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Cell[CellGroupData[{
Cell["Averaging Schemes for Polyhedral Meshes", "Chapter",
  CounterAssignments->{{"Chapter", 6}},
  CellTags->"CHA polyhedral"],

Cell[TextData[{
  "To evaluate a particular cell, simply hit \"Shift+Enter\" with the cursor \
positioned in that cell.  If ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " asks to evaluate initialization cell, click \"Yes\".  See the ",
  StyleBox["Mathematica ",
    FontSlant->"Italic"],
  "Help browser or wolfram.com for more information on using ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "."
}], "Text"],

Cell["\<\
This notebook and those for the other seven chapters use a common stylesheet \
that has been imported into the notebook.  This style sheet supports \
chapters, section, subsections, numbered equations, numbered figures and \
citations.  A separate copy of the stylesheet can be download from the same \
page where this notebook was downloaded.  \
\>", "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 polyhedral meshes in \
chapter 7) 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[CellGroupData[{

Cell["Helper functions from previous chapters", "Subsubsection"],

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Cell[CellGroupData[{

Cell["Topological properties of polyhedral meshes", "Section"],

Cell[CellGroupData[{

Cell["Polyhedral meshes", "Subsection"],

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Cell["Definition and display of monochromatic polyhedral meshes", \
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Cell[CellGroupData[{

Cell["Definition and display of color meshes", "Subsubsection"],

Cell[TextData[{
  "Example of mesh where vertex information includes positions and RGB \
values.  The meshes are of the form ",
  Cell[BoxData[
      \({T, p}\)]],
  " where ",
  Cell[BoxData[
      \(p\)]],
  " is a six\[Hyphen]tuple including three coordinates of position and three \
coordinates of vertex color.  Note that subdividing the color information \
leads to smooth colors over the mesh.  Note that ",
  Cell[BoxData[
      \(colorRing\)]],
  " and ",
  Cell[BoxData[
      \(colorAxe\)]],
  " include creases edges and vertices.  See section 7.2.4 for details."
}], "Text"],

