---
tag: XD8ZB63H78
kind: definition
name: Affine plane
msc: [51A15, "51E15"]
defines: EC7XT2ZM5S
uses: [FGKNAFNHA3, 0AFNNTRTJE, PR7Q3QMEHZ]
sources:
  - url: https://en.wikipedia.org/wiki/Affine_plane_(incidence_geometry)
---

## Definition

An *affine plane* is an [[incidence structure|FGKNAFNHA3]] of points and lines such that:

1. Any two distinct points lie on exactly one common line.
2. For every line $\ell$ and every point $p$ not on $\ell$, exactly one line through $p$ has no point in common with $\ell$.
3. There are three points that are not [[collinear|0AFNNTRTJE]].

Two lines are *parallel* if they are equal or have no common point.

## 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|DANFNN8C4Y]] 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|PR7Q3QMEHZ]].
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.