Let l be a line in a projective plane . The affine derivative of at the line l is defined to be
Then the incidence structure is an affine plane. The following observations are pretty obvious.
Lemma. if and only if there exists an isomorphism with .
Lemma. The map
is an isomorphism of groups.
Lemma. Let (o,e,u,v) be a quadrangle in with . Then is a coordinate frame of the affine plane , and the corresponding ternary fields of and , respectively, are equal.