= Hyperbolic circle in the upper half-plane
{title2=$(x-u)^2+(y-v\cosh R)^2=v^2\sinh^2R$}
In the <Poincaré half-plane model>, a <hyperbolic circle> of radius $R$ centred at $u+iv$ is the Euclidean circle $(x-u)^2+(y-v\cosh R)^2=v^2\sinh^2R$. Its Euclidean centre is shifted upward from its hyperbolic centre, and its Euclidean radius is $v\sinh R$. The lower height $ve^{-R}$ is positive. The formula follows by mapping the centre to zero in the <Poincare disc model> and using radial distance $2\operatorname{artanh}\rho$.
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