Editing Projective plane (section)

==Definition==

A '''projective plane''' is a rank 2 [[incidence structure]] <math>({\mathcal P},{\mathcal L},I)</math> consisting of a set of '''points''' <math>{\mathcal P}</math>, a set of '''lines''' <math>{\mathcal L}</math>, and a symmetric relation <math>I</math> on the set <math>{\mathcal P}\cup{\mathcal L}</math> called '''incidence''', having the following properties:<ref>In a more formal version of the definition it is pointed out that the terms ''point, line'' and ''incidence'' are [[primitive notion]]s (undefined terms). This formal viewpoint is needed to understand the concept of [[duality (projective geometry)|duality]] when applied to projective planes.</ref>
<div id="axioms-of-projective-planes">
#Given any two distinct points, there is exactly one line incident with both of them.
#Given any two distinct lines, there is exactly one point incident with both of them.
#There are four points such that no line is incident with more than two of them.
</div>
The second condition means that there are no [[Parallel (geometry)|parallel lines]]. The last condition excludes the so-called '''''degenerate''''' cases (see [[#Degenerate planes|below]]). The term "incidence" is used to emphasize the symmetric nature of the relationship between points and lines. Thus the expression "point ''P'' is incident with line ''ℓ''" is used instead of either "''P'' is on ''ℓ''" or "''ℓ'' passes through ''P''".

It follows from the definition that the number of points <math>s+1</math> incident with any given line in a projective plane is the same as the number of lines incident with any given point. The (possibly infinite) cardinal number <math>s</math> is called '''order''' of the plane.