What is The Poincare Disk

Author:Caitlyn Aimer, Heather Pierce, Aidan

The Poincare Disk: What is it?

The Poincare Disk is one of many models of hyperbolic space. The four most common models of Hyperbolic space are: the Beltrami-Klein model, the Poincaré disk model, the Poincaré half-plane model, and the Lorentz or hyperboloid model The Poincare Disk is a model of 2D hyperbolic geometry in which all points are inside the unit disk. Straight lines are represented either by circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.

Below is a Display of Hyperbolic Space

The large blue portion is a hyperboloid used to display hyperbolic space The orange ring is the Beltrami-Klein Disk The purple disk is our Poincare Disk

The parabolas created in the blue hyperbolic space are projected as straight lines on the orange Beltrami-Klein Disk, but they are projected as arcs of a circle on the purple Poincare Disk

Below is an Image Depicting the Models of Hyperbolic Space

Now Let's Look at What The Poincare Disk has:

A line is represented as an arc of a circle
Diameters are also used inside the disk

Part 1

A line is represented as an arc of a circle

Part 2

Diameters are also used inside the disk

Construct a Diameter Using the Tool

This can be created from any point along the boundary of the disk

Use the Hyperbolic Line Tool

Considering what you have learned so far, what do you think is the purpose of using the Poincare Disk?

What questions do you still have about the Poincare Disk?