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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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Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics3D  \[SkeletonIndicator]\),
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}, Closed]],

Cell["Here is simple model of a sword", "Text"],

Cell[BoxData[
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                8}, {4, 9}}, 1, 6, 7, 
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                17}}, {{14, 13}, {17, 18}}, 10, 15, 16, 
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              15}, {15, 16}, {16, 17}, {17, 18}, {18, 
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                26}, {21, 22}}, {{19, 20}, {22, 21}}, {{26, 25}, {23, 
                24}}, {19, 22}, {22, 26}, {26, 23}, {23, 
              19}, {{28, 21}, {27, 20}}, {{30, 27}, {24, 20}}, {{24, 
                25}, {30, 29}}, {{25, 21}, {29, 28}}, {27, 28}, {28, 
              29}, {29, 30}, {30, 
              27}, {{27, 31}, {28, 32}}, {{28, 32}, {29, 33}}, {{29, 
                33}, {30, 34}}, {{30, 34}, {27, 31}}, {{34, 33}, {31, 
                32}}, {31, 32}, {32, 33}, {33, 34}, {34, 31}}, {{\(-1.`\), 
              0.`, 0.`}, {\(-1.`\), 13.5`, 0.`}, {\(-1.`\), 15.`, 0.`}, {0.`, 
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00=O6`05Fa]O6elKGa]K6`03Ga/01E/KGa]O6elKFa/00elK00=K6`00Ool0:Goo00<007ooOol04Woo
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5Goo001TOol00`00Oomoo`0GOol00`00Oomoo`0UOol00`00?[hn_@02?[d01Cjn?[dn_Cjm?[h00cjm
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?[h005lK019O6`03001oogoo02Qoo`03001oogoo019oo`03001oogoo01=oo`00Igoo00<007ooOol0
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06Qoo`03001oogoo01Moo`03001oogoo02Eoo`03000n_Sjm008n_@05?[hn_Cjm?[dn_P03?[d00`00
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6`03Ga/01E/KGa]O6elKFa/00elK00<007ooOol0:7oo00<007ooOol04Woo00<007ooOol047oo001Z
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0elK00EK6elKGa]O6e/K00=O6`03001oogoo02Qoo`03001oogoo015oo`03001oogoo00moo`00K7oo
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019O6`03001oogoo02Qoo`03001oogoo015oo`03001oogoo00ioo`00KGoo00<007ooOol05goo00<0
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001oogoo02Qoo`03001oogoo015oo`03001oogoo00Uoo`00Lgoo00<007ooOol05Woo00<007ooOol0
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ogoo02Eoo`0@000n_D;N?[e2gSjm@]hn_D;N?[e2gSjm@]h005lKFa/3Ga/01E/KGa/005lKFa/00elK
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0`00Oomoo`0UOol01000?]hngS[N0ckN00LjgSkN?]hngS[N?]h0000=Ga/00`00Oomoo`0POol40008
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gSkN?]hjgP03?]h00c[N?]h0000<Ga/00`00Oomoo`0HOol4000@Ool00`00Oomoo`0:Ool10001Ool1
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>]hngS[N?]hjgSkN>]h005/K0elK00IK6elKGa]O6e/K000?Ool3000NOol00`00Oomoo`07Ool10001
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D_aFo5;kE_aBo00300001E;lEaaBo5LMD_`00P0000MG7E;l001G6e/LEa]K7002000WOol00`00Oomo
o`0BOol00`00Oomoo`1ROol000eoo`03001oogoo01Qoo`03001oogoo00Aoo`08000XK2a/:6`/K000
>>hhcP800004?A0i3cd?0004?@l300000cd??A0m3`02?@l00cd@0000000>00000dii001BV@020002
D]/300000e;lE_aBo002D_`300000e;lE_aBo002D_`300001e;lEaaBo5<L001K75LL008002Uoo`03
001oogoo01=oo`03001oogoo065oo`003Woo00<007ooOol067oo00<007ooOol00goo00D002a/:6`/
K2Q[00<00008>@lm43T??A0i3cd@000m3`800007?@m1<3d?@C0m3d4`?@l00P0000=1DCe@@E400T5A
00<mD0000000100000E>NTiiCWY>NDij00800009D]]Fg000001Bo5KlD_]Fo5;l00<00006D_aG75;l
EaeBo5LL0P0000MBo5LMD_aG7E;lEaeBo003000/Ool00`00Oomoo`0COol00`00Oomoo`1POol000mo
o`03001oogoo01Qoo`09001oogooOol002a/:6`/K2Q/0080008m3`06?A0i3cd??@lm43T?0`000cd?
