Help to understand Bézier outline shader in d2d1

[Giovanni Mascellani]

Hi,

I am trying to understand how the vertex shader for the Bézier curve
outline works in d2d1. The relevant code is the bytecode in
vs_code_bezier_outline[] in dlls/d2d1/device.c.

In particular, I have problems with this snippet:
>         if (abs(dot(mul(rt, p1), v_p)) < 1.0f)
>         {
>             o.b.xzw = float3(0.0f, 0.0f, 0.0f);
>             o.b.y = dot(mul(rt, p), v_p);
>         }
>         else
>         {
>             o.b.zw = sign(dot(mul(rt, p1), v_p)) * v_p;
>             v_p = -float2(-p.y, p.x) / dot(float2(-p1.y, p1.x), p2);
>             o.b.x = dot(v_p, p1 - 0.5f * p2);
>             o.b.y = dot(v_p, p1);
>         }

The idea as I understand is that o.b.xy should contain the u and v
coordinates over which the implicit curve u^2-v is evaluated. They are
defined to be (0, 0), (1/2, 0) and (1,1) on the vertices of the
"original" Bézier triangle, as in the filling case. Since now the
original triangle has to be enlarged by a stroke width, the u and v
coordinates should be extended by linearity, I believe.

However, I don't see why this is what is happening here. To extend by
linearity one should invert the 2x2 matrix given by p1-p0 and p2-p0 or
something similar. And I see no reason to have an "if" branch.

Also, I completely do not understand what the coordinates o.b.zw are
for. On the other hand, I understand (or at least I think I do) the
screen-space derivative trick used to stroke a line of the right width.
It is probably a rather rough approximation, but it is clever!

If someone (probably Henri Verbeet, I think) can help to understand the
missing bits I would be very thankful!

Giovanni.