How can find the 2D Voronoi cell area distribution?How can Mathematica be used to detect an area surrounded...
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How can find the 2D Voronoi cell area distribution?
How can find the 2D Voronoi cell area distribution?
How can Mathematica be used to detect an area surrounded by the most lines?Find area of polygon on a sphere from a set of latitude-longitude pointsMeshRegion: How to take out a subregionColor code Voronoi cell areas depending on number of verticesRemoving cells from Voronoi mesh if they exceed a certain area or circumferenceHyperbolic Voronoi diagram for the Poincaré model, using RegionPlotHow to find the Convex Hullcolor voronoi cell based on areaHow can I efficiently generate regions from long lists of conditions?How to get size of each polygon of a Voronoi diagram using Shoelace formula?
$begingroup$
I need to find the area distribution function of the 2D Voronoi cells in Mathematica version 11 and later. My old instructions for Mathematica 9 don't work anymore. How can I do it?
computational-geometry
edited 1 hour ago
C. E.
50.6k397204
asked 1 hour ago
massimomassimo
111
New contributor
massimo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I need to find the area distribution function of the 2D Voronoi cells in Mathematica version 11 and later. My old instructions for Mathematica 9 don't work anymore. How can I do it?
computational-geometry
edited 1 hour ago
C. E.
50.6k397204
asked 1 hour ago
massimomassimo
111
New contributor
massimo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
1
$begingroup$
If you were to provide your old instructions, we might be able to fix them.
$endgroup$
– C. E.
1 hour ago
add a comment |
$begingroup$
I need to find the area distribution function of the 2D Voronoi cells in Mathematica version 11 and later. My old instructions for Mathematica 9 don't work anymore. How can I do it?
computational-geometry
edited 1 hour ago
C. E.
50.6k397204
asked 1 hour ago
massimomassimo
111
New contributor
massimo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I need to find the area distribution function of the 2D Voronoi cells in Mathematica version 11 and later. My old instructions for Mathematica 9 don't work anymore. How can I do it?
computational-geometry
computational-geometry
edited 1 hour ago
C. E.
50.6k397204
asked 1 hour ago
massimomassimo
111
New contributor
massimo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 1 hour ago
C. E.
50.6k397204
asked 1 hour ago
massimomassimo
111
New contributor
massimo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 1 hour ago
C. E.
50.6k397204
edited 1 hour ago
C. E.
50.6k397204
edited 1 hour ago
C. E.
50.6k397204
50.6k397204
asked 1 hour ago
massimomassimo
111
New contributor
massimo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
massimomassimo
111
asked 1 hour ago
massimomassimo
111
111
New contributor
massimo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
massimo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
massimo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
$begingroup$
If you were to provide your old instructions, we might be able to fix them.
$endgroup$
– C. E.
1 hour ago
add a comment |
1
$begingroup$
If you were to provide your old instructions, we might be able to fix them.
$endgroup$
– C. E.
1 hour ago
1
1
$begingroup$
If you were to provide your old instructions, we might be able to fix them.
$endgroup$
– C. E.
1 hour ago
$begingroup$
If you were to provide your old instructions, we might be able to fix them.
$endgroup$
– C. E.
1 hour ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
First compute the Voronoi mesh:
pts = RandomReal[{-1, 1}, {25, 2}];
mesh = VoronoiMesh[pts]
Then compute the areas of the mesh primitives (MeshPrimitives yields a polygon for each Voronoi region):
areas = Area /@ MeshPrimitives[mesh, 2];
Histogram[areas]
answered 1 hour ago
C. E.C. E.
50.6k397204
$endgroup$
1
$begingroup$
We can also query for the cell areas viaPropertyValue[{mesh, 2}, MeshCellMeasure].
$endgroup$
– Chip Hurst
47 mins ago
$begingroup$
This could give wrong stats, C.E. and @ChipHurst. I posted a bit different version.
$endgroup$
– Vitaliy Kaurov
12 mins ago
add a comment |
$begingroup$
I think it is a GammaDistribution and we can show it, also see reference:
http://www.scipress.org/journals/forma/pdf/1804/18040221.pdf
@C.E. answer is a good start, but without careful consideration might yield wrong stats. To gather good statistics let's build a "large", 5000 cells VoronoiMesh within a unit Disk:
pts = RandomPoint[Disk[], 5000];
mesh = VoronoiMesh[pts, Axes -> True]
We see the stats are obviously offset by the presence of "border" cells of much larger area than regular inner cells have.
