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.

36 pages, 9 figures