Counting closed billiard paths

Document Type : Original Article

Authors

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

2 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 $\it{closed~billiard~paths}$. 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 ($45^{\circ}$) is at most $1.5n−6$.

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References

[1] d. B. Mark, C. Otfried, v. K. Marc, and O. Mark, Computational geometry algorithms and applications, Spinger, 2008.
[2] J. O’Rourke, Open problems from CCCG 2017, in Proceedings of the 30th Annual Canadian Conference on Computational Geometry, CCCG 2018, Winnipeg, Canada, 2018, Computer Science: Faculty Publications, Smith College, Northampton, MA, pp. 149–154.
[3] d. B. Mark, C. Otfried, v. K. Marc, and O. Mark, Computational geometry algorithms and applications, Spinger, 2008.
[4] [2] J. O’Rourke, Open problems from CCCG 2017, in Proceedings of the 30th Annual Canadian Conference on Computational Geometry, CCCG 2018, Winnipeg, Canada, 2018, Computer Science: Faculty Publications, Smith College, Northampton, MA, pp. 149–154.
AUT Journal of Mathematics and Computing
Volume 1, Issue 2
September 2020
Pages 171-177

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