Related pagesTranslations, contractions and shears
Homologies and elations of P(K,T)
Groups of central-axial collineations
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