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MathGroup Archive 2000

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point inside torus

  • To: mathgroup at smc.vnet.net
  • Subject: [mg25261] point inside torus
  • From: Wouter Meeussen <eu000949 at pophost.eunet.be>
  • Date: Sun, 17 Sep 2000 04:47:48 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

see notebook (12 kB)


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Cell[BoxData[
    \(<< Graphics`Shapes`\)], "Input"],

Cell[BoxData[
    \(plane2 = 
      Compile[{{u, _Real, 2}, {v, _Real, 1}}, 
        Det[{\((v - u[\([1]\)])\), \((u[\([1]\)] - 
                u[\([2]\)])\), \ \((u[\([2]\)] - u[\([3]\)])\)}]]\)], "Input"],

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    \(\(?Torus\)\)], "Input"],

Cell[BoxData[
    \(\(tor = Torus[1.6, 1.3, 29, 23];\)\)], "Input"],

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Cell[BoxData[{
    \(Position[tor, tor[\([1, 1, 1]\)]]\), "\[IndentingNewLine]", 
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Cell[TextData[{
  "the torus consits of (",
  StyleBox["segmentlength",
    FontSlant->"Italic"],
  " * ",
  StyleBox["segmentcount",
    FontSlant->"Italic"],
  ") polygons, or  ",
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    FontSlant->"Italic"],
  " convex polyhedrons, each consisting of ",
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    FontSlant->"Italic"],
  " polygons plus 2 capping lids.\nThis is one lid-less segment :"
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    \(<< Realtime3D`\)], "Input"],

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this is one of the lids : the set of first points in each polygon;
the other lid is the set of last points. (Other programs might have \
differently structured data for a torus)\
\>", "Text"],

Cell[BoxData[
    \(Polygon[\ 
      Part[Take[tor, segmentlength], All, All, 1] /. 
        Polygon \[Rule] Identity]\)], "Input"],

Cell["\<\
the top lid needs reversing in order to get the correct circulation;
below is the entire torus as a list of well oriented convex polyhedrons:\
\>", "Text"],

Cell[BoxData[
    \(\(mytorus = \(Flatten[{Polygon[
                  Reverse at \ Part[#, All, All, 1] /. 
                    Polygon \[Rule] Identity], #, 
                Polygon[\ 
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                    Polygon \[Rule] Identity]}] &\) /@ \ 
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            segmentlength]\ ;\)\)], "Input"],

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for any segment, the circulation of the polygons con be checked from its \
barycentre:\
\>", "Text"],

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    \(loci = \(\((Plus @@ #)\)/
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    \(\(a =  .5;\)\)], "Input"],

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for any segment, the above checks wether the point 'loci' is outside (as \
opposed to inside or on) the segment :\
\>", "Text"],

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    \(FreeQ[\(Sign[
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programming for speed instead of clarity, we try to abort the calculation as \
soon as the testpoint lies on the wrong side of any polygon :\
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and finally, we can verify if if the point loci is inside *any* of the convex \
polyhedrons :\
\>", "Text"],

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we can abort as soon as the testpoint lies inside any polyhedron:\
\>", "Text"],

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  "Now we calculate a \"ray\" through the torus : (0.22 seconds per point on \
average,",
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  ValueBox["$MinorReleaseNumber"],
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  ")\nIf the testpoint lies outside, all 29 (=segmentcount) polyhedral \
segments have to be checked;\nif it's inside, less segments until a hit, but \
then a hit is only shure after checking all 25 (=segmentlength+2) polygons :"
}], "Text"],

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Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be


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