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.

We have the following basic results.

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.

Contributed by Hauke Klein
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