How to draw "cut" areas in a contour plot

Question

I use these commands

Show[ContourPlot[ PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x, y}],
 {x, -2, 2}, {y, -2, 2}, Contours -> {0.05, 0.09}, ContourShading -> {None, Green},    
ContourStyle -> {{Thickness[0.004], Opacity[1]}, {Thickness[0.004], Opacity[1]}}], 
ContourPlot[PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x, y}], 
{x, -2, 2}, {y, -2, 2}, Contours -> {0.06, 0.07}, ContourShading -> {None, Yellow}, 
ContourStyle -> {{Thickness[0.005], Dashed, 
Opacity[1]}, {Thickness[0.005], Dashed, Opacity[1]}}]]

to plot the figure on the left

I want to "cut" the area corresponding to $\mathbb{R}^2-\{x\geq 0 \&\& y\geq 0\}$ to obtain something like the figure on the right, i.e. covering the cutted area with a texture made of diagonal lines.

How can I do? I tried with "Mesh" but the lines appear in all the Frame and not only in the interested areas.

Answer 1

dist = MultinormalDistribution[
   {0, 0}, {{2, 1/2}, {1/2, 1}}];

rgn = ImplicitRegion[{0.05 <= PDF[dist, {x, y}] <= 0.09,
    -2 <= x <= 2, -2 <= y <= 2, ! (x >= 0 && y >= 0)}, {x, y}];

Show[
 ContourPlot[
  PDF[dist, {x, y}], {x, -2, 2}, {y, -2, 2},
  Contours -> {0.05, 0.09},
  ContourShading -> {None, Green},
  ContourStyle -> Thickness[0.004]],
 ContourPlot[
  PDF[dist, {x, y}], {x, -2, 2}, {y, -2, 2},
  Contours -> {0.06, 0.07},
  ContourShading -> {None, Yellow},
  ContourStyle ->
   Directive[Thickness[0.005], Dashed]],
 ContourPlot[(x + y) Boole[! (x >= 0 && y >= 0)],
  {x, -2, 2}, {y, -2, 2},
  Contours -> 50,
  ContourShading -> None,
  RegionFunction -> ({#1, #2} ∈ rgn &)]]

EDIT: A variation using RegionPlot

{outerRgn, innerRgn, rgn} =
  ImplicitRegion[
     {#[[1]] <= PDF[dist, {x, y}] <= #[[2]],
      -2 <= x <= 2, -2 <= y <= 2, #[[3]]}, {x, y}] & /@
   {{0.05, 0.09, 
     True}, {0.06, 0.07, True}, {0.05, 0.09, ! (x >= 0 && y >= 0)}};

Show[
 RegionPlot[{outerRgn, innerRgn},
  PlotRange -> {{-2, 2}, {-2, 2}},
  PlotStyle -> {Green, Yellow}],
 ContourPlot[(x + y) Boole[! (x >= 0 && y >= 0)],
  {x, -2, 2}, {y, -2, 2},
  Contours -> 50,
  ContourShading -> None,
  RegionFunction -> ({#1, #2} \[Element] rgn &),
  BoundaryStyle -> Thick]]

Answer 2

Add a RegionFunction:

Show[
 ContourPlot[
  PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x, 
    y}], {x, -2, 2}, {y, -2, 2}, Contours -> {0.05, 0.09}, 
  ContourShading -> {None, Green}, 
  ContourStyle -> {{Thickness[0.004], Opacity[1]}, {Thickness[0.004], 
     Opacity[1]}}, 
  RegionFunction -> Function[{x, y, z}, x >= 0 && y >= 0]], 
 ContourPlot[
  PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x, 
    y}], {x, -2, 2}, {y, -2, 2}, Contours -> {0.06, 0.07}, 
  ContourShading -> {None, Yellow}, 
  ContourStyle -> {{Thickness[0.005], Dashed, 
     Opacity[1]}, {Thickness[0.005], Dashed, Opacity[1]}}, 
  RegionFunction -> Function[{x, y, z}, x >= 0 && y >= 0]]]

