Tumbling Downhill along a Given Curve

TL;DR

This paper investigates the conditions under which a shape can be carved to roll downhill along a prescribed curve on an inclined plane, revealing that such shapes generally exist and often require multiple repetitions of the curve.

Contribution

It provides a theoretical framework for designing shapes that roll along specific curves on inclined planes, addressing a classical inverse problem in geometric mechanics.

Findings

Solutions exist for most curves on the plane.

Most shapes return to initial orientation after crossing two copies of the curve.

Some curves require arbitrarily many copies for the shape to realign.

Abstract

A cylinder will roll down an inclined plane in a straight line. A cone will roll around a circle on that plane and then will stop rolling. We ask the inverse question: For which curves drawn on the inclined plane $\mathbb{R}^2$ can one carve a shape that will roll downhill following precisely this prescribed curve and its translationally repeated copies? This simple question has a solution essentially always, but it turns out that for most curves, the shape will return to its initial orientation only after crossing a few copies of the curve - most often two copies will suffice, but some curves require an arbitrarily large number of copies.