec0;Matrix2x2 m2x2I = m2x2.GetInverseMatrix();Debug.LogErrorFormat($"m2x2I = {m2x2I}");Vector2 vec2 = m2x2I * vec1;Debug.LogErrorFormat($"vec0 = {vec0} vec1 = {vec1} vec2 = {vec2}");

      结果：

      接下来扩展Matrix3x3：

using NPOI.SS.Formula.Functions;
using System.Collections;
using System.Collections.Generic;
using UnityEngine;[System.Serializable]
public class Matrix3x3
{#region ///propertiespublic const int ROW = 3;public const int COLUMN = 3;public float M00{get { return dataArr[0, 0]; }set { dataArr[0, 0] = value; }}public float M01{get { return dataArr[0, 1]; }set { dataArr[0, 1] = value; }}public float M02{get { return dataArr[0, 2]; }set { dataArr[0, 2] = value; }}public float M10{get { return dataArr[1, 0]; }set { dataArr[1, 0] = value; }}public float M11{get { return dataArr[1, 1]; }set { dataArr[1, 1] = value; }}public float M12{get { return dataArr[1, 2]; }set { dataArr[1, 2] = value; }}public float M20{get { return dataArr[2, 0]; }set { dataArr[2, 0] = value; }}public float M21{get { return dataArr[2, 1]; }set { dataArr[2, 1] = value; }}public float M22{get { return dataArr[2, 2]; }set { dataArr[2, 2] = value; }}public float[,] dataArr = new float[ROW, COLUMN];public Vector3 Row0 { get { return new Vector3(M00, M01, M02); } }public Vector3 Row1 { get { return new Vector3(M10, M11, M12); } }public Vector3 Row2 { get { return new Vector3(M20, M21, M22); } }public Vector3 Column0 { get { return new Vector3(M00, M10, M20); } }public Vector3 Column1 { get { return new Vector3(M01, M11, M21); } }public Vector3 Column2 { get { return new Vector3(M02, M12, M22); } }#endregionpublic float this[int row, int col]{get { return dataArr[row, col]; }set { dataArr[row, col] = value; }}public Matrix3x3() { }public Matrix3x3(float m00, float m01, float m02, float m10, float m11, float m12, float m20, float m21, float m22){M00 = m00;M01 = m01;M02 = m02;M10 = m10;M11 = m11;M12 = m12;M20 = m20;M21 = m21;M22 = m22;}/// <summary>/// xyz基向量排列/// </summary>/// <param name="ax">x基向量</param>/// <param name="ay">y基向量</param>/// <param name="az">z基向量</param>public Matrix3x3(Vector3 ax, Vector3 ay, Vector3 az){M00 = ax.x;M10 = ax.y;M20 = ax.z;M01 = ay.x;M11 = ay.y;M21 = ay.z;M02 = az.x;M12 = az.y;M22 = az.z;}/// <summary>/// 数组排列/// </summary>/// <param name="arr"></param>public Matrix3x3(float[,] arr){M00 = arr[0, 0];M01 = arr[0, 1];M02 = arr[0, 2];M10 = arr[1, 0];M11 = arr[1, 1];M12 = arr[1, 2];M20 = arr[2, 0];M21 = arr[2, 1];M22 = arr[2, 2];}/// <summary>/// 矩阵*vector2/// </summary>/// <param name="m3x3"></param>/// <param name="v2"></param>/// <returns></returns>public static Vector2 operator *(Matrix3x3 m3x3, Vector2 v2){Vector3 v3 = new Vector3(v2.x, v2.y, 1);v3 = m3x3 * v3;v2 = new Vector2(v3.x, v3.y);return v2;}/// <summary>/// 矩阵*vector3/// </summary>/// <param name="m3x3"></param>/// <param name="v3"></param>/// <returns></returns>public static Vector3 operator *(Matrix3x3 m3x3, Vector3 v3){float x = Vector3.Dot(m3x3.Row0, v3);float y = Vector3.Dot(m3x3.Row1, v3);float z = Vector3.Dot(m3x3.Row2, v3);return new Vector3(x, y, z);}/// <summary>/// 矩阵*矩阵/// </summary>/// <param name="m3x3a"></param>/// <param name="m3x3b"></param>/// <returns></returns>public static Matrix3x3 operator *(Matrix3x3 m3x3a, Matrix3x3 