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Notebook[{

Cell[CellGroupData[{
Cell["Bonus material:  Subdivision of tensioned cubic meshes", "Chapter",
  CounterAssignments->{{"Chapter", 6}},
  CellTags->"CHA polyhedral"],

Cell[TextData[{
  StyleBox["OVERVIEW: ",
    FontWeight->"Bold"],
  "This notebook outlines a simple method for implementing both weighted \
averaging and creases in a single framework that can handle cubic meshes of \
arbitrary dimension.  The notebook includes code for linear subdivision of \
these meshes and weighted averaging based on tensions assigned to these \
meshes.  We have include many interesting meshes in this format including \
those that appear in figures 7.15-7.18 and figures 7.21-7.25.  In particular, \
Scott Schaefer has modeled the chess pieces that appear on the cover of the \
book in this form."
}], "Text"],

Cell[TextData[{
  StyleBox["COPYRIGHT ISSUES:",
    FontWeight->"Bold"],
  "  The authors reserve all copyrights associated with this work.  Any of \
the material appearing in these notebooks (such as the meshes) can be used \
and modified without restriction as long as the use is \
non\[Hyphen]commercial.  We simply ask that you acknowledge the authors when \
using material from these notebooks. For those readers interested in \
commercial use of the material in these notebooks, please contact \
jwarren@cs.rice.edu.  "
}], "Text"],

Cell[TextData[StyleBox["IGNORE THE SECTION NUMBERS BELOW, THEY SIMPLY SERVE \
AS AN OUTLINE FOR THE MATERIAL",
  FontWeight->"Bold",
  FontColor->RGBColor[1, 0, 0]]], "Text"],

Cell[CellGroupData[{

Cell["A topological representation for tensioned cubic meshes", "Section"],

Cell[TextData[{
  "A tensioned cubic mesh is a triple ",
  Cell[BoxData[
      \({M, p, \[CapitalSigma]}\)]],
  " where ",
  Cell[BoxData[
      \(M\)]],
  " is a list of topological cubes, ",
  Cell[BoxData[
      \(p\)]],
  " is a list of vertex positions and ",
  Cell[BoxData[
      \(\[CapitalSigma]\)]],
  " is a list whose entries are themselves a list of tensions assigned to \
each dimension of the corresponding cube in ",
  Cell[BoxData[
      \(M\)]],
  ".  As opposed to the flat representation used in the rest of chapter 7, \
the topological cubes comprising ",
  Cell[BoxData[
      \(M\)]],
  " are recursively defined as follows:  a topological ",
  Cell[BoxData[
      \(0\)]],
  "\[Hyphen]cube is simply a vertex index, a topological ",
  Cell[BoxData[
      \(d\)]],
  "\[Hyphen]cube is a pair of topological ",
  Cell[BoxData[
      \(\((d - 1)\)\)]],
  "\[Hyphen]cubes comprising two opposite faces of the ",
  Cell[BoxData[
      \(d\)]],
  "\[Hyphen]cube."
}], "Text"],

Cell[TextData[{
  "For example, an edge connecting the vertex ",
  Cell[BoxData[
      \(1\)]],
  " to vertex ",
  Cell[BoxData[
      \(2\)]],
  " is the list ",
  Cell[BoxData[
      \({1, 2}\)]],
  ".  The quad face connecting the vertices ",
  Cell[BoxData[
      \({1, 2, 3, 4}\)]],
  " has the form ",
  Cell[BoxData[
      \({{1, 2}, {4, 3}}\)]],
  ".  A topological cube might have the form ",
  Cell[BoxData[
      \({{{1, 2}, {3, 4}}, {{5, 6}, {7, 8}}}\)]],
  ".  Each entry in ",
  Cell[BoxData[
      \(\[CapitalSigma]\)]],
  " corresponding to a ",
  Cell[BoxData[
      \(d\)]],
  "\[Hyphen]cube in ",
  Cell[BoxData[
      \(T\)]],
  " is a list of ",
  Cell[BoxData[
      \(d\)]],
  " tensions as specified in section 7.2.3."
}], "Text"],

