### Definition of Distance

Recall that the rigid motions in the Poincaré half-plane are given by the linear fractional transformations that take the upper half-plane to itself. In addition to using these motions to define congruence (two objects are congruent if there is a rigid motion that takes one to the other), we can also use them to define distance. We pick a particular segment and define it to have unit length 1. We can then define distances of any integer length n by taking n copies of the given unit length. By subdividing the unit length and taking multiples of the subdivisions, we can define any rational length. Finally we can take limits to define the length of any segment. The choice of the unit length can be arbitrary, but just as the choice of measuring a circle as 2 pi radians makes formulas simpler in calculus, we can make later formulas simpler by making a careful choice of unit length. Specifically, we will pick our unit length so that "the ratio of the length of corresponding arcs on concentric horocycles is equal to e when the distance between the horocycles is 1." (See Greenberg, Euclidean and Non-Euclidean Geometries.)