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ContourPlot — How do I color by contour curvature?

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4

$begingroup$

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]

plotting color

share|improve this question

edited 10 mins ago

m_goldberg

87.7k872198

asked 2 hours ago

dpholmesdpholmes

19610

$endgroup$

add a comment | 

4

$begingroup$

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]

plotting color

share|improve this question

edited 10 mins ago

m_goldberg

87.7k872198

asked 2 hours ago

dpholmesdpholmes

19610

$endgroup$

add a comment | 

$begingroup$

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]

plotting color

share|improve this question

edited 10 mins ago

m_goldberg

87.7k872198

asked 2 hours ago

dpholmesdpholmes

19610

$endgroup$

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]

share|improve this question

edited 10 mins ago

m_goldberg

87.7k872198

asked 2 hours ago

dpholmesdpholmes

19610

share|improve this question

edited 10 mins ago

m_goldberg

87.7k872198

asked 2 hours ago

dpholmesdpholmes

19610

edited 10 mins ago

m_goldberg

87.7k872198

edited 10 mins ago

m_goldberg

87.7k872198

asked 2 hours ago

dpholmesdpholmes

19610

asked 2 hours ago

dpholmesdpholmes

19610

1 Answer
1

active

oldest

votes

3

$begingroup$

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
 ]

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

edited 1 hour ago

answered 1 hour ago

Henrik SchumacherHenrik Schumacher

57.2k577157

$endgroup$

add a comment | 

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3

$begingroup$

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
 ]

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

edited 1 hour ago

answered 1 hour ago

Henrik SchumacherHenrik Schumacher

57.2k577157

$endgroup$

add a comment | 

3

$begingroup$

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
 ]

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

edited 1 hour ago

answered 1 hour ago

Henrik SchumacherHenrik Schumacher

57.2k577157

$endgroup$

add a comment | 

$begingroup$

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
 ]

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

edited 1 hour ago

answered 1 hour ago

Henrik SchumacherHenrik Schumacher

57.2k577157

$endgroup$

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
 ]

share|improve this answer

edited 1 hour ago

answered 1 hour ago

Henrik SchumacherHenrik Schumacher

57.2k577157

answered 1 hour ago

Henrik SchumacherHenrik Schumacher

57.2k577157

answered 1 hour ago

Henrik SchumacherHenrik Schumacher

57.2k577157