  Given a light source, a spherical reflector, and an observer, where on the
surface of the sphere will the light be directly reflected to the observer,
i.e. where is the the specular point? This is known as the Alhazen-Ptolemy
problem, and finding this specular point for spherical reflectors is useful in
applications ranging from computer rendering to atmospheric modeling to GPS
communications. Existing solutions rely upon finding the roots of a quartic
equation and evaluating numerically which root provides the real specular
point. We offer a formulation, and two solutions thereof, for which the correct
root is predeterminable, thereby allowing the construction of the fully
analytical solutions we present. Being faster to compute, our solutions should
prove useful in cases which require repeated calculation of the specular point,
such as Monte-Carlo radiative transfer, including reflections off of Titan's
hydrocarbon seas.
