Skip to main content

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]

Comments

Post a Comment

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more