Additional translation and scaling

After $T_p$ is applied, the $8$ corners of the frustum are transformed into the corners of a rectangular box, shown in Figure 3.19. The following performs a simple translation of the box along the $z$ axis and some rescaling so that it is centered at the origin and the coordinates of its corners are $(\pm 1,\pm 1,\pm 1)$:

$$T_{st} = \begin{bmatrix}\frac{2}{r-\ell} & 0 & 0 & -\frac{r+\ell}... ... 0 & 0 & \frac{2}{n-f} & -\frac{n+f}{n-f} 0 & 0 & 0 & 1 \end{bmatrix} .$$ (3.47)

If the frustum is perfectly centered in the $xy$ plane, then the first two components of the last column become 0. Finally, we define the canonical view transform $T_{can}$ from (3.41) as

$$T_{can} = T_{st} T_p .$$ (3.48)