There are a number of models of hyperbolic geometry. The concept behind the Poincar disk is that a plane is not infinite.
Lines are arcs which are perpendicular to the circle at both ends.
The big catch is in how distances are measured. The distance between two points is based on standard distances and logarithms. You can see that n and m are arcs which terminate into the side of the circle. This is because l includes the center of the Poincar disk. In order to include the center of the disk and be perpendicular to the circle on both ends, l must be a diameter of the circle.