Real Affine Plane: Definition & Properties

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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.
 

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tgt said:
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.
I'm not familiar with this definition. What is a vertical line? What is a non-trivial line?
Perhaps you can tell us where you found this definition?

Looking here https://en.wikipedia.org/wiki/Affine_plane it seems an affine plane is the usual Euclidean plane minus the metric.
If we're considering only real affine planes, then this is correct.

There are many other things to talk about as well like parabolas, circles etc.
Sure, but parabolas and circles arise from the Euclidean metric, so there's no need. Also, if someone is interested in a different metric on ##\mathbb{R}^2##, then it might not make sense to add either of those to the definition.
 
So it seems all geometrical objects in the affine plane is defined in terms of points and lines?