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]
plotting color
edited 4 hours ago
m_goldberg
87.7k872198
asked 6 hours ago
dpholmesdpholmes
301110
$endgroup$
add a comment |
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]
plotting color
edited 4 hours ago
m_goldberg
87.7k872198
asked 6 hours ago
dpholmesdpholmes
301110
$endgroup$
add a comment |
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]
plotting color
edited 4 hours ago
m_goldberg
87.7k872198
asked 6 hours ago
dpholmesdpholmes
301110
$endgroup$
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]
plotting color
plotting color
edited 4 hours ago
m_goldberg
87.7k872198
asked 6 hours ago
dpholmesdpholmes
301110
edited 4 hours ago
m_goldberg
87.7k872198
asked 6 hours ago
dpholmesdpholmes
301110
edited 4 hours ago
m_goldberg
87.7k872198
edited 4 hours ago
m_goldberg
87.7k872198
edited 4 hours ago
m_goldberg
87.7k872198
87.7k872198
asked 6 hours ago
dpholmesdpholmes
301110
asked 6 hours ago
dpholmesdpholmes
301110
asked 6 hours ago
dpholmesdpholmes
301110
301110
add a comment |
add a comment |
1 Answer
1
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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
]
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.
answered 6 hours ago
Henrik SchumacherHenrik Schumacher
57.2k577157
$endgroup$
add a comment |
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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
]
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.
answered 6 hours ago
Henrik SchumacherHenrik Schumacher
57.2k577157
$endgroup$
add a comment |
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
]
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.
answered 6 hours ago
Henrik SchumacherHenrik Schumacher
57.2k577157
$endgroup$
add a comment |
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
]
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.
answered 6 hours 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
]
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.
answered 6 hours ago
Henrik SchumacherHenrik Schumacher
57.2k577157
edited 6 hours ago
answered 6 hours ago
Henrik SchumacherHenrik Schumacher
57.2k577157
answered 6 hours ago
Henrik SchumacherHenrik Schumacher
57.2k577157
answered 6 hours ago
Henrik SchumacherHenrik Schumacher
57.2k577157
57.2k577157
add a comment |
add a comment |
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