Assume a parametric curve can be denoted by P(t), any attribute T can be interpolated linearly with:
T = (1 - t) * T0 + t * T1 (1)
T0 is the attribute at t=0, and T1 is the attribute at t=1.
Differentiate equation (1) we have:
dTdx = (T1 - T0) * dtdx (2)
Now the problem turns into how to compute dtdx.
Using Chain Rule, we have:
dPdt * dtdx = dPdx (3)
We already know how to compute dPdx as described in Tracing Ray Differentials. Homan Igehy. Computer Science Department. Stanford University., so we solve dtdx by:
dtdx = dPdx / dPdt (4)
dPdt can also be easily computed since it's just the tangent of the curve. Substituting (4) into (2) we got the final form:
dTdx = (T1 - T0) * dPdx / dPdt (5)
This is the ray differential of any curve attribute T along raster x.
