PROJECTIVE GEOMETRY COURSE

§ 2: *Models of a projective geometry*

*Definition:* a model of a (plane) projective geometry is a triple that consists of:

a non-empty set *P* whose elements we call points;

a non-empty set *L* whose elements we call lines;

an incidence relation to which every un-ordered pair of point and line either belongs or does not belong;

whilst the following *axioms* are satisfied:

A1 each line is incident with at least three points;

A2 each point is incident with at least three lines;

A3 with each pair of points there is exactly one line that is incident with both points;

A4 with each pair of lines there is exactly one point that is incident with both lines.

We call the axioms A1 and A2 *dual* to each other; likewise the axioms A3 and A4 are dual.

As a consequence of the fact that each axiom has a dual axiom, we find that, with each proposition we deduce from the axioms, a dual propsition can likewise be deduced.
This is the principle of duality.

By interchanging the roles of *P* and *L* in the definition above, we get the dual model.

The most important model of a projective geometry is the *real projective plane*.

We are given space with in it a fixed point O.
*P* is the set of straight lines through O, *L* is the set of planes through O.

A point *p* belonging to the set *P* is incident with the line *l* belonging to the set *L* if the straight line *p*
lies in the plane *l*.

Check that the axioms are satisfied.

Now consider ℜ^{3} with in it the plane *x _{3}=1*.

Let

This is not a model of a projective geometry, because A4 is not satisfied. To repair this, we consider the so-called points at infinity and the line at infinity.

*Definition :* A *point at infinity* is an equivalence class of parallel lines. The *line at infinity* is the set of points at infinity.

*O5* Let *P"* := *P'*∪{points at infinity}. Let *L"* := *L'*∪{line at infinity}.

How must we define an incidence relation so that the triple of *P"*, *L"* and this incidence relation is a model of a projective
geometry?

First define a bijection A:*P*→*P"* and a bijection B:*L*→*L"*, and make *p* from *P* incident
with *l* from *L* if and only if A*p* is incident is with B*l* (we call the models isomorphic).

*O6* Let *B* be a sphere with centre O as above.

Let *P'''* be the set of the *pairs of antipodal points* on *B*, and *L'''* the set of *great circles* on *B*, with the
common incidence relation.

Show that this model is isomorphic with the real projective plane.

*O7* A model of the *seven points geometry*:

This consists of seven beads *P _{1}*,

Make an incidence matrix, and check the axioms. Why do we call this seven points geometry selfdual?

Prove that each model of a projective geometry contains at least seven points and at least seven lines.