Let’s look at the super cool ISOCHRONOUS CURVE!
This video does a good job of demonstrating that the periodic motion of an object on a (frictionless) isochronous curve has a period that is independent of where the object starts on the curve. It’s a pretty cool little thing.
Edit: OH MY GOD, A WHOLE WEBSITE ABOUT CURVES AND SHAPES
(I’m going to link to one of the pages, ‘cause it looks like the majority of the site is in French, but some of it has been translated to English. I’ve clicked through a lot of the curves using the links at the bottom and I’ve only hit English pages, so if you want to look at some curves, that might be the best way to go (unless you know French)).