Re: Interior of a polygon

*To*: mathgroup at smc.vnet.net*Subject*: [mg28767] Re: Interior of a polygon*From*: "Mariusz Jankowski" <mjkcc at usm.maine.edu>*Date*: Fri, 11 May 2001 20:00:45 -0400 (EDT)*Organization*: University of Southern Maine*References*: <9d85uu$862@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Thank you to all who responded! Andrzej's suggestion to use a polygon triangulation package by Martin Kraus immediately led me to a solution, since I already had a solution for convex polygons. However, in the meantime, Alan quickly came up with a solution that does not require an explicit triangulation step (see his earlier post). It works and is faster than the example I will show below. Therefore the only reason I am demonstrating an alternative solution is that it may be easier for most to understand. Sincerely, Mariusz Load PolygonTriangulation package. <<PolygonTriangulation`SimplePolygonTriangulation` The following code finds all points along the border of a polygon: borderPoints[v:{{_, _}..}] := Union[Join @@ getPoints /@ Partition[v, 2, 1, 1, v]] getPoints[{{a_, b_}, {c_, d_}}] := Transpose[If[Abs[a - c] >= Abs[b - d], ({#1, Round[Interpolation[{{a, b}, {c, d}}, InterpolationOrder -> 1] /@ #1]} & )[ Range[a, c, If[a <= c, 1, -1]]], ({Round[Interpolation[{{b, a}, {d, c}}, InterpolationOrder -> 1] /@ #1], #1} & )[ Range[b, d, If[b <= d, 1, -1]]]]]; Given the function borderPoints the interior (and border) points of a convex polygon can be obtained using the following: regionPointsConvex[v:{{_, _}..}] := Join @@ fillPoints /@ findExtremalPoints[ borderPoints[v]]; findExtremalPoints[pts:{{_, _}..}] := ({#1[[1]], #1[[-1]]} & ) /@ Split[pts, #1[[1]] === #2[[1]] & ]; fillPoints[{{x_, y0_}, {x_, y1_}}] := ({x, #1} & ) /@ Range[y0, y1]; To find the points in any simple polygon use: regionPoints[v:{{_, _}..}] := Union @@ regionPointsConvex /@ (v[[#1]] & ) /@ SimplePolygonTriangulation[v]