4x4矩阵教程

4x4矩阵教程

1. 简介

四维矩阵是计算机图形学和3D变换中的重要工具，用于表示三维空间中的仿射变换。本教程将介绍如何使用C++实现四维矩阵的基本运算和变换。

2. 代码实现

2.1 头文件 (matrix4x4.h)

#ifndef MATRIX4X4_H
#define MATRIX4X4_H#include <array>
#include <stdexcept>
#include <iostream>namespace math {
namespace linear_algebra {/*** @brief 四维矩阵类* * 这个类实现了四维矩阵的基本运算，包括：* - 矩阵加减* - 矩阵乘法* - 标量乘法* - 行列式计算* - 矩阵求逆* - 矩阵转置* - 特征值和特征向量计算* - 矩阵性质检查（可逆性、对称性、正交性）* - 特殊矩阵生成（旋转矩阵、缩放矩阵、平移矩阵、投影矩阵等）*/
class Matrix4x4 {
public:// 构造函数Matrix4x4();  // 默认构造函数，初始化为单位矩阵Matrix4x4(const std::array<std::array<double, 4>, 4>& data);  // 从二维数组初始化// 基本运算Matrix4x4 operator+(const Matrix4x4& other) const;  // 矩阵加法Matrix4x4 operator-(const Matrix4x4& other) const;  // 矩阵减法Matrix4x4 operator*(const Matrix4x4& other) const;  // 矩阵乘法Matrix4x4 operator*(double scalar) const;           // 标量乘法Matrix4x4 operator/(double scalar) const;           // 标量除法// 矩阵运算double determinant() const;                         // 计算行列式Matrix4x4 inverse() const;                          // 计算逆矩阵Matrix4x4 transpose() const;                        // 计算转置矩阵std::array<double, 4> eigenvalues() const;          // 计算特征值std::array<Matrix4x4, 4> eigenvectors() const;      // 计算特征向量// 矩阵性质bool isInvertible() const;                          // 检查是否可逆bool isSymmetric() const;                           // 检查是否对称bool isOrthogonal() const;                          // 检查是否正交// 特殊矩阵static Matrix4x4 identity();                        // 创建单位矩阵static Matrix4x4 rotation(double theta, char axis); // 创建旋转矩阵static Matrix4x4 scaling(double sx, double sy, double sz); // 创建缩放矩阵static Matrix4x4 translation(double tx, double ty, double tz); // 创建平移矩阵static Matrix4x4 perspective(double fov, double aspect, double near, double far); // 创建透视投影矩阵static Matrix4x4 orthographic(double left, double right, double bottom, double top, double near, double far); // 创建正交投影矩阵// 输出运算符friend std::ostream& operator<<(std::ostream& os, const Matrix4x4& m);private:std::array<std::array<double, 4>, 4> data;  // 4x4矩阵数据
};} // namespace linear_algebra
} // namespace math#endif // MATRIX4X4_H

2.2 实现文件 (matrix4x4.cpp)

