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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.  "
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Cell[BoxData[
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                p\[LeftDoubleBracket]n\/2 + 
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                p\[LeftDoubleBracket]n\/2 + 
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                p\[LeftDoubleBracket]n\/2 + 
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Cell["Make a figure showing subdivision with this scheme", "Text"],

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Cell["Eigenvalues and eigenvectors for running example", "Subsubsection"],

Cell["\<\
Extract central portion of subdivision matrix and compute eigenvalues\
\>", "Text"],

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Cell[TextData[{
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Cell["Exact evaluation near extraordinary vertices", "Subsection"],

Cell[CellGroupData[{

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Cell[TextData[{
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  Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[TextData[{
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Cell[CellGroupData[{

Cell["Smoothness analysis at extraordinary vertices", "Section"],

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Cell["The characteristic map", "Subsection"],

Cell["\<\
Plots of the effect of the characteristic map on the running example\
\>", "Text"],

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Cell["See text", "Text"]
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Cell[CellGroupData[{

Cell[TextData[{
  "Sufficient conditions for ",
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Cell[TextData[{
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-6.25387, 0.0353317, 0.499771}, {{126.75, 232.188}, {66.75, 1.625}} -> \
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}, Open  ]],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  GraphicsArray  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Closed]]
}, Closed]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Verifying the smoothness of a subdivision scheme", "Section",
  CellTags->"SEC computing spectral"],

Cell[CellGroupData[{

Cell["Computing eigenvalues of circulant matrices", "Subsection"],

Cell[CellGroupData[{

Cell["Eigenvalues of circulant matrices", "Subsubsection",
  CellTags->"SUBSEC eigenvalues of circulant"],

Cell[TextData[{
  "Given matrix of generating functions ",
  Cell[BoxData[
      \(\((c\_ij[x])\)\)]],
  ", build ",
  Cell[BoxData[
      \(n\[Times]n\)]],
  " circulant matrix ",
  Cell[BoxData[
      \(\((C\_ij)\)\)]],
  " whose first column is coefficients of generating function"
}], "Text"],

Cell[BoxData[
    \(circulant[c_, 
        n_] := \[IndentingNewLine]With[{cc = 
            c /. {x\^k_ \[Rule] 
                  x\^Mod[k, n]}}, \[IndentingNewLine]With[{coef = 
              Table[Coefficient[cc, x, j], {j, 0, 
                  n - 1}]}, \[IndentingNewLine]Table[
            RotateRight[coef, i], {i, 0, n - 1}]]]\)], "Input",
  InitializationCell->True],

Cell["Example of circulant matrix", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(circulant[2 + x + x\^\(-1\), 6] // MatrixForm\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {"2", "1", "0", "0", "0", "1"},
            {"1", "2", "1", "0", "0", "0"},
            {"0", "1", "2", "1", "0", "0"},
            {"0", "0", "1", "2", "1", "0"},
            {"0", "0", "0", "1", "2", "1"},
            {"1", "0", "0", "0", "1", "2"}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell["Compute eigenvalues of circulant matrix", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Eigenvalues[circulant[2 + x + x\^\(-1\), 6]]\)], "Input"],

Cell[BoxData[
    \({0, 1, 1, 3, 3, 4}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Evaluate ",
  Cell[BoxData[
      \(2 + x + x\^\(-1\)\)]],
  " at ",
  Cell[BoxData[
      \({x \[Rule] \[Omega]\_n\^i}\)]],
  " for ",
  Cell[BoxData[
      \(i \[Equal] 0  \[Ellipsis]n - 1\)]],
  " where ",
  Cell[BoxData[
      \(n \[Equal] 6\)]],
  ".  Observe that ",
  Cell[BoxData[
      \(\[Omega]\_n\^i == 
        Cos[\(2  \[Pi]\ i\)\/n] + Sin[\(2  \[Pi]\ i\)\/n] \[ImaginaryI]\)]],
  "."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(Table[
          2 + x + x\^\(-1\) /. {x -> 
                Cos[\(2  \[Pi]\ i\)\/6] + 
                  Sin[\(2  \[Pi]\ i\)\/6] \[ImaginaryI]}, {i, 0, 6 - 1}] // 
        Simplify\) // Sort\)], "Input"],