00<m43d?00000P0000=1D@000000200000E>NTiiCWY>NDij00800003C]aBg00000800005E_aBo5;l
D_aFo00200000e;lE_aBo002D_`02UKlD_`00000EaaBo5<LD_aG75;l0`000elK00<007ooOol0:Woo
00<007ooOol04goo00<007ooOol0Ggoo000@Ool00`00Oomoo`0HOol01P00Ool00000:6`/K0800008
=?0003T??A0i3cd@>@lm40800003?E51DSeA00T00003@KA5]D6d00@00003CWY>N@0000<00003D]a>
fe;L00<00005D_e>g000D_aFo00300001ULLD_aG7E;lEaaBo0<00006D_aG7E;lEaeBo5LM0P0000EK
6`00Ga]O6e/K009O6`03001oogoo02Yoo`03001oogoo019oo`03001oogoo05moo`004Goo00<007oo
Ool067oo0P0000TXK000000e43C`=A0003d@>@l00Sd?0P000SeA00A1DCeA?E4mD@8000=5]003@KA5
]00000<00003CWU>NP0000<00004C]]>g4kLD]`300001U;lC_a>o4klD_a>o0800005E_aBo5;lD_aF
o002D_`200001U;lDaaBo5LLD_aC70<000AO6`03001O6elK00=O6`03001oogoo02Yoo`03001oogoo
019oo`03001oogoo05ioo`004Woo00<007ooOol05goo0`0000Pdl3TA=?0e43C`>A4003d@0P0000/m
DD5B?E51DSeA@E8mD@00@KA5]D6d00@00003BST0000000<00005D]a>fe;LC]]Bg00300001U;mC]aB
oDkLD_e>g0800007D_e>o000D_aG75;lEad00`0000EG7E;lEaeBo5LM00<0009O6`03001K6elK009O
6`05Fa/005lKGa]K6`02Ga/00`00Oomoo`0ZOol00`00Oomoo`0BOol00`00Oomoo`1MOol001=oo`03
001oogoo01Eoo`8000089?8003D@=?0e43C`=A0dl0@0008mD@03@E4mDCeA00H00003BSQ:>DXh00<0
0003C]aBg4kK009>g005D]`00000001Bo003C_`02U;l00000000D_e>o4kmC_aBo@000U;l0P0000EC
75;lEaaBo5<L00<00003Ga/005lK00=O6`03001O6elK00=O6`03001O6elK00=O6`03001oogoo02Uo
o`03001oogoo01=oo`03001oogoo05aoo`0057oo00<007ooOol057oo00P002Sc9?8Xl`00=A0dl3TA
0P0000@mU46E?I@mU0L00003BSU6>4Xi00@00006BYX004kKD]a>fe;L0`0000EBoDkLD_e>g5;m00<0
0008D_e>o5;mC_aBoDklD_e>o0<00005D_aG7E;lEaeBo0020003Ga/01E/KGa/005lKFa/00UlK00<0
05/KGa/00UlK00EK6`00Ga]O6e/K009O6`03001oogoo02Uoo`03001oogoo01=oo`03001oogoo05]o
o`005Goo00<007ooOol04Woo00L002Cb9?8XlbCb000dl0020009?I@00`00BSU:>0040005BY/00`00
D]`00002000015;lC_a>o4kl100000MBoDklC_e>o5;mC_a>o@0400001DkmD_d005<LD_`00`0000<n
_Cjn00001ElK00<005lKGa/00UlK00<005lKGa/00elK00<005lKGa/00UlK00<007ooOol0:Woo00<0
07ooOol04goo00<007ooOol0FWoo000FOol00`00Oomoo`0AOol2000012Sc9?8XlbCb1@000cfD00=1
UCfD0000100000U:VTZKBYY:VdZJBY]:VTZKBYX00P0000ABoDkLD_e>g0<00007C_aBoDklD_e>o5;m