#include "matrix4x4.h"
#include <cmath>namespace math {
namespace linear_algebra {// 默认构造函数：初始化为单位矩阵
Matrix4x4::Matrix4x4() {for (int i = 0; i < 4; ++i) {for (int j = 0; j < 4; ++j) {data[i][j] = (i == j) ? 1.0 : 0.0;}}
}// 从二维数组初始化
Matrix4x4::Matrix4x4(const std::array<std::array<double, 4>, 4>& data) : data(data) {}// 矩阵加法实现
Matrix4x4 Matrix4x4::operator+(const Matrix4x4& other) const {Matrix4x4 result;for (int i = 0; i < 4; ++i) {for (int j = 0; j < 4; ++j) {result.data[i][j] = data[i][j] + other.data[i][j];}}return result;
}// 矩阵减法实现
Matrix4x4 Matrix4x4::operator-(const Matrix4x4& other) const {Matrix4x4 result;for (int i = 0; i < 4; ++i) {for (int j = 0; j < 4; ++j) {result.data[i][j] = data[i][j] - other.data[i][j];}}return result;
}// 矩阵乘法实现
Matrix4x4 Matrix4x4::operator*(const Matrix4x4& other) const {Matrix4x4 result;for (int i = 0; i < 4; ++i) {for (int j = 0; j < 4; ++j) {result.data[i][j] = 0.0;for (int k = 0; k < 4; ++k) {result.data[i][j] += data[i][k] * other.data[k][j];}}}return result;
}// 标量乘法实现
Matrix4x4 Matrix4x4::operator*(double scalar) const {Matrix4x4 result;for (int i = 0; i < 4; ++i) {for (int j = 0; j < 4; ++j) {result.data[i][j] = data[i][j] * scalar;}}return result;
}// 标量除法实现
Matrix4x4 Matrix4x4::operator/(double scalar) const {if (scalar == 0.0) {throw std::runtime_error("Division by zero");}return *this * (1.0 / scalar);
}// 行列式计算实现
double Matrix4x4::determinant() const {// 使用拉普拉斯展开计算4x4矩阵的行列式double det = 0.0;for (int i = 0; i < 4; ++i) {double cofactor = 0.0;for (int j = 0; j < 4; ++j) {if ((i + j) % 2 == 0) {cofactor += data[0][j] * minor(0, j);} else {cofactor -= data[0][j] * minor(0, j);}}det += data[0][i] * cofactor;}return det;
}// 逆矩阵计算实现
Matrix4x4 Matrix4x4::inverse() const {double det = determinant();if (det == 0.0) {throw std::runtime_error("Matrix is not invertible");}Matrix4x4 result;double invDet = 1.0 / det;// 计算伴随矩阵for (int i = 0; i < 4; ++i) {for (int j = 0; j < 4; ++j) {result.data[i][j] = cofactor(j, i) * invDet;}}return result;
}// 转置矩阵实现
Matrix4x4 Matrix4x4::transpose() const {Matrix4x4 result;for (int i = 0; i < 4; ++i) {for (int j = 0; j < 4; ++j) {result.data[i][j] = data[j][i];}}return result;
}// 特征值计算实现
std::array<double, 4> Matrix4x4::eigenvalues() const {// 计算特征多项式系数// 这里使用简化的方法，实际应用中可能需要更复杂的数值方法std::array<double, 4> roots;// ... 求解四次方程的代码 ...return roots;
}// 特征向量计算实现
std::array<Matrix4x4, 4> Matrix4x4::eigenvectors() const {std::array<double, 4> eigenvals = eigenvalues();std::array<Matrix4x4, 4> eigenvecs;// ... 计算特征向量的代码 ...return eigenvecs;
}// 可逆性检查实现
bool Matrix4x4::isInvertible() const {return determinant() != 0.0;
}// 对称性检查实现
bool Matrix4x4::isSymmetric() const {for (int i = 0; i < 4; ++i) {for (int j = i + 1; j < 4; ++j) {if (data[i][j] != data[j][i]) {return false;}}}return true;
}// 正交性检查实现
bool Matrix4x4::isOrthogonal() const {Matrix4x4 product = *this * transpose();Matrix4x4 identity = Matrix4x4::identity();for (int i = 0; i < 4; ++i) {for (int j = 0; j < 4; ++j) {if (std::abs(product.data[i][j] - identity.data[i][j]) > 1e-10) {return false;}}}return true;
}// 单位矩阵创建实现
Matrix4x4 Matrix4x4::identity() {return Matrix4x4();
}// 旋转矩阵创建实现
Matrix4x4 Matrix4x4::rotation(double theta, char axis) {Matrix4x4 result;double cos_theta = std::cos(theta);double sin_theta = std::sin(theta);switch (axis) {case 'x':result.data[1][1] = cos_theta;result.data[1][2] = -sin_theta;result.data[2][1] = sin_theta;result.data[2][2] = cos_theta;break;case 'y':result.data[0][0] = cos_theta;result.data[0][2] = sin_theta;result.data[2][0] = -sin_theta;result.data[2][2] = cos_theta;break;case 'z':result.data[0][0] = cos_theta;result.data[0][1] = -sin_theta;result.data[1][0] = sin_theta;result.data[1][1] = cos_theta;break;default:throw std::runtime_error("Invalid rotation axis");}return result;
}// 缩放矩阵创建实现
Matrix4x4 Matrix4x4::scaling(double sx, double sy, double sz) {Matrix4x4 result;result.data[0][0] = sx;result.data[1][1] = sy;result.data[2][2] = sz;return result;
}// 平移矩阵创建实现
Matrix4x4 Matrix4x4::translation(double tx, double ty, double tz) {Matrix4x4 result;result.data[0][3] = tx;result.data[1][3] = ty;result.data[2][3] = tz;return result;
}// 透视投影矩阵创建实现
Matrix4x4 Matrix4x4::perspective(double fov, double aspect, double near, double far) {Matrix4x4 result;double tan_half_fov = std::tan(fov / 2.0);result.data[0][0] = 1.0 / (aspect * tan_half_fov);result.data[1][1] = 1.0 / tan_half_fov;result.data[2][2] = -(far + near) / (far - near);result.data[2][3] = -1.0;result.data[3][2] = -(2.0 * far * near) / (far - near);result.data[3][3] = 0.0;return result;
}// 正交投影矩阵创建实现
Matrix4x4 Matrix4x4::orthographic(double left, double right, double bottom, double top, double near, double far) {Matrix4x4 result;result.data[0][0] = 2.0 / (right - left);result.data[1][1] = 2.0 / (top - bottom);result.data[2][2] = -2.0 / (far - near);result.data[0][3] = -(right + left) / (right - left);result.data[1][3] = -(top + bottom) / (top - bottom);result.data[2][3] = -(far + near) / (far - near);return result;
}// 输出运算符实现
std::ostream& operator<<(std::ostream& os, const Matrix4x4& m) {for (int i = 0; i < 4; ++i) {os << "[ ";for (int j = 0; j < 4; ++j) {os << m.data[i][j] << " ";}os << "]" << std::endl;}return os;
}} // namespace linear_algebra
} // namespace math

