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algorithm - Behavior of Graphics`Mesh`InPolygonQ with self-intersecting polygons


Playing around with some of the answers in the question How to check if a 2D point is in a polygon? I noticed that:


Graphics`Mesh`InPolygonQ[poly,pt]


Displays different behavior than the procedure that explicitly uses winding numbers (where a non-zero winding number implies polygon inclusion), for example the function in rm -rf♦'s answer to the aforementioned question:


inPolyQ[poly_, pt_] := Graphics`Mesh`PointWindingNumber[poly, pt] =!= 0

We can see this in the case where we define a self-intersecting polygon that has a "hole" in it:


PointWindingNumberInPolygonQ[poly_, pt_] := Graphics`Mesh`PointWindingNumber[poly, pt] =!= 0;

numRandPoints = 10^4;

testPolygon = {{65.4`, 439.5`}, {233.4`, 524.5`}, {364.40000000000003`, 433.5`}, {382.40000000000003`, 377.5`}, {354.40000000000003`, 293.5`}, {258.40000000000003`, 239.5`}, {94.4`, 207.5`}, {40.400000000000006`, 271.5`}, {18.400000000000002`, 356.5`}, {149.4`, 383.5`}, {187.4`, 330.5`}, {199.4`, 258.5`}, {136.4`, 130.5`}};


boundingBoxCoordinates = {{Min[testPolygon[[All, 1]]], Min[testPolygon[[All, 2]]]}, {Min[testPolygon[[All, 1]]], Max[testPolygon[[All, 2]]]}, {Max[testPolygon[[All, 1]]], Max[testPolygon[[All, 2]]]}, {Max[testPolygon[[All, 1]]], Min[testPolygon[[All, 2]]]}};

randomPoints = Table[{RandomReal[{Min[boundingBoxCoordinates[[All, 1]]], Max[boundingBoxCoordinates[[All, 1]]]}], RandomReal[{Min[boundingBoxCoordinates[[All, 2]]], Max[boundingBoxCoordinates[[All, 2]]]}]}, {k, 1, numRandPoints}];


windingNumPointsInPolygon = {};
inPolygonQPointsInPolygon = {};

For[i = 1, i <= Length[randomPoints], i++,


  If[PointWindingNumberInPolygonQ[testPolygon, randomPoints[[i]]] == True,
    windingNumPointsInPolygon = Append[windingNumPointsInPolygon, randomPoints[[i]]];
  ];

 If[Graphics`Mesh`InPolygonQ[testPolygon, randomPoints[[i]]] == True,
    inPolygonQPointsInPolygon = Append[inPolygonQPointsInPolygon, randomPoints[[i]]];
 ];

];



Graphics[Polygon[testPolygon]]
ListPlot[windingNumPointsInPolygon]
ListPlot[inPolygonQPointsInPolygon]

Notice how -


inPolyQ[poly_, pt_] := Graphics`Mesh`PointWindingNumber[poly, pt] =!= 0

Counts points inside the polygon "hole" as being inside the hole, while -


Graphics`Mesh`InPolygonQ[poly,pt]


Counts points inside the polygon "hole" as being outside the polygon, consistent with the shading for -


Graphics[Polygon[testPolygon]]

How can we characterize the behavior/algorithm of InPolygonQ? While I can understand how the winding number test method works, the trouble is that inPolygonQ is an undocumented function that doesn't give me any hints as to what it is doing.



Answer



The winding number is 0 outside the polygon, -1 in the non-self-intersecting region and -2 in the hole. So to exclude points inside the hole you would just test for the winding number being odd, rather than not equal to zero:


inPolyQ[poly_, pt_] := OddQ @  Graphics`Mesh`PointWindingNumber[poly, pt]

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