# Coordinatizing an affine plane

### Related pages

Coordinatization of projective planes
Affine coordinate planes
Let be an affine plane. We define

Definition. A coordinate frame of is a tupel (o,e,U,V) where o,e are two points of and U,V are two parallel classes of such that .

The line is called the diagonal of the coordinate frame (o,e,U,V). We have the following simple lemma.

Lemma. The map

is a bijection, called the coordinate map corresponding to the coordinate frame (o,e,U,V).

Moreover, we have the following theorem.

Theorem. Let K be the diagonal of a coordinate frame (o,e,U,V). Then the map

defines a ternary field (K,T), and the coordinate map is an isomorphism of onto the affine plane over (K,T).

In particular, each affine plane is isomorphic to the affine plane over some ternary field. But note that distinct coordinate frames of an affine plane usually yield non isomorphic ternary fields.

Contributed by Hauke Klein
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