  Reflections from hypersurfaces act by symplectomorphisms on the space of
oriented lines with respect to the canonical symplectic form. We consider an
arbitrary $C^{\infty}$-smooth hypersurface $\gamma\subset\mathbb R^{n+1}$ that
is either a global strictly convex closed hypersurface, or a germ of
hypersurface. We deal with the pseudogroup generated by compositional ratios of
reflections from $\gamma$ and of reflections from its small deformations. In
the case, when $\gamma$ is a global convex hypersurface, we show that the
latter pseudogroup is dense in the pseudogroup of Hamiltonian diffeomorphisms
between subdomains of the phase cylinder: the space of oriented lines
intersecting $\gamma$ transversally. We prove an analogous local result in the
case, when $\gamma$ is a germ. The derivatives of the above compositional
differences in the deformation parameter are Hamiltonian vector fields
calculated by Ron Perline. To prove the main results, we find the Lie algebra
generated by them and prove its $C^{\infty}$-density in the Lie algebra of
Hamiltonian vector fields. We also prove analogues of the above results for
hypersurfaces in Riemannian manifolds.
