# An affine model of a Riemann surface associated to a Schwarz-Christoffel mapping

## TL;DR

This paper constructs an affine model of a Riemann surface linked to Schwarz-Christoffel mappings, revealing connections between geodesics and billiard motions in complex polygons.

## Contribution

It introduces a novel affine model of a Riemann surface associated with Schwarz-Christoffel mappings and explores its geometric and dynamical properties.

## Key findings

- Established a flat Riemannian metric on the surface
- Linked geodesics to billiard trajectories in polygons
- Provided insights into the surface's geometric structure

## Abstract

In this paper we construct an affine model of a Riemann surface with a flat Riemannian metric associated to a Schwarz-Christoffel mapping of the upper half plane onto a rational triangle. We explain the relation between the geodesics on this Riemann surface and billiard motions in a regular stellated $n$-gon in the complex plane.