Introduction

Geodesic flows are natural dynamical systems that arise in the context of Riemannian geometry. There is a long history of academic work dedicated to studying these systems and how the dynamics of these flows are related to the underlying geometry of the space in consideration. Our aim in this project is to understand these flows by building visualizations. Below, you will find some of our favorite visualizations along with links explaining the math involved, approach for coding, and more related animations.

Visualizations

Each of the following subsections corresponds to a set of animations we made to depict a particular topic, and we display one example of an animation from each subsection. To see more animations from a subsection and read about the underlying theory and programming, click on the corresponding subsection’s header.

🔗 Geodesics in the Hyperbolic Plane

Example:

🔗 Toral Automorphisms

Example:

🔗 Hyperbolic \(4g\)-gons

Example:

🔗 Flowing a Hyperbolic Tiling

Example:

🔗 Geodesic Flow on a Hyperbolic Surface

Example:

🔗 Fundamental Domains in \(\text{PSL}(2, \mathbb{R})\)

Example: