Chapter 4  The upper half-plane model

A second model of the hyperbolic plane is the upper half-plane model. Its points consist in the complex numbers with positive imaginary parts. The geodesics are the open vertical half-lines and semicircles that are orthogonal to the real axis. As for the conformal disk model, there is a different metric than in the Euclidean plane, i.e. if we consider two isometric figures, one close to the boundary of the upper half-plane and the other a bit further, the one closer to the boundary will look smaller than the other.

Escher also created a picture that illustrates how the metric in the upper half-plane works.

Figure 4.1: Escher’s Reducing lizards
extracted from [9]