This is quite a complicated approach that misses out on most of the basic numerical and geometric methods that will make the problem simpler. I would recommend looking outside of recent SIGGRAPH papers. Brushing up on the basics of Bernstein polynomials, B-splines, and rootfinding methods will lead you to develop simpler algorithms which are likely faster.
I'm extremely curious what those basic methods are. We're in the process of replacing the higher order rootfinding in kurbo with a new solver based on Yuksel's method[1]. If you know of simpler, faster techniques that would be quite interesting.
