= Calibration for semicircular hyperbolic geodesics
{title2=$\Phi(x,y)=\operatorname{arsinh}(x/y),\quad|\nabla\Phi|=1/y$}
In the upper half-plane $y>0$, the <hyperbolic length> element is $ds/y$. The function $\Phi=\operatorname{arsinh}(x/y)$ has <gradient> $(1,-x/y)/\sqrt{x^2+y^2}$ and norm $1/y$. The <Cauchy-Schwarz inequality> gives $\int ds/y\geq\Phi(B)-\Phi(A)$ for any oriented path from $A$ to $B$. Equality holds along a semicircle $x^2+y^2=R^2$ traversed with increasing $x$, because its tangent is parallel to that gradient. This is a <calibration> proof of global minimality, beyond the stationary-path conclusion of the <Beltrami identity>. Translating $x$ gives the analogous proof for semicircles with other boundary centers.
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