How to do a ParametricPlot within a subregion

Question

I currently have the following parametric plot.

ParametricPlot[{(b - a) x/b + a x/r, (b - a) y/b + a y/r} /. {a -> 2, 
   b -> 3}, {x, -4, 4}, {y, -4, 4}]

where r=Sqrt[x^2+y^2].

I am trying to plot this coordinate system only within the region

a < Sqrt[((b - a) x/b + a x/r)^2 + ((b - a) y/b + a y/r)^2] < b,

the region where Sqrt[((b - a) x/b + a x/r)^2 + ((b - a) y/b + a y/r)^2] > b should be cartesian.

I am unsure how to achieve this. Does anyone have a solution? I apologize if this has already been asked, but I couldn't find a previous question when I looked.

Cheers, Stephen

Answer 1

You need to use a RegionFunction . Based on the look of your plot, I assume you are using version 9, so that's what I'll use here. In version 10.3, the mesh lines look different.

Not sure which of these you are going for, whether you want, in the region definition, for x and y to refer to the Cartesian axes or the ParametricPlot parameters, so take your pick

ParametricPlot[{(b - a) x/b + a x/r, (b - a) y/b + a y/r}, {x, -4, 4}, {y, -4, 4}, 
   RegionFunction -> 
    Function[#,
      a < (((b - a) x/b + a x/(x^2 + y^2))^2 + ((b - a) y/b + 
            a y/(x^2 + y^2))^2) < b]] & /@ {{x, y, u, v}, {u, v, x, y}}