How to understand the expressiong in vulkan spec about perspective interpolation

[willdla]

in specs/1.3/html/chap25.html#triangle_perspective_interpolation

First,
f = {{ a {f_a / w_a} + b {f_b / w_b} + c {f_c / w_c} } \over
{ {a / w_a} + {b / w_b} + {c / w_c} }}
what is f stands for, in my understanding f can stands for normal or z or anything else that need interpolation of the fragment, is that right?

Second, but in spec we see z = a za + b zb + c zc
in my mind depth interpolation should be 1/P.z = a * 1/V0.z + b * 1/V1.z + c * 1/V2.z
why spec say Depth values for triangles must be determined using linear interpolation?

help, can not understand the two questions above.

[willdla]

Questiong 1 formulation here

what is f stands for, in my understanding f can stands for normal or z or anything else that need interpolation of the fragment, is that right?

[willdla]

Question 2 spec saying here

in my mind depth interpolation should be 1/P.z = a * 1/V0.z + b * 1/V1.z + c * 1/V2.z
why spec say its linear interpolation?

[krOoze]

@willdla

what is f stands for

⇒

@teh Specification

The value of an associated datum f for a fragment produced by rasterizing a triangle, whether it be a shader output or the clip w coordinate

[…]

The NoPerspective and Flat interpolation decorations can be used with fragment shader inputs to declare how they are interpolated. When neither decoration is applied, perspective interpolation is performed as described above.

“Shader output” is whatever you put in the out block in the shader (in a shader stage that preceeds the rasterization stage). Takes the shader outputs f_a, f_b, and f_c and during rasterization stage produce value f as a shader input for the fragment shader.

@willdla

in my mind depth interpolation should be 1/P.z = a * 1/V0.z + b * 1/V1.z + c * 1/V2.z

That’s linear over 1/z. Depth in screenspace is sort of a 1/z after projection. You can find plentiful articles online in context of DirectX or OpenGL about this searching for something like “linear vs non-linear depth”.

[willdla]

did you mean that z in vulkan spec is 1/z not z?
Vulkan spec says:
Depth values for triangles must be determined using linear interpolation:

[krOoze]

Well, you provide z_a, z_b, and z_c, so z is whatever you make it to be. Spec just says it will assume that it is linear and interpolate it as such. For depth buffer, it just matters that it is order preserving. Anyway…

With perspective division, z_NDC = z_clip / w_clip. In typical perspective projection matrix, you would have w_clip = z_world. And you would have something like z_clip = (far × z_world - far × near) / (far - near). So if you divide that by the w_clip, you get:
z_NDC = -1/z_world × far × near/(far - near) + far/(far - near).