We also showed that the similarity group S(2) is a subgroup of the inversive
group I(2). In fact, S(2) consists of elements fixing ∞, i.e. S(2) = S({∞),I(2)).
This defines similariy geometry on the set obtained by removing ∞. Observe
that elements of S(2) map ilines through ∞ only to ilines through ∞. Thus the
family of ilines splits in this geometry. The lines of similarity geometry come
from the ilines through ∞, with ∞ deleted. Thes are, of course the euclidean
lines in the familiar plane model.
We defined the hyperbolic group H(2) as the subgroup of I(2) consisting of
the elements mapping the disk D to itself, i.e. H(2) = S(D,I(2)). This is a
subgroup of S(C,I(2)), the inversive symmetry group of the unit circle C.
The subgroup H(2) defines a geometry on D. But, we do not define the lines
of the geometry as all the nonempty sets
L_{n}D , where L is an iline. Observe
that inversion in an iline L maps D to itself, i.e. belongs to H(2), if and only if
L is orthogonal to C. The hlines are the intersections
L_{n}D for such ilines L.
One of the key features of inversive geometry is the property of angle. Thus
similarity (and hence euclidean) and inversive geometries inherit the property
of angle. In iversive geometry, the ilines include all circles and lines. In
similarity geometry, those which do not include ∞, i.e. the circles form
a single conjugacy class. In euclidean geometry, this class splits into infinitely
many classes, diferentiated by a property we christen "euclidean length". In the
hyperbolic case, a circle within D must map to another such circle.
Such circles
are called hyperbolic circles.Again, these
fall into infinitely many classes. The distinguishing property is called "hyperbolic length".
Indeed, this is the way we
defined the hyperbolic length.
Problems involving circles in these geometries are often consequences of
results from inversive geometry. For example, in inversive geometry, there
is a unique iline through any three points. In euclidean geometry, the iline
may give a line or circle. We can state this as
Given three points in the plane, either the points are collinear or they
lie on a unique circle.
In hyperbolic geometry, the iline may be neither orthogonal to C nor within
D, so lie on at most one hline or hyperbolic circle.


We saw that E(2) < S(2) < A(2) , relating euclidean, similarity and affine geometries.
We also showed that A(2) is isomorphic to a subgroup of P(2).In detail, we showed
that A(2) is isomorphic to the subgroup of P(2) fixing the pline Z : z=0, i.e. to the
projective symmetry group of Z, S(Z,P(2)). The elements of S(Z,P(2)) map ppoints
on Z to other ppoints on Z. The subgoup therefore defines a rather trivial geometry
on the set Z. On the other hand, for ppoints P,Q not on Z, then there is an element
of S(Z,P(2)) mapping P to Q. Thus the subgroup defines a geometry on the rest of
the projective plane. We get a nice model by observing that a ppoint not on Z meets
the plane Z* : z=1 in a unique point. The restrictions to Z* give the usual model of
affine geometry. If L and M are plines other than Z, then they define lines of affine
geometry. As plines, they meet in a ppoint R. If this is not on Z, the affine lines do
meet. If R is on Z, then the affine lines do not meet. This
explains the appearance
of the concept of parallelism in affine and hence in euclidean
geometry. Indeed, the
model can be used to prove the fundamental result which we shall call
The Parallel Theorem
If L is a line in affine geometry and P is a point not on L,
then there is a unique line through P not meeting L.
proof


We have shown that the hyperbolic group H(2) is isomorphic to a subgroup of P(2).
Although we could use any conic to obtain this result, we chose the minkowsi model
since this is related to special elativity. The conic is K :
F(x,y) = x^{2}+y^{2}z^{2} = 0. Then
H(2) is isomorphic to S(K,P(2)). We obtain the model be looking at the positive sheet
of the hyperboloid H : F(x,y) = 1.
The hyperbolic lines correspond to the intersections
of H with the plines which actually cut H.Given a pline L and a ppoint P inside
K,
there are many plines M through P (and hence cutting H) which meet L outside K.
The limiting positions of this collection are those joining P to the ppoints where the
pline L meets K. Thus, we see that there is no Parallel Theorem here.
The sketch on the right shows the cone K and a pline M through P which meets L
outside K.
The principal features of projective geometry are incidence (including tangency)
and crossratio. Provided that we take care about ppoints of RP^{2} which do not
give rise to affine or hyperbolic points, we get corresponding results for these
geometries. For example, we have in either geometry
Pappus's Theorem
Suppose that points A,B,C lie on a line L, and A',B',C' on a line L'
and that AB',A'B meet in P, BC',B'C in Q, and CA',C'A in R, then
P,Q,R are collinear.
The italicised condition is vital, ther will be cases where the lines fail to meet.
In affine (and hence euclidean) geometry, lines which fail to meet are parallel,
so we can recover variants for such cases. As an example, we have
Euclidean Variant
Suppose that points A,B,C lie on a line L, and A',B',C' on a line L'
are such that AB'and A'B are parallel, and BC'and B'C are parallel,
then CA',C'A are parallel.
This is easy to see in the projective setting. Two euclidean lines are parallel
corresponds to the statement that they meet at an ideal point. If AB',A'B and
BC',B'C are parallel, then they meet at points on the ideal line. From the
projective version of Pappus, CA',C'A also meet at an ideal point, so that the
euclidean lines are parallel.

