= Equality of horizontal hyperbolic and Euclidean distances
{title2=$b=a/(2\sinh(a/2))$}
For points $ib$ and $a+ib$ with $a,b>0$, the <hyperbolic distance> is $2\operatorname{arsinh}(a/(2b))$. It equals their Euclidean distance $a$ exactly when the displayed relation holds. The function of $a$ decreases continuously from one to zero: its derivative has the sign of $\sinh(a/2)-(a/2)\cosh(a/2)<0$. Hence every height $b\in(0,1)$ has exactly one positive matching horizontal separation.
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