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 infinityalong $AT$ the line $MD$ meets the line $MS$ which is said to beparallel to $AT$. The angle $\widehat{SMT}$ is called theangle of parallelism for perpendicular distance $d$, and isgiven by

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

which is calledLobachevsky’s formula.