Periodical reflection conditions in a sphere. A perfect mirror covered the inside surface of a sphere  (assumption: there is no any loss during reflection and reflections continue endless) and there is a very small laser on point $A$ in the surface of the sphere and the direction of the laser light goes to inside and  reflects from other point $B$ in the surface of the sphere.
What are the whole possible mathematical conditions to get periodical reflections. (Passing again from $A$ to $B$) 

Thanks a lot for answers

Without loss of generality, we can rotate such that $A$ is at the top and the original laser beam lies in the $x$-$z$ plane. Then all the reflected beams will also lie in that plane, and the reflections will all take place on a great circle. Thus the problem is in fact two-dimensional, and we're just asking how a laser beam must be pointed in a circle such that it hits the same spot again. Since all the reflections are at the same angle, the laser beam traverses the same angle $\alpha$ between any two reflections, so the answer is that it must traverse a rational multiple of $\pi$ between any two reflections. Since the angle of incidence it forms with the surface normal on the sphere is $(\pi-\alpha)/2$, an equivalent condition is that the angle of incidence must be a rational multiple of $\pi$.