Cell[BoxData[{
    \(\(\(colorCube = {{{1, 4, 3, 2}, {1, 2, 6, 5}, {2, 3, 7, 6}, {3, 4, 8, 
              7}, {4, 1, 5, 8}, {5, 6, 7, 8}}, {{0, 0, 0, 1, 0, 0}, {1, 0, 0, 
              1, 0, 0}, {1, 1, 0, 1, 0, 0}, {0, 1, 0, 1, 0, 0}, {0, 0, 1, 0, 
              1, 0}, {1, 0, 1, 0, 1, 0}, {1, 1, 1, 0, 1, 0}, {0, 1, 1, 0, 1, 
              0}}};\)\(\n\)
    \)\), "\[IndentingNewLine]", 
    \(\(\(colorRing = {{{5, 6, 2, 1}, {6, 7, 3, 2}, {7, 8, 4, 3}, {8, 5, 1, 
              4}, {13, 9, 10, 14}, {14, 10, 11, 15}, {15, 11, 12, 16}, {16, 
              12, 9, 13}, {2, 3, 11, 10}, {3, 4, 12, 11}, {4, 1, 9, 12}, {6, 
              5, 13, 14}, {7, 6, 14, 15}, {8, 7, 15, 16}, {5, 8, 16, 13}, {1, 
              20, 17, 9}, {17, 18, 10, 9}, {18, 19, 2, 10}, {19, 20, 1, 
              2}, {26, 25, 21, 22}, {27, 26, 22, 23}, {28, 27, 23, 24}, {25, 
              28, 24, 21}, {24, 23, 22, 21}, {17, 18}, {18, 19}, {19, 
              20}, {20, 17}, {25, 26}, {26, 27}, {27, 28}, {28, 
              25}, {24}, {23}, {22}, {21}}, {{\(-3\), 3, 1, 1, 1, 0}, {3, 3, 
              1, 1, 1, 0}, {3, \(-3\), 1, 1, 1, 0}, {\(-3\), \(-3\), 1, 1, 1, 
              0}, {\(-2.5\), 2.5, 1, 1, 1, 0}, {2.5, 2.5, 1, 1, 1, 
              0}, {2.5, \(-2.5\), 1, 1, 1, 0}, {\(-2.5\), \(-2.5\), 1, 1, 1, 
              0}, {\(-3\), 3, \(-1\), 1, 1, 0}, {3, 3, \(-1\), 1, 1, 
              0}, {3, \(-3\), \(-1\), 1, 1, 0}, {\(-3\), \(-3\), \(-1\), 1, 
              1, 0}, {\(-2.5\), 2.5, \(-1\), 1, 1, 0}, {2.5, 2.5, \(-1\), 1, 
              1, 0}, {2.5, \(-2.5\), \(-1\), 1, 1, 
              0}, {\(-2.5\), \(-2.5\), \(-1\), 1, 1, 0}, {\(- .5\), 
              3, \(- .5\), 1, 1, 0}, { .5, 3, \(- .5\), 1, 1, 0}, { .5, 
              3,  .5, 1, 1, 0}, {\(- .5\), 3,  .5, 1, 1, 0}, {\(-1\), 
              4, \(-1\), 1, 1, 1}, {1, 4, \(-1\), 1, 1, 1}, {1, 4, 1, 1, 1, 
              1}, {\(-1\), 4, 1, 1, 1, 1}, {\(- .5\), 3, \(- .5\), 1, 1, 
              1}, { .5, 3, \(- .5\), 1, 1, 1}, { .5, 3,  .5, 1, 1, 
              1}, {\(- .5\), 3,  .5, 1, 1, 1}}};\)\(\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
    \(\(colorAxe = {{{1, 2, 3, 10}, {8, 9, 10, 3}, {2, 3, 8, 9}, {3, 4, 7, 
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              14}, {5, 4, 14, 15}, {21, 30, 23, 22}, {28, 23, 30, 29}, {22, 
              29, 28, 23}, {23, 28, 27, 24}, {24, 27, 26, 
              25}, {22}, {29}, {22, 21}, {21, 30}, {30, 29}, {22, 23}, {23, 
              24}, {24, 25}, {25, 26}, {26, 27}, {27, 28}, {28, 
              29}, {25}, {26}, {31, 32, 33, 40}, {38, 39, 40, 33}, {32, 33, 
              38, 39}, {33, 34, 37, 38}, {34, 35, 36, 37}, {32}, {39}, {32, 
              31}, {31, 40}, {40, 39}, {32, 33}, {33, 34}, {34, 35}, {35, 
              36}, {36, 37}, {37, 38}, {38, 39}, {35}, {36}, {29, 39, 38, 
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              44}, {44, 41}, {45, 46}, {46, 47}, {47, 48}, {48, 45}, {45, 41, 
              42, 46}, {46, 42, 43, 47}, {47, 43, 44, 48}, {48, 44, 41, 
              45}, {48, 45, 46, 47}, {41, 44, 43, 42}}, {{\(-4.0\), \(-2.0\), 
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              0.2,  .5,  .5,  .5}, {\(-3.0\), 3.0, 
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.5}, {\(-2.0\), \(-1.0\), \(-0.3\),  .5,  .5,  .5}, {0.0, \(-1.0\), \(-0.5\), \
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              1.0, \(-0.3\),  .5,  .5,  .5}, {\(-3.1\), 
              2.0, \(-0.2\),  .5,  .5,  .5}, {\(-3.0\), 3.0, 
              0.0,  .5,  .5,  .5}, {\(-4.0\), 2.0, 
              0.0,  .5,  .5,  .5}, {4.0, \(-2.0\), 
              0.0,  .5,  .5,  .5}, {3.0, \(-3.0\), 
              0.0,  .5,  .5,  .5}, {3.1, \(-2.0\), 
              0.2,  .5,  .5,  .5}, {2.0, \(-1.0\), 
              0.3,  .5,  .5,  .5}, {\(-0.0\), \(-1.0\), 
              0.5,  .5,  .5,  .5}, {\(-0.0\), 1.0, 0.5,  .5,  .5,  .5}, {2.0, 
              1.0, 0.3,  .5,  .5,  .5}, {3.1, 2.0, 0.2,  .5,  .5,  .5}, {3.0, 
              3.0, 0.0,  .5,  .5,  .5}, {4.0, 2.0, 
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{2.0, \(-1.0\), \(-0.3\),  .5,  .5,  .5}, {\(-0.0\), \(-1.0\), \(-0.5\),  .5, \
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              2.0, \(-0.2\),  .5,  .5,  .5}, {3.0, 3.0, 
              0.0,  .5,  .5,  .5}, {4.0, 2.0, 
              0.0,  .5,  .5,  .5}, {\(- .75\), 2, \(- .75\),  .5,  .25, 
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              0}, {\(- .75\), \(-10\), \(- .75\),  .5,  .25, 
              0}, { .75, \(-10\), \(- .75\),  .5,  .25, 0}, { .75, \(-10\), 
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  InitializationCell->True],

Cell[TextData[{
  Cell[BoxData[
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  " takes mesh whose vertex information includes an RGB value and rendering \
the mesh using the average of the RGB values at the corner of each polygonal \
face."
}], "Text"],

Cell[BoxData[{
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    \(\(\(Protect[RGBColor];\)\(\[IndentingNewLine]\)
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                RGBColor[\(1\/4\) 
                    Apply[Plus, 
                      G\[LeftDoubleBracket]T\[LeftDoubleBracket]
                          i\[RightDoubleBracket], {4, 5, 
                          6}\[RightDoubleBracket]]]], \
\[IndentingNewLine]Polygon[
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                    3}\[RightDoubleBracket]]}, {i, Length[T]}]], 
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Cell["Examples", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
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Cell["Orientation testing for polyhedral meshes", "Subsubsection"],

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  " draws front (counterclockwise) faces white and back (clockwise) faces as \
black.  Note that all of the meshes defined above are oriented so that \
counterclockwise faces point outward."
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