Let be an affine plane. Let be the set of all parallel classes of . We define a new incidence structure by
Then is a projective plane called the projective closure of . The new line is the so called line at infinity of . Obviously, the affine derivative on the line at infinity equals the original affine plane .
Lemma. Each isomorphism extends to a unique isomorphism .
Lemma. If (o,e,U,V) is a coordinate frame of , then (o,e,U,V) is a quadrangle in and the corresponding ternary fields of and , respectivly, are equal.