Cell[TextData[{
  "Below is a definition of a profile curve for a pawn as a tensioned cubic \
mesh.  Note that ",
  Cell[BoxData[
      \(M\)]],
  " consists of edges and vertices.  "
}], "Text"],

Cell[BoxData[
    \(\(pawn\  = \ {{{2, 1}, 
            2, {3, 2}, {4, 3}, {5, 4}, {6, 5}, {7, 6}, {8, 7}, {9, 8}, {10, 
              9}, {11, 10}, 11, 10, {12, 11}, {13, 12}, 
            13, {14, 13}, {15, 14}, {16, 15}, 4, 
            1}, {{0.`, 0.`}, {1.4814816`, \(-0.014306068`\)}, {1.5740738`, 
              0.39999962`}, {1.3888893`, 0.47500038`}, {1.4506168`, 
              0.6000004`}, {1.4814816`, 0.9250002`}, {1.0802469`, 
              1.0749998`}, {1.1111107`, 1.6999998`}, {0.6172838`, 
              1.8999996`}, {0.4320984`, 3.`}, {1.0802469`, 3.`}, {1.0802469`, 
              3.475`}, {0.4320984`, 3.625`}, {0.80246925`, 
              3.9499998`}, {0.7407408`, 4.7`}, {0.`, 
              5.1359086`}}, {{1}, {}, {1}, {1}, {1}, {1}, {1}, {1}, {1}, {1}, \
{1}, {}, {}, {1}, {1}, {}, {1}, {1}, {1}, {}, {}}};\)\)], "Input",
  InitializationCell->True],

Cell["Here is a plot of this profile curve", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(curvePlot[pawn, 0.03, Frame \[Rule] True, 
      AspectRatio \[Rule] Automatic, 
      PlotRange \[Rule] {{\(-0.01\), 2}, Automatic}]\)], "Input"],

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Cell["Drawing functions for cubic meshes.", "Subsubsection"],

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Cell["\<\
Linear subdivision and weighted averaging for tensioned cubic meshes\
\>", "Section"],

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recurrence ",
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              verts[currIndex] = v; idx[v] = currIndex, 
              idx[v]]]; \[IndentingNewLine]\[IndentingNewLine] (*\ 
          recursive\ helper\ function\ that\ splits\ a\ single\ cubical\ cell\
\ and\ returns\ a\ list\ of\ 2\^d\ subcells\ *) \[IndentingNewLine]splitCube[
            C_] := With[{d = Length[Dimensions[C]]}, \[IndentingNewLine]If[
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                        middle\[LeftDoubleBracket]
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\[IndentingNewLine] (*\ \ apply\ splitCell\ to\ every\ cell\ in\ M, \ 
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Cell[TextData[{
  Cell[BoxData[
      \(weightCellAverage[{M, \[CapitalSigma], p}]\)]],
  " applies weighted cell averaging to a tensioned cubic mesh.  One of the \
main advangtages of representing the topological cubes in the mesh \
recursively is that weighted averaging is particular simple (see the helper \
function ",
  Cell[BoxData[
      \(weightedCentroid\)]],
  ").  Note that ",
  Cell[BoxData[
      \(weightedCellAverage\)]],
  " also handles embedded creases."
}], "Text"],

