Q: Angle but not straight line preserving mapping from $\mathbb{R}^2\rightarrow\mathbb{R}^2$ I am considering mappings $f\colon\mathbb{R}^2\rightarrow\mathbb{R}^2$. I am looking for a mapping $f$ that preserves angles but not straight lines. The inversion with respect to a circle has this property but is not defined on the whole plane.  

A: You are looking for conformal maps, as lisyarus already pointed out in a comment.
Note that you might be better off considering your plane as $\mathbb C$, the plane of complex numbers, instead of $\mathbb R^2$. That's because many basic complex-valued functions are conformal by default; see Wikipedia for details. So depending on what exactly you want to achieve, there is quite a wide variety of functions to choose from. On the other hand, if you have very specific requirements in a small area (a topological disk, as opposed to e.g. a few isolated points), then thanks to analytic continuation that small area will likely already define your mapping for the whole plane.
