坐标变换

重要

本文内容默认以左手笛卡尔坐标系为基础
虽然右手坐标系更为常见

多维坐标变换

考虑三维空间中的一点 P
$x_i^{\prime }$ 和 $x_j$轴的关系如下

$${\lambda }_{ij}=\cos (x_i^{\prime },x_j)$$

此处$x^{\prime }$轴就是$x$轴根据一定角度变换得到的

$$\lambda ={\lambda }_{ij}=\left(\begin{array}{ccc}{\lambda }_{11}& {\lambda }_{12}& {\lambda }_{13}\\ {\lambda }_{21}& {\lambda }_{22}& {\lambda }_{23}\\ {\lambda }_{31}& {\lambda }_{32}& {\lambda }_{33}\end{array}\right)\overrightarrow{x^{\prime }}=\left(\begin{array}{c}{x}_1^{\prime }\\ x_2^{\prime }\\ x_3^{\prime }\end{array}\right)\overrightarrow{x}=\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)$$

变换方程就是

$$\overrightarrow{x^{\prime }}=\lambda \overrightarrow{x}$$$$\overrightarrow{x}={\lambda }^T\overrightarrow{x^{\prime }}\mathrm{或}\overrightarrow{x}={\lambda }^{-1}\overrightarrow{x^{\prime }}$$

二维坐标变换

公式：

$$\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)=\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$$

import numpy as np
theta = np.radians(30)
x,y = 5, 3
rotation_matrix = np.array([
    [np.cos(theta), np.sin(theta)], 
    [-np.sin(theta), np.cos(theta)]
])
point = np.array([x,y])
rotation_matrix.dot(point)

array([5.83012702, 0.09807621])

三维坐标变换

左手笛卡尔坐标系

绕x轴公式：

$$Z(\gamma )=\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \gamma & \sin \gamma \\ 0& -\sin \gamma & \cos \gamma \end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$$

绕y轴公式：

$$Y(\beta )=\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{ccc}\cos \beta & 0& -\sin \beta \\ 0& 1& 0\\ \sin \beta & 0& \cos \beta \end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$$

绕z轴公式：

$$X(\theta )=\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{ccc}\cos \theta & \sin \theta & 0\\ -\sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$$

右手笛卡尔坐标系

绕x轴公式：

$$X(\gamma )=\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \gamma & -\sin \gamma \\ 0& \sin \gamma & \cos \gamma \end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$$

绕y轴公式：

$$Y(\beta )=\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{ccc}\cos \beta & 0& \sin \beta \\ 0& 1& 0\\ -\sin \beta & 0& \cos \beta \end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$$

绕z轴公式：

$$Z(\theta )=\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$$

import numpy as np
alpha = np.radians(30)  # 绕z轴旋转
beta = np.radians(30)  # 绕y轴旋转
gamma = np.radians(30)  # 绕x轴旋转
x, y, z = 3, 4, 5
rotation_matrix_z = np.array([
    [np.cos(alpha), np.sin(alpha), 0],
    [-np.sin(alpha), np.cos(alpha), 0],
    [0, 0, 1]
])
rotation_matrix_y = np.array([
    [np.cos(beta), 0, -np.sin(beta)],
    [0, 1, 0],
    [np.sin(beta), 0, np.cos(beta)]
])
rotation_matrix_x = np.array([
    [1, 0, 0],
    [0, np.cos(gamma), np.sin(gamma)],
    [0, -np.sin(gamma), np.cos(gamma)]
])
point = np.array([x, y, z])
z_transformed_point = rotation_matrix_z.dot(point)
y_transformed_point = rotation_matrix_y.dot(point)
x_transformed_point = rotation_matrix_x.dot(point)
print(z_transformed_point)
print(y_transformed_point)
print(x_transformed_point)

[4.59807621 1.96410162 5.        ]
[0.09807621 4.         5.83012702]
[3.         5.96410162 2.33012702]

参考链接

坐标转换