Solving the Alhazen-Ptolemy Problem: Determining Specular Points on Spherical Surfaces for Radiative Transfer of Titan's Seas

TL;DR

This paper presents fully analytical, faster solutions for the Alhazen-Ptolemy problem, enabling efficient determination of specular points on spherical surfaces, with applications in radiative transfer modeling of Titan's seas.

Contribution

It introduces new analytical solutions that predetermine the correct root for the specular point, improving computational speed and accuracy over existing numerical methods.

Findings

Solutions are faster than numerical root-finding methods.

Analytical formulation allows for deterministic identification of the specular point.

Applicable to Monte-Carlo radiative transfer simulations of Titan's seas.

Abstract

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.