This Week

• Transformation!
• Today
  • Why study transformation
  • 2D transformations: rotation, scale, shear
  • Homogeneous coordinates
  • Composing transforms
  • 3D transformations

Why study transformation

• Modeling → 模型变换 translation, rotation, scaling
• Viewing → 视图变换 (3D to 2D) projection

2D transformations

• Scale: $\begin{bmatrix}s&0\\0&s\end{bmatrix}$
• Scale (Non-Uniform): $\begin{bmatrix}s_x&0\\0&s_y\end{bmatrix}$
• Reflection Matrix: $\begin{bmatrix}-1&0\\0&1\end{bmatrix}$
• Shear Matrix: $\begin{bmatrix}1&a\\0&1\end{bmatrix}$
• Rotate (about the origin (0, 0), CCW by default)
• Rotation Matrix: $\begin{bmatrix}cos\theta&-sin\theta\\sin\theta&cos\theta\end{bmatrix}$
• Linear Transforms = Matrices (of the same dimension)

Homogeneous coordinates

• Translation?

Why Homogenous Coordinates

• Translation cannot be represented in matrix form
  • $\begin{bmatrix}x'\\y'\end{bmatrix} = \begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}+\begin{bmatrix}t_x\\t_y\end{bmatrix}$ → So, translation is NOT linear transform!
• But we don't want translation to be a special case
• Is there a unified way to represent all transformations? (and what's the cost?)