Elementary Functions 243these are the lines parallel to C, namely, C 1 and C 2 in Fig. 5.15, C 1 being
the parallel to the right, C2 the parallel to the left; and (3) those that do
not meet C, neither at a finite point nor at infinity, as, for example, C 4
in Fig. 5.15. Lines of the third type are sometimes called hyperparallels
to the given line.
We note that in hyperbolic geometry there is exactly one perpendicular
to a given line through a given point. In Poincare's model that i~ the unique
semicircle C' passing through P and orthogonal to C. In fact, the center of
this semicircle lies in the intersection of the x-axis and the perpendicular
bisector of the segment joining P to its inverse P' with respect to C, except
when P is a point of C, in which case the center of C' is the point ofintersection of the x-axis with the tangent to C at P. If C happens to be
a perpendicular line to the x-axis, the center of C' is just the foot of that
perpendicular (Fig. 5.16).
The group of non-Euclidean motions is represented in this model by the
group of all bilinear transformations
az +b
w= ---cz+d
(5.11-1)with a, b, c, d real and ad - be = 1. It is easy to see that transformations
of this particular form are automorphisms of the upper plane, i.e., they all
map Imz > 0 onto Imw > 0 (and the real axis onto the real axis); see
Exercises 5.1, problem (6).
In hyperbolic geometry the measure of the angle between two half-lines
with a common origin may be defined to be its ordinary Euclidean measure.
In fact, since the transformation (5.11-1) is directly isogonal, the angle be-
tween two half-lines, as just defined, will be invariant under non-Euclidean
motions.
Next we wish to arrive at a suitable definition of a hyperbolic distance
between two points in such a way as to maintain the usual properties of
a distance (Section 2.5) and also to have invariance of the distance under
the motions represented by (5.11-1).
P'c C'
c
'
' \
- -'-' - - - - -
0 O'
- -'-' - - - - -
Fig. 5.16