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collineation
In Euclidean geometry, translations and dilations are both examples of onetoone onto functions that map straight lines to straight lines. Collineation is a generalization of this notion to more abstract geometric structures.
Definition. Let $\mathscr{S}=(\mathcal{P},\mathcal{L})$ be a nearlinear space. A collineation on $\mathscr{S}$ is an onetoone onto linear function on $\mathscr{S}$ such that its inverse is also linear. A collineation on $\mathscr{S}$ is also called an automorphism on $\mathscr{S}$.
It is easy to see that a collineation preserves collinearity: if three points are collinear, so are their images under a collineation.
Example 1. let $\mathscr{S}$ be the nearlinear space consisting of four points $P,Q,R,S$, and four lines $PQ,QR,RS$, and $SP$.
\pspicture(0,0)(2,2) \pssetunit=2cm \psdots[linecolor=blue,dotsize=5pt](0,0) \psdots[linecolor=blue,dotsize=5pt](0,1) \psdots[linecolor=blue,dotsize=5pt](1,1) \psdots[linecolor=blue,dotsize=5pt](1,0) \pspolygon(0,0)(0,1)(1,1)(1,0) \rput[r](0.125,0)$S$ \rput[r](0.125,1)$P$ \rput[l](1.125,1)$Q$ \rput[l](1.125,0)$R$
There are six collineations of $\mathscr{S}$:

$\sigma_{1}$ is the identity function,

$\sigma_{2}=(PQRS)$,

$\sigma_{3}=\sigma_{2}^{2}=(PR)(QS)$,

$\sigma_{4}=(PSRQ)=\sigma_{2}^{{1}}$,

$\sigma_{5}=(PR)$,

$\sigma_{6}=(QS)$.
It is easy to see that this is the alternating group on four letters $A_{4}$.
If $PR$ is added as another line of $\mathscr{S}$,
\pspicture(0,0)(2,2) \pssetunit=2cm \psdots[linecolor=blue,dotsize=5pt](0,0) \psdots[linecolor=blue,dotsize=5pt](0,1) \psdots[linecolor=blue,dotsize=5pt](1,1) \psdots[linecolor=blue,dotsize=5pt](1,0) \pspolygon(0,0)(0,1)(1,1)(1,0) \psline(0,1)(1,0) \rput[r](0.125,0)$S$ \rput[r](0.125,1)$P$ \rput[l](1.125,1)$Q$ \rput[l](1.125,0)$R$
then $\sigma_{2}$ and $\sigma_{4}$ above are no longer collineations of $\mathscr{S}$. The resulting set is the Klein 4group.
What we have seen is true in general: the set of all collineation on a nearlinear space $\mathscr{S}$ is a group under functional composition. This group is sometimes written $\operatorname{Aut}(\mathscr{S})$.
Example 2. Let $V$ be an $n$dimensional vector space over some division ring $D$, with $n\geq 3$. One may form the affine space $A(V)$ over $V$. The points and lines of $A(V)$ are respectively cosets of zero and one dimensional subspaces of $V$, and $A(V)$ is a linear space. Collineations of $A(V)$ are called affinities. Common examples of affinities are translations, dilations, and reflections (with respect to lines).
Example 3. Again let $V$ and $k$ be as in the last example. One may form the projective space $P(V)$ over $V$. The points and lines of $P(V)$ are one and two dimensional subspaces of $V$, and $P(V)$ is a linear space. Collineations of $P(V)$ are called projectivities.
A point $P\in\mathcal{P}$ is called a fixed point of $\sigma$ if $\sigma(P)=P$. A line $\ell\in\mathcal{L}$ is a fixed line of $\sigma$ if $\sigma(\ell)=\ell$. Note that $\sigma$ fixes line $\ell$ merely means that $\sigma$ permutes the points on $\ell$, and does not necessarily fix them.
Definition. Given a collineation $\sigma$ on $\mathscr{S}$, a center of $\sigma$ is a point $P$ of $\mathscr{S}$ such that $\sigma$ fixes all lines passing through $P$. An axis of $\sigma$ is a hyperplane $\pi$ of $\mathscr{S}$ such that $\sigma$ fixes all points in $\pi$. A collineation is said to be central if it has a center, and axial if it has an axis.
It can be shown that, in a projective space, central collineations are precisely the same as axial collineations. Furthermore, if the (central) collineation is not the identity, then it has a unique center and a unique axis. Given a projective space, nonidentity central collineations can further be classified: those with centers lying in their axes are called elations, and homologies otherwise.
References
 1 L. M. Batten, Combinatorics of Finite Geometries, 2nd edition, Cambridge University Press (1997)
 2 A. Beutelspacher, U. Rosenbaum Projective Geometry, From Foundations to Applications, Cambridge University Press (2000)
Mathematics Subject Classification
51A45 no label found51A05 no label found05C65 no label found Forums
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