Counting closed billiard paths

Document Type : Original Article

Authors

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

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

3 Malek-Ashtar University of Technology, Tehran, Iran

Abstract

Given a pool table enclosing a set of axis-aligned rectangles, with a total of n edges, this paper studies 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 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 (45◦ ) is at most 1.5n − 6.

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