Incidence (geometry)

Incidence (geometry)

In geometry, the relations of incidence are those such as 'lies on' between points and lines (as in 'point P lies on line L'), and 'intersects' (as in 'line L₁ intersects line L₂', in three-dimensional space). That is, they are the binary relations describing how subsets meet. The propositions of incidence stated in terms of them are statements such as 'any two lines in a plane meet'. This is true in a projective plane, though not true in Euclidean space of two dimensions where lines may be parallel.

Historically, projective geometry was introduced in order to make the propositions of incidence true (without exceptions such as are caused by parallels). From the point of view of synthetic geometry it was considered that projective geometry should be developed using such propositions as axioms. This turns out to make a major difference only for the projective plane (for reasons to do with Desargues' theorem).

The modern approach is to define projective space starting from linear algebra and homogeneous co-ordinates. Then the propositions of incidence are derived from the following basic result on vector spaces: given subspaces U and V of a vector space W, the dimension of their intersection is at least dim U + dim V − dim W. Bearing in mind that the dimension of the projective space P(W) associated to W is dim W − 1, but that we require an intersection of subspaces of dimension at least 1 to register in projective space (the subspace {0} being common to all subspaces of W), we get the basic proposition of incidence in this form: linear subspaces L and M of projective space P meet provided dim L + dim M is at least dim P.