Counting closed billiard paths

Document Type : Original Article

Authors

1 Department of Mathematics and Computer Science, Amirkabir University of Technology

2 Malek-Ashtar University of Technology, Tehran, Iran

3 Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)

Abstract

Given a pool table enclosing a set of axis-aligned rectangles, with a total of $n$ edges, this paper studies {it closed billiard paths}. A closed billiard path is formed by following the ball shooting from a starting point into some direction, such that it doesn't touch any corner of a rectangle, doesn't visit any point on the table twice, and stops exactly at the starting position.
The {it signature} of a billiard path is the sequence of the labels of edges in the order that are touched by the path, while repeated edge reflections like $abab$ are replaced by $ab$.

We prove that the length of a signature is at most $4.5n-9$, and we show that there exists an arrangement of rectangles where the length of the signature is $1.25n+2$. We also prove that the number of distinct signatures for fixed shooting direction (ang{45}) is at most $1.5n-6$.

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Volume 1, Issue 2
Summer and Autumn 2020
Pages 171-177
  • Receive Date: 04 November 2019
  • Revise Date: 08 February 2020
  • Accept Date: 17 February 2020