Specular reflection on the surface of a sphere: compass and ruler constructions

We provide an explicit geometric algorithm involving only ruler and compass constructions in order to specify the specular reflection point on the surface of a reflecting sphere of radius $r$ given two focal points $A$ and $B$ lying outside of it. By numerically implementing the algorithm we compute the point in question for a number of cases. We conclude by discussing how the first iteration of the algorithm constitutes a first order approximation to the real solution by providing a closed expression for it as well as the error involved in doing so, as a function of the distances of the two focal points from the origin and the angle formed between them.