208 IV Continued Fractions and Their Uses
6 Non-EuclideanGeometry.....................................
There is an important connection between the modular group and the non-Euclidean
geometry of Bolyai (1832) and Lobachevski (1829). It was first pointed out by
Beltrami (1868) that theirhyperbolic geometryis the geometry on a manifold of con-
stant curvature. In the model of Poincar ́e (1882) for two-dimensional hyperbolic geom-
etry the underlying space is taken to be the upper half-planeH. A ‘line’ is either a
semi-circle with centre on the real axis or a half-line perpendicular to the real axis. It
follows that through any two distinct points there passes exactly one ‘line’. However,
through a given point not on a given ‘line’ there passes more than one ‘line’ having no
point in common with the given ‘line’.
Although Euclid’s parallel axiom fails to hold, all the other axioms of Euclidean
geometry are satisfied. Poincar ́e’s model shows that if Euclidean geometry is free from
contradiction, then so also is hyperbolic geometry. Before the advent of non-Euclidean
geometry there had been absolute faith in Euclidean geometry. It is realized today that
it is a matter for experiment to determine what kind of geometry best describes our
physical world.
Poincar ́e’s model will now be examined in more detail (with the constant
curvature normalized to have the value−1). A curveγinHis specified by a continu-
ously differentiable functionz(t)=x(t)+iy(t)(a≤t≤b). The (hyperbolic)length
ofγis defined to be
(γ)=∫bay(t)−^1 |dz/dt|dt.It follows from this definition that the ‘line’ segment joining two pointsz,wofHhas
length
d(z,w)=ln|z− ̄w|+|z−w|
|z− ̄w|−|z−w|.
It may be shown that any other curve joiningzandwhas greater length. Thus the
‘lines’ aregeodesics.
For anyz 0 ∈H, there is a unique geodesic throughz 0 in any specified direction.
Also, for any distinct real numbersξ,η, there is a unique geodesic which intersects the
real axis atξ,η, namely the semicircle with centre at(ξ+η)/2. (By abuse of language
we say ‘ξ’, for example, when we mean the point(ξ, 0 ).)
A linear fractional transformation
z′=f(z)=(az+b)/(cz+d),wherea,b,c,d∈Randad−bc=1, maps the upper half-planeH onto itself and
maps ‘lines’ onto ‘lines’. Moreover, if the curveγis mapped onto the curveγ′,then
(γ)=(γ′),sinceIf(z)=Iz/|cz+d|^2 anddf/dz= 1 /|cz+d|^2. In particular,
d(z,w)=d(z′,w′).Thus a linear fractional transformation of the above form is anisometry.Itmaybe
shown that any isometry is either a linear fractional transformation of this form or is