I'm partial to the piriform curve [1], which I call the "gumdrop curve." I had a lot of fun trying to make a game where the characters were gumdrops. I could change either parameter (width or height) while keeping area constant, so I could have procedural animations of them stretching to give the impression of falling from a great height and such. (Keeping area constant is a basic principle in making animations believable.) I didn't end up finishing the game but I learned a lot about programming UIs and using canvas-like interfaces.

If you turn up such a paper I'd love to take a look. It does seem like something that you could solve with gradient descent once you've guessed the right family of curves (eg Lissajous curves).

[1] https://mathworld.wolfram.com/PiriformCurve.html

A true classic.