Trouble with the Z axis in Orthographic projection

Question

I've been having trouble understanding why the transformation of the Z coordinate in orthographic projection is the way it is in a right-handed coordinate system.

The "function" for Z using the matrix everyone uses is:

f(z) = -2*z/(far-near) - (far+near)/(far-near)

As far as I know the way orthographic projection works is that it should map the left/right, top/bottom, near/far coordinates to 1/-1.

But if we substitute the near and far coordinates in we get:

f(far) = -2*far/(far-near) - (far+near)/(far-near) = (-3*far - near)/(far - near)
f(near) = -2*near/(far-near) - (far+near)/(far-near) = (-far - 3*near)/(far-near)

Which in most cases won't give back -1 and +1.

Basically my assumption would be that the entire function should be negated (compared to how the X and Y coordinates are computed), but instead the "translation" part stays the same.

Answer 1

The near and far values are actually the distances to the corresponding depth clipping planes and are thus positive values. The coordinate frame to consider is (e_x, e_y, e_z) with e_x pointing to the right, e_y pointing upwards and e_z pointing towards the camera / eye. In this coordinate frame, the z coordinates of the depth clipping planes end up being -near and -far which make the math check out as you noticed.