Editing Projective plane (section)

===Definition===
More formally an '''[[Affine plane (incidence geometry)|affine plane]]''' consists of a set of '''lines''' and a set of '''points''', and a relation between points and lines called '''incidence''', having the following properties:
<div id="axioms-of-affine-planes">
#Given any two distinct points, there is exactly one line incident with both of them.
#Given any line ''l'' and any point ''P'' not incident with ''l'', there is exactly one line incident with ''P'' that does not meet ''l''.
#There are four points such that no line is incident with more than two of them.
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The second condition means that there are [[Parallel (geometry)|parallel lines]] and is known as [[John Playfair|Playfair's]] axiom. The expression "does not meet" in this condition is shorthand for "there does not exist a point incident with both lines".

The Euclidean plane and the Moulton plane are examples of infinite affine planes. A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed. The '''order''' of a finite affine plane is the number of points on any of its lines (this will be the same number as the order of the projective plane from which it comes). The affine planes which arise from the projective planes PG(2,&nbsp;''q'') are denoted by AG(2,&nbsp;''q'').

There is a projective plane of order ''N'' if and only if there is an [[affine plane (incidence geometry)|affine plane]] of order ''N''.  When there is only one affine plane of order ''N'' there is only one projective plane of order ''N'', but the converse is not true. The affine planes formed by the removal of different lines of the projective plane will be isomorphic if and only if the removed lines are in the same orbit of the collineation group of the projective plane. These statements hold for infinite projective planes as well.