Why depth values must be interpolated directly by barycentric coordinates in openGL?

Question

OpenGL spec:

It says: However, depth values for polygons must be interpolated by (14.10).

Why? Are the z coordinates depth values in camera space? If so, we should use perspective correctly barycentric coordinates to interpolate them, isn't it?(like equation 14.9)

Update:
So the z coordinates are NDC coordinates(which already divided by w). I have a small demo which implement a rasterizer. When I use linear interpolation of the NDC z coordinates, the result is a bit unusual(image below). While I use perspective correctly interpolation of camera z coordinates, the result is ok.

This is the perspective projection matrix I use:

Answer 1

Why? Are the z coordinates depth values in camera space? If so, we should use perspective correctly barycentric coordinates to interpolate them, isn't it?

No, they are not. They are in window space, meaning they already have been divided by w. It is correct that if you wanted to interpolate camrea space z, you would have to apply perspective correction. But for NDC and window space Z this would be wrong - after all, the perspective transformation (as achieved by perspective projection matrix and perspective divide) still maps straight lines to straight lines, and flat trinagles to flat triangles. That's why we use the hyperbolically distorted Z values as depth in the first place. This is also a property that is exploited for the hierarchical depth test optimization. Have a look at my answer here for some more details, including a few diagrams.