OpenGL How to calculate worldspace coordinates from frustum aligned vectors?

Question

I am a graphics programming beginner working on my own engine and tried to implement frustum-aligned volume rendering.

The idea was to render multiple planes as vertical slices across the view frustum and then use the world coordinates of those planes for procedural volumes.

Rendering the slices as a 3d model and using the vertex positions as worldspace coordinates works perfectly fine:

//Vertex Shader  
gl_Position = P*V*vec4(vertexPosition_worldspace,1);
coordinates_worldspace = vertexPosition_worldspace;

Result:

However rendering the slices in frustum-space and trying to reverse engineer the world space coordinates doesent give expected results. The closest i got was this:

//Vertex Shader
gl_Position = vec4(vertexPosition_worldspace,1);
coordinates_worldspace = (inverse(V) * inverse(P) * vec4(vertexPosition_worldspace,1)).xyz;

Result:

My guess is, that the standard projection matrix somehow gets rid of some crucial depth information, but other than that i have no clue what i am doing wrong and how to fix it.

Answer 1

Well, it is not 100% clear what you mean by "frustum space". I'm going to assume that it does refer to normalized device coordinates in OpenGL, where the view frustum is (by default) the axis-aligned cube -1 <= x,y,z <= 1. I'm also going to assume a perspective projection, so that NDC z coordinate is actually a hyperbolic function of eye space z.

My guess is, that the standard projection matrix somehow gets rid of some crucial depth information, but other than that i have no clue what i am doing wrong and how to fix it.

No, a standard perspective matrix in OpenGL looks like

( sx   0   tx   0  ) 
(  0  sy   ty   0  )
(  0   0    A   B  )
(  0   0   -1   0  )

When you multiply this by a (x,y,z,1) eye space vector, you get the homogenous clip coordinates. Consider only the last two lines of the matrix as separate equations:

z_clip = A * z_eye + B
w_clip = -z_eye

Since we do the perspective divide by w_clip to get from clip space to NDC, we end up with

z_ndc = - A - B/z_eye

which is actually the hyperbolically remapped depth information - so that information is completely preserved. (Also note that we do the division also for x and y).

When you calculate inverse(P), you only invert the 4D -> 4D homogenous mapping. But you will get a resulting w that is not 1 again, so here:

coordinates_worldspace = (inverse(V) * inverse(P) * vec4(vertexPosition_worldspace,1)).xyz;
                                                                                       ^^^

lies your information loss. You just skip the resulting w and use the xyz components as if it were cartesian 3D coordinates, but they are 4D homogenous coordinates representing some 3D point.

The correct approach would be to divide by w:

vec4 coordinates_worldspace = (inverse(V) * inverse(P) * vec4(vertexPosition_worldspace,1));
coordinates_worldspace /= coordinates_worldspace.w