4.II.12H

Describe the hyperbolic lines in both the disc and upper half-plane models of the hyperbolic plane. Given a hyperbolic line $l$ and a point $P \notin l$, we define

$d(P, l):=\inf _{Q \in l} \rho(P, Q)$

where $\rho$ denotes the hyperbolic distance. Show that $d(P, l)=\rho\left(P, Q^{\prime}\right)$, where $Q^{\prime}$ is the unique point of $l$ for which the hyperbolic line segment $P Q^{\prime}$ is perpendicular to $l$.

Suppose now that $L_{1}$ is the positive imaginary axis in the upper half-plane model of the hyperbolic plane, and $L_{2}$ is the semicircle with centre $a>0$ on the real line, and radius $r$, where $0<r<a$. For any $P \in L_{2}$, show that the hyperbolic line through $P$ which is perpendicular to $L_{1}$ is a semicircle centred on the origin of radius $\leqslant a+r$, and prove that

$d\left(P, L_{1}\right) \geqslant \frac{a-r}{a+r} .$

For arbitrary hyperbolic lines $L_{1}, L_{2}$ in the hyperbolic plane, we define

$d\left(L_{1}, L_{2}\right):=\inf _{P \in L_{1}, Q \in L_{2}} \rho(P, Q)$

If $L_{1}$ and $L_{2}$ are ultraparallel (i.e. hyperbolic lines which do not meet, either inside the hyperbolic plane or at its boundary), prove that $d\left(L_{1}, L_{2}\right)$ is strictly positive.

[The equivalence of the disc and upper half-plane models of the hyperbolic plane, and standard facts about the metric and isometries of these models, may be quoted without proof.]

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