Re: Point inside a plygon? and PointInsidePolyhedronQ

• To: mathgroup at smc.vnet.net
• Subject: [mg25353] Re: [mg25239] Point inside a plygon? and [mg24992] PointInsidePolyhedronQ
• From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
• Date: Sat, 23 Sep 2000 03:36:35 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```I am really to busy at this time to do so myself but I hope other people
will test this. I must admit however that I am sceptical. The reason is
this. Take almost any book on combinatorial or piecewise linear topology
e.g. Rourke and Sanderson, "Piecewise-Linear Topology" and look at the proof
of the theorem (Theorem 2.11 of R&S) , which says that any compact
polyhedron can be triangulated. Admittedly this is a lot more general than
what you are trying to do. Still, you will see that the proof is
non-constructive and much more complicated than the one you are proposing.
Of course I am not saying that it is impossible to find simple and
constructive proofs of classical results that earlier had complicated
non-constructive ones, but it does not happen terribly often. On the other
hand finding mistakes in other people proofs is generally time consuming and
not very rewarding, unless it concerns your own work. Of course I wish you
luck and I hope an interesting Mathematica package will arise out of this
but I suggest you write out the mathematical part of your argument and get
some specialists on combinatorial topology and geometry to have a look at
it.

Andrzej

on 00.9.22 5:32 AM, David Park at djmp at earthlink.net wrote:

> Dear MathGroup and All,
>
> Here is another algorithm for determining if a point is within a polygon. It
> works for any polygon defined by a sequence of points, and is approximately
> 100 times faster than the complex contour integral method. The method was
> suggested to me by Richard I. Pelletier.
>
> The idea is to draw a line from the point in question to a point that is
> known to be outside the polygon. We then find out how many times that line
> intersects the edge segments of the polygon. If it is an odd number of
> times, the point is within the polygon, otherwise it is outside. We can
> easily generate a point outside the polygon by finding the maximum and
> minimum coordinates of the vertices.
>
> The following routine tests if two line segments intersect. The first
> segment would be from the test point to the outside point. The second
> segment would be an edge of the polygon. The expressions were obtained by
> parametrizing the two line segments so that their parameters run from 0 to 1
> over the segment, and solving for the intersection point. d is a denominator
> and is 0 for parallel lines. The test for the edges is designed to avoid
> double counting intersections at vertices.
>
>
> SegmentIntersection::usage = "SegmentIntersection[{p1,p2},{p3,p4}] \
> determine if the line segment {p1,p2} intersects the line segment {p3,p4}."\
> ;
> SegmentIntersection[{{x1_, y1_}, {x2_, y2_}}, {{x3_, y3_}, {x4_, y4_}}] :=
> With[{d = x4*(y1 - y2) + x3*(-y1 + y2) + (x1 - x2)*(y3 - y4)},
> d != 0 && 0 <= (x4*(y1 - y3) + x1*(y3 - y4) + x3*(-y1 + y4))/d <= 1 &&
> Inequality[0, LessEqual, (x3*(-y1 + y2) + x2*(y1 - y3) + x1*(-y2 + y3))/
> d, Less, 1]]
>
> This is the routine that tests if a point is within a polygon.
>
> IntersectionPointInPolygon::usage =
> "IntersectionPointInPolygon[point, polygonvertices] will determine if a
> \
> point is within a polygon defined by its vertices. Its method is to define a
> \
> line segment between the point and an outside point, and count how many
> times \
> it intersects the edge segments of the polygon. An odd count puts the point
> \
> within the polygon.";
> IntersectionPointInPolygon[pt : {_, _}, polypoints : {{_, _} ..}] :=
> Module[{xmin, xmax, ymin, ymax, xpts, ypts, outpoint, edges, count},
> {xpts, ypts} = Transpose[polypoints];
> {xmax, ymax} = Max /@ {xpts, ypts};
> {xmin, ymin} = Min /@ {xpts, ypts};
> outpoint = {(3xmax - xmin)/2, (ymin + ymax)/2};
> edges = Partition[Join[polypoints, {First[polypoints]}], 2, 1];
> count = Count[SegmentIntersection[{pt, outpoint}, #] & /@ edges, True];
> OddQ[count]
> ]
>
> Needs["Graphics`Colors`"]
>
> This tests the routine with a moderately complex polygon and random points.
>
> Module[
> {pt = {Random[Real, {-5, 5}], Random[Real, {-5, 5}]},
> poly = {{-3.72826, -3.17076}, {-0.801402, -1.84671}, {1.70733, \
> -3.79795}, {5.26138, -1.21952}, {2.4042, 0.801401}, {4.00701,
> 4.14638}, {0.243904, 3.24045}, {-3.65857, 5.05231}, {-5.88856,
> 1.56796}, {-4.0767, -0.174218}},
> linepts},
> linepts = Join[poly, {First[poly]}];
> Show[Graphics[
> {Line[linepts],
> AbsolutePointSize[4], Point /@ poly,
> Red, Point[pt],
> Black,
> Text[IntersectionPointInPolygon[pt, poly], {-9, 9}, {-1, 0}]}],
>
> AspectRatio -> Automatic, PlotRange -> {{-10, 10}, {-10, 10}},
> Background -> Linen]];
>
> David Park
>

--
Andrzej Kozlowski
Toyama International University
JAPAN

http://platon.c.u-tokyo.ac.jp/andrzej/
http://sigma.tuins.ac.jp/

```

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