= 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.
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