You will, doubtless, be surprised to learn that these two models are isomorphic;, i.e., there is a one-to-one correspondence between the points and lines of the two models such that the relations of incidence, betweenness, and congruence are preserved.
Let 
be a sphere of the same radius as the radius of 
, tangent to
at the origin. Use orthogonal
projection of the Klein
model in upward onto the lower hemisphere of the sphere and then
stereographic projection from the north pole back to the plane. The image will
be the Poincaré model.
Let 
. Orthogonal projection maps (x,y) onto the bottom
hemisphere of 
by
Stereographic projection of onto the coordinate plane is the analogous
map to that described for mapping the punctured circle onto the line. A point
is sent to the point on the xy-plane at which the line
through (0,0,2a) and (x,y,z) intersects the plane. We can parameterize the
line by:
passes through the xy-plane when
Thus, stereographic projection sends (x,y,z) to
.
We need a shrinking map to shrink the image back into . The equator of
the sphere (x,y,a) maps to (2x,2y)=(x',y'). Thus, 
.
Shrinking back into it is only necessary to divide by 2.
Combining the three maps, we have a mapping from the Klein model to the
Poincaré model given by
It is easy to check that this is a continuous and differentiable map from
to itself. It sends k-lines to p-lines and is an isomorphism
of geometries.