ContourPlot — How do I color by contour curvature?












5












$begingroup$


I'm plotting the stream lines of fluid flow past a cylinder, and I would like the colors to increase with contour curvature (i.e. increase as the velocity of the flow increases. Here's a MWE that seems to color it based on the the y-axis value:



ψ[r_, θ_] := U (r - a^2/r) Sin[θ]
r = Sqrt[x^2 + y^2];
θ = ArcSin[y/r];

stream = ContourPlot[
ψ[r, θ] /. {U -> 10, a -> 1},
{x, -5,5}, {y, -5, 5},
Contours -> 10 Table[i, {i, -10, 10, 0.025}]
];

cyl = Graphics[Disk[{0, 0}, 1]];

Show[stream, cyl]


stream lines around a cylinder










share|improve this question











$endgroup$

















    5












    $begingroup$


    I'm plotting the stream lines of fluid flow past a cylinder, and I would like the colors to increase with contour curvature (i.e. increase as the velocity of the flow increases. Here's a MWE that seems to color it based on the the y-axis value:



    ψ[r_, θ_] := U (r - a^2/r) Sin[θ]
    r = Sqrt[x^2 + y^2];
    θ = ArcSin[y/r];

    stream = ContourPlot[
    ψ[r, θ] /. {U -> 10, a -> 1},
    {x, -5,5}, {y, -5, 5},
    Contours -> 10 Table[i, {i, -10, 10, 0.025}]
    ];

    cyl = Graphics[Disk[{0, 0}, 1]];

    Show[stream, cyl]


    stream lines around a cylinder










    share|improve this question











    $endgroup$















      5












      5








      5


      2



      $begingroup$


      I'm plotting the stream lines of fluid flow past a cylinder, and I would like the colors to increase with contour curvature (i.e. increase as the velocity of the flow increases. Here's a MWE that seems to color it based on the the y-axis value:



      ψ[r_, θ_] := U (r - a^2/r) Sin[θ]
      r = Sqrt[x^2 + y^2];
      θ = ArcSin[y/r];

      stream = ContourPlot[
      ψ[r, θ] /. {U -> 10, a -> 1},
      {x, -5,5}, {y, -5, 5},
      Contours -> 10 Table[i, {i, -10, 10, 0.025}]
      ];

      cyl = Graphics[Disk[{0, 0}, 1]];

      Show[stream, cyl]


      stream lines around a cylinder










      share|improve this question











      $endgroup$




      I'm plotting the stream lines of fluid flow past a cylinder, and I would like the colors to increase with contour curvature (i.e. increase as the velocity of the flow increases. Here's a MWE that seems to color it based on the the y-axis value:



      ψ[r_, θ_] := U (r - a^2/r) Sin[θ]
      r = Sqrt[x^2 + y^2];
      θ = ArcSin[y/r];

      stream = ContourPlot[
      ψ[r, θ] /. {U -> 10, a -> 1},
      {x, -5,5}, {y, -5, 5},
      Contours -> 10 Table[i, {i, -10, 10, 0.025}]
      ];

      cyl = Graphics[Disk[{0, 0}, 1]];

      Show[stream, cyl]


      stream lines around a cylinder







      plotting color






      share|improve this question















      share|improve this question













      share|improve this question




      share|improve this question








      edited 4 hours ago









      m_goldberg

      87.7k872198




      87.7k872198










      asked 6 hours ago









      dpholmesdpholmes

      301110




      301110






















          1 Answer
          1






          active

          oldest

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          6












          $begingroup$

          f = {ψ[r, θ]} /. {U -> 10, a -> 1};
          gradf = D[f, {{x, y}, 1}];
          Hessf = D[f, {{x, y}, 2}];
          normal = gradf[[1]]/Sqrt[gradf[[1]].gradf[[1]]];
          secondfundamentalform = -PseudoInverse[gradf].Hessf // ComplexExpand // Simplify;
          tangent = RotationMatrix[Pi/2].normal // Simplify;
          curvaturevector = Simplify[(secondfundamentalform.tangent).tangent];
          signedcurvature = curvaturevector.normal;

          stream = ContourPlot[
          ψ[r, θ] /. {U -> 10, a -> 1}, {x, -5, 5}, {y, -5, 5},
          Contours -> 10 Table[i, {i, -10, 10, 0.2}],
          ContourShading -> None
          ];
          curvatureplot = DensityPlot[signedcurvature, {x, -5, 5}, {y, -5, 5},
          ColorFunction -> "DarkRainbow",
          PlotPoints -> 50,
          PlotRange -> {-1, 1} 2
          ];
          Show[
          curvatureplot,
          stream,
          cyl
          ]


          enter image description here



          The white regions are peaks in the curvature distribution. You may increase PlotRange to make the white regions smaller, however, at the price of less contrast.






          share|improve this answer











          $endgroup$













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            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            6












