Transformation

2D 变换

缩放

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}{s}_x& 0\\ 0& s_y\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$

齐次坐标形式：

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\underset{\mathbf{R}(\alpha )}{\underbrace{\left[\begin{array}{ccc}{s}_x& 0& 0\\ 0& s_y& 0\\ 0& 0& 1\end{array}\right]}}\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$$$\mathbf{R}(-\alpha )=\mathbf{R}(\alpha {)}^{\mathrm{\top }}=\mathbf{R}(\alpha {)}^{-1}$$

切变

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}1& a\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$

旋转

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

齐次坐标形式：

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\underset{\mathbf{S}(s_x,s_y)}{\underbrace{\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]}}\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

平移

仿射变换（Affine Transformations）形式：

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}{t}_x\\ t_y\end{array}\right]$$

齐次坐标形式：

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ w^{\prime }\end{array}\right]=\underset{\mathbf{T}(t_x,t_y)}{\underbrace{\left[\begin{array}{ccc}1& 0& t_x\\ 0& 1& t_y\\ 0& 0& 1\end{array}\right]}}\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

齐次坐标（Homogenous coordinates）

在齐次坐标中，一个 2D 点可以表示为 $(x,y,1{)}^{\mathrm{\top }}$，一个 2D 向量可以表示为 $(x,y,0{)}^{\mathrm{\top }}$;

在齐次坐标中，$\left[\begin{array}{c}x\\ y\\ w\end{array}\right]$ 代表的是一个 2D 点 $\left[\begin{array}{c}x/w\\ y/w\\ 1\end{array}\right]$，$w\ne 0$

这样就有：

• vector + vector = vector
• point - point = vector
• point + vector = point
• point - point = point (中点)

Complex Transforms

如何绕指定的点旋转？

$$\mathbf{T}(c)\cdot \mathbf{R}(\alpha )\cdot \mathbf{T}(-c)$$

3D 变换

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{cccc}a& b& c& t_x\\ d& e& f& t_y\\ g& h& i& t_z\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\\ 1\end{array}\right]$$

缩放

$$\mathbf{S}(s_x,s_y,s_z)=\left[\begin{array}{cccc}{s}_x& 0& 0& 0\\ 0& s_y& 0& 0\\ 0& 0& s_z& 0\\ 0& 0& 0& 1\end{array}\right]$$

平移

$$\mathbf{T}(t_x,t_y,t_z)=\left[\begin{array}{cccc}1& 0& 0& t_x\\ 0& 1& 0& t_y\\ 0& 0& 1& t_z\\ 0& 0& 0& 1\end{array}\right]$$

旋转

$${\mathbf{R}}_x(\alpha )=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \alpha & -\sin \alpha & 0\\ 0& \sin \alpha & \cos \alpha & 0\\ 0& 0& 0& 1\end{array}\right]$$$${\mathbf{R}}_y(\alpha )=\left[\begin{array}{cccc}\cos \alpha & 0& \sin \alpha & 0\\ 0& 1& 0& 0\\ -\sin \alpha & 0& \cos \alpha & 0\\ 0& 0& 0& 1\end{array}\right]$$$${\mathbf{R}}_z(\alpha )=\left[\begin{array}{cccc}\cos \alpha & -\sin \alpha & 0& 0\\ \sin \alpha & \cos \alpha & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

基于欧拉角定义：

$${\mathbf{R}}_{xyz}(\alpha ,\beta ,\gamma )={\mathbf{R}}_x(\alpha ){\mathbf{R}}_y(\beta ){\mathbf{R}}_z(\gamma )$$

绕任意轴 $\mathbf{n}$ 旋转，根据罗德里格斯旋转公式 ‌（Rodrigues' rotation formula）：

$$\mathbf{R}(\mathbf{n},\alpha )=\cos (\alpha )\mathbf{I}+(1-\cos (\alpha ))\mathbf{n}{\mathbf{n}}^{\mathrm{\top }}+\sin (\alpha )\underset{\mathbf{N}}{\underbrace{\left[\begin{array}{ccc}0& -n_z& n_y\\ n_z& 0& n_x\\ -n_y& n_x& 0\end{array}\right]}}$$