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. Do not confuse these new tools (in the “disk” menu near the right) with their Euclidean analogs! 3
The new menu includes tools for drawing elliptic lines and line segments and measuring their length, an elliptic circle tool, an elliptic compass, and elliptic angle measurement. (The disk tool itself merely places points and should be avoided; use the Euclidean point tool instead.) These tools are labeled with icons that more-or-less match their Euclidean analogs, and should be used accordingly; compare Footnote 7.6.2 (but note that the hyperbolic tools do not require points to exist before being selected).