3. 使用示例

3.1 基本运算

#include "matrix4x4.h"
#include <iostream>using namespace math::linear_algebra;int main() {// 创建矩阵Matrix4x4 m1({{{1.0, 2.0, 3.0, 4.0},{5.0, 6.0, 7.0, 8.0},{9.0, 10.0, 11.0, 12.0},{13.0, 14.0, 15.0, 16.0}}});Matrix4x4 m2({{{16.0, 15.0, 14.0, 13.0},{12.0, 11.0, 10.0, 9.0},{8.0, 7.0, 6.0, 5.0},{4.0, 3.0, 2.0, 1.0}}});// 基本运算Matrix4x4 sum = m1 + m2;      // 矩阵加法Matrix4x4 diff = m1 - m2;     // 矩阵减法Matrix4x4 prod = m1 * m2;     // 矩阵乘法Matrix4x4 scaled = m1 * 2.0;  // 标量乘法// 输出结果std::cout << "m1:\n" << m1 << std::endl;std::cout << "m2:\n" << m2 << std::endl;std::cout << "m1 + m2:\n" << sum << std::endl;std::cout << "m1 - m2:\n" << diff << std::endl;std::cout << "m1 * m2:\n" << prod << std::endl;std::cout << "m1 * 2:\n" << scaled << std::endl;return 0;
}

3.2 3D变换

#include "matrix4x4.h"
#include <iostream>
#include <cmath>using namespace math::linear_algebra;int main() {// 创建变换矩阵Matrix4x4 rotation = Matrix4x4::rotation(M_PI / 4.0, 'z');  // 绕Z轴旋转45度Matrix4x4 scaling = Matrix4x4::scaling(2.0, 3.0, 4.0);      // 缩放Matrix4x4 translation = Matrix4x4::translation(1.0, 2.0, 3.0); // 平移// 组合变换（注意顺序：先缩放，再旋转，最后平移）Matrix4x4 transform = translation * rotation * scaling;// 创建投影矩阵Matrix4x4 perspective = Matrix4x4::perspective(M_PI / 4.0, 16.0/9.0, 0.1, 100.0);Matrix4x4 ortho = Matrix4x4::orthographic(-1.0, 1.0, -1.0, 1.0, 0.1, 100.0);// 输出结果std::cout << "旋转矩阵:\n" << rotation << std::endl;std::cout << "缩放矩阵:\n" << scaling << std::endl;std::cout << "平移矩阵:\n" << translation << std::endl;std::cout << "组合变换:\n" << transform << std::endl;std::cout << "透视投影:\n" << perspective << std::endl;std::cout << "正交投影:\n" << ortho << std::endl;return 0;
}

4. 编译和运行

4.1 编译

使用提供的Makefile进行编译：

make        # 编译所有目标
make test   # 运行测试
make demo   # 运行示例
make clean  # 清理编译文件

4.2 运行测试

./matrix4x4_test

4.3 运行示例

./matrix4x4_demo

5. 注意事项

1. 数值精度

   • 在比较浮点数时使用适当的误差范围
   • 例如：std::abs(det) < 1e-10 判断是否可逆

2. 异常处理

   • 除以零的情况
   • 不可逆矩阵求逆的情况
   • 无效的旋转轴
   • 投影参数的有效性检查

3. 性能考虑

   • 矩阵乘法的时间复杂度为O(n³)
   • 特征值计算可能需要迭代方法
   • 考虑使用SIMD指令优化计算
   • 投影矩阵的预计算

4. 使用建议

   • 优先使用成员函数而不是全局函数
   • 保持接口的一致性
   • 提供清晰的错误信息
   • 注意变换矩阵的组合顺序

6. 扩展阅读

1. 线性代数基础

   • 矩阵运算
   • 行列式
   • 特征值和特征向量

2. 3D变换

   • 旋转矩阵
   • 缩放矩阵
   • 平移矩阵
   • 投影矩阵

3. 应用领域

   • 计算机图形学
   • 3D游戏开发
   • 计算机视觉
   • 机器人学
   • 物理模拟

   声明

该文章为学习过程中的笔记，目的是防止自己忘记，也为了方便随时随地查阅。其中大部分内容收集于互联网。