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I want to create the following 3D solid object, which is an octahedron, cutting by six spheres centered at its vertices.

The code is as:

RegionPlot3D[-2 x + 2 y - 2 z + Sqrt[2] >= 
   0 && -2 x - 2 y - 2 z + Sqrt[2] >= 0 && 
  2 x - 2 y - 2 z + Sqrt[2] >= 0 && 2 x + 2 y - 2 z + Sqrt[2] >= 0 && 
  2 x + 2 y + 2 z + Sqrt[2] >= 0 && -2 x + 2 y + 2 z + Sqrt[2] >= 0 &&
   2 x - 2 y + 2 z + Sqrt[2] >= 0 && -2 x - 2 y + 2 z + Sqrt[2] >= 0
  && (-(1/Sqrt[2]) - x)^2 + y^2 + z^2 > 1/4 && 
  x^2 + (1/Sqrt[2] - y)^2 + z^2 > 1/4 && 
  x^2 + y^2 + (-(1/Sqrt[2]) - z)^2 > 1/4 && 
  x^2 + y^2 + (1/Sqrt[2] - z)^2 > 1/4 && 
  x^2 + (-(1/Sqrt[2]) - y)^2 + z^2 > 
   1/4 && (1/Sqrt[2] - x)^2 + y^2 + z^2 > 1/4, {x, -1, 1}, {y, -1, 
  1}, {z, -1, 1}, PlotPoints -> 250, Mesh -> None, 
 PlotTheme -> "Classic"]

and the output looks like: enter image description here

There are moustache like black lines on the edge. Increase the PlotPoints option to 250 cannot remove it.

Update:

Outputs with BoundaryStyle->None: enter image description here

How can I obtain desired smooth version of the solid? Why such black lines happen?

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  • 1
    $\begingroup$ Try BoundaryStyle -> None. $\endgroup$ Commented Jul 27, 2017 at 10:11
  • $\begingroup$ The lines still exist but turn into white or something transparent. $\endgroup$ Commented Jul 27, 2017 at 10:17
  • 1
    $\begingroup$ I've found Regions to be a bit unreliable when they get really thin like that. An adaptive mesh approach, using DiscretizeRegion with an appropriate MeshRefinementFunction could do the trick. Have a look at 105756, and see if it helps. $\endgroup$ Commented Jul 27, 2017 at 11:10
  • 2
    $\begingroup$ Using contourRegionPlot3D[] on this problem gives this. $\endgroup$ Commented Jul 27, 2017 at 12:52
  • $\begingroup$ Thank you @J.M. , this new function via ContourPlot3D solves my problem. By increasing PlotPoints into 80 or so, the output graph looks pretty good. contourRegionPlot3D is much better than the built-in RegionPlot3D in these aspects if 3D solid model is not needed $\endgroup$ Commented Jul 28, 2017 at 2:22

1 Answer 1

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  • Since
ineqs=-2  x + 2  y - 2  z + Sqrt[2] >= 
    0 && -2  x - 2  y - 2  z + Sqrt[2] >= 0 && 
   2  x - 2  y - 2  z + Sqrt[2] >= 0 && 
   2  x + 2  y - 2  z + Sqrt[2] >= 0 && 
   2  x + 2  y + 2  z + Sqrt[2] >= 
    0 && -2  x + 2  y + 2  z + Sqrt[2] >= 0 && 
   2  x - 2  y + 2  z + Sqrt[2] >= 
    0 && -2  x - 2  y + 2  z + Sqrt[2] >= 0;
RegionEqual[
 ImplicitRegion[ineqs, {x, y, z}], Octahedron[]]

True

So the solid is the Octanhedron[] remove 6 balls which centers is the vertexs of such Octanhedron[] with 1/2 radius

  • For version 14.1.0, another method work.
$Version

centers = PolyhedronData["Octahedron", "Points"][[;; , 1]];
balls = BoundaryDiscretizeGraphics@Ball[#, 1/2] & /@ centers;
Fold[RegionDifference[#1, #2] &, 
 BoundaryDiscretizeGraphics@Octahedron[], balls]

enter image description here

Or

$Version

centers = PolyhedronData["Octahedron", "Points"][[;; , 1]];
regs = RegionIntersection[
     BoundaryDiscretizeGraphics@
      Octahedron[], #] & /@ (RegionDifference[
       BoundaryDiscretizeRegion@Cuboid[5  {-1, -1, -1}, 5  {1, 1, 1}],
        BoundaryDiscretizeGraphics@Ball[#, 1/2]] & /@ centers);
reg = RegionIntersection @@ regs // BoundaryDiscretizeRegion;
RegionPlot3D[reg]

enter image description here

Or

Needs["OpenCascadeLink`"];
Needs["NDSolve`FEM`"];
centers = PolyhedronData["Octahedron", "Points"][[;; , 1]]; balls = 
 OpenCascadeShapeUnion[
  OpenCascadeShape /@ (Ball[#, 1/2] & /@ centers)]; reg = 
 OpenCascadeShape[
  ToBoundaryMesh@PolyhedronData["Octahedron", "BoundaryMeshRegion"]];
shape = OpenCascadeShapeDifference[reg, balls];
bm = OpenCascadeShapeSurfaceMeshToBoundaryMesh[shape, 
  "ShapeSurfaceMeshOptions" -> {"AngularDeflection" -> 
     0.05}]; Graphics3D[{FaceForm[Orange], EdgeForm[], 
  BoundaryMeshRegion[bm], {FaceForm[], 
   EdgeForm[Directive@{Thick, Cyan}], Octahedron[]}}, Boxed -> False]

enter image description here

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