Hyperbolic circle

A hyperbolic circle of centre $o$ and radius $R>0$ is the level set $\{x:d_{\mathbb{H}}(o,x)=R\}$ for the hyperbolic distance. Equally spaced directions from its centre give vertices of a regular geodesic polygon.

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