Affine plane

Defintion

An affine plane is defined as an incidence structure satisfying the following properties (note that blocks are now referred to as lines):

1. To any two distinct points, there exists a unique line incident with both of them
2. Given a line and a point not incident on it, there exist a unique line through that point that is parallel to the given line (viz, does not have any common point with the given line)
3. There exist three non-collinear points

A set of three non-collinear points is termed a triangle and a set of three non-concurrent lines is termed a trilateral.

Facts

Equivalence relation of parallelism

Define two lines to be parallel if they are equal or have no points in common. The relation is reflexive and symmetric; transitivity follows by an easy argument (For full proof, refer: parallelism is transitive)

Thus, parallelism is an equivalence relation, and the equivalence classes under this relation are termed ideal points.

Projectivizing an affine plane

The steps are:

• Add ideal points for each equivalence class of parallel lines, to the point set
• Make each ideal point incident on each parallel line in its equivalence class
• Add an ideal line to the line set, which is incident precisely with all the ideal points