Screen-space orthographic projection

This matrix is used to convert from screen-space (pixel) coordinates to normalized device coordinates (NDC) for Vulkan.

$P=\left[\begin{array}{cccc}\frac{2}{w}& 0& 0& \mathrm{-1}\\ 0& \frac{2}{h}& 0& \mathrm{-1}\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$

This matrix is useful for displaying text in 3D engines.

Where w is the width of the screen in pixels, h is the height of the screen in pixels, this matrix maps X coordinates in the range 0…w to the range -1…1 and Y coordinates in the range 0…h to the same range. The Z-coordinate is discarded since it is unnecessary for displaying text. The W-coordinate is kept the same for the perspective division. Proof.

Here are some examples:

• map from the top-left of the screen to the top-left of NDC
• map from the bottom-right of the screen to the bottom-right of NDC