On curves and polygons with the equiangular chord property

Tarik Aougab, Xidian Sun, Serge Tabachnikov and Yuwen Wang

Vol. 274 (2015), No. 2, 305–324

DOI: 10.2140/pjm.2015.274.305

Abstract

Let $C$ be a smooth, convex curve on either the sphere ${\mathbb{S}}^2$, the hyperbolic plane ${ℍ}^2$ or the Euclidean plane ${\mathbb{E}}^2$ with the following property: there exists $\alpha$ and parametrizations $x\left(t\right)$ and $y\left(t\right)$ of $C$ such that, for each $t$, the angle between the chord connecting $x\left(t\right)$ to $y\left(t\right)$ and $C$ is $\alpha$ at both ends.

Assuming that $C$ is not a circle, E. Gutkin completely characterized the angles $\alpha$ for which such a curve exists in the Euclidean case. We study the infinitesimal version of this problem in the context of the other two constant curvature geometries, and in particular, we provide a complete characterization of the angles $\alpha$ for which there exists a nontrivial infinitesimal deformation of a circle through such curves with corresponding angle $\alpha$. We also consider a discrete version of this property for Euclidean polygons, and in this case, we give a complete description of all nontrivial solutions.

To the memory of Eugene Gutkin

Keywords

mathematical billiards, capillary floating problem, geometric rigidity

Mathematical Subject Classification 2010

Primary: 37A45, 37E10, 52A10

Milestones

Received: 5 November 2013

Accepted: 15 July 2014

Published: 1 April 2015