Ellipsoid Mesh
Ellipsoid Mesh
Ellipsoid
Ellipsoid
In[]:=
makeEllipsoidCode[a_,b_,c_,resolution_]:=StringTemplate["cd(\"`6`\")\n"<>"fd=@(p) p(:, 1).^2 / `1`^2 + p(:, 2).^2 / `2`^2 + p(:, 3).^2 / `3`^2 - 1;\n"<>"[p, t] = distmeshsurface(fd, @huniform, `4`, [-1.1 * `1`, -1.1 * `2`, -1.1 * `3`; 1.1 * `1`, 1.1 * `2`, 1.1 * `3`]);\n"<>"writematrix(p, \"`5`/points.txt\");\n"<>"writematrix(t, \"`5`/triangles.txt\");"][a,b,c,resolution,$TemporaryDirectory,FileNameJoin[{$Dropbox,"Physics/CodeDevelopment/ExternalCode/distmesh"}]]
In[]:=
makeEllipsoidCode[1,3,3,0.1]
Out[]=
cd("/Users/maxitg/Dropbox (Wolfram)/Physics/CodeDevelopment/ExternalCode/distmesh")fd=@(p) p(:, 1).^2 / 1^2 + p(:, 2).^2 / 3^2 + p(:, 3).^2 / 3^2 - 1;[p, t] = distmeshsurface(fd, @huniform, 0.1, [-1.1 * 1, -1.1 * 3, -1.1 * 3; 1.1 * 1, 1.1 * 3, 1.1 * 3]);writematrix(p, "/private/var/folders/pz/q1ty3f9x1yq5f2ppwrkhgz4r0000gn/T/points.txt");writematrix(t, "/private/var/folders/pz/q1ty3f9x1yq5f2ppwrkhgz4r0000gn/T/triangles.txt");
In[]:=
getEllipsoid[]:=With[{points=Import[FileNameJoin[{$TemporaryDirectory,"points.txt"}],"CSV"],triangles=Import[FileNameJoin[{$TemporaryDirectory,"triangles.txt"}],"CSV"]},With[{simpleGraph=SimpleGraph[UndirectedEdge@@@Catenate[Partition[#,2,1,-1]&/@triangles]]},Graph3D[simpleGraph,VertexCoordinatespoints〚VertexList[simpleGraph]〛]]]
In[]:=
ellipsoid113=getEllipsoid[];
In[]:=
sphere=;
In[]:=
ellipsoid123=;
In[]:=
ellipsoid113=;
Ricci Scalar
Ricci Scalar
In[]:=
ricciScalar[graph_]:=With{volumes=GraphNeighborhoodVolumes[graph]},1-&/@(Take[#,Min[Length/@volumes]]&)/@volumes
6(2+2)
2
Range[Length[#]-1]
Rest[#]
π
2
Range[Length[#]-1]
In[]:=
2//N
2
(GraphDiameter[sphere]/π)
Out[]=
0.0152309
In[]:=
ricciScalarPlot[graph_,opts___]:=ListPlot[MeanAround/@Transpose[Values[ricciScalar[graph]]],opts]
In[]:=
ricciScalarPlot[sphere,PlotRange{All,{-0.01,0.02}}]
Out[]=
In[]:=
2//N
2
(GraphDiameter[ellipsoid113]/π)
Out[]=
0.00360468
In[]:=
2//N
2
(GraphDiameter[ellipsoid113]/3/π)
Out[]=
0.0324421
In[]:=
ricciScalarEllipsoid=N[ricciScalar[ellipsoid113]];
In[]:=
ricciScalarSphere=N[ricciScalar[sphere]];
In[]:=
Take[ReverseSort[Max/@ricciScalarEllipsoid],10]
Out[]=
300.0475295,270.0412991,220.0411476,210.0392825,10.037806,190.0366416,290.0354772,280.0354772,200.0348024,150.0343129
In[]:=
ListPlot[ricciScalarEllipsoid[30],JoinedTrue,PlotRange{All,{-0.01,All}}]
Out[]=
In[]:=
Take[Sort[ricciScalarEllipsoid〚All,10〛],10]
Out[]=
3365-0.0396034,3541-0.0396034,616-0.0388395,1106-0.0388395,923-0.0380755,3373-0.0380755,3616-0.0380755,3914-0.0380755,704-0.0373116,1295-0.0373116
In[]:=
VertexCount[ellipsoid113]
Out[]=
4310
In[]:=
Normal[Sort[ricciScalarEllipsoid〚All,10〛]]〚4310/2〛
Out[]=
1852-0.0205048
The position on the ellipsoid with the largest curvature
In[]:=
HighlightGraph[ellipsoid113,NeighborhoodGraph[ellipsoid113,30,1],GraphHighlightStyle"Thick"]
Out[]=
The smallest curvature
In[]:=
HighlightGraph[ellipsoid113,NeighborhoodGraph[ellipsoid113,3365,1],GraphHighlightStyle"Thick"]
Out[]=
In the middle
Orthogonal Directions
Orthogonal Directions
Geodesic Bundles in orthogonal directions [old]
Geodesic Bundles in orthogonal directions [old]
Cylinder growth [old]
Cylinder growth [old]
Longer tubes [old]
Longer tubes [old]
Ball growth
Ball growth