            $begingroup$

            f = {ψ[r, θ]} /. {U -> 10, a -> 1};
            gradf = D[f, {{x, y}, 1}];
            Hessf = D[f, {{x, y}, 2}];
            normal = gradf[[1]]/Sqrt[gradf[[1]].gradf[[1]]];
            secondfundamentalform = -PseudoInverse[gradf].Hessf // ComplexExpand // Simplify;
            tangent = RotationMatrix[Pi/2].normal // Simplify;
            curvaturevector = Simplify[(secondfundamentalform.tangent).tangent];
            signedcurvature = curvaturevector.normal;

            stream = ContourPlot[
            ψ[r, θ] /. {U -> 10, a -> 1}, {x, -5, 5}, {y, -5, 5},
            Contours -> 10 Table[i, {i, -10, 10, 0.2}],
            ContourShading -> None
            ];
            curvatureplot = DensityPlot[signedcurvature, {x, -5, 5}, {y, -5, 5},
            ColorFunction -> "DarkRainbow",
            PlotPoints -> 50,
            PlotRange -> {-1, 1} 2
            ];
            Show[
            curvatureplot,
            stream,
            cyl
            ]


            enter image description here



            The white regions are peaks in the curvature distribution. You may increase PlotRange to make the white regions smaller, however, at the price of less contrast.






            share|improve this answer











            $endgroup$


















              6












              $begingroup$

              f = {ψ[r, θ]} /. {U -> 10, a -> 1};
              gradf = D[f, {{x, y}, 1}];
              Hessf = D[f, {{x, y}, 2}];
              normal = gradf[[1]]/Sqrt[gradf[[1]].gradf[[1]]];
              secondfundamentalform = -PseudoInverse[gradf].Hessf // ComplexExpand // Simplify;
              tangent = RotationMatrix[Pi/2].normal // Simplify;
              curvaturevector = Simplify[(secondfundamentalform.tangent).tangent];
              signedcurvature = curvaturevector.normal;

              stream = ContourPlot[
              ψ[r, θ] /. {U -> 10, a -> 1}, {x, -5, 5}, {y, -5, 5},
              Contours -> 10 Table[i, {i, -10, 10, 0.2}],
              ContourShading -> None
              ];
              curvatureplot = DensityPlot[signedcurvature, {x, -5, 5}, {y, -5, 5},
              ColorFunction -> "DarkRainbow",
              PlotPoints -> 50,
              PlotRange -> {-1, 1} 2
              ];
              Show[
              curvatureplot,
              stream,
              cyl
              ]


              enter image description here



              The white regions are peaks in the curvature distribution. You may increase PlotRange to make the white regions smaller, however, at the price of less contrast.






              share|improve this answer











              $endgroup$
















                6












                6








                6





                $begingroup$

                f = {ψ[r, θ]} /. {U -> 10, a -> 1};
                gradf = D[f, {{x, y}, 1}];
                Hessf = D[f, {{x, y}, 2}];
                normal = gradf[[1]]/Sqrt[gradf[[1]].gradf[[1]]];
                secondfundamentalform = -PseudoInverse[gradf].Hessf // ComplexExpand // Simplify;
                tangent = RotationMatrix[Pi/2].normal // Simplify;
                curvaturevector = Simplify[(secondfundamentalform.tangent).tangent];
                signedcurvature = curvaturevector.normal;

                stream = ContourPlot[
                ψ[r, θ] /. {U -> 10, a -> 1}, {x, -5, 5}, {y, -5, 5},
                Contours -> 10 Table[i, {i, -10, 10, 0.2}],
                ContourShading -> None
                ];
                curvatureplot = DensityPlot[signedcurvature, {x, -5, 5}, {y, -5, 5},
                ColorFunction -> "DarkRainbow",
                PlotPoints -> 50,
                PlotRange -> {-1, 1} 2
                ];
                Show[
                curvatureplot,
                stream,
                cyl
                ]


                enter image description here



                The white regions are peaks in the curvature distribution. You may increase PlotRange to make the white regions smaller, however, at the price of less contrast.






                share|improve this answer











                $endgroup$



                f = {ψ[r, θ]} /. {U -> 10, a -> 1};
                gradf = D[f, {{x, y}, 1}];
                Hessf = D[f, {{x, y}, 2}];
                normal = gradf[[1]]/Sqrt[gradf[[1]].gradf[[1]]];
                secondfundamentalform = -PseudoInverse[gradf].Hessf // ComplexExpand // Simplify;
                tangent = RotationMatrix[Pi/2].normal // Simplify;
                curvaturevector = Simplify[(secondfundamentalform.tangent).tangent];
                signedcurvature = curvaturevector.normal;

                stream = ContourPlot[
                ψ[r, θ] /. {U -> 10, a -> 1}, {x, -5, 5}, {y, -5, 5},
                Contours -> 10 Table[i, {i, -10, 10, 0.2}],
                ContourShading -> None
                ];
                curvatureplot = DensityPlot[signedcurvature, {x, -5, 5}, {y, -5, 5},
                ColorFunction -> "DarkRainbow",
                PlotPoints -> 50,
                PlotRange -> {-1, 1} 2
                ];
                Show[
                curvatureplot,
                stream,
                cyl
                ]


                enter image description here



                The white regions are peaks in the curvature distribution. You may increase PlotRange to make the white regions smaller, however, at the price of less contrast.







                share|improve this answer














                share|improve this answer



                share|improve this answer








                edited 5 hours ago

























                answered 6 hours ago









                Henrik SchumacherHenrik Schumacher

                57.2k577157




                57.2k577157






























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