# Straightening Billiard Trajectories in Flat Disks and Katok-Zemlyakov Construction

## TL;DR

This paper introduces a method to straighten billiard trajectories in flat disks, analyzes their closures, and generalizes the Katok-Zemlyakov construction to rational flat disks, providing new insights into their geometric structure.

## Contribution

It presents a novel method for straightening billiard trajectories and extends the Katok-Zemlyakov construction to rational flat disks, including calculating associated Euler characteristics.

## Key findings

- Trajectory closures contain a vertex for almost all directions.
- A translation surface is constructed for each rational flat disk.
- Euler characteristics of these surfaces are explicitly calculated.

## Abstract

We provide a method to straighten each billiard trajectory in a flat disk. As an application, we show that for each point on the disk and for almost all directions, the closure of the corresponding billiard trajectory contains a vertex. We generalize Katok-Zemlyakov construction to the flat disks with rational angle data: for each rational flat disk we obtain a translation surface and a projection to the doubling of the disk. We calculate the Euler characteristics of this translation surface.