Pixel-perfect projection matrix in OpenGL?

Question

I'm using OpenTK and I would like to be able to render polygons and textures pixel perfect when their Z is a certain number, anywhere else in the projection doesn't matter (this is for a UI). Is there a way to set up a projection matrix that does this?

Answer 1

First off, forget about that "special" Z value. When you want to draw a UI you switch to a orthographic projection, so that you can use Z for layering your graphics elements.

So the basic problem boils down to finding a orthographic projection that identity maps view space to viewport space. So let's see what OpenGL specifies about that:

If a vertex in clip coordinates is given by

 xc 
 yc 
 zc 
 wc 

then the vertex’s normalized device coordinates are

 xd     xc/wc 
 yd  =  yc/wc 
 zd     zc/wc 

13.6.1 Controlling the Viewport

The viewport transformation is determined by the selected viewport’s width and height in pixels, px and py, respectively, and its center (ox,oy) (also in pixels).

The vertex’s window coordinates are given by

 xw      px·xd/2  +   ox   
 yw  =   py·yd/2  +   oy   
 zw     (f−n)zd/2 + (n+f)/2 

(…) separately for each primitive. The factor and offset applied to zdfor each viewport encoded by n and f are set using glDepthRange… (…)

So there you have it: To become pixel perfect you must find a projection that's exactly the inverse of the composition of vertex clip coordinates to vertex normalized coordinates to vertex window coordinates.

Let w = 1 then you can substitute normalized for clip vertex coordinates and hence

  xw      px·xc/2  +   ox   
  yw  =   py·yc/2  +   oy   
  zw     (f−n)zc/2 + (n+f)/2 

Since we're not interested in the z coordinate we can rewrite this slightly

  xw     px·xc/2 + ox     px/2     0  ox   xc 
  yw  =  py·yc/2 + oy  =     0  py/2  oy   yc  
  ……           …             …     …   …   ……  
   1           1             0     0   1    1 

i.e. we have changed this into a homogenous matrix transformation

v_w = V · v_c

where v_c is the vertex position after projection from eye space to clip space and where V is the viewport transformation matrix

      px/2     0  ox 
 V =     0  py/2  oy 
         …     …   … 
         0     0   1 

So we can rewrite this again as

v_w = V · P · v_e

We're getting close. We want that

v_w = v_e

so V·P must be identity

I = V · P = V · V^-1

Hence we know that the projection matrix P must be the inverse of the viewport matrix

                  px/2     0  ox 
 P = inv V = inv     0  py/2  oy 
                     …     …   … 
                     0     0   1 

      2/px     0  -ox 
   =     0  2/py  -oy 
         …     …    … 
         0     0    1 

For the sake of non singularity we choose the Z row as (0,0,1,0) for vertex eye space Z in the range [0, 1]. So this is the projection matrix to use for pixel perfect mapping from eye space to glViewport(ox, oy, px, py)