  We show that a bijection $f:\mathbb{H}^2\rightarrow\mathbb{H}^2$ of the
hyperbolic plane that sends horocycles to horocycles (respectively hypercycles
to hypercycles) is an isometry. This extends a previous result of J. Jeffers on
geodesics to all curves with constant curvature in $\mathbb{H}^2$. We go beyond
by showing that every abstract automorphism of the geodesic graph (respectively
horocycles and hypercycles graphs) is induced by an earthquake map
(respectively an isometry) of $\mathbb{H}^2$. This shadowed the difference
between the geometry of geodesics and that of horocycles/hypercycles.
