In incidence geometry, an affine plane is a system of points and lines that satisfy the following axioms :

• Any two distinct points lie on a unique line.
• Given a point and line there is a unique line which contains the point and is parallel to the line
• There exist three non-collinear points.

In an affine plane, two lines are called parallel if they are equal or disjoint.

The familiar Euclidean plane is an affine plane in this sense. As well as affine planes over fields (or division ring

Division ring

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...

s), there are also many non-Desarguesian plane

Non-Desarguesian plane

In mathematics, a non-Desarguesian plane, named after Gérard Desargues, is a projective plane that does not satisfy Desargues's theorem, or in other words a plane that is not a Desarguesian plane...

s satisfying these axioms. gives axioms for higher dimensional affine spaces.

An affine plane can be obtained from any projective plane

Projective plane

In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...

 by removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane as a closure

Closure (mathematics)

In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...

by adding a "line at infinity" whose points correspond to equivalence classes of parallel lines.