2nd Stage, Lecture 4: The incidence and projective geometries

Incidence Geometry
Ax1. For each two distinct points, there exists a unique line on both of them.
Ax2. For every line, there exist at least two distinct points on it.
Ax3. There exists at least three distinct points.
Ax4. Not all points lie on the same line.

A geometry that satisfies all four axioms will be called incidence geometry.
Examples: 4-line geometry, Fano’s geometry and Young’s geometry.
Incidence geometry has one of the following parallel conditions:
Parallel conditions: Given a line l and a point P not lies on l, then three possibilities exist for a parallel axiom:
PP1. There exist no lines on P that are parallel to l.
PP2. There exists exactly one line on P that is parallel to l, or
PP3. There exists more than one line on P parallel to l.
An incidence geometry satisfying PP2 is said to be Euclidean, otherwise it is said to be non-Euclidean.

4.2 A projective geometry
A projective geometry is an incidence geometry having no parallel lines (parallel condition 1) and in which each line has at least three points.
Ax1 For every two distinct points, there exists a unique line on both of them.
Ax2 Every line is on at least three distinct points.
Ax3 Not all the points are on the same line.
Ax4 Any two distinct lines are on at least one point (There are no parallel lines).

Examples. The Four-Line, Five-Line and Fano’s geometries are projective.