Up: Models of Hyperbolic Geometry

# The Poincaré Disk

The Poincaré Model is defined to be the set of all points is an open disk D. The "lines" of hyperbolic space are represented by Euclidean arcs that intersect the boundary circle of D perpendicularly. The figure below illustrates a few possible lines in the Poincare model.

Observe that l, m and n intersect the boundary circle orthogonally. One may notice that l, spanning the diameter, is a line, not an arc. However, we can think of l as an arc of a circle with infinite radius.

Lines m and n are not parallel as they share a common point. Notice that l and n are divergently parallel. Finally l and m provide us with an example of two lines asymptotically parallel.

In this model we measure angles by taking the Euclidean measurements of the angles of the tangents to the arcs. For example, if we wanted to know the angle of intersection between m and n we would take the Euclidean tangents to the arcs at the point of intersection and measure the angle with a protractor. Because the angles are measured in an Euclidean fashion this model is called conformal. You may click on the "Equations" button below for further discussion on measurement and distance.

Interactive Java Applet

Equations

Poincaré-Draw: Design your own Poincaré disk.

Next: The Upper Half Plane