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Poincaré half-plane model
In non-Euclidean geometry, the Poincaré model model of the hyperbolic plane regards it as a homogeneous space for the group of Möbius transformations. This can be done in one of two ways, that are closely related:
These are related by a conformal mapping.
In the unit disc model, the geodesics are circular arcs orthogonal to the boundary of the unit discs. In the half plane model, the geodesics are either circular arcs or straight lines, orthogonal to the boundary (in both cases).
The famous circle limit III http://www.mcescher.com/Gallery/recogn-bmp/LW434.jpg and IV http://www.mcescher.com/Gallery/recogn-bmp/LW436.jpg drawings of M. C. Escher illustrate the unit disc version of the model quite well. In both one can clearly see the geodesics (in III they appear explicitly). It is also possible to see quite plainly the negative curvature of the hyperbolic plane, via its effect on the sum of angles in triangles and squares.
For example, in III every vertex is the intersection of three triangles and three squares. In normal Euclidean plane, this would sum up to 450°, leading to a contradiction. Hence we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is the fact that the hyperbolic plane has exponential growth. In IV, for example, one can see that the number of angels with a distance of n from the center rises exponentially. The angels have equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n.
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To view or edit this article at Wikipedia go to http://www.wikipedia.org/wiki/Poincaré_half-plane_model
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