Labeling coordinate curves in ParametricPlot

Question

I want to display a coordinate grid in parabolic coordinates:
ParametricPlot[{-parabolic[r, phi], parabolic[r, phi]}, {r,0,1}, {phi,-1, 1},
  PlotStyle -> {Gray, Gray}, 
  BoundaryStyle -> Dashed, Mesh -> 9, Frame -> False, Ticks -> None]

where
parabolic[r_,phi_] := {1/2 (r^2 - phi^2), phi r};
Is there a relatively simply way to include labels for a selection of coordinate curves within the plot (essentially a 'non-dynamic' Tooltip)?

Bob Hanlon · Accepted Answer

To see Tooltips it is easier to use ContourPlot Clear["Global`*"] parabolic[r_, phi_] := {1/2 (r^2 - phi^2), phi r}; sol[1] = Solve[{x, y} == parabolic[r, phi], {r, phi}, Reals]; sol[2] = Solve[{x, y} == -parabolic[r, phi], {r, phi}, Reals]; {cpr1, cpr2} = ContourPlot[r /. sol[#], {x, -0.5, 0.5}, {y, -1, 1}, Contours -> Range[0, 1, 0.2], ContourStyle -> Directive[{Red, Blue}[[#]], AbsoluteThickness[1.5]], ContourShading -> None, ContourLabels -> All, PlotPoints -> 50, AspectRatio -> 2, ImageSize -> Medium , RegionFunction -> Function[{x, y, r}, Evaluate[(And @@ Thread[-1 <= (phi /. sol[#]) <= 1])]] ] /. {Tooltip[expr_, tooltip_] :> Tooltip[expr, StringForm["r``=``", #, NumberForm[tooltip, 1]]]} & /@ {1, 2}; Show[cpr1, cpr2, PlotRange -> {{-0.55, 0.55}, {-1, 1}}] {cpp1, cpp2} = ContourPlot[phi /. sol[#], {x, -0.5, 0.5}, {y, -1, 1}, Contours -> Range[0, 1, 0.2], ContourStyle -> Directive[{Red, Blue}[[#]], AbsoluteThickness[1.5]], ContourShading -> None, ContourLabels -> All, PlotPoints -> 50, AspectRatio -> 2 , ImageSize -> Medium , RegionFunction -> Function[{x, y, r}, Evaluate[(And @@ Thread[-1 <= (r /. sol[#]) <= 1])]] ] /. {Tooltip[expr_, tooltip_] :> Tooltip[expr, StringForm["phi``=``", #, NumberForm[tooltip, 1]]]} & /@ {1, 2}; Show[cpp1, cpp2, PlotRange -> {{-0.55, 0.55}, {-1, 1}}] Row[{ Show[cpr1, cpp1, PlotRange -> {{-0.55, 0.55}, {-1, 1}}], Show[cpr2, cpp2, PlotRange -> {{-0.55, 0.55}, {-1, 1}}]}]

Labeling coordinate curves in ParametricPlot

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