How exactly does the Sigma parameter affect the Velvet shader?

Question

How does the Sigma value of the Cycles Velvet BSDF shader work? Intuitively, it seems to serve as some sort of ceiling value for the shading. How does it affect the light bouncing? I've looked at the ray diagram on the shaders page, which seems to show an away-from-center spread:

Is the sigma value related to the variance (in statistics, σ means variance) of the spread? Comparing the diagram to that of the normal distribution seems to indicate that this may be the case:

This would seem to mean that a σ-value of 0 makes the shader behave like a glossy shader, while a σ-value of 1 would make it behave like a diffuse shader; however, this does not appear to be the case.

I've prepared a video demonstrating a changing σ-value, from 0 to 1, on an actual velvety object for added realism.

Could one of the developers who understands what is going on identify the meaning here, mathematically or otherwise?

Answer 1

It rather seems to define the response curve towards the view angle.

The velvet shader makes a material look brighter the smaller the angle between the view vector and the surface tangent vector is, it's bright if you look on a surface in the tangent direction and dark if you look at it from it's surface normal direction.

With sigma=0, you seem to have to look at a surface perfectly in tangential direction to receive light, while higher values spread it across a wider range of angles so that with sigma=1, the distribution goes all the way to a normal view angle and is relatively even.