The Ptolemy-Alhazen problem and quadric surface mirror reflection

TL;DR

This paper explores the reflection properties of spherical and quadric surface mirrors, deriving algebraic solutions for the classical Alhazen problem and analyzing caustic curves related to elliptical reflections.

Contribution

It introduces an algebraic equation to solve Alhazen's problem for quadric mirrors and links elliptical reflection properties to caustic curve formation.

Findings

Derived algebraic solution for Alhazen's problem on quadric mirrors

Connected ellipse reflection properties to caustic curves

Provided insights into mirror reflection behavior on quadric surfaces

Abstract

We discuss the problem of the reflection of light on spherical and quadric surface mirrors. In the case of spherical mirrors, this problem is known as the Alhazen problem. For the spherical mirror problem, we focus on the reflection property of an ellipse, and show that the catacaustic curve of the unit circle follows naturally from the equation obtained from the reflection property of an ellipse. Moreover, we provide an algebraic equation that solves Alhazen's problem for quadric surface mirrors.