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I have the following overlapped contour plot

cplot=Show[
  ContourPlot[
    f[x, y], {x, -1, 1}, {y, -1, 1},  
    Contours->15, ContourShading -> None,ContourLabels -> None,
    ContourStyle ->Blue,
    PlotLegends->{"\[ScriptL]₁(x, y) = x\.b3+xy"}
  ],
  ContourPlot[
    g[x, y], {x, -1, 1}, {y, -1, 1}, 
    Contours->15,ContourShading -> None,  ContourLabels -> None,
    ContourStyle ->Red, 
    PlotLegends->{"\[ScriptL]₂(x, y) = \[Minus]xy"}
  ]
]

I want to add information about the value of the level sets by either:

  • changing the alpha value of the contour lines (more transparent = larger value)
  • changing the thickness of the contour lines (thicker=smaller value)
  • changing the style: thick|normal|dashed|dotted (small -> large)

enter image description here

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1 Answer 1

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You can use a custom DisplayFunction to modify styling of contour lines based on contour levels (using the tooltip labels to identify the contour level for each line):

ClearAll[displayFunction]
displayFunction[ordering_: ReverseSort, range_: {.1, 1}, 
   directives_: {AbsoluteThickness[3 #] &, AbsoluteDashing[3 #] &, Opacity}] := 
  Module[{labels = Rescale[#, MinMax @ #, range] & @
       ordering@Cases[#, Tooltip[a_, l_] :> l, All]}, 
    ReplaceAll[l_Line :> With[{lbls = Last[labels = RotateLeft[labels]]}, 
     {## & @@ Through[directives @ lbls], l}]] @ #] &;

Examples:

ClearAll[f, g]
f[x_, y_] := x^3 + x y;
g[x_, y_] := - x y;

With default arguments displayFunction[] modifies thickness, dashing and opacity directives based on contour levels in descending order:

Show[MapThread[
  ContourPlot[#[x, y], {x, -1, 1}, {y, -1, 1}, 
    Contours -> 15, 
    ContourStyle -> #2, 
    PlotLegends -> #3,
    ContourShading -> None,
    DisplayFunction -> displayFunction[]] &, 
 {{f, g}, {Red, Blue}, {{x^3 + x y}, {- x y}}}]]

enter image description here

Use displayFunction[Sort] to make higher contour levels rendered with larger thickness/dashing/opacity:

enter image description here

Use displayFunction[ReverseSort, {.1, 1}, {Opacity}] to change only opacity:

enter image description here

Use displayFunction[ReverseSort, {.1, 1}, {AbsoluteThickness[5 #] &}] to change only thickness:

enter image description here

Use displayFunction[ReverseSort, {.1, .8}, {AbsoluteDashing[10 #] &}] to modify dashing only:

enter image description here

You can use the last argument of displayFunction to inject arbitrary styling directives. For example, use

displayFunction[ReverseSort, {.1, .8}, {AbsoluteThickness[3] &, Dashed &,
  Dynamic @ Blend[{CurrentValue["Color"], Yellow}, #] &}]

to make the contour lines thick and dashed and blend the contour color with Yellow based on contour levels:

enter image description here

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