Thanks, that the same approach I was suggesting in my other comment in this thread https://news.ycombinator.com/item?id=45628245 . But couldn't find literature specific to the Bezier curves to help break the communication gap, and my specific knowledge of Bezier Curve isn't deep enough.
I am happy to see other people use the approach I consider more natural.
It's a generic global optimization approach and geometry is always full of pathological edge cases, so it's hard to tell if you miss any. Getting to work in the average case is usually easy, but to be sure it always work is much harder.
Your comment motivated me to also comment :) I agree: Bézier curves and b-splines have a lot of rich geometry baked it, so it makes sense to use it, especially if you can avoid second derivatives. I think I misunderstood your comment about the extra control point work to find an initial guess. It looks like we’re saying something pretty close though: you can split the curve smartly to remove the single crossing in this case, which skips all the recursion that I’m suggesting, which is probably better.
Yes the real trouble is true optimality guarantees. I remember that there were edge cases of the above approach needed that a lot of subdivision steps to succeed for general degree curves, so it might end up worse than a more rigorously justified approach in these cases.
But I mostly want to say thanks for everything you do. Your good vibes are deeply appreciated and you are an inspiration.