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) (*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info at wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 9657, 297]*) (*NotebookOutlinePosition[ 10339, 321]*) (* CellTagsIndexPosition[ 10295, 317]*) (*WindowFrame->Normal*) Notebook[{ 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"], Cell[BoxData[ \(\(?Torus\)\)], "Input"], Cell[BoxData[ \(\(tor = Torus[1.6, 1.3, 29, 23];\)\)], "Input"], Cell[BoxData[ \(Length[tor]\)], "Input"], Cell[BoxData[{ \(Position[tor, tor[\([1, 1, 1]\)]]\), "\[IndentingNewLine]", \(segmentlength = \(Position[tor, tor[\([1, 1, 1]\)]]\)[\([2, 1]\)]\)}], "Input"], Cell[BoxData[ \(segmentcount = Length[tor]/segmentlength\)], "Input"], Cell[TextData[{ "the torus consits of (", StyleBox["segmentlength", FontSlant->"Italic"], " * ", StyleBox["segmentcount", FontSlant->"Italic"], ") polygons, or ", StyleBox["segmentcount", FontSlant->"Italic"], " convex polyhedrons, each consisting of ", StyleBox["segmentlength", FontSlant->"Italic"], " polygons plus 2 capping lids.\nThis is one lid-less segment :" }], "Text"], Cell[BoxData[ \(<< Realtime3D`\)], "Input"], Cell[BoxData[ \( (*\ << Default3D`\ *) \)], "Input"], Cell[BoxData[ \(\(Take[tor, segmentlength];\)\)], "Input"], Cell[BoxData[ \(Show[Graphics3D[%], PlotRange \[Rule] All]\)], "Input"], Cell["\<\ 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[\ Part[#, All, All, \(-1\)] /. Polygon \[Rule] Identity]}] &\) /@ \ Partition[Take[tor, segmentlength*\((segmentcount)\)], segmentlength]\ ;\)\)], "Input"], Cell[BoxData[ \(Length[mytorus]\)], "Input"], Cell[BoxData[ \(Show[Graphics3D[mytorus], PlotRange \[Rule] All]\)], "Input"], Cell["\<\ for any segment, the circulation of the polygons con be checked from its \ barycentre:\ \>", "Text"], Cell[BoxData[ \(\(segmi = mytorus[\([5]\)];\)\)], "Input"], Cell[BoxData[ \(loci = \(\((Plus @@ #)\)/ Length[#] &\)@\((Flatten[\((segmi /. Polygon \[Rule] Identity)\), 1] // Union)\)\)], "Input"], Cell[BoxData[ \(\(a = .5;\)\)], "Input"], Cell[BoxData[ \(Show[ Graphics3D[{segmi, Cuboid[\(-a\)/2 {1, 1, 1} + loci, a/2 {1, 1, 1} + loci]}], PlotRange \[Rule] All]\)], "Input"], Cell[BoxData[ \(\(Sign[Chop[plane2[Take[#, 3]\ , \ loci]]] &\) /@ \((segmi /. Polygon \[Rule] Identity)\)\)], "Input"], Cell["\<\ for any segment, the above checks wether the point 'loci' is outside (as \ opposed to inside or on) the segment :\ \>", "Text"], Cell[BoxData[ \(FreeQ[\(Sign[ Chop[plane2[ Take[#, 3]\ , \ {\(-8. \), 0. , 0. } + 0*loci]]] &\) /@ \((segmi /. Polygon \[Rule] Identity)\), \(-1\)]\)], "Input"], Cell[BoxData[ RowBox[{"Timing", "[", RowBox[{\(z = \(q = 0\)\), ";", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Do", "[", RowBox[{ RowBox[{"it", "=", RowBox[{"FreeQ", "[", RowBox[{ RowBox[{\(z++\), ";", StyleBox[\(\(Sign[\(q++\); Chop[plane2[Take[#, 3]\ , 1*\ \ {0. , 0. , 8. } + 0*loci]]] &\) /@ \((segmi /. Polygon -> Identity)\)\), FontColor->RGBColor[0, 0, 1]]}], ",", \(-1\)}], "]"}]}], ",", \({100}\)}], "]"}], ";", "it"}], ",", "q", ",", "z"}], "}"}]}], "]"}]], "Input"], Cell["\<\ 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 :\ \>", "Text"], Cell[BoxData[ RowBox[{"Timing", "[", RowBox[{\(z = \(q = 0\)\), ";", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Do", "[", RowBox[{ RowBox[{"it", "=", RowBox[{ RowBox[{ StyleBox[\(FreeQ[\(z++\); Sign[Chop[ plane2[Take[#, 3]\ , 1*\ {0. , 0. , 8. } + 0*loci]]], \(-1\)]\), FontColor->RGBColor[0, 0, 1]], "&"}], "/@", \((\(q++\); And @@ \ \((segmi /. Polygon -> Identity)\))\)}]}], ",", \({100}\)}], "]"}], ";", "it"}], " ", ",", "q", ",", "z"}], "}"}]}], "]"}]], "Input"], Cell["\<\ and finally, we can verify if if the point loci is inside *any* of the convex \ polyhedrons :\ \>", "Text"], Cell[BoxData[{ \(z = \(q = 0\); Timing[Or @@ \((\(\(FreeQ[\(z++\); Sign[Chop[ plane2[Take[#, 3]\ , \ 0*\ \ {\(-8. \), 0. , 0. } + 1*loci]]], \(-1\)] &\) /@ \((\(q++\); And @@ \((# /. Polygon :> Identity)\))\) &\) /@ mytorus)\)]\), "\[IndentingNewLine]", \({z, q}\)}], "Input"], Cell["\<\ we can abort as soon as the testpoint lies inside any polyhedron:\ \>", "Text"], Cell[BoxData[{ \(z = \(q = 0\); Timing[\ \((\(\(FreeQ[\(z++\); Sign[Chop[ plane2[Take[#, 3]\ , 0*{\(-8. \), 0. , 0. } + 1*loci]]], \(-1\)] &\) /@ \((\(q++\); And @@ \((# /. Polygon :> Identity)\))\) &\) /@ \((Or @@ mytorus)\))\)]\), "\[IndentingNewLine]", \({z, q}\)}], "Input"], Cell[TextData[{ "Now we calculate a \"ray\" through the torus : (0.22 seconds per point on \ average,", " Version Number: ", ValueBox["$VersionNumber"], ".", ValueBox["$ReleaseNumber"], ".", ValueBox["$MinorReleaseNumber"], ", Win'98, 32Mb, 100MHz", ")\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"], Cell[BoxData[ \(Timing[ Table[z = \(q = 0\); {\((\(\((\(FreeQ[\(z++\); Sign[Chop[ plane2[ Take[#, 3]\ , \ {w/4, w, w/10}]]], \(-1\)] &\) /@ \((\(q++\); And @@ \ \((# /. 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J. MEEUSSEN w.meeussen.vdmcc at vandemoortele.be eu000949 at pophost.eunet.be