On curves and polygons with the equiangular chord property

Abstract

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

Revision: better figures, acknowledgments added

References in corpus (2)

• Capillary floating and the billiard ball problem
• On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces

Cited by in corpus (1)

• Magnetic billiards: Non-integrability for strong magnetic field; Gutkin type examples