  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.
