Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Correspondence between Billiards and Geodesics

Daniele Giannetto

TL;DR

The paper studies a geometric correspondence between billiard trajectories and geodesics by constructing fold-type hypersurfaces $M_\\lambda^{p_0}$ over a billiard table $K$ and proving that, under curvature bounds, sequences of arclength-parameterized geodesics on these folds converge (uniformly on a fixed interval $[-T,T]$) to a billiard trajectory inside $K$. For convex Euclidean tables with $C^2$ boundary, the folds converge to $K\cap U$ in the Hausdorff sense as $\\lambda\to 0^+$, and every sequence of geodesic segments on the folds has a subsequence converging to a billiard trajectory; this extends to a broader class of Riemannian billiard tables under an assumption (H) ensuring uniform lower curvature bounds. The approach relies on κ-quasigeodesic theory, Hausdorff convergence, and compactness arguments, producing explicit non-Euclidean examples in constant curvature geometries. The results unify geodesic approximation and billiard dynamics, enabling new insights into billiard trajectories in curved spaces and broadening the scope of geometric optics and dynamical systems applications. All mathematical constructs are carefully framed with curvature bounds, convergence notions, and reflection laws, providing a robust bridge between interior geodesic flows and boundary billiard dynamics.

Abstract

From a geometric viewpoint, billiard trajectories and geodesics are related by mutual approximation results. In one direction, it is known that every geodesic curve in the boundary of a smooth convex body can be approximated by a sequence of billiard trajectories inside of it. We establish the other direction by proving that, for Riemannian billiard tables (under mild assumptions), there exists a family of fold-type surfaces such that every sequence of geodesic segments on these surfaces has a subsequence that converges to a billiard trajectory in the table. In particular, this is true for convex Euclidean tables. We also describe a more general class of tables to which this result applies and present explicit non-Euclidean examples.

A Correspondence between Billiards and Geodesics

TL;DR

The paper studies a geometric correspondence between billiard trajectories and geodesics by constructing fold-type hypersurfaces over a billiard table and proving that, under curvature bounds, sequences of arclength-parameterized geodesics on these folds converge (uniformly on a fixed interval ) to a billiard trajectory inside . For convex Euclidean tables with boundary, the folds converge to in the Hausdorff sense as , and every sequence of geodesic segments on the folds has a subsequence converging to a billiard trajectory; this extends to a broader class of Riemannian billiard tables under an assumption (H) ensuring uniform lower curvature bounds. The approach relies on κ-quasigeodesic theory, Hausdorff convergence, and compactness arguments, producing explicit non-Euclidean examples in constant curvature geometries. The results unify geodesic approximation and billiard dynamics, enabling new insights into billiard trajectories in curved spaces and broadening the scope of geometric optics and dynamical systems applications. All mathematical constructs are carefully framed with curvature bounds, convergence notions, and reflection laws, providing a robust bridge between interior geodesic flows and boundary billiard dynamics.

Abstract

From a geometric viewpoint, billiard trajectories and geodesics are related by mutual approximation results. In one direction, it is known that every geodesic curve in the boundary of a smooth convex body can be approximated by a sequence of billiard trajectories inside of it. We establish the other direction by proving that, for Riemannian billiard tables (under mild assumptions), there exists a family of fold-type surfaces such that every sequence of geodesic segments on these surfaces has a subsequence that converges to a billiard trajectory in the table. In particular, this is true for convex Euclidean tables. We also describe a more general class of tables to which this result applies and present explicit non-Euclidean examples.
Paper Structure (16 sections, 18 theorems, 70 equations, 4 figures)

This paper contains 16 sections, 18 theorems, 70 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Strategy of proof of Theorem \ref{['maintheorem']}
  3. Acknowledgements
  4. Preliminaries
  5. Notation and conventions
  6. Comparison theory in complete Riemannian manifolds and quasigeodesic curves
  7. Hausdorff convergence and quasigeodesic curves
  8. Geodesics approximating Billiards
  9. Billiard tables, billiard trajectories and quasigeodesics
  10. Families of folds and flattening
  11. Proof of the main theorem
  12. Examples in constant curvature geometries
  13. Flat geometry
  14. Negatively curved geometry
  15. Positively curved geometry
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $K$ be a convex body in $\mathbb{R}^n$ whose boundary is of class $C^{2,1}$ and has a positive definite second fundamental form at every point. Let $p_0\in \partial K$ and $\{\gamma_k\}_{k\in \mathbb{N}}$ be a sequence of billiard trajectories of $K$ starting from $p_0$. Then, if the initial dir

Figures (4)

  • Figure 1: Ellipsoids in $\mathbb{R}^3$ with the x and y semiaxes of unit length and with the z-semiaxis of length: 1, 0.7, 0.4 respectively.
  • Figure 2: $M=\{x_2\leq 0\} = C_{(0,0)}M\subset \mathbb{R}^2$. On the left, the pair of polar vectors $u_1 = (-1,0)$, $v_1 = (1,0)$ and, on the right, the pair of polar vectors $u_2 = (-\sqrt{2}/2,-\sqrt{2}/2)$, $v_2 = (\sqrt{2}/2,-\sqrt{2}/2)$.
  • Figure 3: On the left, the smooth arc $c_1(t)$ of Example (\ref{['example1']}) and, on the right, the piecewise straight curve $c_2(t)$ of Example (\ref{['example2']}).
  • Figure 4: A portion of the parabolic cylinders $\{x_3^2 = \lambda^2 x_1\}\subset \mathbb{R}^3$ for $\lambda = 0.5,0.2,0.06$ respectively.

Theorems & Definitions (52)

  • Theorem 1.1: Theorem B of Lange
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 42 more