Periodic Orbits on Obtuse Edge Tessellating Polygons

TL;DR

This paper classifies and derives formulas for periodic billiard orbits on specific obtuse edge tessellating polygons, using plane tilings to analyze their trajectories and periods.

Contribution

It introduces a method to classify and compute periods of periodic billiard orbits on certain obtuse polygons via edge tessellations and unfolding techniques.

Findings

Derived formulas for orbit periods in specific polygons

Classified periodic orbits based on initial conditions

Used tessellation to simplify orbit analysis

Abstract

A periodic orbit on a frictionless billiard table is a piecewise linear path of a billiard ball that begins and ends at the same point with the same angle of incidence. The period of a primitive periodic orbit is the number of times the ball strikes a side of the table as it traverses its trajectory exactly once. In this paper we find and classify the periodic orbits on a billiard table in the shape of a 120-isosceles triangle, a 60-rhombus, a 60-90-120-kite, and a 30-right triangle. In each case, we use the edge tessellation (also known as tiling) of the plane generated by the figure to unfold a periodic orbit into a straight line segment and to derive a formula for its period in terms of the initial angle and initial position.