Definition of a real affine plane is the incidence structure with point set R^2 and line set the union of the vertical lines and the non-trivial lines, with the natural incidence relation. Looking here https://en.wikipedia.org/wiki/Affine_plane it seems an affine plane is the usual Euclidean plane minus the metric. My question why in the above definition talk specifically about vertical lines and non trivial lines? Why isolate these objects? There are many other things to talk about as well like parabolas, circles etc.