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 S2, the hyperbolic plane 2 or the Euclidean plane E2 with the following property: there exists α and parametrizations x(t) and y(t) of C such that, for each t, the angle between the chord connecting x(t) to y(t) and C is α at both ends.

Assuming that C is not a circle, E. Gutkin completely characterized the angles α 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 α for which there exists a nontrivial infinitesimal deformation of a circle through such curves with corresponding angle α. 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