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Outer length billiards on polygons

Lael Edwards-Costa

TL;DR

The paper introduces the outer length billiard on polygons as a variational, length-extremizing analogue of outer billiards and investigates far-field dynamics. It develops a polygonal framework (singularity diagrams, 2-gon and triangle cases), proves a universal far-field confinement for distant orbits, and constructs a circle-center map that ties center dynamics to the symplectic polar dual, establishing structural links to the usual outer billiard. It proves that every triangle has a 3-periodic outer length orbit and provides extensive experimental evidence, including a square-specific conjecture for an escaping orbit and explicit piecewise-map decompositions. The work broadens polygonal outer billiard theory, complements smooth-table results, and connects geometric constructions with duality concepts in a dynamical setting.

Abstract

The classical inner and outer billiards can be formulated in variational terms, with length and area as the respective generating functions. The other two combinations, ``inner with area'' and ``outer with length,'' are more recently described. Here, we consider the latter system in the special case of polygonal tables. We describe the behavior of orbits far away from the table and pose conjectures regarding an escaping orbit when the table is a square. This paper is meant to complement arXiv:2510.08370 which handles smooth tables with positive curvature.

Outer length billiards on polygons

TL;DR

The paper introduces the outer length billiard on polygons as a variational, length-extremizing analogue of outer billiards and investigates far-field dynamics. It develops a polygonal framework (singularity diagrams, 2-gon and triangle cases), proves a universal far-field confinement for distant orbits, and constructs a circle-center map that ties center dynamics to the symplectic polar dual, establishing structural links to the usual outer billiard. It proves that every triangle has a 3-periodic outer length orbit and provides extensive experimental evidence, including a square-specific conjecture for an escaping orbit and explicit piecewise-map decompositions. The work broadens polygonal outer billiard theory, complements smooth-table results, and connects geometric constructions with duality concepts in a dynamical setting.

Abstract

The classical inner and outer billiards can be formulated in variational terms, with length and area as the respective generating functions. The other two combinations, ``inner with area'' and ``outer with length,'' are more recently described. Here, we consider the latter system in the special case of polygonal tables. We describe the behavior of orbits far away from the table and pose conjectures regarding an escaping orbit when the table is a square. This paper is meant to complement arXiv:2510.08370 which handles smooth tables with positive curvature.

Paper Structure

This paper contains 16 sections, 16 theorems, 47 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Familiar billiard systems
  3. Variational formulations
  4. Main definition
  5. Structure of the paper
  6. Basic analysis
  7. Singularity diagrams
  8. The 2-gon
  9. Triangles
  10. Behavior at infinity
  11. The outer area billiard
  12. The outer length billiard
  13. Circle centers
  14. Experimental results and conjectures
  15. Orbit and singularity structure
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $Q$ be a 2-gon and let $x$ be a point not collinear with $Q$. Then $x$ and $T(x)$ lie on an ellipse $E$ with foci at the endpoints of $Q$. The orbit under $T$ can be identified with a usual (inner) billiard orbit in $E$, and in particular, the forward and reverse images are attracted to the majo

Figures (20)

  • Figure 1: The construction of the outer length billiard about the convex body $K$.
  • Figure 2: The singularity portrait of the equilateral triangle under the outer area (left) and outer length (right) billiards.
  • Figure 3: The construction of the outer length billiard about a 2-gon and the ellipse guaranteed by Proposition \ref{['prop:2-gon']}.
  • Figure 4: The triangle $\triangle ABC$ is the extouch triangle of $\triangle XYZ$.
  • Figure 5: The virtual table is shown as a thick red line segment for a steady (left) and an unsteady (right) point.
  • ...and 15 more figures

Theorems & Definitions (41)

  • Remark 1.1
  • Definition 1.2
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Theorem 1
  • Corollary 2.3
  • proof : Proof of Theorem \ref{['thm:extouch']}
  • Remark 2.4
  • Definition 3.1
  • ...and 31 more