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computational geometry - Inflate and unite a list of 0D to 2D regions


I've got a List of BoundaryMeshRegions, created via ConvexHullMesh:


hulls0 = ConvexHullMesh /@ RandomReal[{-10, 10}, {3, 1, 2}];
hulls1 = ConvexHullMesh /@ RandomReal[{-10, 10}, {3, 2, 2}];
hulls2 = ConvexHullMesh /@ RandomReal[{-10, 10}, {3, 3, 2}];

hulls = Flatten[List[hulls0, hulls1, hulls2]];

Show[hulls]

regions example


Question


I now want to extend every region, to include also all points within a given distance d. Afterwards I want to obtain the union of all extended regions.


A 0D region (point) will therefore become a circle, a 1D region (line) will become two half circles with a rectangle in between, and so on.


My simple approach using


infReg[d_,regs_] := ImplicitRegion[RegionDistance[#, {x, y}] < d, {x, y}] & /@ regs
RegionUnion[infReg[2,hulls]]


doesn't work...



Real test case


You can take these hulls to test a solution with one of my real cases: PasteBin - Testcase


Minimal test case (take d=1)


poly1 = ConvexHullMesh[{{0, 0}, {1, 1}, {2, 0}, {1, -1}}];
poly2 = ConvexHullMesh[{{0, 0}, {2, 2}, {2, 0}, {0, -2}}];
hulls = {poly1, poly2}

Answer



Here we run up against the slowness of ImplicitRegion and RegionPlot when compared to ContourPlot. It's the same thing that led to this fantastic post.



Here is the first instinct,


SeedRandom[42];
hulls0 = ConvexHullMesh /@ RandomReal[{-10, 10}, {3, 1, 2}];
hulls1 = ConvexHullMesh /@ RandomReal[{-10, 10}, {3, 2, 2}];
hulls2 = ConvexHullMesh /@ RandomReal[{-10, 10}, {3, 3, 2}];
hulls = Flatten[List[hulls0, hulls1, hulls2]];

ImplicitRegion[
  RegionDistance[hulls1[[1]], {x, y}] <= 2, {x, y}] // RegionPlot


Mathematica graphics


That was really slow and not accurate! Basically, the framework underlying ImplicitRegion are not optimized for what we want to do. But we can take advantage of another function, which is optimized for this task. What we want is the boundary to the region, where the RegionDistance is equal to some number d, and finding this line boundary line can be done easily and quickly by ContourPlot, and the result can be fed directly to BoundaryDiscretizeGraphics to create the MeshRegion


{#, BoundaryDiscretizeGraphics@#} &@
 ContourPlot[
  RegionDistance[hulls1[[1]], {x, y}] == 2, {x, -7, -1}, {y, -8, 12}, AspectRatio -> Automatic]

Mathematica graphics


Now all that's left is to wrap this up in a function, taking care to figure out the bounds for the ContourPlot first.


Now consider


ClearAll[expandedMeshRegion];

expandedMeshRegion[x_MeshRegion | x_BoundaryMeshRegion, d_] := 
 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]

It works super fast and seems to work with any RegionDimension less than 3. It works on the whole list quickly as well


expandedMeshRegion[#, 2] & /@ hulls


Mathematica graphics


You can combine or show the regions however you like


RegionUnion[expandedMeshRegion[#, 2] & /@ hulls]

Mathematica graphics


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