The Klein View of Geometry 
We use the Poincare disc model of the hyperbolic plane. This has as its set of points the disc D = { z : z < 1 }. The disc has boundary the unit circle C = { z : z = 1}. The points of C do not belong to the geometry, but play a role similar to the "points at infinity" in euclidean geometry.
We derive the hyperbolic group from a subgroup of the inversive group I(2),
 
Let H(2) be the subgroup of I(2) consisting of those elements which map D to D and C to C.
In fact it would be enough to demand that they map D to D, but this
Since the elements of H(2) map D to D, we can
consider their
Definition The hyperbolic group H(2) is the group
Clearly H(2) is isomorphic to H(2). We shall usually ignore the distinction
As inversive transformations, elements of H(2) may be direct
Suppose that L is an extended line passing through O.
A knowledge of inversive geometry, and of inverse points in particular,


Theorem H1
If tεH(2) is direct, then t(z) = κ(zc)/(c*z1),


In the hyperbolic geometry pages, we observed that,
if L is an iline orthogonal to C, then i_{L}, inversion with respect to L, maps C to C and D to D. Thus i_{L}εH(2). Here, we establish a result stated on the hyperbolic geometry pages, namely that these maps (known as hinversions) generate H(2). The proof requires the Origin Lemma from the hyperbolic geometry pages.
Theorem H2 
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