Chapter 4 - Projective Planes

Abstract

This chapter discusses projective planes. A projective plane is an incidence structure, “P = (P, L, I).” The structure consists of a set “P” of points,a set “L” of lines, and an incidence relation “I”, in such a manner that the following three axioms are fulfilled. — Any two distinct points are incident with precisely one common line; any two distinct lines are incident with a common point; and there exist at least two lines; any line is incident with at least three points. Projective planes can be described as diagram geometries. Projective planes are one of the most classical objects in geometry and the most classical projective planes, namely the Desarguesian ones. The fact that a projective plane is Desarguesian is intimately related to the existence of all possible central collineations. To be precise, it can be said that P is (P, l)-transitive for a point-line pair (P, l) if for some line m ≠ l through P and any two points X, Y on m with X, Y ≠ P, l ∩m there exists a central collineation with axis l and centre P mapping X onto Y.