# 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 $\u2102$ rather than as subsets of ${\mathbb{R}}^{2}$ (that is, the Poincaré disc model is $$ and the upper half plane model is $\{z\in \u2102:\mathrm{Im}(z)>0\}$), then one can use Möbius transformations^{} to convert between the two models. The entry unit disk upper half plane conformal equivalence theorem yields that $f:\u2102\cup \{\mathrm{\infty}\}\to \u2102\cup \{\mathrm{\infty}\}$ 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^{}, ${f}^{-1}:\u2102\cup \{\mathrm{\infty}\}\to \u2102\cup \{\mathrm{\infty}\}$ 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 |