 Affine maps can collapse lines

An affine map need not be injective. For $G(x,y)=(x+y,0)$, the direction $(1,-1)$ belongs to the kernel of its linear part of an affine map. Consequently the whole line $y=-x$ maps to $(0,0)$, and every parallel line $y=-x+c$ maps to $(c,0)$. These are collapsed lines, not images with a surviving direction. Assertions that an affine map preserves distinct parallel lines require the appropriate noncollapse or injectivity hypothesis.