Lec2 Transformation¶
Note
- Model transformation
- View( 视图 )/Camera transformation
- Projection( 投影 ) transformation
- Orthographic( 正交 ) projection
- Perspective( 透视 ) projection
Model transformation¶
2D transformations¶
Scale Matrix
\[\begin{bmatrix}x' \\ y' &\end{bmatrix}=\begin{bmatrix}s & 0\\ 0 & s\end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix}\]
Scale Matrix(Non-Uniform)
\[\begin{bmatrix}x' \\ y' &\end{bmatrix}=\begin{bmatrix}s_x & 0\\ 0 & s_y\end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix}\]
Reflection Matrix
\[\begin{bmatrix}x' \\ y' &\end{bmatrix}=\begin{bmatrix}-1 & 0\\ 0 & 1\end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix}\]
Shear Matrix
\[\begin{bmatrix}x' \\ y' &\end{bmatrix}=\begin{bmatrix}1 & a\\ 0 & 1\end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix}\]
Rotation Matrix
\[\begin{bmatrix}x' \\ y' &\end{bmatrix}=\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix}\]
Homogeneous coordinates¶
Translation
\[\begin{bmatrix}x' \\ y' &\end{bmatrix}=\begin{bmatrix}a & b\\ c & d \end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix}+\begin{bmatrix}t_x \\ t_y\end{bmatrix}\]
Add a third coordinate
2D point \((x,y,1)^T\)
2D vector \((x,y,0)^T\)
Affine map = linear map + translation
\[\begin{pmatrix}x' \\ y' &\end{pmatrix}=\begin{pmatrix}a & b\\ c & d \end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix}+\begin{pmatrix}t_x \\ t_y\end{pmatrix}\]
Using Homogeneous coordinates
\[\begin{pmatrix}x' \\ y' \\ 1 &\end{pmatrix}=\begin{pmatrix}a & b & t_x\\ c & d & t_y \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix}x \\ y \\1\end{pmatrix}\]
3D Transformation¶
!!! note rotation around x-, y-, z-axis
$$\bm{R}_x(\alpha)=\begin{pmatrix} 1 & 0 & 0 & 0\\0 & \cos \alpha & -\sin \alpha & 0 \\ 0 & \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 0 & 1\end{pmatrix}$$
$$\bm{R}_y(\alpha)=\begin{pmatrix} \cos \alpha & 0 & \sin \alpha & 0\\0 & 1 & 0 & 0 \\ -\sin \alpha & 0 & \cos \alpha & 0 \\ 0 & 0 & 0 & 1\end{pmatrix}$$
$$\bm{R}_z(\alpha)=\begin{pmatrix} \cos \alpha & -\sin \alpha & 0 & 0\\\sin \alpha & \cos \alpha & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix}$$
\[\bm{R}_{xyz}(\alpha,\beta,\gamma)=\bm{R}_x(\alpha)\bm{R}_y(\beta)\bm{R}_z(\gamma)\]
Note
often used in flight simulators: roll, pitch, yaw
!!! note Rodrigues' Rotation Formula rotation by angle alpha around axis n
$$\bm{R}(\bm{n},\alpha)=\cos(\alpha)\bm{I}+(1-\cos(\alpha))\bm{n}\bm{n}^T + \sin(\alpha) \begin{pmatrix}0 & -n_z & n_y \\ n_z & 0 & -n_x \\ -n_y & n_x & 0\end{pmatrix}$$
View transformation¶
Note
- Postion \(\overrightarrow{e}\)
- Look-at/gaze direction \(hat{g}\)
- Up derection \(\hat{t}\)
Model View Transformation
-
Transformation objects together with the camera
-
Until camera's at the origin, up at Y, look at -Z
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