Cell[BoxData[
    \({0, 1, 1, 3, 3, 4}\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Eigenvalues for block circulant matrices", "Subsubsection",
  CellTags->"SUBSEC eigenvalues of block circulant"],

Cell["<< LinearAlgebra`MatrixManipulation`", "Input",
  InitializationCell->True,
  CellTags->"S5.48.1"],

Cell[TextData[{
  "Given matrix of generating functions in ",
  Cell[BoxData[
      \(x\)]],
  ", construct block circulant matrix with consisting of ",
  Cell[BoxData[
      \(n\[Times]n\)]],
  " matrices"
}], "Text"],

Cell[BoxData[
    \(blockCirculant[C_, n_] := \[IndentingNewLine]BlockMatrix[
        Table[\[IndentingNewLine]circulant[
            C\[LeftDoubleBracket]i, j\[RightDoubleBracket], 
            n], {i, \(Dimensions[
                C]\)\[LeftDoubleBracket]1\[RightDoubleBracket]}, {j, \
\(Dimensions[C]\)\[LeftDoubleBracket]2\[RightDoubleBracket]}]]\)], "Input",
  InitializationCell->True],

Cell["Example", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{
      RowBox[{"blockCirculant", "[", 
        RowBox[{
          RowBox[{"(", GridBox[{
                {\(2 + x + x\^\(-1\)\), "1"},
                {"0", \(4 + x + x\^\(-1\)\)}
                }], ")"}], ",", "6"}], "]"}], "//", "MatrixForm"}]], "Input"],

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            {"0", "1", "2", "1", "0", "0", "0", "0", "1", "0", "0", "0"},
            {"0", "0", "1", "2", "1", "0", "0", "0", "0", "1", "0", "0"},
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\[Omega]\^i], coord[z\[LeftDoubleBracket]5\[RightDoubleBracket] \[Omega]\^i], 
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\[Omega]\^i], 
                    coord[z\[LeftDoubleBracket]6\[RightDoubleBracket] \
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Cell[CellGroupData[{

Cell["Display characteristic map for Catmull\[Hyphen]Clark", "Subsubsection"],

Cell[TextData[{
  "Define generating function matrix for CC scheme over three ring, vertex \
numbering is ",
  Cell[BoxData[
      \(10, 11, 20, 12, 21, 22, 30, 13, 31, 23, 32, 33\)]],
  "."
}], "Text"],

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                  "0", "0", "0", "0", "0"},
                {\(9\/16 + \(1\/64\) x + \(1\/64\) 
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                      x\^\(-1\)\), \(3\/32\), \(\(1\/64\) 
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                {\(\(1\/16\) x\), \(3\/8\), \(\(1\/16\) 
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                  "0", "0"},
                {\(1\/16\), \(3\/8\), \(1\/16\), \(1\/16\), \(3\/8\), \(1\/16\
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            GridBoxOptions->{ColumnAlignments->{Center}}]}]}], 
      StyleBox[";",
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  InitializationCell->True,
  CellTags->"EQN catmull clark gen fun"],

Cell[BoxData[
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            N[Cos[\(\(2\)  \(\[Pi]\)\(\ \)\)\/n] + 
                Sin[\(\(2\)  \(\[Pi]\)\(\ \)\)\/n] \[ImaginaryI]], ans, z, 
          c}, \[IndentingNewLine]ans = \(Eigensystem[
              ccFun3 /. {x \[Rule] \[Omega]}]\)\[LeftDoubleBracket]2, 
            1\[RightDoubleBracket]; \[IndentingNewLine]z = 
          ans\/ans\[LeftDoubleBracket]1\[RightDoubleBracket]; \
\[IndentingNewLine]c[x_] := {Re[x], Im[x]}; \[IndentingNewLine]Show[
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                    coord[z\[LeftDoubleBracket]1\[RightDoubleBracket] \
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                    coord[z\[LeftDoubleBracket]10\[RightDoubleBracket] \
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                    coord[z\[LeftDoubleBracket]12\[RightDoubleBracket] \
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                Line[{coord[
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          AspectRatio \[Rule] Automatic]]\)], "Input",
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}, Closed]],

Cell[CellGroupData[{

Cell["Display annulus for triangular schemes", "Subsubsection"],

Cell["\<\
This code constructs and displays the annulus (for triangular meshes) used in \
regularity analysis.\
\>", "Text"],

