GAMES 101 学习记录

Review of Linear Algebra

叉乘

$$\vec a \times \vec b = A * b = \begin{pmatrix} 0 & -z_{a} & y_{a} \\ z_{a} & 0 & -x_{a} \\ -y_{a} & x_{a} & 0 \end{pmatrix} \begin{pmatrix} x_{b} \\ y_{b} \\ z_{b} \end{pmatrix}$$

Transformation

• Modeling 模型变换
• Viewing 视图变换

Linear Transformation = Matrices

Scale Matrix

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} s & 0 \\ 0 & s \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$$

Shear Matrix

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$$

Rotation Matrix

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$$

Translation

$$\begin{pmatrix} x' \\ y' \\ 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 & t_{x} \\ 0 & 1 & t_{y} \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ 1 \end{pmatrix}$$

Affine Transformations

$$\begin{pmatrix} x' \\ y' \\ 1 \end{pmatrix} = \begin{pmatrix} a & b & t_{x} \\ c & d & t_{y} \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ 1 \end{pmatrix}$$

Transformation Cont.

3D Transformations

$4 \times 4$

3D Rotations

$R{xyz}(\alpha,\beta,\gamma)=R_x(\alpha)R_y(\beta)R{z}(\gamma)$

View Transformation

$ \vec e $

$ \hat g $

$\hat t$