models of hyperbolic geometry
In the hyperbolic pages, we used the poincare disk model of hyperbolic geometry.
Here, we look at two other models. We have the klein-beltrami model. This again
uses the unit disk, but now the hyperbolic lines are segments of euclidean lines.
We also introduce the minkowski model. This uses a surface in three dimensions.
The model is related to special relativity. It is also a key step in understanding the
relation between hyperbolic and projective geometries.

hyperbolic and projective geometries
This is one of the main features of the klein view of geometry
We show that the hyperbolic group is isomorphic to a subgroup of the projective
group. In fact, it is isomorphic to the symmetry group of a conic. The link is the
minkowski model of hyperbolic geometry. Unfortunately, the proof is complicated.

plane conics

some special conics and their tangents
We look at the tangents to some special conics. We conider one parabola and
one hyperbola, and the circle. These are the key to the investigation of conics
and tangents in affine geometry.

affine geometry

the classification of conics in affine geometry
We define the conics as non-degenerate loci with equations quadratic in x and y.
This is necessary as we do not have the concept of length. We show that there
are exactly three affine congruence classes, the parabolas, thy hyperbolas, and
the ellipses (including circles). For each class, we give a member with a simple
equation.

affine symmetry groups of conics
We investigate the groups of affine symmetries of conics. Of course, the symmetry
groups for cojugate conics are conjugate, so we need only look at one from each
class. In contrast with the euclidean case, all the groups are infinite. In particular,
we show that the group acts transitively on the conic. this shows that there are no
"special" points - unlike, say the vertex of a parabola in euclidean geometry. Further,
the group of a parabola is actually doubly transitive. the structure of the groups are
sufficient to distinguish the classes.

tangents of affine conics
With a slightly surprising definition, we show that tangency is an affine concept.
As with the symmetry groups, the study of tangents allows us to distinguish the
classes. For an ellipse, every family of parallel lines includes two tangents, for a
parabola, there is one family with no tangents. For a hyperbola, there are two
exceptional families, and each containa an asymptote. In this way, we conclude
that asymptotes are affine objects.

projective conics

the three point theorem
In affine geometry, we have one- and two-point theorems describing the transitivity
of the affine symmetry groups of conics. In projective geometry, there is only one
congruence class of conics. We show that the projective symmetry group is triply
transitive. This is useful in proving theorems about conics. An additional result is the
Parametrisation Theorem which gives a parametric description of a conic.

the projective symmetry group of a conic
Since all conics are projective congruent, all the projective symmetry groups are
conjugate, so it is enough to study just one. We give an algebraic description of
the elements of the projective symmetry group of one particular conic.

the five point theorem
In euclidean geometry, three non-collinear poits determine a unique circle. Here,
we show that five p-points, no three of which are collinear, determine a unique
projective conic. This is illustrated by a CabriJava applet.