 Hyperbolic translation length (source code)

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= Hyperbolic translation length
{title2=$\ell(A)=2\operatorname{arcosh}(|\operatorname{tr}A|/2)$}

For a <hyperbolic Möbius transformation> represented by $A\in\mathrm{SL}_2(\mathbb R)$, the infimum of $d(z,Az)$ is attained on its axis and equals $2\operatorname{arcosh}(|\operatorname{tr}A|/2)$. It is the length of the corresponding closed <geodesic> in a quotient.

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