6 Non-Euclidean Geometry 209obtained by composing such a transformation with the (orientation-reversing) trans-
formationx+iy→−x+iy. For any two ‘lines’LandL′, there is an isometry which
mapsLontoL′.
We m a y d e fi n eanglesto be the same as in Euclidean geometry, since any linear
fractional transformation is conformal. The (hyperbolic)areaof a domainD⊂H,
defined by
μ(D)=∫∫
Dy−^2 dxdy,is invariant under any isometry. In particular, this givesπ−(α+β+γ)for the area
of a ‘triangle’ with anglesα,β,γ. Since the angles are non-negative, the area of a ‘tri-
angle’ is at mostπand, since the area is necessarily positive, the sum of the angles of
a ‘triangle’ is less thanπ.
For example, ifFis the fundamental domain of the modular groupΓ,thenF ̄is a
‘triangle’ with anglesπ/ 3 ,π/ 3 ,0 and hence the area ofF ̄isπ− 2 π/ 3 =π/3. For
any fixedz 0 ∈Fon the imaginary axis, we may characterizeFas the set of allz∈H
such that, for everyg∈Γwithg=I,
d(z,z 0 )<d(z,g(z 0 ))=d(g−^1 (z),z 0 ).By identifying two pointsz,z′ofH ifz′=g(z)for someg∈Γwe obtain the
quotient spaceM=H/Γ. Equivalently, we may regardMas the closureF ̄of the
fundamental domainFwith the boundary point− 1 / 2 +iyidentified with the bound-
ary point 1/ 2 +iy( 1 ≤y<∞)and the boundary point−e−iθidentified with the
boundary pointeiθ( 0 <θ<π/ 2 ).
Since the elements ofΓare isometries ofH, the metric onHinduces a metric on
Min which the geodesics are the projections ontoMof the geodesics inH. Thus if
we regardMasF ̄with appropriate boundary points identified, then a geodesic inM
will be a sequence of geodesic arcs inF, each with initial point and endpoint on the
boundary ofF, so that the initial point of one arc is the point identified to the endpoint
of the preceding arc.
LetLbe a geodesic inHwhich intersects the real axis in irrational pointsξ,η
such thatξ> 1 ,− 1 <η<0andlet
ξ=[a 0 ,a 1 ,a 2 ,...], − 1 /η=[a− 1 ,a− 2 ,...]be the continued fraction expansions ofξand− 1 /η. If we chooseξandη=ξ′to be
conjugate quadratic irrationals then, by Proposition 7, the doubly-infinite sequence
[...,a− 2 ,a− 1 ,a 0 ,a 1 ,a 2 ,...]is periodic and it is not difficult to see that the geodesic inMobtained by projection
fromLis closed. Artin (1924) showed that there are other geodesics which behave
very differently. Let the convergents ofξbepn/qnand put
ξ=(pn− 1 ξn+pn− 2 )/(qn− 1 ξn+qn− 2 ), η=(pn− 1 ηn+pn− 2 )/(qn− 1 ηn+qn− 2 ).