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

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

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