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.

- Two lines a,b in are parallel if and only if .
- If a and p are a line and a point of , then the line through p parallel with a is exactly .
- The parallel classes of correspond to the points on l.
- There exists a unique isomorphism with for each point .

**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.

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