The following is found in Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry, by Jürgen Richter-Gebert.
Definition 2.1. A projective plane is a triple $(\mathcal{P},\mathcal{L}, \mathrm{I})$. The set $\mathcal{P}$ consists of the points, and the set $\mathcal{L}$ consists of the lines of the geometry. The inclusion $\mathrm{I} \subseteq \mathcal{P}\times\mathcal{L}$ is an incidence relation satisfying the following three axioms:
- (i) For any two distinct points, there is exactly one line incident with both of them.
- (ii) For any two distinct lines, there is exactly one point incident with both of them.
- (iii) There are four distinct points such that no line is incident with more than two of them.
This is apparently typical of "incidence geometry". From what I have gleaned from various discussions, this approach to geometry does not define lines as collections of points. As I discussed in Is Richter-Gebert's Theorem: "A projective transformation maps collinear points to collinear points." really a theorem? It is conceivable that components of points could transform under the same law as those of lines. Line space and point space would continue to coincide, but incident relations would not be preserved. The axioms (i),(ii) and (iii) would still hold, just not between the images of the objects being incident prior to transformation.
It is natural that we would typically want incidence to be preserved under transformations, but I have not seen that issue addressed in the various discussions I have examined.
Does the concept of point-line incidence assume the relation is preserved under transformations?
I will add that in case of transformations of the real projective plane $\mathbb{RP}^2$ Richter-Gebert asserts:
The homogeneous coordinates of a line must be mapped in such a way that incidences of points and lines are preserved under the mapping.
But that does not appear to follow from the given definitions and axioms. The assertion follows the proof discussed at the link above, and was apparently used in the proof. I expect that the assertion is entirely general. What I want to know is where it enters into the logical structure.