Affine plane

Remarks

Example: for a field $k$, the points of $k^2$ and the lines $\set{(x, y) : ax + by = c}$ with $(a, b) \ne (0, 0)$ form an affine plane. Check: two distinct points determine one such equation up to a factor. The parallels to $ax + by = c$ are the lines $ax + by = c'$, and exactly one of them passes through a given point. Finally, $(0,0)$, $(1,0)$, $(0,1)$ are not collinear. For $k = \F_2$ this is the smallest affine plane, with $4$ points and $6$ lines.

Non-example: the real projective plane satisfies clauses 1 and 3 but not clause 2, because any two of its lines meet.

Clause 2 is the parallel axiom in the form of Playfair's axiom. Parallelism is an equivalence relation on the lines of an affine plane. In a finite affine plane all lines have the same number $n$ of points, called the order, and there are $n^2$ points.