# Region computations on FilledCurve (with hole)

How can I obtain a precompiled RegionMemberFunction for this geometry? It contains a hole, and the hole's boundary intersects the outer boundary at one point.

region = FilledCurve[{Line[{{34.5222,-86.2377},{34.5229,-86.2345},{34.5228,-86.2332},{34.53,-86.233},{34.532,-86.2337},{34.5322,-86.2249},{34.5326,-86.2228},{34.534,-86.2209},{34.5379,-86.2186},{34.5395,-86.2165},{34.5423,-86.2108},{34.5465,-86.2056},{34.5491,-86.2016},{34.5509,-86.2003},{34.5551,-86.1993},{34.5571,-86.1975},{34.5635,-86.1896},{34.5662,-86.1842},{34.5675,-86.1806},{34.5681,-86.1753},{34.5677,-86.1733},{34.5694,-86.1725},{34.575,-86.1678},{34.5796,-86.1666},{34.581,-86.1667},{34.5823,-86.1679},{34.5863,-86.1688},{34.5935,-86.169},{34.5955,-86.1703},{34.5993,-86.1675},{34.5991,-86.1485},{34.5883,-86.1487},{34.5497,-86.1494},{34.5506,-86.1543},{34.5502,-86.1551},{34.5412,-86.1433},{34.5343,-86.1441},{34.5343,-86.1445},{34.5348,-86.1498},{34.5336,-86.1498},{34.5335,-86.1455},{34.5333,-86.1398},{34.5261,-86.1439},{34.5209,-86.1478},{34.52,-86.1456},{34.5175,-86.1479},{34.5147,-86.1522},{34.5237,-86.1622},{34.5247,-86.1638},{34.5256,-86.1677},{34.5246,-86.1709},{34.5221,-86.174},{34.5219,-86.1764},{34.509,-86.1859},{34.5106,-86.1899},{34.5128,-86.191},{34.5129,-86.1922},{34.511,-86.1938},{34.5103,-86.1956},{34.5125,-86.1953},{34.5154,-86.1975},{34.5157,-86.198},{34.5163,-86.2004},{34.5161,-86.2056},{34.5162,-86.2082},{34.5169,-86.2101},{34.5174,-86.2111},{34.519,-86.2178},{34.5199,-86.2195},{34.5199,-86.2201},{34.5127,-86.2203},{34.513,-86.2264},{34.5129,-86.2277},{34.513,-86.2379},{34.5222,-86.2377}}],Line[{{34.5336,-86.1498},{34.5336,-86.1499},{34.5338,-86.1542},{34.5301,-86.1543},{34.5302,-86.1498},{34.5333,-86.1498},{34.5336,-86.1498}}]}];
region // Graphics


RegionQ[region]


False

Intersection @@ Cases[region, {{__?NumberQ} ..}, Infinity];


{{34.5336, -86.1498}}

RegionMember[region]


RegionMember::regp: A correctly specified region expected at position 1...

I tried this with no luck:

diff = RegionDifference @@ First[region /. Line -> Polygon];


which gives

diff // RegionQ


True

RegionMember[diff]


RegionMember[...] (* inert *)

diff // DiscretizeRegion


DiscretizeRegion::drf: DiscretizeRegion was unable to discretize the region RegionDifference[<<2>>]

I noticed if the hole and outer boundary do not intersect, there is no problem.

l1 = Line@{{0, 0}, {0, 10}, {10, 10}, {10, 0}, {0, 0}};
l2 = l1 /. {0 -> 4, 10 -> 6};
c = FilledCurve@{l1, l2};
c // Graphics


RegionDifference @@ First[c /. Line -> Polygon] // RegionMember


Is there something better than this approach?

outer = Polygon @@ region[[1, 1]] // RegionMember
inner = Polygon @@ region[[1, 2]] // RegionMember
member[x_] := outer@x && ! inner@x;


## 1 Answer

Discretizing polygons first then difference facilitates. Apologies if I have misunderstood issue.

pol = Cases[region, Line[x__] :> Polygon[x], Infinity];
rm = RegionDifference @@ (DiscretizeRegion /@ pol);
rmf = RegionMember[rm]
mx = Max[pol[[1, All, #]]] & /@ {1, 2};
mn = Min[pol[[1, All, #]]] & /@ {1, 2};
Manipulate[
Column[{Show[Graphics[Point[p]], RegionPlot[rm, PlotStyle -> c],
PlotRange -> Transpose[{mn, mx}], ImageSize -> 400],
If[rmf[p], c = Red; "True", c = Black;
"False"]}], {{p, {34.56, -86.2}}, Locator}]


• Perfect, thank you. – mfvonh Jun 22 '15 at 2:42
• @mfvonh thank you...I made slightly more concise... – ubpdqn Jun 22 '15 at 2:55