converting between the Poincaré disc model and the upper half plane model
If both the Poincaré disc model and the upper half plane model are considered as subsets of rather than as subsets of (that is, the Poincaré disc model is and the upper half plane model is ), then one can use Möbius transformations to convert between the two models. The entry unit disk upper half plane conformal equivalence theorem yields that defined by maps the upper half plane model to the Poincaré disc model, and thus its inverse, defined by , maps the Poincaré disc model to the upper half plane model.
Note that the Möbius transformation gives another justification of including in the boundary of the upper half plane model (see the entry on parallel lines in hyperbolic geometry for more details): (or the ordered pair ) is on the boundary of the Poincaré disc model and .
|Title||converting between the Poincaré disc model and the upper half plane model|
|Date of creation||2013-03-22 17:07:43|
|Last modified on||2013-03-22 17:07:43|
|Last modified by||Wkbj79 (1863)|