Year: 2012
Type: preprint
Abstract: We consider a billiard table rectangle. If a billiard ball is sent out from position F(1) at the angle of $π/4$, then the ball will rebound against the sides of the rectangle consecutively in points $F(2),F(3),...$. Let $n\geq5$ and $Φ= \{F(j): 1\leq j\leq n \}$ be the set of different points. An open connected subset of the perimeter of the billiard rectangle with different endpoints from the set $Φ$ is called \textit{segment}. \textit{Length} of a segment is a distance along the perimeter between its endpoints. A segment with endpoints $F(k)$, F(l), $1\le k,l\le n$, is called \textit{even} (or \textit{odd}), and has \textit{weight} $|k-l|$ (or $k+l$) if $k$, $l$ are of the same (or different) parity. A segment is called \textit{elementary} if there are no points of the set $Φ$ between its endpoints. Suppose $\emptyset \neq V\subseteq\{F(1),F(n)\}$. A segment $I$ is \textit{associated} with $V$ if $I$ is an elementary segment incident with an element of $V$ or $I\capΦ$ is nonempty set contained in $V$. Let $ω_1
Source: arXiv (Cornell University)
Author: Florek, Jan
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2168128901