I am attempting to approximate intersections of Bézier curves using iterative clipping.
This common method is described here and here.
It basically works like this:
Find bounding lines outside one curve.
Clip the other curve to these lines.
Alternate and repeat.
There are some extra cases to handle (multiple intersections), but that's the idea.
I get stuck on step (2). In a one sentence explanation, both sources say to use de Casteljau's algorithm to clip the curve $P$ to between $t_1$ and $t_2$.
Bézier clipping is completed by subdividing $P$ twice using the de Casteljau algorithm, such that portions of $P$ over parameter values $t < 0.25$ and $t > 0.75$ are removed.
I understand how to use de Casteljau's to split a curve on a single $t$ value, but not multiple.
How is this intended to work?
Lets call the original curve $C1$, and suppose we want to clip it to the parameter interval $[0.25, 0.75]$. We do this by applying de Casteljau splitting twice.
First, split $C1$ at $t=0.75$. Keep the "left" portion (the one corresponding to $[0,0.75]$) and throw away the right portion. Call the resulting curve $C2$.
Now we want to split $C2$. The split has to happen at the point $P$ that corresponds to $t=0.25$ on the original curve $C1$. The only subtlety is that the point $P$ corresponds to $t=\tfrac13$ on the curve $C2$, because $0.25 = \tfrac13*0.75$. So, split $C2$ at $t=\tfrac13$, and keep the right-hand portion, $C3$. The curve $C3$ is the one you want.