Cell[BoxData[
    \(triAnnulus[
        n_] := \[IndentingNewLine]Module[{\[Omega] = 
            N[Cos[\(\(2\)  \(\[Pi]\)\(\ \)\)\/n] + 
                Sin[\(\(2\)  \(\[Pi]\)\(\ \)\)\/n] \[ImaginaryI]], ans, 
          z}, \[IndentingNewLine]ans = \(Eigensystem[
              loopFun3 /. {x \[Rule] \[Omega]}]\)\[LeftDoubleBracket]2, 
            1\[RightDoubleBracket]; \[IndentingNewLine]z = 
          ans\/ans\[LeftDoubleBracket]1\[RightDoubleBracket]; \
\[IndentingNewLine]Show[
          Graphics[\[IndentingNewLine]{Table[\[IndentingNewLine]Polygon[{\
coord[z\[LeftDoubleBracket]1\[RightDoubleBracket] \[Omega]\^i], 
                    coord[z\[LeftDoubleBracket]3\[RightDoubleBracket] \
\[Omega]\^i], coord[z\[LeftDoubleBracket]2\[RightDoubleBracket] \[Omega]\^i], 
                    coord[z\[LeftDoubleBracket]3\[RightDoubleBracket] \
\[Omega]\^\(i + 1\)], 
                    coord[z\[LeftDoubleBracket]1\[RightDoubleBracket] \
\[Omega]\^\(i + 1\)]}], \[IndentingNewLine]{i, 0, 
                  n - 1}], \[IndentingNewLine]{GrayLevel[0.5], 
                Polygon[{coord[z\[LeftDoubleBracket]1\[RightDoubleBracket]], 
                    coord[z\[LeftDoubleBracket]3\[RightDoubleBracket]], 
                    coord[z\[LeftDoubleBracket]2\[RightDoubleBracket]], 
                    coord[z\[LeftDoubleBracket]1\[RightDoubleBracket] \
\[Omega]]}]}, \[IndentingNewLine]{GrayLevel[0], 
                Table[{\[IndentingNewLine]Line[{coord[0], 
                        coord[z\[LeftDoubleBracket]1\[RightDoubleBracket] \
\[Omega]\^i], coord[z\[LeftDoubleBracket]3\[RightDoubleBracket] \[Omega]\^i], 
                        coord[z\[LeftDoubleBracket]6\[RightDoubleBracket] \
\[Omega]\^i]}], \[IndentingNewLine]Line[{coord[
                          z\[LeftDoubleBracket]1\[RightDoubleBracket] \
\[Omega]\^\(i + 1\)], coord[
                          z\[LeftDoubleBracket]2\[RightDoubleBracket] \
\[Omega]\^i], 
                        coord[z\[LeftDoubleBracket]5\[RightDoubleBracket] \
\[Omega]\^i]}], \[IndentingNewLine]Line[{coord[
                          z\[LeftDoubleBracket]3\[RightDoubleBracket] \
\[Omega]\^\(i + 1\)], coord[
                          z\[LeftDoubleBracket]4\[RightDoubleBracket] \
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                          z\[LeftDoubleBracket]1\[RightDoubleBracket] \
\[Omega]\^i], coord[z\[LeftDoubleBracket]2\[RightDoubleBracket] \[Omega]\^i], 
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                          z\[LeftDoubleBracket]4\[RightDoubleBracket] \
\[Omega]\^i]}], \[IndentingNewLine]Line[{coord[
                          z\[LeftDoubleBracket]3\[RightDoubleBracket] \
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                        coord[z\[LeftDoubleBracket]5\[RightDoubleBracket] \
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                          z\[LeftDoubleBracket]1\[RightDoubleBracket] \
\[Omega]\^\(i + 1\)], coord[
                          z\[LeftDoubleBracket]1\[RightDoubleBracket] \
\[Omega]\^i]}], \[IndentingNewLine]Line[{coord[
                          z\[LeftDoubleBracket]3\[RightDoubleBracket] \
\[Omega]\^i], coord[z\[LeftDoubleBracket]2\[RightDoubleBracket] \[Omega]\^i], 
                        coord[z\[LeftDoubleBracket]3\[RightDoubleBracket] \
\[Omega]\^\(i + 1\)]}], \[IndentingNewLine]Line[{coord[
                          z\[LeftDoubleBracket]6\[RightDoubleBracket] \
\[Omega]\^i], coord[z\[LeftDoubleBracket]5\[RightDoubleBracket] \[Omega]\^i], 
                        coord[z\[LeftDoubleBracket]4\[RightDoubleBracket] \
\[Omega]\^i], 
                        coord[z\[LeftDoubleBracket]6\[RightDoubleBracket] \
\[Omega]\^\(i + 1\)]}]}, {i, 0, n - 1}]}}], 
          AspectRatio \[Rule] Automatic]]\)], "Input",
  InitializationCell->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(Show[GraphicsArray[Table[triAnnulus[i], {i, 3, 7}]]]\)], "Input",
  CellTags->"FIG annulus for triangular meshes"],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  GraphicsArray  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Extracting submeshes for regularity testing on Loop's scheme", \
"Subsubsection"],