C_`0100000MBoDklD_e>o5;mC_aBo@0200001cjm@]h004;N?[d005/K00=O6`05Fa]O6`00Ga]K6`02
Ga/00`00Fa]O6`02Ga/01e/K001O6elKFa]O6`0002aoo`03001oogoo019oo`03001oogoo05Yoo`00
5goo00<007ooOol047oo00<000BE00000RCb00<003hI?QT00chI1P001DZK1P000dZl00<004kl0000
0P0000M>o5;mC_a>oDklD_e>o00300001DklC_e>oE;mC_`00Tkm0`0000<n_Cjn00000Sjm00Dn_P00
?[dn_@0000IO6`03001O6elK009O6`03001O6elK009O6`03001O6elK00=O6`03001oogoo02Yoo`03
001oogoo019oo`03001oogoo05Uoo`0067oo00<007ooOol03goo00D000BD000TlP00008n6@05?QPn
6ChI?QTn6002?QT00`00BYX0000500001d[LA[a:gDZlB]a6_4[M00@00006C_aBoDklD_e>o5;m1000
00M>o5;mC_aBoDklD_e>o00300003cjm@]h004;N?[e2gP00@]hn_D;N0012gSjm001K6`03Ga/01E/K
Ga/005lKFa/00UlK00<005/KGa/00UlK00L005lKGa]O6e/KGa/0000/Ool00`00Oomoo`0BOol00`00
Oomoo`1HOol001Uoo`03001oogoo00moo`030004U00000`000U:_0@000=:g@03001>oDkl00@00006
D_e>o4kmC_eBoDkl0Tkm0`000Woo00D003jm?[dn_P00008n_@06?[h003jm?[dn_P000Sjm00<005lK
Ga/015lK00D005lKGa]O6`0000EO6`03001O6elK00=O6`03001oogoo02Yoo`03001oogoo019oo`03
001oogoo05Moo`006Woo00<007ooOol03goo0P00014jG3IK>U`jFcYL=U/jG3YK>U`004[MB[a:g4Jl
B]e:_4[L00@00005A]a:gD[MB]e6g003B]d2000024klD_e>o5;mC_aBoDklD_d40006Ool04@00@]hn
_D;N0012gSjm@]h004;N?[e2gP00@]hn_@00Fa/00elK00UK6elK001O6e/KGa/005lKFa/00elK00L0
05lKGa]O6e/KGa/0000[Ool00`00Oomoo`0COol00`00Oomoo`1FOol001]oo`03001oogoo00moo`H0
0003>U`jFcYL008jG003001:_4Zl00@000M:g@<00004B_e:gD[m0002C_d00e;mC_a>o@03000;Ool0
1@00?[dn_Cjn00000Sjm00Hn_P00?[dn_Cjn0002?[d00`00Ga]O6`03Ga/00`00Ga]O6`02Ga/00`00
Ga]O6`03Ga/00`00Ga]O6`03Ga/00`00Oomoo`0YOol00`00Oomoo`0COol00`00Oomoo`1EOol001ao
o`03001oogoo00ioo`03014003JM008fW@03=Yh0000000H0009:g@03A]a:gD[M00D0000:A]e:oTKM
B_e6gD[nA]d004klD_d3000?Ool04@00@]hn_D;N0012gSjm@]h004;N?[e2gP00@]hn_@00Fa/00elK
00UK6`00Ga]O6e/KGa/005lKFa/00elK00L005lKGa]O6e/KGa/0000[Ool00`00Oomoo`0BOol00`00
Oomoo`1EOol001eoo`03001oogoo00ioo`03000fWCJM00PfW@03001:gD[M00@0000;B]e:oD[MB_e:
gD[mB]e:oD[MB_e:g@04000COol01@00?[dn_Cjn00000Sjm00Hn_P00?[dn_Cjn0002?[d00`00Ga]O
6`03Ga/00`00Ga]O6`02Ga/00`00Ga]O6`03Ga/00`00Ga]O6`03Ga/00`00Oomoo`0YOol00`00Oomo
