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 $ℂ$ rather than as subsets of ${ℝ}^2$ (that is, the Poincaré disc model is and the upper half plane model is $\{z\in ℂ:\mathrm{Im}(z)>0\}$), then one can use Möbius transformations to convert between the two models. The entry unit disk upper half plane conformal equivalence theorem yields that $f:ℂ\cup \{\mathrm{\infty }\}\to ℂ\cup \{\mathrm{\infty }\}$ defined by $f(z)={\displaystyle \frac{z-i}{z+i}}$ maps the upper half plane model to the Poincaré disc model, and thus its inverse, $f^{-1}:ℂ\cup \{\mathrm{\infty }\}\to ℂ\cup \{\mathrm{\infty }\}$ defined by $f^{-1}(z)={\displaystyle \frac{-iz-i}{z-1}}$, maps the Poincaré disc model to the upper half plane model.

Note that the Möbius transformation $f^{-1}$ gives another justification of including $\mathrm{\infty }$ in the boundary of the upper half plane model (see the entry on parallel lines in hyperbolic geometry for more details): $1$ (or the ordered pair $(1,0)$) is on the boundary of the Poincaré disc model and $f^{-1}(1)=\mathrm{\infty }$.

Note also that lines in the Poincaré disc model passing through $1$ (or the ordered pair $(1,0)$) are in one-to-one correspondence with the lines that are vertical rays in the upper half plane model.

Title	converting between the Poincaré disc model and the upper half plane model
Canonical name	ConvertingBetweenThePoincareDiscModelAndTheUpperHalfPlaneModel
Date of creation	2013-03-22 17:07:43
Last modified on	2013-03-22 17:07:43
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	7
Author	Wkbj79 (1863)
Entry type	Topic
Classification	msc 51M10
Classification	msc 51-00
Related topic	PoincareDiscModel
Related topic	UpperHalfPlaneModel
Related topic	UnitDiskUpperHalfPlaneConformalEquivalenceTheorem
Related topic	PoincareUpperHalfPlaneModel
Related topic	UpperHalfPlane