The most up-to-date version of the associated paper, entitled Solving the Alhazen-Ptolemy Problem: Determining Specular Points on Spherical Surfaces for Radiative Transfer of Titan's Seas, can be found here.

Implementations of the solutions for determining the specular point on a spherical surface given an arbitrary spatial configuration of source and observer (known as the Alhazen-Ptolemy problem).

Alhazen-Ptolemy.c++ has three use modes: benchmarking, one-finite, and both-finite.

The benchmarking usage is

alhazen-ptolemy

The one-finite usage is

alhazen-ptolemy <observer_angle> <c>

And the both-finite usage is

alhazen-ptolemy <observer_angle> <c> <b>

observer_angle corresponds to θ_obs from the paper (or θ_src, if the observer was rotated to π / 2 instead). Note that the branch-deductions assume the point with the smaller radius has been rotated to π / 2. observer_angle is specified in degrees as the angle from the positive x-axis. b is the ratio between the radius of the sphere and the radius of either the observer or the source -- whichever is larger -- and c is the other ratio. In the associated paper, b is the ratio between the radius of the sphere and the radius of the observer, i.e. R_sph / R_obs, and c is the ratio between the radius of the sphere and the radius of the source, i.e. R_sph / R_src.

alhazenptolemy.py has the same usage as Alhazen-Ptolemy.c++ except it does not feature any benchmarking. This directory is also an installable python module. Installing with

python -m pip install Alhazen-Ptolemy

or

pip install Alhazen-Ptolemy

will make the alhazenptolemy.py module importable via

import alhazenptolemy