Angle of parallelism

The angle between two d-lines in the model, absolute in size, are measured by us to be the angle between the tangents to the circular arc at their point of intersection and hence this is the angle as measured by the inhabitants of the disc as well.

If we construct the d-line from the point P and perpendicular to the fixed d-line, then the angles this makes with the two parallels are equal as shown. Lobachevsky called this value  the angle of parallelism and showed that it is determined by the distance MP. The angle takes all values between 0 and 90. At 0 we have the trebly asymptotic triangle, all 3 lines parallel to each other, and as the angle approaches 90, we see that we approach the Euclidean case of just one parallel. For a very small distance from M to P it is difficult to distinguish between the two geometries.