projectiveplane

A projective plane is an incidence structure consisting of a set of points and a set of lines, together with an incidence relation, that satisfies three axioms: any two distinct points lie on a unique line; any two distinct lines meet in a unique point; and there exist four points, no three of which are collinear. A basic feature is duality: points and lines may be interchanged in the axioms to obtain the dual statement.

Finite projective planes of order n have n^2+n+1 points and the same number of lines. Each line contains n+1 points, and each point lies on n+1 lines. Such planes exist for every prime power n, and they can be constructed as PG(2,q), the projective plane over the finite field GF(q), arising from the 3-dimensional vector space GF(q)^3. The smallest example is the Fano plane, order 2.

The real projective plane RP^2 is the set of lines through the origin in R^3, with homogeneous coordinates [x:y:z]. A line in RP^2 can be written as ax+by+cz=0, and a point [x:y:z] lies on that line precisely when ax+by+cz=0. Geometrically, RP^2 can also be described as the quotient of the 2-sphere S^2 by identifying antipodal points.

Topologically and algebraically, RP^2 is a non-orientable surface with Euler characteristic 1 and fundamental group isomorphic to Z/2. In projective geometry, duality asserts that many theorems remain valid when the roles of points and lines are interchanged, producing dual statements. Non-Desarguesian planes are examples of projective planes not arising from a field-theoretic construction, illustrating the diversity of incidence geometries beyond PG(2,q).