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A region in 3D space has been discretized to obtain a list of points.

The resulting ListPointPlot3D plot is:

enter image description here

I will like to obtain a mesh from these points, but the mesh should not connect points that are too far away from each other.

Using ConvexHullMesh and BoundingRegion directly gives the following result:

enter image description here

The mesh connects points that are far away, closing a part of the volume which must be concave. Is there a way to set the maximum distance between the nodes of the mesh?

The list of 3D points is available at: Points

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  • $\begingroup$ You asked for ConvexHullMesh and you got it. You can of course refine it by applying DiscretizeRegion with a small setting for MaxCellMeasure. Want you obtain is the remeshed convex hull. But maybe you want to compute an $\alpha$-shape? $\endgroup$
    – Henrik Schumacher
    Jul 4, 2019 at 0:36
  • $\begingroup$ In general, it is a very nontrivial problem to construct a mesh from just a point cloud. $\endgroup$
    – Henrik Schumacher
    Jul 4, 2019 at 0:46
  • $\begingroup$ Don't know much about meshing, but I guess I want to compute an alpha shape. How can I do this in Mathematica? $\endgroup$
    – Guillermo Oliver
    Jul 4, 2019 at 12:19
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    $\begingroup$ mathematica.stackexchange.com/a/8812 $\endgroup$
    – Henrik Schumacher
    Jul 4, 2019 at 12:57

1 Answer 1

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Thank you for pointing me in the right direction, Henrik Schumacher!

These two links helped me to solve this problem:

Finding a concave hull

Random discrete data 3D plot

Adequately modifying this code by Taiki and Simon Woods was just what I needed:

tetrahedra = Level[MeshPrimitives[DelaunayMesh[data], 3], {-3}];
radius[p_] := Circumsphere[p][[2]];
radii = radius /@ tetrahedra;
alphashape[rmax_] := Pick[tetrahedra, radii, r_ /; r < rmax]
faces[tetras_] := Flatten[
   tetras /. {a_, b_, c_, d_} :> {{a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}}, 1];
externalfaces[faces_] := Cases[Tally[Sort /@ faces], {face_, 1} :> face];
radiusmax = 150;
polys = externalfaces@faces@alphashape[radiusmax];
Graphics3D[
  {EdgeForm[{Thin, Opacity[1/30, Black]}], Polygon@polys},
  AxesLabel -> {"x", "y", "z"},
  Axes -> True,
  Boxed -> False,
  BoxRatios -> {1, 1, 1},
  ViewPoint -> 1000 {-1, 1, -1}
]

The result is: enter image description here

Further playing with the rmax parameter can yield even more refined results.

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