Skip to content

Lec2 Transformation

MVP transformation

  • Model transformation
  • View( 视图 )/Camera transformation
  • Projection( 投影 ) transformation
    • Orthographic( 正交 ) projection
    • Perspective( 透视 ) projection

Review of Linear Algebra

  • Dot Product
    • Forward/backward(positive/negative)
  • Cross Product
    • Left/Right(outward/inward)

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

rotation around x-, y-, z-axis

\[\bm{R}_x(\alpha)=\begin{pmatrix} 1 & 0 & 0 & 0\\0 & \cos \alpha & -\sin \alpha & 0 \\ 0 & \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 0 & 1\end{pmatrix}\]
\[\bm{R}_y(\alpha)=\begin{pmatrix} \cos \alpha & 0 & \sin \alpha & 0\\0 & 1 & 0 & 0 \\ -\sin \alpha & 0 & \cos \alpha & 0 \\ 0 & 0 & 0 & 1\end{pmatrix}\]
\[\bm{R}_z(\alpha)=\begin{pmatrix} \cos \alpha & -\sin \alpha & 0 & 0\\\sin \alpha & \cos \alpha & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix}\]
\[\bm{R}_{xyz}(\alpha,\beta,\gamma)=\bm{R}_x(\alpha)\bm{R}_y(\beta)\bm{R}_z(\gamma)\]

Euler angles

often used in flight simulators: roll, pitch, yaw 无法显示

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

define the camera

  • Postion \(\vec{e}\)
  • 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

评论区

如果有什么问题或想法，欢迎大家在下方留言~