GAMES101 Lecture 04 - Transformation Cont.

I. Rodrigues' Rotation Formula

$\alpha$ $\textbf{n}$

\begin{matrix} R (n, α) = \cos (α) I + (1 - \cos (α)) n n^{T} + \sin (α) \underset{N}{\underset{⏟}{[\begin{matrix} 0 & - n_{z} & n_{y} \\ n_{z} & 0 & - n_{x} \\ - n_{y} & n_{x} & 0 \end{matrix}]}} \end{matrix}

Model-Viewing-Projection Transformation.

View/Camera Transformation: Converts points in canonical coordinates (or world coordinates) to camera coordinates (or places them in camera space).
- Camera Space - Eye Space
Projection Transformation: Moves points from camera space to the canonical view volume.
- Sometimes called Viewing Transformation
- Canonical View Volume - Clip Space or Normalized Device Coordinates
Viewport Transformation: Maps the canonical view volume to screen space.
- Screen Space - Pixel Coordinates

Let the following attributes be that of the camera:

The target of the view transformation is to transform the coordinates such that:

$x$ $\hat{g} \times \hat{t}$ $y$ $\hat{t}$ $z$ $-\hat{g}$ . (Eases the deduction)

$[-1, 1]^3$ .

\begin{matrix} M_{ortho} = [\begin{matrix} \frac{2}{r - l} & 0 & 0 & 0 \\ 0 & \frac{2}{t - b} & 0 & 0 \\ 0 & 0 & \frac{2}{n - f} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 & 0 & - \frac{r + l}{2} \\ 0 & 1 & 0 & - \frac{t + b}{2} \\ 0 & 0 & 1 & - \frac{n + f}{2} \\ 0 & 0 & 0 & 1 \end{matrix}] \end{matrix}

\begin{matrix} M_{ortho} = [\begin{matrix} \frac{2}{r - l} & 0 & 0 & - \frac{r + l}{r - l} \\ 0 & \frac{2}{t - b} & 0 & - \frac{t + b}{t - b} \\ 0 & 0 & \frac{2}{n - f} & - \frac{n + f}{n - f} \\ 0 & 0 & 0 & 1 \end{matrix}] \end{matrix}

$z_n > z_f$ by convention.

If such an orthographic projection happens after applying a perspective projection, then the matrix can be simplified to:

\begin{matrix} M_{ortho} = [\begin{matrix} \frac{2}{r - l} & 0 & 0 & 0 \\ 0 & \frac{2}{t - b} & 0 & 0 \\ 0 & 0 & \frac{2}{n - f} & - \frac{n + f}{n - f} \\ 0 & 0 & 0 & 1 \end{matrix}] \end{matrix}

\begin{matrix} M_{persp} = M_{ortho} M_{persp \to ortho} = [\begin{matrix} \frac{2 n}{r - l} & 0 & - \frac{r + l}{r - l} & 0 \\ 0 & \frac{2 n}{t - b} & - \frac{t + b}{t - b} & 0 \\ 0 & 0 & \frac{n + f}{n - f} & - \frac{2 n f}{n - f} \\ 0 & 0 & 1 & 0 \end{matrix}] \end{matrix}

where

\begin{matrix} M_{persp \to ortho} = [\begin{matrix} n & 0 & 0 & 0 \\ 0 & n & 0 & 0 \\ 0 & 0 & n + f & - n f \\ 0 & 0 & 1 & 0 \end{matrix}] \end{matrix}