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Section 5.1 The Poincaré Disk

One model for hyperbolic geometry is the Poincaré Disk. In this model, one starts with a disk \(D\) in the Euclidean plane, consisting of a boundary circle and its interior.  1 

  • The points of the Poincaré disk are all Euclidean points in the interior of \(D\text{.}\)
  • The lines of the Poincaré disk are all (arcs of) Euclidean circles that meet \(D\) at right angles (and lie within \(D\)). Diameters of \(D\) are included as a special case, and can be thought of as arcs of Euclidean circles of infinite radius.
  • The angles of the Poincaré disk are Euclidean angles in \(D\text{.}\)  2 
The disk \(D\)is often taken to have unit radius, but this assumption is not necessary.
Angles between curves are of course measured using their tangent lines.

You can explore constructions in the Poincaré disk using the new tools shown in Figure 5.1.1. Don't confuse these new tools (in the “disk” menu near the right) with their Euclidean analogs!

Figure 5.1.1. A GeoGebra interface for the Poincaré disk. (You may need to click on the menu icon in the top right in order to gain access to the new menus.)
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