# Lobachevsky’s formula

Let $AB$ be a line. Let $M,T$ be two points so that $M$ not lies on $AB$,
$T$ lies on $AB$, and $MT$ perpendicular^{} to $AB$. Let $MD$ be any other line who meets
$AT$ in $D$.In a hyperbolic geometry, as $D$ moves off to infinity
along $AT$ the line $MD$ meets the line $MS$ which is said to be
parallel^{} to $AT$. The angle $\widehat{SMT}$ is called the
*angle of parallelism* for perpendicular distance $d$, and is
given by

$$P(d)=2{\mathrm{tan}}^{-1}({e}^{-d}),$$ |

which is called
*Lobachevsky’s formula.*

Title | Lobachevsky’s formula |
---|---|

Canonical name | LobachevskysFormula |

Date of creation | 2013-03-22 14:05:53 |

Last modified on | 2013-03-22 14:05:53 |

Owner | vmoraru (1243) |

Last modified by | vmoraru (1243) |

Numerical id | 6 |

Author | vmoraru (1243) |

Entry type | Definition |

Classification | msc 51M10 |

Defines | angle of parallelism |