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

Note that the Möbius transformation $f^{{-1}}$ gives another justification of including $\infty$ 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 $f^{{-1}}(1)=\infty$.

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.

## Mathematics Subject Classification

51M10*no label found*51-00

*no label found*

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