Cell[TextData[{
  Cell[BoxData[
      \(submesh\)]],
  " defines the ",
  Cell[BoxData[
      \(4\[Times]4\)]],
  " submesh centered over the two gray triangles above."
}], "Text"],

Cell[BoxData[
    RowBox[{\(subMesh[x_]\), ":=", 
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(1\/x\), 
              "0", \(x\^2\), \(\(x\^2\ \((1 + 13\ x + x\^2)\)\)\/\(\((2 + 
                        x)\)\ \((1 + 2\ x)\)\)\)},
            {\(3\/\(1 + x\)\), "1", "x", \(\(3\ x\^2\)\/\(1 + x\)\)},
            {\(\(2 + 32\ x + 87\ x\^2 + 55\ x\^3 + 
                    7\ x\^4\)\/\(\((1 + x)\)\ \((2 + x)\)\ \((1 + 
                        2\ x)\)\ \((1 + 3\ x + x\^2)\)\)\), \(\(1 + 13\ x + 
                    x\^2\)\/\(\((2 + x)\)\ \((1 + 
                        2\ x)\)\)\), \(\(3\ x\)\/\(1 + 
                    x\)\), \(\(x\ \((1 + 13\ x + x\^2)\)\)\/\(\((2 + 
                        x)\)\ \((1 + 2\ x)\)\)\)},
            {\(\(x\ \((25 + 195\ x + 459\ x\^2 + 355\ x\^3 + 89\ x\^4 + 
                        5\ x\^5)\)\)\/\(\((1 + x)\)\ \((2 + x)\)\ \((1 + 
                        2\ x)\)\ \((1 + 3\ x + x\^2)\)\^2\)\), \(\(3\ x\ \((5 \
+ 23\ x + 5\ x\^2)\)\)\/\(\((2 + x)\)\ \((1 + 2\ x)\)\ \((1 + 3\ x + 
                        x\^2)\)\)\), \(\(x\ \((7 + 55\ x + 87\ x\^2 + 
                        32\ x\^3 + 2\ x\^4)\)\)\/\(\((1 + x)\)\ \((2 + 
                        x)\)\ \((1 + 2\ x)\)\ \((1 + 3\ x + 
                        x\^2)\)\)\), \(\(x\ \((2 + 32\ x + 87\ x\^2 + 
                        55\ x\^3 + 7\ x\^4)\)\)\/\(\((1 + x)\)\ \((2 + 
                        x)\)\ \((1 + 2\ x)\)\ \((1 + 3\ x + x\^2)\)\)\)}
            }], "\[NoBreak]", ")"}]}]], "Input",
  InitializationCell->True],

Cell[TextData[{
  Cell[BoxData[
      \(normMesh\)]],
  " separates ",
  Cell[BoxData[
      \(submesh\)]],
  " into real and imaginary parts using ",
  Cell[BoxData[
      \(x = \[Omega]\_n\)]],
  ".   Note that ",
  Cell[BoxData[
      \(normMesh\)]],
  " takes a few seconds to compute!"
}], "Text"],

