Affine plane


In geometry, an affine plane is a two-dimensional affine space.

Definitions

There are two ways to formally define affine planes, which are equivalent for affine planes over a field.
The first way consists in defining an affine plane as a set on which a vector space of dimension two acts simply transitively. Intuitively, this means that an affine plane is a vector space of dimension two in which one has "forgotten" where the origin is.
The second way occurs in incidence geometry, where an affine plane is defined as an abstract system of points and lines satisfying a system of axioms.

Coordinates and isomorphism

All the affine planes defined over a field are isomorphic. More precisely, the choice of an affine coordinate system for an affine plane over a field induces an isomorphism of affine planes between and.
In the more general situation, where the affine planes are not defined over a field, they will in general not be isomorphic. Two affine planes arising from the same non-Desarguesian projective plane by the removal of different lines may not be isomorphic.

Examples

Typical examples of affine planes are
In the applications of mathematics, there are often situations where an affine plane without the Euclidean metric is used instead of the Euclidean plane. For example, in a graph, which can be drawn on paper, and in which the position of a particle is plotted against time, the Euclidean metric is not adequate for its interpretation, since the distances between its points or the measures of the angles between its lines have, in general, no physical importance.