Usually, if we want to interpolate some vertex attributes, say $A_1$, $A_2$ and $A_3$, we have to first divide them by their corresponding $z$ in view space, then we interpolate them using barycentric weights and finally we multiply them by the interpolated $z$ at the current pixel. To interpolate this $z$, we have to inverse the 3 $z$ of each vertex, then interpolate them and finally inverse the result to have the interpolated $z$.
In other words, the interpolated attribute $A$ is equal to: $$ A = z \left(w_0 \times \frac{A_1}{Z_1} + w_1 \times \frac{A_2}{Z_2} + w_2 \times \frac{A_3}{Z_3}\right) $$
where $w_i$ are the barycentric weights, $Z_i$ are the depths of each vertex and $z$ is the interpolated depth: $$ z = \frac{1}{w_0 \times \frac{1}{Z_1} + w_1 \times \frac{1}{Z_2} + w_2 \times \frac{1}{Z_3}} $$
Now my question is: instead of going back and forth between attributes and their reciprocals, can we just use the vertices'$z$ in normalized device coordinates, that is, the $z$s that have been divided by $w$, after the perspective projection?
If I understand correctly, we interpolate the reciprocal of vertex attributes, because it's linear in screen space, but what about the $z$ in NDC space, after the homogeneous divide? If we can manage to use it, we already have a $z$ that we can linearly interpolate, because it has been homogenized?
In the book "Game Engine Architecture", page 667, they say:
with w-buffering, we cannot linearly interpolate depths directly. Depths must be inverted prior to interpolation and the re-inverted prior to being stored the w-buffer.
In other words, we don't need to invert $z$ back and forth to find the interpolated depth: we just have to use the $z$ in NDC space, that has been divided by view-space $z$. So we can directly interpolate the $z$ in NDC space to have perspective-correct depth, but how can I use this interpolated depth to interpolate other vertex attributes?
