Is there an $f:\mathbb{R}^2 \to \mathbb{R}^2$ such that:
- $(0,0)\mapsto (0,0)$; and
- for any $a,b,c$ with $a^2 + b^2 >0$, the set $A=\{(x,y):ax+by=c\}$ is mapped onto $f(A)=\{(x,y):a'x+b'y=c'\}$ for some $a',b',c'$ with $a'^2 + b'^2 >0$; and
- $f$ non-linear?