denote the circle of radius 1 centered at the point (0,1) in
the real plane. Let N denote the point (0,2). Then N and each point on
the circle determine a line which has non-zero slope. It must then
intersect the x-axis in some point. On the other hand, given any point on
the x-axis, it and the point N determine a line which must intersect the
circle in some point. Thus, there is a one-to-one correspondence
between the points of the real line (here, the x-axis) and the points of the
circle , less N. The mappings are
What this means to us is that we may add a point at infinity to the real line and we get a circle, in some sense. Think of taking the Poincaré half-plane model and adding a point at infinity to the real axis. We will then have the interior of a circle sitting in the real plane. The vertical lines will correspond to diameters of the circle and the circles in the half-plane model will correspond to circles which are orthogonal to the boundary circle.
To make the setting more precise, let 
. A p-line in this model is either
, or
. A circle 
is
orthogonal to if at each point of intersection of and
the radii of and through that point are
perpendicular. 
, an open arc which represents a
p-line in 
.
Figure 17.4: A p-line in the Poincaré disk model
A point in lies on a p-line if and only if it lies on it in
the Euclidean sense. Betweenness has the same interpretation, though if A,
B, and C are on an open arc from with center P then 
if and only if 
is between 
and 
.
If 
and if P and Q are the ends of the p-line
through A and B, then we define the Poincaré distance from A to B to
be
As with the above model, this Poincaré model is conformal. If two directed circular arcs meet at A then the measure of the angle they form is the measure of the angle between their tangent rays at A.
Figure 17.5: Angle measure in the Poincaré Disk
Model
Question: IF and A and b do not lie
on a diameter of , how do you construct the p-line through A and
B?
Figure 17.6: Construction of the p-line through A and B
We will verify these steps later.
at A, intersecting
in P and Q.
at P and Q. These tangents
intersect at a point A'.
, containing A, A', and B.
is orthogonal to and 
is the p-line through A and B.
Let us compare the two disk models by looking at certain familiar objects in each of them.
Figure 17.7: Limiting Parallel Rays
Figure 17.8: Hyperparallel Lines
Figure 17.9: Horoparallel Lines
Klein ModelPoincaré Model
Figure 17.10: Lambert Quadrilateral
Figure 22.1: Saccheri Quadrilateral
Klein ModelPoincaré Model