Cell[BoxData[
    \(weightedCellAverage[{M_, p_, \[CapitalSigma]_}] := 
      Module[{weightedCentroid, flatM = Map[Flatten, Map[List, M]], 
          newp = 0*p, \ val = Table[0, {Length[p]}], 
          d = Table[\[Infinity], {Length[
                  p]}]}, \[IndentingNewLine]\[IndentingNewLine]\
weightedCentroid = 
          Function[{C, \[Sigma]}, \[IndentingNewLine]If[
              Length[\[Sigma]] \[Equal] 0, 
              p\[LeftDoubleBracket]
                C\[RightDoubleBracket], \
\[IndentingNewLine]\(First[\[Sigma]]*weightedCentroid[C\[LeftDoubleBracket]1\
\[RightDoubleBracket], Rest[\[Sigma]]] + weightedCentroid[C\
\[LeftDoubleBracket]2\[RightDoubleBracket], Rest[\[Sigma]]]\)\/\(First[\
\[Sigma]] + 1\)]]; \[IndentingNewLine]\[IndentingNewLine] (*\ 
          compute\ table\ of\ vertex\ valences\ and\ dimensions\ \
*) \[IndentingNewLine]Do[
          With[{dim = 
                Length[Dimensions[
                    M\[LeftDoubleBracket]
                      l\[RightDoubleBracket]]]}, \[IndentingNewLine]Do[
              With[{idx = 
                    flatM\[LeftDoubleBracket]l, 
                      i\[RightDoubleBracket]}, \[IndentingNewLine]If[
                  dim < d\[LeftDoubleBracket]
                      idx\[RightDoubleBracket], \[IndentingNewLine]d\
\[LeftDoubleBracket]idx\[RightDoubleBracket] = dim; 
                  val\[LeftDoubleBracket]idx\[RightDoubleBracket] = 
                    1, \[IndentingNewLine]If[
                    dim \[Equal] 
                      d\[LeftDoubleBracket]
                        idx\[RightDoubleBracket], \(++val\[LeftDoubleBracket]
                        idx\[RightDoubleBracket]\)]]], \[IndentingNewLine]{i, 
                Length[flatM\[LeftDoubleBracket]
                    l\[RightDoubleBracket]]}]], \[IndentingNewLine]{l, 
            Length[M]}]; \[IndentingNewLine]\[IndentingNewLine] (*\ 
          apply\ smoothing\ mask\ to\ each\ cell\ and\ accumulate\ results\
*) \[IndentingNewLine]Do[
          With[{dim = 
                Length[Dimensions[
                    M\[LeftDoubleBracket]l\[RightDoubleBracket]]], 
              cent = weightedCentroid[
                  M\[LeftDoubleBracket]
                    l\[RightDoubleBracket], \[CapitalSigma]\
\[LeftDoubleBracket]l\[RightDoubleBracket]]}, \[IndentingNewLine]Do[
              With[{idx = 
                    flatM\[LeftDoubleBracket]l, 
                      i\[RightDoubleBracket]}, \[IndentingNewLine]If[
                  dim \[Equal] d\[LeftDoubleBracket]idx\[RightDoubleBracket], 
                  newp\[LeftDoubleBracket]idx\[RightDoubleBracket] += 
                    cent\/val\[LeftDoubleBracket]idx\[RightDoubleBracket]]], \
\[IndentingNewLine]{i, 
                Length[flatM\[LeftDoubleBracket]
                    l\[RightDoubleBracket]]}]], \[IndentingNewLine]{l, 
            Length[M]}]; \[IndentingNewLine]\[IndentingNewLine]{M, 
          newp, \[CapitalSigma]}]\)], "Input",
  InitializationCell->True],

Cell["An examples of using weighted cell averaging in 2D", "Text"],

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\>"],
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Cell[BoxData[
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      False,
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Cell[CellGroupData[{

Cell["\<\
Interesting examples of tensioned cubic meshes under subdivision\
\>", "Section"],

Cell[CellGroupData[{

Cell["Man\[Hyphen]made shapes", "Subsubsection"],

Cell["\<\
Here is a definition of the classic mechanical part. Note the \
cross\[Hyphen]section of the shaft is not circular (but could be made so if \
the tensions were adjust properly).  This model is the only \
non\[Hyphen]original model in this notebook and is probably due to Tony \
DeRose's old research group at the University of Washington.\
\>", "Text"],

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