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.

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

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