Affine Plane (incidence Geometry)

Affine Plane (incidence Geometry)

In 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 (Playfair's Axiom).
• There exist three non-collinear points (points not on a single line).

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

Since no concepts other than those involving the relationship between points and lines are involved in the axioms, an affine plane is an object of study belonging to incidence geometry.

The familiar Euclidean plane is an affine plane. There are many finite and infinite affine planes. As well as affine planes over fields (and division rings), there are also many non-Desarguesian planes, not derived from coordinates in a division ring, satisfying these axioms. The Moulton plane is an example of one of these.

An affine plane can be obtained from any projective plane by removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane by adding a line at infinity, each of whose points is that point at infinity where an equivalence class of parallel lines meets. If the projective plane is non-Desarguesian, the removal of different lines could result in non-isomorphic affine planes.