Very interesting! I explored this method and compared it to Newton's method, you may like it: https://github.com/sharnett/point_to_ellipse/blob/master/poi...

I found a paper using a similar idea (but using a different method to step the parameter) to find the orthogonal projection onto curves. It also converges quadraticaly, and seems a little faster through my experiments. http://www.geometrie.tugraz.at/wallner/sproj.pdf

OP here

This blog post was subject of yesterday's Ask HN (https://news.ycombinator.com/item?id=15110162)

I haven't written something like this up in a long time, so feedback appreciated.

Thank you

Congrats on the find, although I'm not familiar with any of the material presented it's a clear and nicely written Blog Post.