On curves and polygons with the equiangular chord property

Let C be a smooth, convex curve on either the sphere S², the hyperbolic plane H² 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.