Cell[BoxData[
    \(\(normMesh[
          n_] = \[IndentingNewLine]\(Map[coord, 
              subMesh[Cos[\(2  \[Pi]\)\/n] + \[ImaginaryI]\ Sin[\(2\ \
\[Pi]\)\/n]] // FullSimplify, {2}] // ComplexExpand\) // 
          FullSimplify;\)\)], "Input"],

Cell[TextData[{
  Cell[BoxData[
      \(displaySubmesh\)]],
  " display the ",
  Cell[BoxData[
      \(4\[Times]4\)]],
  " submesh"
}], "Text"],

Cell[BoxData[
    \(displaySubMesh[net_] := \[IndentingNewLine]Show[
        Graphics[{{GrayLevel[0.5], 
              Polygon[{net\[LeftDoubleBracket]2, 2\[RightDoubleBracket], 
                  net\[LeftDoubleBracket]3, 2\[RightDoubleBracket], 
                  net\[LeftDoubleBracket]3, 3\[RightDoubleBracket], 
                  net\[LeftDoubleBracket]2, 
                    3\[RightDoubleBracket]}]}, \[IndentingNewLine]Table[\
\[IndentingNewLine]Line[{net\[LeftDoubleBracket]i, j\[RightDoubleBracket], 
                  net\[LeftDoubleBracket]i + 1, j\[RightDoubleBracket], 
                  net\[LeftDoubleBracket]i + 1, j + 1\[RightDoubleBracket], 
                  net\[LeftDoubleBracket]i, j\[RightDoubleBracket], 
                  net\[LeftDoubleBracket]i, j + 1\[RightDoubleBracket], 
                  net\[LeftDoubleBracket]i + 1, 
                    j + 1\[RightDoubleBracket]}], \[IndentingNewLine]{i, \
\(Dimensions[net]\)\[LeftDoubleBracket]1\[RightDoubleBracket] - 
                  1}, {j, \(Dimensions[
                      net]\)\[LeftDoubleBracket]2\[RightDoubleBracket] - 
                  1}]\[IndentingNewLine]}], 
        AspectRatio \[Rule] Automatic]\)], "Input",
  InitializationCell->True],

Cell["Draw some example submeshes", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Show[
      GraphicsArray[
        Table[displaySubMesh[normMesh[i]], {i, 3, 7}]]]\)], "Input",
  CellTags->"FIG 3x3 submesh"],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  GraphicsArray  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Code for constructing limit meshes", "Subsubsection"],

Cell[TextData[{
  "The imaginary parts is divided by ",
  Cell[BoxData[
      \(Sin[\(2  \[Pi]\)\/n]\)]],
  ". Note that this is done symbolically and allows direct evaluation at ",
  Cell[BoxData[
      \(n = \[Infinity]\)]],
  ". (Be patient, it takes a while to do this symbolically!)"
}], "Text"],

Cell[BoxData[
    \(\(limitMesh[
          n_] = \[IndentingNewLine]\(Map[
              Function[x, coord[x]*{1, Csc[\(2  \[Pi]\)\/n]}], 
              subMesh[Cos[\(2  \[Pi]\)\/n] + \[ImaginaryI]\ Sin[\(2\ \
\[Pi]\)\/n]] // FullSimplify, {2}] // ComplexExpand\) // 
          FullSimplify;\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Show[
      GraphicsArray[{displaySubMesh[limitMesh[8]], 
          displaySubMesh[limitMesh[16]], displaySubMesh[limitMesh[32]], 
          displaySubMesh[limitMesh[\[Infinity]]]}]]\)], "Input",
  CellTags->"FIG limit meshes"],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  GraphicsArray  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
Code for testing the regularity of a uniform subdivision schemes on a given \
mesh\
\>", "Subsubsection",
  CellTags->"SUBSEC regularity test"],

Cell[TextData[{
  Cell[BoxData[
      \(jacobian[net]\)]],
  " takes a tensor product mesh in the plane and determines whether the \
corresponding surface is regular under the assumption that derivative schemes \
for the two mesh directions have non\[Hyphen]negative basis functions and \
satisfy the convex hull property.  ",
  Cell[BoxData[
      \(jacobian\)]],
  " returns an interval bounding the value of the Jacobian of the domain."
}], "Text"],

