The Poincaré Half-Plane Model

In this model take to be the set of all points in the coordinate Euclidean plane that satisfy

We interpret lines in this model to be one of the following:

1. the intersection of a Euclidean vertical line with , or
2. the intersection of a Euclidean circle, whose center lies on the x-axis, with .

Once again, our concept of congruence must change. As before, we must define a new measure for segments. However, the manner of measuring angles  is exactly the method that is used in Euclidean geometry. To measure the angle formed by two of these lines, we simply measure the Euclidean angle formed by the tangent lines to the two Poincaré lines.

 
Figure 17.3: Measuring angles in the Poincaré Half-Plane Model

The method of measuring segments is more easily described in the next model, which is derived from this model.