Coordinate transformation that would transform an doughnut inside out (in 2D) Sorry for the non-precise lingo, but not really a mathematician here.
I am looking for a transformation that given a set of points in a carthesian plane, would move the outermosts points closer to the origin, and the points originally closer to the origin towards infinit. This is basically getting a 2D doughnut and making the inner boundary becoming the outer part, and vice versa.
The illustration below shows what I want:
Example of the transformation
My constrains are:

*

*I'm doing it in a 2D plane

*The transformation can be non-linear

*It has to be reversible. i.e. I also need a anti-transformation.


The function $f(x,y) = (\frac{x}{x^2+y^2}, \frac{y}{x^2+y^2})$ is it's own inverse.
The points of distance $d$ to the origin get mapped to points of distance $\frac{1}{d}.$
