  We consider the control problem where, given an orthonormal tangent frame in
the hyperbolic plane or three dimensional hyperbolic space, one is allowed to
transport the frame a fixed distance $r > 0$ along the geodesic in direction of
the first vector, or rotate it in place a right angle. We characterize the
values of $r > 0$ for which the set of orthonormal frames accessible using
these transformations is discrete.
  In the hyperbolic plane this is equivalent to solving the discreteness
problem for a particular one parameter family of two-generator subgroups of
$PSL_2(\mathbb{R})$. In the three dimensional case we solve this problem for a
particular one parameter family of subgroups of the isometry group which have
four generators.
