仿射变换原理及图解：从数学本质到计算机图形学实战

import numpy as np
import matplotlib.pyplot as plt
def affine_transform(points, matrix):
    """
    对点集应用仿射变换
    :param points: Nx2 数组，每行是一个点 (x, y)
    :param matrix: 3x3 仿射变换矩阵
    :return: 变换后的 Nx2 数组
    """
    # 将点集转换为齐次坐标 (添加一列 1)
    ones = np.ones((points.shape[0], 1))
    homogeneous_points = np.hstack((points, ones))
    # 矩阵乘法
    transformed_points = np.dot(homogeneous_points, matrix.T)
    # 转换回笛卡尔坐标
    return transformed_points[:, :2]
示例：定义一个三角形
triangle = np.array([[0, 0], [1, 0], [0.5, 1]])
定义一个旋转30度并平移的矩阵
theta = np.pi / 6  # 30度
matrix = np.array([
    [np.cos(theta), -np.sin(theta), 0.5],  # 旋转 + x平移
    [np.sin(theta),  np.cos(theta), 0.5],  # 旋转 + y平移
    [0,             0,               1]    # 齐次坐标
])
应用变换
rotated_triangle = affine_transform(triangle, matrix)
绘图对比
plt.figure(figsize=(10, 5))
plt.subplot(1, 2, 1)
plt.plot(zip(triangle), 'o-')
plt.title("Original Triangle")
plt.axis('equal')
plt.grid(True)
plt.subplot(1, 2, 2)
plt.plot(zip(rotated_triangle), 'o-', color='red')
plt.title("Transformed Triangle")
plt.axis('equal')
plt.grid(True)
plt.show()

// 伪代码示例
glm::mat4 model = glm::mat4(1.0f);
// 1. 平移
model = glm::translate(model, glm::vec3(1.0f, 0.0f, 0.0f));
// 2. 旋转
model = glm::rotate(model, glm::radians(45.0f), glm::vec3(0.0f, 0.0f, 1.0f));
// 3. 缩放
model = glm::scale(model, glm::vec3(0.5f, 0.5f, 0.5f));
// 将模型矩阵传递给着色器
unsigned int modelLoc = glGetUniformLocation(shader.ID, "model");
glUniformMatrix4fv(modelLoc, 1, GL_FALSE, glm::value_ptr(model));