The first way consists in defining an affine plane as a set on which a vector space of dimension two acts simply transitively.
Intuitively, this means that an affine plane is a vector space of dimension two in which one has "forgotten" where the origin is.
The second way occurs in incidence geometry, where an affine plane is defined as an abstract system of points and lines satisfying a system of axioms.
More precisely, the choice of an affine coordinate system (or, in the real case, a Cartesian coordinate system) for an affine plane
For example, in a graph, which can be drawn on paper, and in which the position of a particle is plotted against time, the Euclidean metric is not adequate for its interpretation, since the distances between its points or the measures of the angles between its lines have, in general, no physical importance (in the affine plane the axes can use different units, which are not comparable, and the measures also vary with different units and scales[a]).