  There are non-radial null geodesics emanating from the shell focusing
singularity formed at the symmetric center in a spherically symmetric dust
collapse. In this article, assuming the self-similarity in the region filled
with the dust fluid, we study these singular null geodesics in detail. We see
the time evolution of the angular diameter of the central naked singularity and
show that it might be bounded above by the value corresponding to the circular
null geodesic in the Schwarzschild spacetime. We also investigate the angular
frequency of a physical field which propagates along the singular null geodesic
and find that it depends on the impact parameter. Further, we comment on the
non-uniformity of the topology of the central naked singularity.
