  We describe the methodology and compute the illumination of geometrically
thin accretion disks around black holes of arbitrary spin parameter $a$ exposed
to the radiation of a point-like, isotropic source at arbitrary height above
the disk on its symmetry axis. We then provide analytic fitting formulae for
the illumination as a function of the source height $h$ and the black hole
angular momentum $a$. We find that for a source on the disk symmetry axis and
$h/M > 3$, the main effect of the parameter $a$ is allowing the disk to extend
to smaller radii (approaching $r/M \to 1$ as $a/M \to 1$) and thus allow the
illumination of regions of much higher rotational velocity and redshift. We
also compute the illumination profiles for anisotropic emission associated with
the motion of the source relative to the accretion disk and present the
fractions of photons absorbed by the black hole, intercepted by the disk or
escaping to infinity for both isotropic and anisotropic emission for $a/M=0$
and $a/M=0.99$. As the anisotropy (of a source approaching the disk) increases
the illumination profile reduces (approximately) to a single power-law, whose
index, $q$, because of absorption of the beamed photons by the black hole,
saturates to a value no higher than $q \gtrsim 3$. Finally, we compute the
fluorescence Fe line profiles associated with the specific illumination and
compare them among various cases.
