Comment by MITSardine

I can't help but find this comment a little insulting. It's very similar to saying "if, while, else, malloc. The LLM can figure the rest out!" as if CS were a solved thing and the whole challenge weren't assembling those elementary bricks together in computationally efficient and robust ways.

Also more to the point, I doubt you'll have much success with local optimization on general surfaces if you don't have some kind of tessellation or other spacial structure to globalize that a bit, because you can very easily get stuck in local optima even while doing something as trivial as projecting a point onto a surface. Think of anything that "folds", like a U-shape, a point can be very close to one of the branches, but Newton might still find it on the other side if you seeded the optimizer closer to there. It doesn't matter whether you use vanilla Newton or Newton with tricks up to the gills. And anything to do with matrices will only help with local work as well because, well, these are non-linear things.

"Just work in parameter space" is hardly a solution either, considering many mappings encountered in BREPs are outright degenerate in places or stretch the limits floating point stability. And the same issue with local minima will arise, even though the domain is now convex.

So I might even reduce your list to: Taylor expansion, linear solver. You probably don't need much more than that, the difficulty is everything else you're not thinking of.

And remember, this has to be fast, perfectly robust, and commit error under specified tolerance (ideally, something most CAD shops don't even promise).