Special types of correlations[edit]
Polarity[edit]
If a correlation φ is an involution (that is, two applications of the correlation equals the identity: φ2(P) = P for all points P) then it is called a polarity. Polarities of projective spaces lead to polar spaces, which are defined by taking the collection of all subspace which are contained in their image under the polarity.
Natural correlation[edit]
There is a natural correlation induced between a projective space P(V) and its dual P(V∗) by the natural pairing ⟨⋅,⋅⟩ between the underlying vector spaces V and its dual V∗, where every subspace W of V∗ is mapped to its orthogonal complement W⊥ in V, defined as W⊥ = {v ∈ V | ⟨w, v⟩ = 0, ∀w ∈ W}.[4]
Composing this natural correlation with an isomorphism of projective spaces induced by a semilinear map produces a correlation of P(V) to itself. In this way, every nondegenerate semilinear map V → V∗ induces a correlation of a projective space to itself.