Elementary Functions 241not contain z3), then 0 < Tz < 1; and if z lies between z 2 and z 3 (in the
arc that does not contain z1), then Tz < 0.
If z does not lie on 0, it follows that Im(z, zi, z 2 , z3) f 0. [With respect
to a certain orientation of C we say that such a point z lies to the right of
C if Im(z, zi, z2, z 3 ) > 0, and to the left of C if Im(z, z1, z 2 , z 3 ) < 0. This
characterization corresponds to the usual distinction between right-and
left-handed planes, since in view of (5.10-1) we have, letting Tz = ~ + iry,
Im(z,z1,z2,z3) ='f/and 'f/ > 0 defines the right half-plane determined by the real axis oriented
as in Fig. 5.14, while μ < 0 defines the left half-plane.
On the extended complex plane (or on the Riemann sphere) a positive
orientation can be defined on any proper circle (i.e., not a straight line) by
prescribing that oo be to the right of the circle. Whenever a circle C is
oriented positively, the set of points to the right of the circle is called the
outside of C, whereas the set of points on the left is called its inside.
Example The unit circle as oriented by the ordered triple [1, i, -1] has
a positive orientation since
Im( oo, 1, i, -1) = 1 > 0
5.11 THE POINCARE MODEL OF LOBACHEVSKY
NON-EUCLIDEAN GEOMETRY
In this section we propose to discuss briefly an interesting connection
between certain bilinear transformations and non-Euclidean geometry.
F. Klein, in his famous lecture at the University of Erlangen, known as
the Erlangen Program, was the first to define geometry as the study of the
properties of figures of a certain space S that remains unchanged under a
group of transformations defined on S. According to this, in each space
there are as many geometries as groups of transformations can be defined
among its elements. In particular, two-dimensional Euclidean geometry is
the study of those properties of plane figures that are invariant under the
so-called fundamental group, consisting of all rigid motions, similarities,
and symmetries.
By replacing the fundamental group by the collineation group leaving
a certain conic invariant, A. Cayley in 1857 and F. Klein in 1871 realized
that the axioms of the non-Euclidean geometries, as well as those of the Eu-
clidean case, can be satisfied, the resulting type of geometry depending on
the particular conic, thus bringing all three types of geometries-hyperbolic
(Lobachevsky-Bolyai), elliptic (Riemann), and parabolic (Euclid)-under