Phase Portraits of Hyperbolic Geometry

TL;DR

This paper introduces a novel method for visualizing functions in hyperbolic geometry using phase portraits, leveraging conformal maps to accurately depict geodesics on various hyperbolic surfaces.

Contribution

It reinvents phase plotting for hyperbolic spaces, ensuring visualizations effectively convey geometric information through conformal mappings.

Findings

Effective visualization of hyperbolic functions using phase portraits.

Conformal maps enable unique representation of geodesics.

Method applicable to various hyperbolic surfaces.

Abstract

Phase plotting is a useful way of visualising functions on complex space. We reinvent the method in the context of hyperbolic geometry, and we use it to plot functions on various representative surfaces for hyperbolic space, illustrating with direct motions in particular. The reinvention is nontrivial, and we discuss the essential features for ensuring that such visualisations convey useful information. Our approach is to exploit conformal maps between representative surfaces, in order to uniquely represent the preimages of geodesics. Our considerations and methods are prototypical of what one might consider for adapting similar methods of visualisation in other contexts.