ParametricPlot with condition for parameters

Question

For example

ParametricPlot[{Sin[x] - y, Cos[y] - x}, {x, -\[Pi], \[Pi]}, {y, -\[Pi], \[Pi]}]

gives

What should I do to impose x^2+y^2==2 (actually it could be a more complicated function) and only plot for those parameters that satisfy that condition? Mesh didnt work.

Answer 1

Perhaps:

 Quiet @ ParametricPlot[{Sin[x] - y, Cos[y] - x} , {x, -π, π}, {y, -π, π},  
   BoundaryStyle -> Dotted, 
   PlotStyle -> Opacity[.25, LightBlue], 
   MeshStyle -> Opacity[1, Red], 
   MeshFunctions -> {#3^+ #4^2 &}, 
   Mesh -> {{2}}]

Use BoundaryStyle -> None and PlotStyle -> None to hide the surface and its boundary.