Snell meets Fagnano. Path optimization through an imperfect mirror

TL;DR

This paper explores a modified Fagnano problem involving weighted perimeter minimization, linking classical billiard trajectories with Snell billiards to find optimal inscribed triangles.

Contribution

It introduces a new variant of the Fagnano problem incorporating weights, connecting it to Snell billiards and path optimization.

Findings

Identifies minimal weighted perimeter inscribed triangles

Establishes connection between Fagnano problem and Snell billiards

Provides a framework for path optimization with weights

Abstract

The renowned Fagnano problem asks for the inscribed triangle of minimal perimeter within a given reference triangle. Equivalently, it seeks a billiard trajectory inside the triangle that closes after three reflections. In this note, we consider a modification of this classical problem: finding the inscribed triangle of minimal weighted perimeter -- or, equivalently, the periodic trajectory of a Snell billiard.