Cell[BoxData[
    RowBox[{\(jacobian[net_]\), ":=", "\[IndentingNewLine]", 
      RowBox[{"With", "[", 
        RowBox[{\({d = \(Dimensions[net]\)\[LeftDoubleBracket]{1, 
                  2}\[RightDoubleBracket]}\), ",", "\[IndentingNewLine]", 
          RowBox[{"With", "[", 
            
            RowBox[{\({\[IndentingNewLine]xs = 
                  Table[net\[LeftDoubleBracket]i + 1, j, 
                        1\[RightDoubleBracket] - 
                      net\[LeftDoubleBracket]i, j, 
                        1\[RightDoubleBracket], {i, 
                      d\[LeftDoubleBracket]1\[RightDoubleBracket] - 1}, {j, 
                      d\[LeftDoubleBracket]2\[RightDoubleBracket]}], \
\[IndentingNewLine]xt = 
                  Table[net\[LeftDoubleBracket]i, j + 1, 
                        1\[RightDoubleBracket] - 
                      net\[LeftDoubleBracket]i, j, 
                        1\[RightDoubleBracket], {i, 
                      d\[LeftDoubleBracket]1\[RightDoubleBracket]}, {j, 
                      d\[LeftDoubleBracket]2\[RightDoubleBracket] - 
                        1}], \[IndentingNewLine]ys = 
                  Table[net\[LeftDoubleBracket]i + 1, j, 
                        2\[RightDoubleBracket] - 
                      net\[LeftDoubleBracket]i, j, 
                        2\[RightDoubleBracket], {i, 
                      d\[LeftDoubleBracket]1\[RightDoubleBracket] - 1}, {j, 
                      d\[LeftDoubleBracket]2\[RightDoubleBracket]}], \
\[IndentingNewLine]yt = 
                  Table[net\[LeftDoubleBracket]i, j + 1, 
                        2\[RightDoubleBracket] - 
                      net\[LeftDoubleBracket]i, j, 
                        2\[RightDoubleBracket], {i, 
                      d\[LeftDoubleBracket]1\[RightDoubleBracket]}, {j, 
                      d\[LeftDoubleBracket]2\[RightDoubleBracket] - 1}]}\), 
              ",", "\[IndentingNewLine]", 
              RowBox[{"Det", "[", 
                RowBox[{"(", GridBox[{
                      {\(Interval[{Min[xs], Max[xs]}]\), \(Interval[{Min[xt], 
                            Max[xt]}]\)},
                      {\(Interval[{Min[ys], Max[ys]}]\), \(Interval[{Min[yt], 
                            Max[yt]}]\)}
                      }], ")"}], "]"}]}], "]"}]}], "]"}]}]], "Input",
  InitializationCell->True],

Cell[TextData[{
  Cell[BoxData[
      \(regular\)]],
  " used ",
  Cell[BoxData[
      \(jacobian\)]],
  " to test if ",
  Cell[BoxData[
      \(net\)]],
  " is regular. If the result is indeterminate, i.e. the interval bound on \
the Jacobain contains zero, then ",
  Cell[BoxData[
      \(net\)]],
  " is subdivided using the mask ",
  Cell[BoxData[
      \(s\)]],
  " and the test is applied a maximum of ",
  Cell[BoxData[
      \(level\)]],
  " times before failure is assumed.  (Note that this code is tailored to ",
  Cell[BoxData[
      \(4\[Times]4\)]],
  " input nets and ",
  Cell[BoxData[
      \(5\[Times]5\)]],
  " subdivision masks, i.e. Loop and Catmull\[Hyphen]Clark masks.)"
}], "Text"],

