#### Vol. 274, No. 2, 2015

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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
##### 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
##### Authors
 Tarik Aougab Department of Mathematics Yale University 10 Hillhouse Avenue New Haven, CT 06510 United States Xidian Sun Department of Mathematics Wabash College 301 West Wabash Avenue Crawfordsville, IN 47933 United States Serge Tabachnikov Department of Mathematics Penn State University University Park, PA 16802 United States Institute for Computational and Experimental Research in Mathematics Brown University Box 1995 Providence, RI 02912 United States Yuwen Wang Department of Mathematics Cornell University 310 Malott Hall Ithaca, NY 14853 United States