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Snell meets Fagnano. Path optimization through an imperfect mirror

Maxim Arnold, Jaewoo Park

TL;DR

The paper generalizes Fagnano's minimal-perimeter problem to a weighted, Snell-refraction setting inside a triangle. It proves that every 3-periodic Snell billiard trajectory corresponds to the pedal triangle of a Snell-Fagnano point F_lambda, with its isogonal conjugate F_lambda^* carrying tripolar coordinates tied to the weights. It provides an elementary geometric construction of F_lambda, along with interiority criteria and a Ceva-type framework via Apollonian circles. The work lays groundwork for extensions to other polygons and geometries and highlights rich connections between triangle geometry, billiards, and weighted optimization.

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.

Snell meets Fagnano. Path optimization through an imperfect mirror

TL;DR

The paper generalizes Fagnano's minimal-perimeter problem to a weighted, Snell-refraction setting inside a triangle. It proves that every 3-periodic Snell billiard trajectory corresponds to the pedal triangle of a Snell-Fagnano point F_lambda, with its isogonal conjugate F_lambda^* carrying tripolar coordinates tied to the weights. It provides an elementary geometric construction of F_lambda, along with interiority criteria and a Ceva-type framework via Apollonian circles. The work lays groundwork for extensions to other polygons and geometries and highlights rich connections between triangle geometry, billiards, and weighted optimization.

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.

Paper Structure

This paper contains 11 sections, 10 theorems, 20 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Snell--Fagnano Problem.
  3. Existence of the Snell-Fagnano point.
  4. Main construction.
  5. Discussion and further directions.
  6. Other polygons.
  7. Other geometries.
  8. Outer billiards.
  9. Contractive and dispersive Snell billiards.
  10. Acknowledgements
  11. Conversion from tripolar to barycentric coordiantes.

Key Result

Lemma 1

Let $X$ be the point on the river bank, minimizing the weighted sum Then the angles between the normal to the river bank at $X$ and the rays $AX$ and $XB$ satisfy Snell's law of refraction (see the right panel of Fig. fig: river):

Figures (6)

  • Figure 1: Left: River access optimiaztion problem. Right: Snell's law of refraction.
  • Figure 2: Left: Fagnano orbit. Right: Unwrapping argument.
  • Figure 3: Snell-Fagnano point.
  • Figure 4: Left: Apollonian Circle $\odot_{A,B}(\lambda)$. Right: $F^*$ is the isogonal conjugate of $F$ with respect to triangle $ABC$
  • Figure 5: Proof of the Lemma \ref{['lm:Akopyan']}.
  • ...and 1 more figures

Theorems & Definitions (30)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 1
  • Remark 2
  • Definition 1
  • Definition 2
  • ...and 20 more