Understanding whether the image on a camera is a projective plane or not 
I know that projective geometry began to develop from art at the start . So suppose if we take a picture of a rails of a station we would get a picture which (i think maybe wrong) is a projective transformation of a $3D$ structure into a $2D$ one from camera perspective.


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*If its correct are both the actual rails structure and the photograph one both are projective planes (treated as being in that only) the actual one would have parallel line which would still meet at infinity point and the other bent one would also meet not at infinity but at some vanishing point ? The camera just convert it to  different projective points in the projective plane ?

 A: 
… both are projective planes …

Für the actual rails that plane would be the ground, so it would be 2D not 3D. Yes.

… the actual one would have parallel line which would still meet at infinity point and the other bent one would also meet not at infinity but at some vanishing point ?

Correct.
I wouldn't call the lines in the photo “bent”, though. If you are assuming a mathematically convenient pin hole camera, the lines in the photo will be straight, too, just not parallel. If you have a photo where the previously straight lines actually look bent, then you have some complicated effects from the camera lenses that break the whole concept of treating that as a projective transformation, since projective transformations always map straight lines to straight lines.

The camera just convert it to different projective points in the projective plane ?

Right again.
General comments
Typically the term “projective transformation” refers to something preserving the dimensionality. So if your operation is going from a 3D structure to a 2D image, you would call that a projection, but not a projective transformation.
Things are different of you focus on the ground plane where the real rails lie. That would make your transformation 2D to 2D. Now you could rightfully call this a projective transformation.
If you go by the axioms of a projective plane, there is nothing in there about “infinity” and therefore nothing about parallelism either. So on a certain level, all points of the projective plane are considered equal. In that sense, the projective transformation doesn't change anything qualitatively about the relationship between the two rails.
For many practical applications, this view is too generic. We are used to parallelism as a concept. And when we match this known concept with the typical models of the projective plane, we find that all the parallels meet on a certain line, and calling it the line at infinity matches intuition.
So the object we are working with here is a bit more than the bare minimum of a projective plane. It is a projective plane with a specific line designated as the line at infinity. And in this view, the transformation changed something significant by taking a point of intersection at infinity in the real world, and turning it to an intersection not at infinity in the photograph.
