(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. 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). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@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[ 1217253, 31575]*) (*NotebookOutlinePosition[ 1220940, 31662]*) (* CellTagsIndexPosition[ 1220368, 31648]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["(* no spelling warnings, set fonts for tick labels, ... *)", "CodeComment"], "\nGet[ToFileName[ReplacePart[\"FileName\" /. \n \ NotebookInformation[EvaluationNotebook[]], \"Initialization.m\", 2]]];" }], "Input", InitializationCell->True, GeneratedCell->False], Cell[BoxData[ RowBox[{\(Get::"noopen"\), \(\(:\)\(\ \)\), "\<\"Cannot open \\!\\(\\\"C:\ \\\\\\\\Documents and Settings\\\\\\\\Luc \ Barthelet\\\\\\\\Desktop\\\\\\\\published notebooks\\\\\\\\Initialization.m\\\ \"\\). \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::noopen\\\"]\\)\"\>"}]], "Message"] }, Open ]], Cell[TextData[{ StyleBox["c)", "EXNUM"], " In the following we will implement the subdivision that runs in ", Cell[BoxData[ \(TraditionalForm\`O(n)\)]], " for ", Cell[BoxData[ \(TraditionalForm\`n\)]], " vertices. We will use hashing techniques to achieve this complexity. \ Using more elaborate data structures would allow for a smaller constant (see \ [571", ButtonBox["\[SixPointedStar]", ButtonData:>"Zorin00", ButtonStyle->"Hyperlink", ButtonNote->"Zorin00"], "], [366", ButtonBox["\[SixPointedStar]", ButtonData:>"Mueller00", ButtonStyle->"Hyperlink", ButtonNote->"Mueller00"], "] for details). The information about the connectivity of our grid \ (simplicial complex) will be stored in the form of all neighbors of all \ vertices. At each stage of the subdivision, we will keep only information \ about the current simplicial complex. This means we drop the index ", Cell[BoxData[ \(TraditionalForm\`l\)]], " from ", Cell[BoxData[ \(TraditionalForm\`V\_\(vertex; \ k\)\%\((l)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`V\_\(edge; \ k\)\%\((l)\)\)]], ". The ", StyleBox["k", "TI"], "th vertex is represented as ", StyleBox["\[ScriptCapitalV][", "MR"], StyleBox["k", "TI"], StyleBox["]", "MR"], ". The edge connecting the vertices ", StyleBox["\[ScriptCapitalV][", "MR"], StyleBox["k", "TI"], StyleBox["]", "MR"], " and ", StyleBox["\[ScriptCapitalV][", "MR"], StyleBox["l", "TI"], StyleBox["]", "MR"], " by ", StyleBox["\[ScriptCapitalE][\[ScriptCapitalV][", "MR"], StyleBox["k", "TI"], StyleBox["], \[ScriptCapitalV][", "MR"], StyleBox["l", "TI"], StyleBox["]]", "MR"], " and the triangle formed by the three vertices ", StyleBox["\[ScriptCapitalV][", "MR"], StyleBox["k", "TI"], StyleBox["]", "MR"], ", ", StyleBox["\[ScriptCapitalV][", "MR"], StyleBox["l", "TI"], StyleBox["]", "MR"], ", and ", StyleBox["\[ScriptCapitalV][", "MR"], StyleBox["m", "TI"], StyleBox["]", "MR"], " by ", StyleBox["\[ScriptCapitalT][\[ScriptCapitalV][", "MR"], StyleBox["k", "TI"], StyleBox["], \[ScriptCapitalV][", "MR"], StyleBox["k", "TI"], StyleBox["], \[ScriptCapitalV][", "MR"], StyleBox["m", "TI"], StyleBox["]]", "MR"], ". The functions ", StyleBox["neighborVertices[", "MR"], StyleBox["k", "TI"], StyleBox["]", "MR"], " gives a list of the vertices that are nearest neighbors to the vertex ", StyleBox["k", "TI"], ". ", StyleBox["allTriangles", "MR"], " is a list of all triangles forming the surface. Newly introduced vertices \ along an edge joining the vertices ", StyleBox["\[ScriptCapitalV][", "MR"], StyleBox["k", "TI"], StyleBox["]", "MR"], " and ", StyleBox["\[ScriptCapitalV][", "MR"], StyleBox["l", "TI"], StyleBox["]", "MR"], " will be denoted by ", StyleBox["\[ScriptCapitalV][", "MR"], StyleBox["k", "TI"], StyleBox[", ", "MR"], StyleBox["l", "TI"], StyleBox["]", "MR"], ". To make sorting these vertices easier, we give the symbol ", StyleBox["\[ScriptCapitalV]", "MR"], " the attribute ", StyleBox["Orderless", "MR"], "." }], "SolutionSubgroup", GeneratedCell->False, CellTags->{ "Index[08, {complexity, of subdivisions}]", "Index[08, {triangulation, of surfaces}]", "Index[08, {smoothing, nonsmooth surfaces}]", "InverseReference[Zorin00, 1]", "InverseReference[Mueller00, 1]"}], Cell["SetAttributes[\[ScriptCapitalV], Orderless];", "Input", GeneratedCell->False], Cell[TextData[{ "The function ", StyleBox["makeTrianglesAtVertexDefinitions", "MR"], " associates the triangles to the vertices." }], "Text", GeneratedCell->False], Cell["\<\ makeTrianglesAtVertexDefinitions := Map[(trianglesAtVertex[#[[1, 1]]] = Last /@ #)&, Split[Sort[Flatten[{{#[[1]], #}, {#[[2]], #}, {#[[3]], #}}& /@ allTriangles, 1]], #1[[1]] === #2[[1]]&], {1}];\ \>", "Input", GeneratedCell->False], Cell[TextData[{ StyleBox["makeAllEdges[]", "MR"], " generates a list of all edges of the network. 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Length[vertices];\n \ (\[Alpha][n] vertexCoordinates[\[ScriptCapitalV][k]] + \n \ \[Beta][n] Plus @@ (vertexCoordinates /@ vertices))/8]" }], "Input", GeneratedCell->False], Cell[TextData[{ StyleBox["(* make new vertices at the existing edges *)", "CodeComment"], " \ngetVertexCoordinates[\[ScriptCapitalE][\[ScriptCapitalV][k_], \ \[ScriptCapitalV][l_]]] := \nWith[{", StyleBox["(* third points of the two triangles sharing edge \ \[ScriptCapitalE][\[ScriptCapitalV][k], \[ScriptCapitalV][l]] *)", "CodeComment"], "\n neighbors = List @@ Complement[Join @@ Select[\n \ trianglesAtVertex[\[ScriptCapitalV][k]], MemberQ[#, \[ScriptCapitalV][l]]&],\n\ \[ScriptCapitalT][\[ScriptCapitalV][\ k], \[ScriptCapitalV][l]]]},\n (3 \ (vertexCoordinates[\[ScriptCapitalV][k]] + \ vertexCoordinates[\[ScriptCapitalV][l]]) +\n Plus @@ (vertexCoordinates \ /@ neighbors))/8.]" }], "Input", GeneratedCell->False], Cell[TextData[{ "Given a triangle ", StyleBox["\[ScriptCapitalT][\[ScriptCapitalV][", "MR"], StyleBox["p", "TI"], StyleBox["], \[ScriptCapitalV][", "MR"], StyleBox["q", "TI"], StyleBox["], \[ScriptCapitalV][", "MR"], StyleBox["r", "TI"], StyleBox["]]", "MR"], ", the two functions ", StyleBox["makeNewEdges", "MR"], " and ", StyleBox["makeNewTriangles", "MR"], " generate the edges and triangles of the next stage of the subdivision \ process." }], "Text", GeneratedCell->False], Cell["\<\ makeNewEdges[\[ScriptCapitalT][\[ScriptCapitalV][p_], \[ScriptCapitalV][q_], \ \[ScriptCapitalV][r_]]] := With[{pq = \[ScriptCapitalV][p, q], pr = \[ScriptCapitalV][p, r], qr = \ \[ScriptCapitalV][q, r]}, Apply[Sort[\[ScriptCapitalE][##]]&, {{\[ScriptCapitalV][p], pq}, {\ \[ScriptCapitalV][p], pr}, \t\t\t\t\t {\[ScriptCapitalV][q], pq}, {\[ScriptCapitalV][q], qr}, {\[ScriptCapitalV][r], pr}, {\[ScriptCapitalV][r], \ qr}, {pq, pr}, {pq, qr},{pr, qr}}, {1}]]\ \>", "Input", GeneratedCell->False], Cell["\<\ makeNewTriangles[\[ScriptCapitalT][\[ScriptCapitalV][p_], \ \[ScriptCapitalV][q_], \[ScriptCapitalV][r_]]] := With[{pq = \[ScriptCapitalV][p, q], pr = \[ScriptCapitalV][p, r], qr = \ \[ScriptCapitalV][q, r]}, Apply[\[ScriptCapitalT][##]&, { \t{\[ScriptCapitalV][p], pq, pr}, \t\t\t\t\t\t{pq, \[ScriptCapitalV][q], qr}, \t {qr, \[ScriptCapitalV][r], pr},(* this one was \ mis-oriented *) \t{pq,qr,pr} (* this one was mis-oriented *) \t}, {1}]]\ \>", "Input", GeneratedCell->False], Cell[TextData[{ "Now it is straightforward to implement the function ", StyleBox["subdivide[]", "MR"], ". ", StyleBox["subdivide[]", "MR"], " carries out one step in the subdivision process. ", StyleBox["subdivide[]", "MR"], " updates the global list ", StyleBox["allTriangles", "MR"], " and the definitions associated with the three symbols ", StyleBox["neighborVertices", "MR"], ", ", StyleBox["vertexCoordinates", "MR"], ", and ", StyleBox["trianglesAtVertex", "MR"], ". For a nicer\[Hyphen]looking result, we number the newly created vertices \ consecutively after each subdivision step." }], "Text", GeneratedCell->False, CellTags->{ "Index[08, {caching, in action}]", "Index[08, {assignments, cached ~}]b"}], Cell[CellGroupData[{ Cell[BoxData[ \(Flatten[\ Apply[test[##] &, {{8, 2, 3}, {4, 5, 6}}, {1}]]\)], "Input"], Cell[BoxData[ \({test[8, 2, 3], test[4, 5, 6]}\)], "Output"] }, Open ]], Cell[TextData[{ "subdivide[] := \nModule[{n, allEdges, newEdges, allNewVertices, \ renumberingRules, aux, \n vertex, newVertexCoordinates},\n", StyleBox["(* update existing vertices; keep their numbering *)\n ", "CodeComment"], "n = Length[DownValues[vertexCoordinates]]", StyleBox[";", "CodeComment"], "\n Do[newVertexCoordinates[\[ScriptCapitalV][k]] = getVertexCoordinates[\ \[ScriptCapitalV][k]], {k, n}];\n", StyleBox["(* all edges *)", "CodeComment"], "\nallEdges = makeAllEdges[]; \n", StyleBox["(* make new vertices along the edges *)", "CodeComment"], " \n(newVertexCoordinates[\[ScriptCapitalV][#[[1, 1]], #[[2, 1]]]] = \n \ getVertexCoordinates[#])& /@ allEdges;\n", StyleBox["(* the new edges *)", "CodeComment"], "\nnewEdges = Union[Flatten[makeNewEdges /@ allTriangles]];\n", StyleBox["(* renumber new vertices *)", "CodeComment"], "\nallNewVertices = Union[Cases[newEdges, \[ScriptCapitalV][_, _], {-2}]];\n\ ", StyleBox["(* create dispatch table for fast renumbering *)", "CodeComment"], "\nrenumberingRules = Dispatch[MapIndexed[#1 -> \[ScriptCapitalV][n + \ #2[[1]]]&, \n allNewVertices]];\n\ ", StyleBox["(* generate new neighbor relations *)\n", "CodeComment"], "Clear[neighborVertices", StyleBox["];\n(* group vertices and neighbors *)", "CodeComment"], "\naux = Split[Sort[Join[newEdges, Reverse /@ newEdges]],\n \ First[#1] === First[#2]&];\nDo[vertex = aux[[k, 1, 1]] /. renumberingRules;\n \ neighborVertices[vertex] = (Union[Last /@ aux[[k]]] /. renumberingRules),\n \ {k, Length[aux]}];\n", StyleBox["(* new coordinates of vertices; \n vertexCoordinates is a \ globally visible variable *) ", "CodeComment"], " \nClear[vertexCoordinates];\nDownValues[vertexCoordinates] = \n \ (DownValues[newVertexCoordinates] /. renumberingRules /.\n \ newVertexCoordinates -> vertexCoordinates);\n", StyleBox["(* new triangles *)", "CodeComment"], " \nallTriangles =(Flatten[makeNewTriangles /@ allTriangles] /. \n \ renumberingRules);\n", StyleBox["(* triangles at a vertex *)", "CodeComment"], " \nmakeTrianglesAtVertexDefinitions]" }], "Input", GeneratedCell->False], Cell["\<\ Now let us apply the routine subdivide to the massive wireframe of a cube. We \ start by constructing the cube wireframe.