Let's exclude large cells by selecting only those within original region of random points distribution - unit Disk:
vor=MeshPrimitives[mesh,2];
vor//Length
5000
vorInner=Select[vor,RegionWithin[Disk[],#]&];
vorInner//Length
4782
We got fewer elements of course and they all are nice regular cells:
Graphics[{FaceForm[None], EdgeForm[Gray], vorInner}]
Now you can see that there is a minimum of distribution at zero area. Close to zero-area cells of course are not present much with the finite number of points per region (or finite density). So there is some prevalent finite mean area there.
areas = Area /@ vorInner;
hist = Histogram[areas, Automatic, "PDF", PlotTheme -> "Detailed"]
You can find a very close simple analytic distribution:
dis=FindDistribution[areas]
GammaDistribution[3.3458431368450587,0.00018456010158014188]
which matches very nicely the empirical histogram:
Show[hist, Plot[PDF[dis, x], {x, 0, .0015}]]
And now it is easy to find thing like:
{Mean[dis], Variance[dis], Kurtosis[dis]}
{0.0006175091492073445
, 1.1396755130437449*^-7, 4.793269963533813`}
Probability[x > .0005, x [Distributed] dis]
0.5740480899719699`
answered 24 mins ago
Vitaliy KaurovVitaliy Kaurov
57.3k6161280
$endgroup$
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
First compute the Voronoi mesh:
pts = RandomReal[{-1, 1}, {25, 2}];
mesh = VoronoiMesh[pts]
Then compute the areas of the mesh primitives (MeshPrimitives yields a polygon for each Voronoi region):
areas = Area /@ MeshPrimitives[mesh, 2];
Histogram[areas]
answered 1 hour ago
C. E.C. E.
50.6k397204
$endgroup$
1
$begingroup$
We can also query for the cell areas viaPropertyValue[{mesh, 2}, MeshCellMeasure].
$endgroup$
– Chip Hurst
47 mins ago
$begingroup$
This could give wrong stats, C.E. and @ChipHurst. I posted a bit different version.
$endgroup$
– Vitaliy Kaurov
12 mins ago
add a comment |
$begingroup$
First compute the Voronoi mesh:
pts = RandomReal[{-1, 1}, {25, 2}];
mesh = VoronoiMesh[pts]
Then compute the areas of the mesh primitives (MeshPrimitives yields a polygon for each Voronoi region):
areas = Area /@ MeshPrimitives[mesh, 2];
Histogram[areas]
answered 1 hour ago
C. E.C. E.
50.6k397204
$endgroup$
1
$begingroup$
We can also query for the cell areas viaPropertyValue[{mesh, 2}, MeshCellMeasure].
$endgroup$
– Chip Hurst
47 mins ago
$begingroup$
This could give wrong stats, C.E. and @ChipHurst. I posted a bit different version.
$endgroup$
– Vitaliy Kaurov
12 mins ago
add a comment |
$begingroup$
First compute the Voronoi mesh:
pts = RandomReal[{-1, 1}, {25, 2}];
mesh = VoronoiMesh[pts]
Then compute the areas of the mesh primitives (MeshPrimitives yields a polygon for each Voronoi region):
areas = Area /@ MeshPrimitives[mesh, 2];
Histogram[areas]
answered 1 hour ago
C. E.C. E.
50.6k397204
$endgroup$
First compute the Voronoi mesh:
pts = RandomReal[{-1, 1}, {25, 2}];
mesh = VoronoiMesh[pts]
Then compute the areas of the mesh primitives (MeshPrimitives yields a polygon for each Voronoi region):
areas = Area /@ MeshPrimitives[mesh, 2];
Histogram[areas]
answered 1 hour ago
C. E.C. E.
50.6k397204
answered 1 hour ago
C. E.C. E.
50.6k397204
answered 1 hour ago
C. E.C. E.
50.6k397204
answered 1 hour ago
C. E.C. E.
50.6k397204
50.6k397204
1
$begingroup$
We can also query for the cell areas viaPropertyValue[{mesh, 2}, MeshCellMeasure].
$endgroup$
– Chip Hurst
47 mins ago
$begingroup$
This could give wrong stats, C.E. and @ChipHurst. I posted a bit different version.
$endgroup$
– Vitaliy Kaurov
12 mins ago
add a comment |
1
$begingroup$
We can also query for the cell areas viaPropertyValue[{mesh, 2}, MeshCellMeasure].
$endgroup$
– Chip Hurst
47 mins ago
$begingroup$
This could give wrong stats, C.E. and @ChipHurst. I posted a bit different version.