o`0BOol00`00Oomoo`1DOol001ioo`03001oogoo00ioo`d00009B_i6gD[mA]e:oTKMB_e6gD[n00H0
01Qoo`0A0012gSjm@]h004;N?[e2gP00@]hn_@00?[e2gSjm001K6`03Ga/02E/K001O6elKFa]O6`00
Ga]K6`03Ga/01`00Ga]O6elKFa]O6`0002]oo`03001oogoo019oo`03001oogoo05=oo`007goo00<0
07ooOol03Woo00D003:N<YhfWS:M008bWP08=YhbWC:N001:oD[MB_e:g@H001moo`05000n_Cjm?[h0
0002?[d01Cjn000n_Cjm00000cjm00<005lKGa/00elK00<005lKGa/00UlK00<005lKGa/00elK00<0
05lKGa/00UlK00<007ooOol0:Woo00<007ooOol04Woo00<007ooOol0DWoo000POol00`00Oomoo`0>
Ool>000VOol04@00@]hn_D;N0012gSjm@]h004;N?[d003jm@]hn_@00Fa/00elK00UK6`00Ga]O6e/K
Ga/005lKFa/00elK00H005lKGa]O6e/K000[Ool00`00Oomoo`0COol00`00Oomoo`1AOol0025oo`03
001oogoo01Qoo`03001oogoo02Moo`05000n_Cjm?[h00002?[d01Cjn000n_Cjm00000cjm00<005lK
Ga/00elK00<005lKGa/00UlK00<005lKGa/00UlK00<005lKGa/00elK00<007ooOol0:Goo00<007oo
Ool04Woo00<007ooOol0DGoo000ROol00`00Oomoo`0HOol00`00Oomoo`0WOol04@00@]hn_D;N0012
gSjm@]h004;N?[d003jm@]hn_@00Fa/00elK00UK6`00Ga]O6e/KGa/005lKFa/00UlK00<005/KGa/0
0UlK00=K6`00Ool0:Woo00<007ooOol04Woo00<007ooOol0D7oo000SOol00`00Oomoo`0HOol00`00
Oomoo`0WOol01@00?[dn_Cjn00000Sjm00Dn_P00?[dn_@0000<n_@03001O6elK00=O6`05001O6elK
Ga/00005Ga/00`00Ga]O6`03Ga/00`00Oomoo`0YOol00`00Oomoo`0BOol00`00Oomoo`1?Ool002Ao
o`03001oogoo01Qoo`03001oogoo02Moo`0A0012gSjm@]h004;N?[e2gP00@]hn_@00?[e2gSjm001K
6`03Ga/01U/K001O6elKFa/0009O6`05Fa]O6elK001K6`02Ga/00`00Fa/0000[Ool00`00Oomoo`0B
Ool00`00Oomoo`1>Ool002Eoo`03001oogoo01Qoo`03001oogoo02Moo`05000n_Cjm?[h00002?[d0
1@00?[dn_Cjm00000Sjm00<005lKGa/00elK00<005lKGa/00UlK00<005lKGa/00elK00<005lK0000
0elK00<007ooOol0:Goo00<007ooOol04Woo00<007ooOol0CGoo000VOol00`00Oomoo`0HOol00`00
Oomoo`0VOol04P00?[e2gSjm000n_D;N?[d003jm@]hn_@00?[e2gP00Ga]K6`=O6`06001O6elKGa]K
6`000UlK00YK6elKGa/005/KGa]K6elKFa/002]oo`03001oogoo019oo`03001oogoo04aoo`009goo
00<007ooOol067oo00<007ooOol09Woo00D003jm?[dn_@0000<n_@05000n_Cjm?[d00002?[d00`00
Ga]O6`03Ga/00`00Ga]O6`02Ga/00`00Ga]O6`020007Ga/00`00Oomoo`0XOol00`00Oomoo`0BOol0
0`00Oomoo`1<Ool002Qoo`03001oogoo01Qoo`03001oogoo02Ioo`0B000n_D;N?[d003jm@]hn_@00
?[e2gSjm000n_D;N001O6e/K0elK00D005lKGa]O6e/K0080000:Ga]K6elKFa]O6e/KGa]K6elK000[