\ \>", "Text", GeneratedCell->False, CellTags->"T[D[Smoothing a Cube Wireframe]]"], Cell["Needs[\"Graphics`Polyhedra`\"]", "Input", GeneratedCell->False], Cell[TextData[{ "solidFrame[Polygon[l_], {f1i_, f2_}] := \nModule[{mp = Plus @@ \ l/Length[l], l1 = Append[l, First[l]], \n pl1i, pl2i, P = Partition},\n\ ", StyleBox["(* points arising from contracting the polygon \n in \ plane and in normal directions *)", "CodeComment"], "\n (* Lucb B removed duplicate line pl1i = (mp + f1i(# - mp))& /@ \ l1; *)\n l2 = f2 l1;\n {pl1i, pl2i} = ((mp + f1i(# - mp))& /@ #)& \ /@ {l1, l2};\n ", StyleBox["(* three new polygons *)", "CodeComment"], "\n {(Polygon[Join[#[[1]], Reverse[#[[2]]]]]& /@ Transpose[#])& /@ \n \ {{P[l1, 2, 1], P[pl1i, 2, 1]},\n {P[pl2i, 2, 1],P[l2, 2, 1]}, (* \ Lucb reversed this duplet *)\n { P[pl1i, 2, 1],P[pl2i, 2, 1]}\n \ }}]" }], "Input", GeneratedCell->False], Cell[CellGroupData[{ Cell[BoxData[ \(Partition[{1, 2, 3, 4}, 2, 1]\)], 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NotebookInformation[ EvaluationNotebook[]], "\", 2]], cubeFrameGraphics];\)\)], "Input"], Cell["\<\ We divide each of the quadrilaterals that form the cube into two triangles \ and extract the connectivity information of the starting complex.\ \>", "Text", GeneratedCell->False], Cell[TextData[{ StyleBox["(* the quadrilaterals forming the cube frame *)", "CodeComment"], "\npolys = Cases[cubeFrame, _Polygon, Infinity];\n", StyleBox["(* the vertices of cubeFrame *)", "CodeComment"], "\nvertices = Union[Level[Cases[cubeFrame, _Polygon, Infinity], {-2}]];" }], "Input", GeneratedCell->False], Cell[TextData[{ StyleBox["(* make triangles *)", "CodeComment"], "\nallTriangles = startTriangles = \nFlatten[Apply[{\[ScriptCapitalT][#1, \ #2, #3], \[ScriptCapitalT][#3, #4, #1]}&, \n Map[\[ScriptCapitalV], \ First /@ Map[Position[vertices, #][[1, 1]]&, \n \ polys, {-2}], {-1}], {1}]];" }], "Input", GeneratedCell->False], Cell[TextData[{ StyleBox["(* index vertices *)", "CodeComment"], "\nMapIndexed[(vertexCoordinates[\[ScriptCapitalV][#2[[1]]]] = #1)&, \ vertices];\nstartVertexCoordinates = DownValues[vertexCoordinates];" }], "Input", GeneratedCell->False], Cell[TextData[{ StyleBox["(* connectivity *)", "CodeComment"], "\nApply[Set[neighborVertices[#1], #2]&, {#[[1, 1]], \nUnion[Flatten[Last \ /@ #]]}& /@ Split[Sort[Flatten[\n Apply[{{#1, {#2, #3}}, {#2, {#1, #3}}, \ {#3, {#1, #2}}}&,\n allTriangles, {1}], 1]], #1[[1]] === #2[[1]]&], \ {1}];\nstartNeighborVertices = DownValues[neighborVertices]; " }], "Input", GeneratedCell->False], Cell["\<\ makeTrianglesAtVertexDefinitions; startTrianglesAtVertex = DownValues[trianglesAtVertex];\ \>", "Input", GeneratedCell->False], Cell[TextData[{ "The following function ", StyleBox["limitSurfaceVertexCoordinates", "MR"], " calculates the positions of the vertices in the limit of infinite \ subdivision." }], "Text", GeneratedCell->False], Cell["\<\ limitSurfaceVertexCoordinates[\[ScriptCapitalV][k_]] := Module[{vertices = neighborVertices[\[ScriptCapitalV][k]], n}, n = Length[vertices]; (3/\[Beta][n] vertexCoordinates[\[ScriptCapitalV][k]] + Plus @@ (vertexCoordinates /@ vertices))/(3/\[Beta][n] + n)]\ \>", "Input", GeneratedCell->False], Cell["\<\ limitSurfacePoints = limitSurfaceVertexCoordinates[\[ScriptCapitalV][#]]& /@ \ Range[144];\ \>", "Input", GeneratedCell->False], Cell["We start with 144 triangles.", "Text", GeneratedCell->False], Cell[BoxData[""], "Input"], Cell[BoxData[ \(<< Default3D`\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(UndividedCube\ = \ Show[Graphics3D[{SurfaceColor[Hue[0.12], Hue[0.12], 2.5], makeAllPolygons[]}], Boxed \[Rule] False]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.0855 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics3D %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % 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1 .472 0 r .25382 .4349 m .22122 .48995 L .23261 .43087 L F .989 .409 0 r .12488 .72803 m .17324 .8242 L .17191 .80359 L F 1 .42 0 r .12979 .71427 m .12488 .72803 L .17191 .80359 L F 1 .356 0 r .17237 .49231 m .1866 .42039 L .20938 .42542 L F .817 .282 0 r .1866 .42039 m .17237 .49231 L .14846 .49752 L F 1 .409 0 r .20938 .42542 m .23261 .43087 L .19714 .49045 L F 1 .422 0 r .23261 .43087 m .22122 .48995 L .19714 .49045 L F 1 .354 0 r .17237 .49231 m .20938 .42542 L .19714 .49045 L F .037 0 0 r .70848 .35405 m .69637 .37394 L .70795 .37015 L F .538 .208 0 r .50675 .19797 m .42923 .20031 L .43959 .17905 L F .543 .193 0 r .52727 .17601 m .50675 .19797 L .43959 .17905 L F .05 .156 0 r .69288 .83891 m .68308 .82856 L .69578 .80463 L F .194 .11 0 r .24555 .60544 m .27329 .63078 L .2504 .5738 L F .132 .108 0 r .24555 .60544 m .27354 .68903 L .27329 .63078 L F 1 1 0 r .86677 .82534 m .8381 .88613 L .8358 .83601 L F .968 1 0 r .8381 .88613 m .80317 .89152 L .8358 .83601 L F .563 .264 0 r 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.53655 .36983 L .53727 .36603 L F .53655 .36983 m .52406 .32085 L .53727 .36603 L F .53727 .36603 m .54059 .34574 L .53806 .36187 L F 1 .795 0 r .53503 .38792 m .54059 .38565 L .53727 .36603 L F .969 .583 0 r .53191 .41827 m .53727 .36603 L .52847 .42296 L F 0 0 0 r .53727 .36603 m .53806 .36187 L .52847 .42296 L F 1 .795 0 r .52962 .44057 m .54059 .38565 L .53503 .38792 L F 0 0 0 r .52847 .42296 m .53096 .40707 L .53436 .40759 L F .53096 .40707 m .54059 .34574 L .53436 .40759 L F 1 .628 0 r .54059 .38565 m .53852 .45746 L .55154 .40059 L F 1 .693 0 r .54059 .38565 m .52962 .44057 L .53852 .45746 L F 0 0 0 r .54442 .39495 m .53771 .40811 L .53436 .40759 L F .759 .619 0 r .52471 .42807 m .52847 .42296 L .53436 .40759 L F .49896 .48273 m .52847 .42296 L .52471 .42807 L F .969 .583 0 r .52962 .44057 m .53191 .41827 L .52847 .42296 L F 1 1 0 r .49694 .49637 m .52962 .44057 L .52847 .42296 L F .96 .749 0 r .49896 .48273 m .49694 .49637 L .50327 .48162 L F .49896 .48273 m .50327 .48162 L 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r .38088 .5695 m .3862 .5634 L .39789 .55972 L F .3283 .57639 m .3862 .5634 L .33818 .57526 L F .37 .315 0 r .3311 .58097 m .2843 .57701 L .32778 .59363 L F 0 0 0 r .3283 .57639 m .33818 .57526 L .31874 .58006 L F .207 0 0 r .29635 .57109 m .28604 .56781 L .3283 .57639 L F 0 .025 0 r .2843 .57701 m .28604 .56781 L .27756 .56511 L F .207 0 0 r .27749 .56796 m .28604 .56781 L .29635 .57109 L F 0 .025 0 r .27756 .56511 m .28604 .56781 L .26197 .53761 L F 1 1 0 r .30746 .87813 m .31729 .82488 L .32855 .86539 L F .31729 .82488 m .33792 .81232 L .32855 .86539 L F .831 .552 0 r .72476 .31616 m .73601 .27881 L .7327 .30312 L F 1 .321 0 r .20762 .54584 m .22122 .48995 L .23002 .5403 L F 1 .307 0 r .19714 .49045 m .22122 .48995 L .20762 .54584 L F .652 .1 0 r .84552 .68294 m .84267 .68421 L .82831 .66787 L F .87294 .70695 m .84267 .68421 L .84552 .68294 L F .764 .247 0 r .84058 .68513 m .84267 .68421 L .85277 .70418 L F 0 .019 0 r .84267 .68421 m .83389 .67234 L .82831 .66787 L F .444 0 0 r 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