The narrow version of the question is whether given distinct points $a,b,c,c' \in \mathbb{R}^2$ there exists a diffeomorphism $\mathbb{R}^2 \to \mathbb{R}^2$ such that $a\mapsto a,\ b\mapsto b,\ c\mapsto c'$. Moreover I can assume that $a = (0,0)$ and $b=(1,0)$. I'm aware that if $c,c'$ are not collinear to $a$ and $b$ then the map can be taken to be the translation of a linear map. However, what if $c$ or $c'$ lies in the $\overline{ab}$ line?
The more general version is whether there exists a diffeomorphism $\mathbb{R}^2 \to \mathbb{R}^2$ such that $a\mapsto a',\ b\mapsto b',\ c\mapsto c'$ for points $a,b,c,a',b',c' \in \mathbb{R}^2$.
I guess an even more general question, which I find interesting, is: Does there exist a diffeomorphism $\mathbb{R}^n \to \mathbb{R}^n$ such that $a_1\mapsto a_1',\,...,\ a_{n+1}\mapsto a_{n+1}'$?
By the way, I suspect Moebius transformations can help (for the $\mathbb{R}^2$ case), but I haven't looked into that possibility yet, and I'm curious to know if the diffeomorphism can be provided by any other means
Appreciate any help, thank you!
if $c=(c_x,0)$ is on the $\overline{ab}$ line, observe that $(x,y)\mapsto (x,y+x^2-x)$ os a diffeomorphism that leaves $a$ and $b$ fixed wile moving $c$ to $(c_x,c_x^2-c_x)$, which is not on the $\overline{ab}$ line. This (and perhaps something similar if $c'$ is on th eline) reduces the situation to the one you already solved.