Related pagesThe projective plane over a vector space
Fundamental theorem of affine geometry
An affine plane is isomorphic to some AG(V) if and only if the projective closure of is desarguesian. Moreover, given two vector space V, V' of dimension two over skewfields K and K', respectively, then the affine planes AG(V) and AG(V') are isomorphic if and only if the skewfields K and K' are isomorphic.
Given a vector space V of dimension 2 over a skewfield K, then
via the isomorphism