Answer 3

If I understand which region you are interested in correctly:

Show[ContourPlot[
  PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x, 
    y}], {x, -2, 2}, {y, -2, 2}, Contours -> {0.05, 0.09}, 
  ContourShading -> {None, Green}, 
  ContourStyle -> {{Thickness[0.004], Opacity[1]}, {Thickness[0.004], 
     Opacity[1]}}, 
  RegionFunction -> Function[{x, y, z}, x <= 0 || y <= 0]], 
 ContourPlot[
  PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x, 
    y}], {x, -2, 2}, {y, -2, 2}, Contours -> {0.06, 0.07}, 
  ContourShading -> {None, Yellow}, 
  ContourStyle -> {{Thickness[0.005], Dashed, 
     Opacity[1]}, {Thickness[0.005], Dashed, Opacity[1]}}, 
  RegionFunction -> Function[{x, y, z}, x <= 0 || y <= 0]]]

yielding:

Edit:

This may not be the most elegant solution, but:

Show[ContourPlot[
  PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x, 
    y}], {x, -2, 2}, {y, -2, 2}, Contours -> {0.05, 0.09}, 
  ContourShading -> {None, Green}, 
  ContourStyle -> {{Thickness[0.004], Opacity[1]}, {Thickness[0.004], 
     Opacity[1]}}], 
 ContourPlot[
  PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x, 
    y}], {x, -2, 2}, {y, -2, 2}, Contours -> {0.06, 0.07}, 
  ContourShading -> {None, Yellow}, 
  ContourStyle -> {{Thickness[0.005], Dashed, 
     Opacity[1]}, {Thickness[0.005], Dashed, Opacity[1]}}], 
 ListLinePlot[{{0, 
    FindRoot[
      Evaluate[
        PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x,
            y}] /. x -> 0] == .05, {y, 1}][[1, 2]]}, {0, 
    FindRoot[
      Evaluate[
        PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x,
            y}] /. x -> 0] == .09, {y, 1}][[1, 2]]}}, 
  PlotStyle -> Directive[Black, AbsoluteThickness[2]]], 
 ListLinePlot[{{FindRoot[
      Evaluate[
        PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x,
            y}] /. y -> 0] == .09, {x, 1}][[1, 2]], 
    0}, {FindRoot[
      Evaluate[
        PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x,
            y}] /. y -> 0] == .05, {x, 1}][[1, 2]], 0}}, 
  PlotStyle -> Directive[Black, AbsoluteThickness[2]]], 
 ContourPlot[Sin[x*10 + y*10] == 0, {x, -2, 2}, {y, -2, 2}, 
  ContourStyle -> Directive[Black, AbsoluteThickness[2]], 
  RegionFunction -> 
   Function[{x, y, 
     z}, .05 <= 
      PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x, 
        y}] <= .09 && (x <= 0 || y <= 0)]]]

gives:

Answer 4

This can all be done with a single call to ContourPlot[] and the clever use of MeshFunctions:

mnpdf[x_, y_] = PDF[MultinormalDistribution[{0, 0}, {{2, 1/2}, {1/2, 1}}], {x, y}];

ContourPlot[mnpdf[x, y], {x, -2, 2}, {y, -2, 2}, Contours -> {0.05, 0.06, 0.07, 0.09}, 
            ContourShading -> {None, Green, Yellow, Green}, 
            ContourStyle -> {Directive[Thickness[0.002], Opacity[1]], 
                             Directive[Thickness[0.005], Dashed, Opacity[1]], 
                             Directive[Thickness[0.005], Dashed, Opacity[1]]}, Mesh -> 25, 
            MeshFunctions -> {(#1 + #2) Boole[! (#1 > 0 && #2 > 0) && 
                              0.05 < mnpdf[#1, #2] < 0.09] &}, MeshStyle -> Opacity[1]]