Cell[BoxData[
    \(regular[net_, s_, level_] := \[IndentingNewLine]If[level < 0, False, 
        If[jacobian[net] > 0, True, 
          False, \[IndentingNewLine]Module[{x, y, d = Dimensions[s], 
              refinedNet}, \[IndentingNewLine]refinedNet = 
              Transpose[
                CoefficientList[\[IndentingNewLine]Sum[
                        net\[LeftDoubleBracket]i, 
                            j\[RightDoubleBracket] \(x\^\(2  i - 2\)\) 
                          y\^\(2  j - 2\), {i, 
                          d\[LeftDoubleBracket]1\[RightDoubleBracket] - 
                            1}, {j, 
                          d\[LeftDoubleBracket]2\[RightDoubleBracket] - 
                            1}]*\[IndentingNewLine]Sum[
                        s\[LeftDoubleBracket]i, 
                            j\[RightDoubleBracket] \(x\^\(i - 1\)\) 
                          y\^\(j - 1\), {i, 
                          d\[LeftDoubleBracket]1\[RightDoubleBracket]}, {j, 
                          d\[LeftDoubleBracket]2\[RightDoubleBracket]}] + \(x\
\^\(2  d\[LeftDoubleBracket]1\[RightDoubleBracket] - 2\)\) 
                      y\^\(2  d\[LeftDoubleBracket]2\[RightDoubleBracket] - 2\
\), {x, y}], {3, 1, 2}]; \[IndentingNewLine]regular[
                refinedNet\[LeftDoubleBracket]
                  Range[d\[LeftDoubleBracket]1\[RightDoubleBracket] - 1, 
                    2  d\[LeftDoubleBracket]1\[RightDoubleBracket] - 3], 
                  Range[d\[LeftDoubleBracket]2\[RightDoubleBracket] - 1, 
                    2  d\[LeftDoubleBracket]2\[RightDoubleBracket] - 
                      3]\[RightDoubleBracket], s, 
                level - 1] && \[IndentingNewLine]regular[
                refinedNet\[LeftDoubleBracket]
                  Range[d\[LeftDoubleBracket]1\[RightDoubleBracket] - 1, 
                    2  d\[LeftDoubleBracket]1\[RightDoubleBracket] - 3], 
                  Range[d\[LeftDoubleBracket]2\[RightDoubleBracket], 
                    2  d\[LeftDoubleBracket]2\[RightDoubleBracket] - 
                      2]\[RightDoubleBracket], s, 
                level - 1] && \[IndentingNewLine]regular[
                refinedNet\[LeftDoubleBracket]
                  Range[d\[LeftDoubleBracket]1\[RightDoubleBracket], 
                    2  d\[LeftDoubleBracket]1\[RightDoubleBracket] - 2], 
                  Range[d\[LeftDoubleBracket]2\[RightDoubleBracket] - 1, 
                    2  d\[LeftDoubleBracket]2\[RightDoubleBracket] - 
                      3]\[RightDoubleBracket], s, 
                level - 1] && \[IndentingNewLine]regular[
                refinedNet\[LeftDoubleBracket]
                  Range[d\[LeftDoubleBracket]1\[RightDoubleBracket], 
                    2  d\[LeftDoubleBracket]1\[RightDoubleBracket] - 2], 
                  Range[d\[LeftDoubleBracket]2\[RightDoubleBracket], 
                    2  d\[LeftDoubleBracket]2\[RightDoubleBracket] - 
                      2]\[RightDoubleBracket], s, level - 1]]]]\)], "Input",
  InitializationCell->True]
}, Closed]],

Cell[CellGroupData[{

Cell["Code for carrying out tests on Loop's scheme", "Subsubsection",
  CellTags->"SUBSEC loop verification"],

Cell[TextData[{
  "Check that all valences from ",
  Cell[BoxData[
      \(3\)]],
  " to ",
  Cell[BoxData[
      \(31\)]],
  " are regular."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{"Do", "[", "\[IndentingNewLine]", 
      RowBox[{
        RowBox[{"Print", "[", 
          RowBox[{"regular", "[", 
            RowBox[{\(normMesh[N[i]]\), ",", 
              RowBox[{\(1\/16\), 
                RowBox[{"(", GridBox[{
                      {"1", "2", "1", "0", "0"},
                      {"2", "6", "6", "2", "0"},
                      {"1", "6", "10", "6", "1"},
                      {"0", "2", "6", "6", "2"},
                      {"0", "0", "1", "2", "1"}
                      }], ")"}]}], ",", "3"}], "]"}], "]"}], 
        ",", \({i, 3, 31}\)}], "]"}]], "Input"],

Cell[BoxData[
    \(True\)], "Print"],

Cell[BoxData[
    \(True\)], "Print"],

Cell[BoxData[
    \(True\)], "Print"],

Cell[BoxData[
    \(True\)], "Print"],

Cell[BoxData[
    \(True\)], "Print"],

Cell[BoxData[
    \(True\)], "Print"],

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(*******************************************************************
End of Mathematica Notebook file.
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