1.I.4B

Write down the Riemannian metric on the disc model $\Delta$ of the hyperbolic plane. What are the geodesics passing through the origin? Show that the hyperbolic circle of radius $\rho$ centred on the origin is just the Euclidean circle centred on the origin with Euclidean radius $\tanh (\rho / 2)$.

Write down an isometry between the upper half-plane model $H$ of the hyperbolic plane and the disc model $\Delta$, under which $i \in H$ corresponds to $0 \in \Delta$. Show that the hyperbolic circle of radius $\rho$ centred on $i$ in $H$ is a Euclidean circle with centre $i \cosh \rho$ and of radius $\sinh \rho$.

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