m3x3b){float c00 = Vector2.Dot(m3x3a.Row0, m3x3b.Column0);float c01 = Vector2.Dot(m3x3a.Row0, m3x3b.Column1);float c02 = Vector2.Dot(m3x3a.Row0, m3x3b.Column2);float c10 = Vector2.Dot(m3x3a.Row1, m3x3b.Column0);float c11 = Vector2.Dot(m3x3a.Row1, m3x3b.Column1);float c12 = Vector2.Dot(m3x3a.Row1, m3x3b.Column2);float c20 = Vector2.Dot(m3x3a.Row2, m3x3b.Column0);float c21 = Vector2.Dot(m3x3a.Row2, m3x3b.Column1);float c22 = Vector2.Dot(m3x3a.Row2, m3x3b.Column2);Matrix3x3 ret = new Matrix3x3(c00, c01, c02, c10, c11, c12, c20, c21, c22);return ret;}/// <summary>/// 矩阵/标量/// </summary>/// <param name="m3x3"></param>/// <param name="f"></param>/// <returns></returns>public static Matrix3x3 operator /(Matrix3x3 m3x3, float f){for (int x = 0; x < ROW; x++){for (int y = 0; y < COLUMN; y++){m3x3[x, y] /= f;}}return m3x3;}/// <summary>/// 求行列式/// </summary>/// <returns></returns>public float GetDeterminant(){float det = M00 * M11 * M22 + M01 * M12 * M20 + M02 * M10 * M21 - M02 * M11 * M20 - M01 * M10 * M22 - M00 * M12 * M21;return det;}/// <summary>/// 求转置矩阵/// </summary>/// <returns></returns>public Matrix3x3 GetTransposeMatrix(){Matrix3x3 m3x3T = new Matrix3x3();for (int x = 0; x < ROW; x++){for (int y = 0; y < COLUMN; y++){m3x3T[x, y] = this[y, x];}}return m3x3T;}/// <summary>/// 求余子式矩阵/// </summary>/// <param name="r"></param>/// <param name="c"></param>/// <returns></returns>public Matrix2x2 GetCofactorMatrix(int r, int c){Matrix2x2 m2x2 = new Matrix2x2();for (int x = 0; x < ROW; x++){for (int y = 0; y < COLUMN; y++){if (x != r && y != c){int row = x > r ? x - 1 : x;int col = y > c ? y - 1 : y;m2x2[row, col] = this[x, y];}}}return m2x2;}/// <summary>/// 求代数余子式（余子式矩阵的行列式）矩阵/// 余子式矩阵行列式正负号/// + - +/// - + -/// + - +/// </summary>/// <returns></returns>public Matrix3x3 GetCofactorDeterminantMatrix(){Matrix3x3 m3x3 = new Matrix3x3();for (int x = 0; x < ROW; x++){for (int y = 0; y < COLUMN; y++){Matrix2x2 m2x2 = GetCofactorMatrix(x, y);float m2x2det = m2x2.GetDeterminant();bool ispostive = (x + y) % 2 == 0;m3x3[x, y] = ispostive ? m2x2det : -m2x2det;}}return m3x3;}/// <summary>/// 求伴随矩阵/// 算法：代数余子式矩阵的转置/// </summary>/// <returns></returns>public Matrix3x3 GetAdjointMatrix(){Matrix3x3 m3x3 = GetCofactorDeterminantMatrix();Matrix3x3 m3x3T = m3x3.GetTransposeMatrix();return m3x3T;}/// <summary>/// 求逆矩阵/// 算法：伴随矩阵/行列式值/// </summary>/// <returns></returns>public Matrix3x3 GetInverseMatrix(){Matrix3x3 m3x3 = GetAdjointMatrix();float det = GetDeterminant();Matrix3x3 m3x3I = m3x3 / det;return m3x3I;}public override string ToString(){string ret = $"换行\nM00:{M00} M01:{M01} M02:{M02} \nM10:{M10} M11:{M11} M12:{M12} \nM20:{M20} M21:{M21} M12:{M22}";return ret;}
}

      还是用逆矩阵验证一下：

Matrix3x3 m3x3 = new Matrix3x3();
m3x3.M00 = -0.3f;
m3x3.M01 = 6.2f;
m3x3.M02 = 12.6f;
m3x3.M10 = 5.2f;
m3x3.M11 = -1.8f;
m3x3.M12 = 7.8f;
m3x3.M20 = -52.2f;
m3x3.M21 = 6.4f;
m3x3.M22 = -70.1f;Debug.LogErrorFormat($"m3x3 = {m3x3}");Vector3 vec0 = new Vector3(20.3f, -54f, 4.4f);Vector3 vec1 = m3x3 * vec0;Matrix3x3 m3x3I = m3x3.GetInverseMatrix();Debug.LogErrorFormat($"m3x3I = {m3x3I}");Vector3 vec2 = m3x3I * vec1;Debug.LogErrorFormat($"vec0 = {vec0} vec1 = {vec1} vec2 = {vec2}");

      结果：

      OK，洗了睡，这里吐槽一下：这些矩阵计算unity应该直接提供，写起来眼睛都要看瞎了。