= Hyperbolic circle in the Poincare disc
{title2=$d_D(p,z)=R$}
In the <Poincare disc model>, a <hyperbolic circle in the Poincare disc> consists of points at one fixed hyperbolic distance from a center. At center zero it is the Euclidean circle $|z|=\tanh(R/2)$, since radial hyperbolic distance is $2\operatorname{artanh}|z|$. A <Poincare disc automorphism> sending a general center to zero transports this description. Conversely every Euclidean circle entirely inside the disc is such a circle: the product of inversion in that circle and unit-circle inversion has reciprocal fixed points, one inside; sending the inner fixed point to zero makes the circle concentric with the unit circle.
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