Viewport transform

The last transform to be applied in the chain (3.41) is the viewport transform $T_{vp}$. After $T_{can}$ has been applied, the $x$ and $y$ coordinates each range from $-1$ to $1$. One last step is required to bring the projected points to the coordinates used to index pixels on a physical display. Let $m$ be the number of horizontal pixels and $n$ be the number of vertical pixels. For example, $n = 1080$ and $m = 1920$ for a 1080p display. Suppose that the display is indexed with rows running from 0 to $n-1$ and columns from 0 to $m-1$. Furthermore, $(0,0)$ is in the lower left corner. In this case, the viewport transform is

$$T_{vp} = \begin{bmatrix}\frac{m}{2} & 0 & 0 & \frac{m-1}{2} 0 ... ...rac{n}{2} & 0 & \frac{n-1}{2} 0 & 0 & 1 & 0 0 & 0 & 0 & 1 \end{bmatrix} .$$ (3.49)