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On cusps of caustics by reflection in two dimensional projective Finsler metrics

Serge Tabachnikov

TL;DR

The paper addresses the problem of lower bounding the number of cusps on caustics by reflection for billiards in the plane endowed with a projective Finsler metric. It extends Euclidean four-cusps results to two-dimensional projective Finsler billiards by leveraging the duality between the space of oriented lines and caustics, together with Segre's spherical inflection theorem. The main result states that the $n$-th caustic by reflection has at least four cusps for all $n \ge 1$, with a proof that recasts the caustic as the projective dual of a line-trajectory curve $C_n$ and analyzes its spherical inflections. This work broadens the 4-vertex-type phenomena to anisotropic optical models and highlights a symplectic structure underpinning Finsler billiards, suggesting avenues for further generalizations and connections to classical geometric statements like Jacobi's Last Geometric Statement.

Abstract

A Finsler, not necessarily symmetric, metric in the plane or its convex subset is called projective if its geodesics are straight segments. We consider Finsler billiards in a convex planar domain endowed with a projective Finsler metric. A caustic by reflection is the envelope of the oriented lines, the billiard trajectories, that start at a point inside the billiard and undergo a fixed number of reflections. We show that such a caustic has at least four cusps. This problem is motivated by the "Last Geometric Statement of Jacobi" that the conjugate locus of a non-umbilic point of a triaxial ellipsoid has exactly four cusps. The present note extends the recent results in this direction concerning Euclidean billiards.

On cusps of caustics by reflection in two dimensional projective Finsler metrics

TL;DR

The paper addresses the problem of lower bounding the number of cusps on caustics by reflection for billiards in the plane endowed with a projective Finsler metric. It extends Euclidean four-cusps results to two-dimensional projective Finsler billiards by leveraging the duality between the space of oriented lines and caustics, together with Segre's spherical inflection theorem. The main result states that the -th caustic by reflection has at least four cusps for all , with a proof that recasts the caustic as the projective dual of a line-trajectory curve and analyzes its spherical inflections. This work broadens the 4-vertex-type phenomena to anisotropic optical models and highlights a symplectic structure underpinning Finsler billiards, suggesting avenues for further generalizations and connections to classical geometric statements like Jacobi's Last Geometric Statement.

Abstract

A Finsler, not necessarily symmetric, metric in the plane or its convex subset is called projective if its geodesics are straight segments. We consider Finsler billiards in a convex planar domain endowed with a projective Finsler metric. A caustic by reflection is the envelope of the oriented lines, the billiard trajectories, that start at a point inside the billiard and undergo a fixed number of reflections. We show that such a caustic has at least four cusps. This problem is motivated by the "Last Geometric Statement of Jacobi" that the conjugate locus of a non-umbilic point of a triaxial ellipsoid has exactly four cusps. The present note extends the recent results in this direction concerning Euclidean billiards.
Paper Structure (7 sections, 3 theorems, 12 equations, 9 figures)

This paper contains 7 sections, 3 theorems, 12 equations, 9 figures.

Table of Contents

  1. Motivation and previous results
  2. Finsler metrics and Finsler billiards
  3. Proof.
  4. Projective Finsler metric in two dimensions
  5. The four cusps theorem
  6. Proof of Theorem.
  7. Acknowledgment.

Key Result

Lemma 2.1

Let $L(x,v)$ be as in (eq:Lagr). Then, for every $x$, the indicatrix $L(x,v)=1$ is a focus-centered ellipse.

Figures (9)

  • Figure 1: First three caustics by reflection in a circle.
  • Figure 2: Finsler billiard reflection.
  • Figure 3: The indicatrix is a Kepler ellipse.
  • Figure 4: Magnetic billiard: the angle of incidence equals the angle of reflection.
  • Figure 5: Hilbert's and Funk's metrics.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Lemma 2.1
  • Theorem 1
  • Theorem 2