Suppose that we define a geometry as a set S and a group G. Another geometry
with set S' and group G' is essentially the same as this if the points of S' are in
oneone correspondence with the points of S in such a way that the action of G'
on S' corresponds to the action of G on S. In such a case, we say that we have
two models of the geoemtry, defined on S and on S'.
Formally, we require a bijection k from S to S', so that the sets correspond. We
also want the actions of the groups G and G' to correspond. Suppose that PεS and
that gεG with g(P) = Q. The element g' of G'corresponding to g must send k(P)
to k(Q), i.e. we must have g'ok(P) = k(Q) = kog(P).
Since we require this for all
P and g, we require that G' = kGk^{1}.
Observe that, if we know S, S' and the bijection k, then we can obtain a model on
the set S'
simply by defining the group G' as kGk^{1}.
Finally, if the geometry on S has "lines", then we can define the "lines" in S' as the
subsets {k(L) : L is a "line" in S}. Of course, the lines in S' will
enjoy geometrical
properties identical to those in S. But note that the two sets of "lines" may look
different, i.e. may have different euclidean properties.
We introduced hyperbolic geometry using the poincare disk model, whose set is the
unit disk D and whose group is H(2). The hyperbolic lines are arcs of euclidean circles
rather than line segments.The advantage is that the angles between hyperbolic lines
are represented by the euclidean angles between the euclidean tangents to the arcs.
There are other models, each with their own useful features. We shall consider two.
To relate the models to the poincare disk, we need an algebraic description of when
two circles are orthogonal.
Lemma
The circle C : x^{2}+y^{2}2ax2by+c = 0 is orthogonal to the circle
C : x^{2}+y^{2} = 1
if and only if c = 1 and a^{2}+b^{2} > 1.
the klein (or kleinbeltrami) model
The set is D, as for the poincare model.
Regard D as lying on the xyplane in threedimensional space. The map k is defined in two stages:
a twodimensional map, i.e. we suppress the zcoordinate which is always 0.
As above, the group is kH(2)k^{1}, though it is difficult to describe explicitly. The PoincareKlein Map
Thus, the klines are the intersections of D with
euclidean lines  i.e. are
Pappus's Theorem in Hyperbolic Geometry
Suppose that A,B,C lie on a hyperbolic line L, and A',B',C' on a hyperbolic line L'.
In the klein model, the hyperbolic lines are euclidean segments. The euclidean


the minkowski model
This model has as its set a certain surface in threedimensional space. This is discussed further in the hyperboloid page.
We map D to H by stereographic projection from Z'(0,0,1). This gives The PoincareMinkowski Map
proof
The minkowski model has the set H. The mlines are the
intersections of H
This model is related to special relativity. It is also of considerable significance The HyperbolicProjective Theorem
If m is the bijection from the hyperbolic model to the minkowski model, 
