4.II.12H

Geometry | Part IB, 2006

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

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

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

Suppose now that L1L_{1} is the positive imaginary axis in the upper half-plane model of the hyperbolic plane, and L2L_{2} is the semicircle with centre a>0a>0 on the real line, and radius rr, where 0<r<a0<r<a. For any PL2P \in L_{2}, show that the hyperbolic line through PP which is perpendicular to L1L_{1} is a semicircle centred on the origin of radius a+r\leqslant a+r, and prove that

d(P,L1)ara+r.d\left(P, L_{1}\right) \geqslant \frac{a-r}{a+r} .

For arbitrary hyperbolic lines L1,L2L_{1}, L_{2} in the hyperbolic plane, we define

d(L1,L2):=infPL1,QL2ρ(P,Q)d\left(L_{1}, L_{2}\right):=\inf _{P \in L_{1}, Q \in L_{2}} \rho(P, Q)

If L1L_{1} and L2L_{2} are ultraparallel (i.e. hyperbolic lines which do not meet, either inside the hyperbolic plane or at its boundary), prove that d(L1,L2)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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