Tumbling Downhill along a Given Curve

A cylinder will roll down an inclined plane in a straight line. A cone will wiggle along a circle on that plane and then will stop rolling.

We ask the inverse question: For which curves drawn on the inclined plane $\real^2$ can one chisel a shape that will roll downhill following precisely this prescribed curve and its translationally repeated copies?

This is a nice, and easy to understand problem, but the solution is quite interesting.

(Based on work mostly with Y. Sobolev and T. Tlusty. After a Nature paper, Solid-body trajectoids shaped to roll along desired pathways, August 2023.)