Counting closed billiard paths
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, do...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Amirkabir University of Technology
2020-09-01
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| Series: | AUT Journal of Mathematics and Computing |
| Subjects: | |
| Online Access: | https://ajmc.aut.ac.ir/article_3821_7338945819e8a369d3c32dde65cdfafb.pdf |
| _version_ | 1797307164511961088 |
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| author | Zahed Rahmati Sina Farahzad Ali Rahmati |
| author_facet | Zahed Rahmati Sina Farahzad Ali Rahmati |
| author_sort | Zahed Rahmati |
| collection | DOAJ |
| description | 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 $1.5n−6$. |
| first_indexed | 2024-03-08T00:52:15Z |
| format | Article |
| id | doaj.art-c674f7710dde416fab791112f80aaa37 |
| institution | Directory Open Access Journal |
| issn | 2783-2449 2783-2287 |
| language | English |
| last_indexed | 2024-03-08T00:52:15Z |
| publishDate | 2020-09-01 |
| publisher | Amirkabir University of Technology |
| record_format | Article |
| series | AUT Journal of Mathematics and Computing |
| spelling | doaj.art-c674f7710dde416fab791112f80aaa372024-02-14T19:35:59ZengAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24492783-22872020-09-011217117710.22060/ajmc.2020.17320.10263821Counting closed billiard pathsZahed Rahmati0Sina Farahzad1Ali Rahmati2Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, IranDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, IranMalek-Ashtar University of Technology, Tehran, IranGiven 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 $1.5n−6$.https://ajmc.aut.ac.ir/article_3821_7338945819e8a369d3c32dde65cdfafb.pdfbilliard pathsmaximum path lengthcomputational geometry |
| spellingShingle | Zahed Rahmati Sina Farahzad Ali Rahmati Counting closed billiard paths AUT Journal of Mathematics and Computing billiard paths maximum path length computational geometry |
| title | Counting closed billiard paths |
| title_full | Counting closed billiard paths |
| title_fullStr | Counting closed billiard paths |
| title_full_unstemmed | Counting closed billiard paths |
| title_short | Counting closed billiard paths |
| title_sort | counting closed billiard paths |
| topic | billiard paths maximum path length computational geometry |
| url | https://ajmc.aut.ac.ir/article_3821_7338945819e8a369d3c32dde65cdfafb.pdf |
| work_keys_str_mv | AT zahedrahmati countingclosedbilliardpaths AT sinafarahzad countingclosedbilliardpaths AT alirahmati countingclosedbilliardpaths |