Thank you! Do we have a guarantee that these subcurves are solvable with Newton's method? The approach with derivatives has this because we know there is one crossing, and also clipping them to the zero-derivatives makes sure there won't be multiple curve "pits".

The trick is that you only solve Newton on a single subcurve: you recursively find the closest control point on all subcurves, split that subcurve and repeat until the closest control point doesn’t move much, or just however many subdivision steps work in practice. So the last curve that you apply Newton to should be smooth enough to succeed. I think there are edge cases with cusps, but I can’t exactly remember the theoretical guarantees anymore.

I think this is the main reference for this algorithm (should be able to search for the pdf): https://www.sciencedirect.com/science/article/abs/pii/S01678...