Tumbling Downhill along a Given Curve

• Published in 2024
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.

Other information

key
TumblingDownhillalongaGivenCurve
type
article
2024-07-24
date_published
2024-07-24

BibTeX entry

@article{TumblingDownhillalongaGivenCurve,
key = {TumblingDownhillalongaGivenCurve},
type = {article},
title = {Tumbling Downhill along a Given Curve},
author = {Jean-Pierre Eckmann and Yaroslav I. Sobolev and Tsvi Tlusty},
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.},
comment = {},
}