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)$:

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

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



Steven M LaValle 2016-12-31