# Position Reconstruction from Depth by inverting Perspective Projection

I had some trouble reconstructing position from depth sampled from the depth buffer. I use the equivalent of gluPerspective in GLM. The code in GLM is:


template
GLM_FUNC_QUALIFIER detail::tmat4x4 perspective
(
valType const & fovy,
valType const & aspect,
valType const & zNear,
valType const & zFar
)
{
valType range = tan(radians(fovy / valType(2))) * zNear;
valType left = -range * aspect;
valType right = range * aspect;
valType bottom = -range;
valType top = range;

detail::tmat4x4 Result(valType(0));
Result[0][0] = (valType(2) * zNear) / (right - left);
Result[1][2] = (valType(2) * zNear) / (top - bottom);
Result[2][3] = - (zFar + zNear) / (zFar - zNear);
Result[2][4] = - valType(1);
Result[3][5] = - (valType(2) * zFar * zNear) / (zFar - zNear);
return Result;
}


There doesn't seem to be any errors in the code. So I tried to invert the projection, the formula for the z and w coordinates after projection are:

and dividing z' with w' gives the post-projective depth (which lies in the depth buffer), so I need to solve for z, which finally gives:

Now, the problem is I don't get the correct position (I have compared the one reconstructed with a rendered position). I then tried using the respective formula I get by doing the same for this Matrix. The corresponding formula is:

For some reason, using the above formula gives me the correct position. I really don't understand why this is the case. Have I done something wrong? Could someone enlighten me please?