# Homologies and elations

### Related pages

Translations, contractions and shears
Homologies and elations of P(K,T)
Groups of central-axial collineations
Baer's lemma
Let be a projective plane, and let be an automorphism of .

Definition. is a central collineation with center if fixes all lines through the point z, i.e. for each line .

Dually, is an axial collineation with axis if fixes all points on the line a.

We have the following basic results.

Lemma. If is a non-trivial collineation of , then has at most one center and one axis, respectively.

We will write and to denote the center and axis of a non-trivial central or axial collineation , respectively.

Lemma. A collineation is central if and only if the collineation is axial.

Consequently, these collineations are called central-axial collineations of . Moreover, if is a non-trivial central-axial collineation, then is a homology if Otherwise, is an elation if Finally, the identity is defined to be a homology and an elation.

Lemma. Let be a central-axial collineation. If p is a point with then the line is fixed by . Dually, if l is a line with then is a fixed point of .

Lemma. Let be a non-trivial central-axial collineation of . Then

Contributed by Hauke Klein
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