Another similar "hidden but intuitive" property is higher order geometric curvature continuity. For example squircles/superellipses have more smoothly changing curve than naive rounded rectangle, or industrial design using Gn continuity/class A surfaces:
https://en.wikipedia.org/wiki/Class_A_surface
https://www.johndcook.com/blog/2018/02/13/squircle-curvature...
I do see quite clear parallels between higher order time derivates and these higher order curvature measures, although I don't know if there is any formal relation here