I think this comment section would be a good place to ask for help with a related problem.

I want to design a "smooth" closed path that fits in a square, as long as I can make it. It needs to smooth in such a way that a constant velocity can be maintained subject to limits on acceleration and jerk. The point is to develop a test for maximum flow rate on a filament 3D printer without the motion system ever slowing the tool-head down. (In reality, the standard smoothed "E" shape is good enough. This is more of an exercise.)

I know roads are designed to limit acceleration and jerk, and someone knowledgeable about road design would know about finding curves with constraints on acceleration and jerk.

Do you know any free tools or resources on theory that I could use?