# parallel lines in hyperbolic geometry

In hyperbolic geometry, there are two kinds of parallel lines^{}. If two lines do not intersect within a model of hyperbolic geometry but they do intersect on its boundary, then the lines are called *asymptotically parallel* or *hyperparallel*. (Note that, in the upper half plane model, any two vertical rays are asymptotically parallel. Thus, for consistency, $\mathrm{\infty}$ is considered to be part of the boundary.) Any other set of parallel lines is called *disjointly parallel* or *ultraparallel*.

Below is an example of asymptotically parallel lines in the Beltrami-Klein model:

Below are some examples of asymptotically parallel lines in the Poincaré disc model:

Below are some examples of asymptotically parallel lines in the upper half plane model:

Below is an example of disjointly parallel lines in the Beltrami-Klein model:

Below is an example of disjointly parallel lines in the Poincaré disc model:

Below are some examples of disjointly parallel lines in the upper half plane model:

Title | parallel lines in hyperbolic geometry |

Canonical name | ParallelLinesInHyperbolicGeometry |

Date of creation | 2013-03-22 17:06:43 |

Last modified on | 2013-03-22 17:06:43 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 11 |

Author | Wkbj79 (1863) |

Entry type | Topic |

Classification | msc 51-00 |

Classification | msc 51M10 |

Defines | asymptotically parallel |

Defines | asymptotically parallel lines |

Defines | hyperparallel |

Defines | hyperparallel lines |

Defines | disjointly parallel |

Defines | disjointly parallel lines |

Defines | ultraparallel |

Defines | ultraparallel lines |