$endgroup$
– Vitaliy Kaurov
12 mins ago
1
1
$begingroup$
We can also query for the cell areas via
PropertyValue[{mesh, 2}, MeshCellMeasure].$endgroup$
– Chip Hurst
47 mins ago
$begingroup$
We can also query for the cell areas via
PropertyValue[{mesh, 2}, MeshCellMeasure].$endgroup$
– Chip Hurst
47 mins ago
$begingroup$
This could give wrong stats, C.E. and @ChipHurst. I posted a bit different version.
$endgroup$
– Vitaliy Kaurov
12 mins ago
$begingroup$
This could give wrong stats, C.E. and @ChipHurst. I posted a bit different version.
$endgroup$
– Vitaliy Kaurov
12 mins ago
add a comment |
$begingroup$
I think it is a GammaDistribution and we can show it, also see reference:
http://www.scipress.org/journals/forma/pdf/1804/18040221.pdf
@C.E. answer is a good start, but without careful consideration might yield wrong stats. To gather good statistics let's build a "large", 5000 cells VoronoiMesh within a unit Disk:
pts = RandomPoint[Disk[], 5000];
mesh = VoronoiMesh[pts, Axes -> True]
We see the stats are obviously offset by the presence of "border" cells of much larger area than regular inner cells have.
Let's exclude large cells by selecting only those within original region of random points distribution - unit Disk:
vor=MeshPrimitives[mesh,2];
vor//Length
5000
vorInner=Select[vor,RegionWithin[Disk[],#]&];
vorInner//Length
4782
We got fewer elements of course and they all are nice regular cells:
Graphics[{FaceForm[None], EdgeForm[Gray], vorInner}]
Now you can see that there is a minimum of distribution at zero area. Close to zero-area cells of course are not present much with the finite number of points per region (or finite density). So there is some prevalent finite mean area there.
areas = Area /@ vorInner;
hist = Histogram[areas, Automatic, "PDF", PlotTheme -> "Detailed"]
You can find a very close simple analytic distribution:
dis=FindDistribution[areas]
GammaDistribution[3.3458431368450587,0.00018456010158014188]
which matches very nicely the empirical histogram:
Show[hist, Plot[PDF[dis, x], {x, 0, .0015}]]
And now it is easy to find thing like:
{Mean[dis], Variance[dis], Kurtosis[dis]}
{0.0006175091492073445
, 1.1396755130437449*^-7, 4.793269963533813`}
Probability[x > .0005, x [Distributed] dis]
0.5740480899719699`
answered 24 mins ago
Vitaliy KaurovVitaliy Kaurov
57.3k6161280
$endgroup$
add a comment |
$begingroup$
I think it is a GammaDistribution and we can show it, also see reference:
http://www.scipress.org/journals/forma/pdf/1804/18040221.pdf
@C.E. answer is a good start, but without careful consideration might yield wrong stats. To gather good statistics let's build a "large", 5000 cells VoronoiMesh within a unit Disk:
pts = RandomPoint[Disk[], 5000];
mesh = VoronoiMesh[pts, Axes -> True]
We see the stats are obviously offset by the presence of "border" cells of much larger area than regular inner cells have.
Let's exclude large cells by selecting only those within original region of random points distribution - unit Disk:
vor=MeshPrimitives[mesh,2];
vor//Length
5000
vorInner=Select[vor,RegionWithin[Disk[],#]&];
vorInner//Length
4782
We got fewer elements of course and they all are nice regular cells:
Graphics[{FaceForm[None], EdgeForm[Gray], vorInner}]
Now you can see that there is a minimum of distribution at zero area. Close to zero-area cells of course are not present much with the finite number of points per region (or finite density). So there is some prevalent finite mean area there.
areas = Area /@ vorInner;
hist = Histogram[areas, Automatic, "PDF", PlotTheme -> "Detailed"]
You can find a very close simple analytic distribution:
dis=FindDistribution[areas]
GammaDistribution[3.3458431368450587,0.00018456010158014188]
which matches very nicely the empirical histogram:
Show[hist, Plot[PDF[dis, x], {x, 0, .0015}]]
And now it is easy to find thing like:
{Mean[dis], Variance[dis], Kurtosis[dis]}
{0.0006175091492073445
, 1.1396755130437449*^-7, 4.793269963533813`}
Probability[x > .0005, x [Distributed] dis]
0.5740480899719699`
answered 24 mins ago
Vitaliy KaurovVitaliy Kaurov
57.3k6161280
$endgroup$
add a comment |
$begingroup$
I think it is a GammaDistribution and we can show it, also see reference:
http://www.scipress.org/journals/forma/pdf/1804/18040221.pdf
@C.E. answer is a good start, but without careful consideration might yield wrong stats. To gather good statistics let's build a "large", 5000 cells VoronoiMesh within a unit Disk:
pts = RandomPoint[Disk[], 5000];
mesh = VoronoiMesh[pts, Axes -> True]
We see the stats are obviously offset by the presence of "border" cells of much larger area than regular inner cells have.
