Coordinate transformation that would transform an doughnut inside out (in 2D)

Question

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:

1. I'm doing it in a 2D plane
2. The transformation can be non-linear
3. It has to be reversible. i.e. I also need a anti-transformation.

Answer 1

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 $\frac{1}{d}.$