I'm going over some lecture notes (http://www.math.umd.edu/~wmg/pgom.pdf, PG.7 definition 1.1 and exercise 1.2) that states that a map is affine if and only if parallel sets are parallel under the map.
The given definition of parallel sets are that $A,B$ are parallel if there exists a translation $g$ such that $g(A) = B$.
The problem I have with this is that affine maps must also preserve lines. Does is necessarily follow from $A||B \implies f(A)||f(B)$ that $f$ preserves lines? I am having trouble finding a counter example for maps between spaces of dimension greater than 1, but intuitively it seems unlikely that this property is strong enough to guarantee preservation of lines.