Let's exclude large cells by selecting only those within original region of random points distribution - unit Disk:
vor=MeshPrimitives[mesh,2];
vor//Length
5000
vorInner=Select[vor,RegionWithin[Disk[],#]&];
vorInner//Length
4782
We got fewer elements of course and they all are nice regular cells:
Graphics[{FaceForm[None], EdgeForm[Gray], vorInner}]
Now you can see that there is a minimum of distribution at zero area. Close to zero-area cells of course are not present much with the finite number of points per region (or finite density). So there is some prevalent finite mean area there.
areas = Area /@ vorInner;
hist = Histogram[areas, Automatic, "PDF", PlotTheme -> "Detailed"]
You can find a very close simple analytic distribution:
dis=FindDistribution[areas]
GammaDistribution[3.3458431368450587,0.00018456010158014188]
which matches very nicely the empirical histogram:
Show[hist, Plot[PDF[dis, x], {x, 0, .0015}]]
And now it is easy to find thing like:
{Mean[dis], Variance[dis], Kurtosis[dis]}
{0.0006175091492073445
, 1.1396755130437449*^-7, 4.793269963533813`}
Probability[x > .0005, x [Distributed] dis]
0.5740480899719699`
answered 24 mins ago
Vitaliy KaurovVitaliy Kaurov
57.3k6161280
$endgroup$
I think it is a GammaDistribution and we can show it, also see reference:
http://www.scipress.org/journals/forma/pdf/1804/18040221.pdf
@C.E. answer is a good start, but without careful consideration might yield wrong stats. To gather good statistics let's build a "large", 5000 cells VoronoiMesh within a unit Disk:
pts = RandomPoint[Disk[], 5000];
mesh = VoronoiMesh[pts, Axes -> True]
We see the stats are obviously offset by the presence of "border" cells of much larger area than regular inner cells have.
Let's exclude large cells by selecting only those within original region of random points distribution - unit Disk:
vor=MeshPrimitives[mesh,2];
vor//Length
5000
vorInner=Select[vor,RegionWithin[Disk[],#]&];
vorInner//Length
4782
We got fewer elements of course and they all are nice regular cells:
Graphics[{FaceForm[None], EdgeForm[Gray], vorInner}]
Now you can see that there is a minimum of distribution at zero area. Close to zero-area cells of course are not present much with the finite number of points per region (or finite density). So there is some prevalent finite mean area there.
areas = Area /@ vorInner;
hist = Histogram[areas, Automatic, "PDF", PlotTheme -> "Detailed"]
You can find a very close simple analytic distribution:
dis=FindDistribution[areas]
GammaDistribution[3.3458431368450587,0.00018456010158014188]
which matches very nicely the empirical histogram:
Show[hist, Plot[PDF[dis, x], {x, 0, .0015}]]
And now it is easy to find thing like:
{Mean[dis], Variance[dis], Kurtosis[dis]}
{0.0006175091492073445
, 1.1396755130437449*^-7, 4.793269963533813`}
Probability[x > .0005, x [Distributed] dis]
0.5740480899719699`
answered 24 mins ago
Vitaliy KaurovVitaliy Kaurov
57.3k6161280
edited 7 mins ago
answered 24 mins ago
Vitaliy KaurovVitaliy Kaurov
57.3k6161280
answered 24 mins ago
Vitaliy KaurovVitaliy Kaurov
57.3k6161280
answered 24 mins ago
Vitaliy KaurovVitaliy Kaurov
57.3k6161280
57.3k6161280
add a comment |
add a comment |
massimo is a new contributor. Be nice, and check out our Code of Conduct.
massimo is a new contributor. Be nice, and check out our Code of Conduct.
massimo is a new contributor. Be nice, and check out our Code of Conduct.
massimo is a new contributor. Be nice, and check out our Code of Conduct.
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1
$begingroup$
If you were to provide your old instructions, we might be able to fix them.
$endgroup$
– C. E.
1 hour ago