# The Ptolemy-Alhazen problem and spherical mirror reflection

## TL;DR

This paper investigates the classical Ptolemy-Alhazen problem involving spherical mirror reflection, solving the related quartic equation using symbolic computation, and connecting it to modern optics and mathematical billiards.

## Contribution

It provides a detailed algebraic solution to the Ptolemy-Alhazen problem and explores its relevance to contemporary topics like ray-tracing and electromagnetic scattering.

## Key findings

- Solved the quartic equation analytically using symbolic software.
- Connected classical optics problem to modern mathematical and physical applications.
- Enhanced understanding of spherical mirror reflection in both historical and modern contexts.

## Abstract

An ancient optics problem of Ptolemy, studied later by Alhazen, is discussed. This problem deals with reflection of light in spherical mirrors. Mathematically this reduces to the solution of a quartic equation, which we solve and analyze using a symbolic computation software. Similar problems have been recently studied in connection with ray-tracing, catadioptric optics, scattering of electromagnetic waves, and mathematical billiards, but we were led to this problem in our study of the so-called triangular ratio metric.