# 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 LobachevskysFormula 2013-03-22 14:05:53 2013-03-22 14:05:53 vmoraru (1243) vmoraru (1243) 6 vmoraru (1243) Definition msc 51M10 angle of parallelism