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Bug introduced in 10.0.0 and fixed in 10.0.2

Consider the following ImplicitRegion:

reg = ImplicitRegion[x^2 + y^2 + z^2 == 16, {x, y, z}];

We can discretize it using DiscretizeRegion

dr = DiscretizeRegion[reg]

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To get finer triangles the option MaxCellMeasure is available, only it does nothing when used.

DiscretizeRegion[reg, MaxCellMeasure -> {"Area" -> #}] & /@ {0.1, 0.05}

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Of course using one of the Graphics primitives

DiscretizeRegion[Ball[], MaxCellMeasure -> {"Area" -> #}] & /@ {0.1, 0.05}

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You can see that it works fine. I'm on Windows 8.1, can anyone confirm this on other platforms? Confirmed on Linux via the Wolfram Programming Cloud.

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  • $\begingroup$ Tip: you can test on Linux using wolframcloud.com $\endgroup$
    – Szabolcs
    Commented Aug 5, 2014 at 3:08
  • $\begingroup$ @Szabolcs, Thanks nice tip. But it's so painful using it :) $\endgroup$
    – RunnyKine
    Commented Aug 5, 2014 at 3:12
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    $\begingroup$ Using the other quality control options, PrecisionGoal, AccuracyGoal,MeshQualityGoal, and MeshRefinementFunction on the ImplicitRegion similarly has no effect. $\endgroup$
    – Michael E2
    Commented Aug 5, 2014 at 12:29
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    $\begingroup$ Have you reported it? $\endgroup$
    – Szabolcs
    Commented Aug 5, 2014 at 21:44
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    $\begingroup$ @Szabolcs. Yes. Hopefully more people will report it, since that'll get their attention. $\endgroup$
    – RunnyKine
    Commented Aug 5, 2014 at 21:53

2 Answers 2

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Edit: Wolfram Technical Support has confirmed this as a bug

The only workaround I know is to turn the MeshRegion into a BoundaryMeshRegion and triangulate the resulting mesh object:

dr = DiscretizeRegion[reg, MaxCellMeasure -> {"Area" -> 0.05}];

Then:

TriangulateMesh[BoundaryMeshRegion[MeshCoordinates[dr], MeshCells[dr, 2]], 
                                               MaxCellMeasure -> #] & /@ {0.1, 0.005}

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It seems like another workaround is to repeatedly apply DiscretizeRegion which is definitely a weird approach. ( Thanks to Michael E2 )

Rest @ NestList[DiscretizeRegion, reg, 2]

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  • $\begingroup$ DiscretizeRegion[dr, MaxCellMeasure -> {"Area" -> 0.01}] produces similar results. Note that the outer function (DiscretizeRegion or TriangulateMesh) subdivides a mesh region and does not attempt to approximate the implicit region. Perhaps someone will come along and let us know whether implicit regions can be approximated with arbitrary precision. $\endgroup$
    – Michael E2
    Commented Aug 5, 2014 at 12:26
  • $\begingroup$ @MichaelE2, That's an interesting find. So basically, One has to discretize the result of DiscretizedRegion to see the effect. That makes no sense. $\endgroup$
    – RunnyKine
    Commented Aug 5, 2014 at 15:20
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    $\begingroup$ @MichaelE2, This bug is also present in DiscretizeGraphics and has been sort-of confirmed by @user21. $\endgroup$
    – RunnyKine
    Commented Aug 5, 2014 at 15:21
  • $\begingroup$ Well, this is a bug (or lack of implementation?) and by all means, you should report this. $\endgroup$
    – user21
    Commented Aug 6, 2014 at 7:17
  • $\begingroup$ @user21, I already reported it and linked to this page. $\endgroup$
    – RunnyKine
    Commented Aug 6, 2014 at 7:27
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To get finer triangles the option MaxCellMeasure is available, only it does nothing when used.

This is fixed in 10.0.2. on windows 7, 64 bit

Mathematica graphics

Mathematica graphics

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