Collineation

Remarks

Equivalently, $f$ is a bijection on points such that the image of every line is a line and every line of $S'$ is such an image. The same definition applies to affine spaces and to other incidence structures whose lines are sets of points.

Example: every projective transformation $[v] \mapsto [Tv]$ of $\mathbb{P}(V)$ is a collineation. Check: the line through $[u]$ and $[w]$ is $\mathbb{P}(W)$ for the plane $W$ spanned by $u, w$. The map sends it onto $\mathbb{P}(T W)$, again a line, and the inverse map comes from $T^{-1}$.

Non-example: in the Fano plane $\mathbb{P}^2(\F_2)$, whose lines are the sets $\set{a, b, a + b}$ of nonzero vectors of $\F_2^3$, swap $e_1$ and $e_2$ and fix all other points. The line $\set{e_1, e_3, e_1 + e_3}$ goes to $\set{e_2, e_3, e_1 + e_3}$, which is not a line, because $e_2 + e_3 \ne e_1 + e_3$.

The collineations of $S$ form a group under composition, the collineation group of $S$.