Why does RegionUnion fail for these Regions?

Question

Bug introduced in v10.0, resolved in v11.0.

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Introduction

Lately, I asked for a solution to inflate and unite a List of BoundaryMeshRegions: Inflate and unite a list of 0D to 2D regions

JasonB came up with a very good approach, using ContourPlot to do the hard work, instead of ImplicitRegion.

Problem

Now it happens, that the solution seems to fail for some cases of my application (I use PasteBin to save my List of regions for convenience reasons):

Quit;
ClearAll[expandedMeshRegion];
expandedMeshRegion[x_MeshRegion | x_BoundaryMeshRegion, d_: 2] := 
 Module[{xmin, xmax, ymin, ymax}, {{xmin, xmax}, {ymin, ymax}} = 
   Plus[#, {-1.1 d, 1.1 d}] & /@ 
    MinMax /@ Transpose[MeshCoordinates[x]];
  ContourPlot[
    RegionDistance[x, {xx, yy}] == d, {xx, xmin, xmax}, {yy, ymin, 
     ymax}] // BoundaryDiscretizeGraphics]

<< http://pastebin.com/raw/Yx4Jbj7F;

expandedMeshRegion[#, 10] & /@ hulls // RegionUnion

Interestingly, it works for the polygons in this case << http://pastebin.com/raw/fMBvnc6G; with double as many regions.

Side Info

• Mathematica 10.4.0.0

Answer 1

A simplified minimal example from {hulls[[6]], hulls[[12]]} (similarly failes if you replace BoundaryMeshRegion with MeshRegion):

SeedRandom@1;
b1 = BoundaryMeshRegion[
    {{142., -82.}, {187., -56.}, {142., -133.8}, {187., -108.}},
    {Line[{{3, 4}, {4, 2}, {2, 1}, {1, 3}}]},
    Properties -> {}, Method -> {}];
b2 = BoundaryMeshRegion[
    {{187., -56.}, {231., -82.2}, {97.7, -108.}, {142., -133.}},
    {Line[{{3, 4}, {4, 2}, {2, 1}, {1, 3}}]},
    Properties -> {}, Method -> {}];
RegionUnion[b1, b2]

These two are not joined. The alarming thing is, that if you evaluate it a couple more times, the order with which the unevaluated RegionUnion returns is not the same, regardless of SeedRandom. I would consider it a bug, and a twofold at that: internal numerical imprecisions (?) inhibit joining and the internal method of RegionUnion is immune to SeedRandom.

RegionBounds indicates that the bounds are almost identical along the y dimension.

RegionBounds /@ test

{{{142.375, 187.061}, {-133.8, -56.4}}, {{97.6877, 231.748}, {-133.8, -56.4}}}

Workaround 1: Coordinate offset

A possible workaround is to slightly offset all or one appropriate coordinate and transform the region a bit. Note, that the visible lip is due to me rounding the original hulls coordinates but is not the effect of the slight coordinate shift.

b2 = BoundaryMeshRegion[
    {{187., -56.}, {231., -82.2}, {97.7, -108.}, {142., -133.}} + 
     10^-6,
    {Line[{{3, 4}, {4, 2}, {2, 1}, {1, 3}}]},
    Properties -> {}, Method -> {}];
 RegionUnion[b1, b2]

Workaround 2: Convert to Polygons

TechSupport kindly suggested the following workaround:

b1 = Polygon[{{142., -82.}, {187., -56.}, {187., -108.}, {142., -133.8}}]
b2 = Polygon[{{187., -56.}, {231., -82.2}, {142., -133.}, {97.7, -108.}}]
b = RegionUnion[b1, b2] (* Note, that this does not really join the polygons *)
RegionPlot[b, AspectRatio -> Automatic]

You can easily extract the Polygon represenations by using MeshPrimitives. Then you can then simply plot the result as a list of graphics primitives:

exp  = expandedMeshRegion[#, 10] &/@ hulls;
poly = Chop@Flatten[MeshPrimitives[#, 2] &/@ exp];
Graphics[{{RandomColor[], #} &/@ poly}]

Or ypu can join individual primitives into one big Polygon - this of course only works so simply if the individual polygons are all convex.

comb = Polygon @ (Join @@ List @@@ poly);  (* would have been nice without the parenthesis *)
Graphics@{Blue, comb}

Don't try to discretize the resulting combined Polygon:

BoundaryDiscretizeGraphics@comb

 BoundaryMeshRegion::bcinsect: -- Message text not found -- ...

Joining the polygons as regions and plotting the result with RegionPlot also fails and on my machine it crashes the kernel.

(* Be advased, this might crash your kernel *)
RegionPlot @ (RegionUnion @@ poly)