Hyperbolic Space
The open unit disk is commonly used as a model for the hyperbolic plane, by introducing a new metric on it, the Poincaré metric. Using the above mentioned conformal map between the open unit disk and the upper half-plane, this model can be turned into the Poincaré half-plane model of the hyperbolic plane. Both the Poincaré disk and the Poincaré half-plane are conformal models of hyperbolic space, i.e. angles measured in the model coincide with angles in hyperbolic space, and consequently the shapes (but not the sizes) of small figures are preserved.
Another model of hyperbolic space is also built on the open unit disk: the Klein model. It is not conformal, but has the property that straight lines in the model correspond to straight lines in hyperbolic space.
Read more about this topic: Unit Disk
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