Today

• 3D Transformations
• Viewing (观测) transformation
  • View (视图) / Camera transformation
  • Projection (投影) transformation
    • Orthographic (正交) projection
    • Perspective (透视) projection

3D Transformations

• Scale: $\mathbf{S}(s_x, s_y, s_z) = \begin{pmatrix}s_x&0&0&0\\0&s_y&0&0\\0&0&s_z&0\\0&0&0&1\end{pmatrix}$
• Translation: $\mathbf{T}(t_x, t_y, t_z) = \begin{pmatrix}1&0&0&t_x\\0&1&0&t_y\\0&0&1&t_z\\0&0&0&1\end{pmatrix}$
• Rotate around x-, y-, or z-axis
  • $\mathbf{R}_x(\alpha) = \begin{pmatrix}1&0&0&0\\0&cos\alpha&-sin\alpha&0\\0&sin\alpha&cos\alpha&0\\0&0&0&1\end{pmatrix}$
  • $\mathbf{R}_y(\alpha) = \begin{pmatrix}cos\alpha&0&sin\alpha&0\\0&1&0&0\\-sin\alpha&0&cos\alpha&0\\0&0&0&1\end{pmatrix}$
  • $\mathbf{R}_z(\alpha) = \begin{pmatrix}cos\alpha&-sin\alpha&0&0\\sin\alpha&cos\alpha&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$

3D Rotations

• Compose any 3D rotation from $\mathbf{R}_x, \mathbf{R}_y, \mathbf{R}_z$?
  • $\mathbf{R}_{xyz}(\alpha,\beta,\gamma) = \mathbf{R}_x(\alpha)\mathbf{R}_y(\beta)\mathbf{R}_z(\gamma)$
• So-called Euler angles
• Often used in flight simulators: roll (左右倾斜/翻滚), pitch (抬头低头/俯仰), yaw (左右转向/偏航)

Rodrigues’ Rotation Formula

• Rotation by angle $\alpha$ around axis $\mathbf{n}$
  • $\mathbf{R}(\mathbf{n}, \alpha) = cos(\alpha)\mathbf{I}+(1-cos(\alpha))\mathbf{nn}^T+sin(\alpha)\begin{pmatrix}0&-n_z&n_y\\n_z&0&-n_x\\-n_y&n_x&0\end{pmatrix}$
• How to prove this magic formula?
  • Checkout the supplementary material on the course website!

View / Camera Transformation

• What is view transformation?
  • Think about how to take a photo
    • Find a good place and arrange people (model transformation)
    • Find a good “angle” to put the camera (view transformation)
    • Cheese! (projection transformation)
• How to perform view transformation?
• Define the camera first
  • Position $\vec{e}$
  • Look-at / gaze direction $\hat{g}$
  • Up direction $\hat{t}$ (assuming perp. to look-at)
• Key observation
  • If the camera and all objects move together, the “photo” will be the same
• How about that we always transform the camera to
  • The origin, up at Y, look at -Z
  • And transform the objects along with the camera