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

Poincaré-Draw: Design your own Poincaré disk.

**Up:** *Models of Hyperbolic Geometry*

Created: Jul 15 1996 --- Last modified: Jul 15 1996