a collineation
Imagine shining a slide projector at a tilted screen, or photographing a flat painting from an angle: straight lines stay straight, but lengths, angles, and ratios get stretched and skewed. A collineation is the precise name for this kind of transformation of the projective plane — a one-to-one mapping of points to points (and lines to lines) that carries straight lines to straight lines. It is exactly the family of motions that projective geometry treats as 'allowed', the symmetries under which its theorems are invariant.
Concretely, a collineation of the real projective plane is given in homogeneous coordinates by an invertible 3-by-3 matrix M: a point (x : y : z) maps to M times the column (x, y, z), read again up to scaling. Because matrix multiplication sends the solutions of one linear equation to the solutions of another, lines go to lines and incidence is preserved — if a point was on a line, its image is on the image line. Such a map can move the line at infinity to an ordinary line, which is why a collineation can turn parallel lines into intersecting ones (the projector tilting the horizon into view). The most important fact is what it preserves: a collineation keeps the cross-ratio of any four collinear points unchanged. So cross-ratio, harmonic conjugacy, and 'being a conic' all survive, while distance, angle, parallelism, and midpoint generally do not.
Collineations form a group (you can compose two and invert any one), and that group is precisely the transformation group whose invariants ARE projective geometry, in the spirit of Klein's classification. The fundamental theorem of projective geometry pins them down: a collineation of the real projective plane is determined by where it sends four points in general position, and conversely any four points in general position can be mapped to any other four. One honest subtlety: over the real numbers every line-preserving bijection is of the matrix form above, but over fields with nontrivial automorphisms (like the complex numbers) there are extra 'semilinear' collineations, so the clean statement 'collineation = matrix' is special to the reals.
The matrix M with rows (1, 0, 0), (0, 1, 0), (1, 0, 1) sends (x : y : z) to (x : y : x + z). The ordinary point (1 : 0 : 1) goes to (1 : 0 : 2) = (1/2 : 0 : 1). The point at infinity (1 : 0 : 0) (slope-0 direction) maps to (1 : 0 : 1), an ordinary point — so this collineation has pulled a point of infinity into the finite plane, exactly what a tilt of perspective does.
An invertible 3-by-3 matrix on homogeneous coordinates: straight lines and cross-ratio survive, distances do not.
A collineation preserves cross-ratio but NOT distance, angle, area, parallelism, or midpoint — those are finer (affine/Euclidean) notions. Over the real plane every collineation is a matrix map (a 'projectivity'), but over other fields semilinear maps appear too, so 'collineation' is slightly more general than 'projective matrix transformation' in full generality.