Ool00`00Oomoo`0BOol00`00Oomoo`1;Ool002Uoo`03001oogoo01Qoo`03001oogoo02Ioo`05000n
_Cjm?[d00003?[d01@00?[dn_Cjm00000Sjm00<005lKGa/00elK00H005lK000005lK000:Ga/00`00
Oomoo`0YOol00`00Oomoo`0BOol00`00Oomoo`1:Ool002Yoo`03001oogoo01Qoo`03001oogoo02Io
o`0B000n_D;N?[d003jm@]hn_@00?[e2gSjm000n_D;N001O6e/K0elK014005lKFa]O6e/K001K6elK
Fa]O6e/KGa]K6elKFa]O6`0002]oo`03001oogoo019oo`03001oogoo04Uoo`00:goo00<007ooOol0
67oo00<007ooOol09Woo00D003jm?[dn_@0000<n_@05000n_Cjm?[d00002?[d00`00Ga]O6`020006
Ga/00`00Ga]O6`03Ga/00`00Ga]O6`02Ga/00`00Oomoo`0YOol00`00Oomoo`0BOol00`00Oomoo`18
Ool002aoo`03001oogoo01Qoo`03001oogoo02Ioo`07000n_D;N?[d003jm@]h02P0001EK6elKFa]O
6e/KGa]K6elKFa/005/KGa]K6elKFa/005/KGa]K6elK0000:goo00<007ooOol04Woo00<007ooOol0
Agoo000]Ool00`00Oomoo`0HOol00`00Oomoo`0VOol600002SkM?[d003jm?]dn_D;N?[dng@002UlK
00<005lKGa/00UlK00<005lKGa/00elK00<007ooOol0:Goo00<007ooOol04Goo00<007ooOol0Agoo
000^Ool2000IOol00`00Oomoo`0VOol09P00?[e2gSjm@]hn_D;N?[d003jm@]hn_D;N?[e2gP00Ga]K
6elKFa]O6e/KGa]K6elKFa/005/KGa]K6elK001O6e/KGa]K6elK000ZOol00`00Oomoo`0BOol00`00
Oomoo`16Ool0031oo`03001oogoo01Moo`03001oogoo02Ioo`0@000n_CkM?[e2gSjm?]dn_@00?[dn
gCjm@]hn_CkM0005Ga/01@00Ga]O6elK00001ElK00<005lKGa/00elK00<007ooOol0:7oo00<007oo
Ool04Woo00<007ooOol0AGoo000aOol00`00Oomoo`0GOol00`00Oomoo`0VOol09P00?[e2gSjm@]hn
_D;N?[d003jm@]hn_D;N?[e2gP00Ga]K6elKFa]O6`00Ga]K6elK001O6e/KGa]K6elK001O6e/KGa]K
6elK000ZOol00`00Oomoo`0BOol00`00Oomoo`14Ool0039oo`03001oogoo01Moo`03001oogoo02Io
o`0@000n_CkM?[e2gSjm?]d004;N?[dng@00@]hn_CkM0004Ga/00`00Ga]O6`02Ga/00`00Ga]O6`03
Ga/00`00Ga]O6`03Ga/00`00Oomoo`0XOol00`00Oomoo`0BOol00`00Oomoo`13Ool003=oo`03001o
ogoo01Moo`03001oogoo02Ioo`0U000n_D;N?[e2gSjm@]h004;N?[e2gP00@]hn_D;N001O6e/KGa]K
6`00Fa]O6e/KGa/005lKFa]O6e/KGa/005lKFa]O6e/K0000:goo00<007ooOol04Woo00<007ooOol0
@Woo000dOol00`00Oomoo`0GOol00`00Oomoo`0VOol03`00?[dngCjm@]hn_CkM0012gSjm?]d004;N
?[d00005Ga/00`00Ga]O6`02Ga/00`00Ga]O6`03Ga/00`00Ga]O6`02Ga/00`00Oomoo`0YOol00`00
Oomoo`0AOol00`00Oomoo`12Ool003Eoo`03001oogoo01Moo`03001oogoo02Eoo`0V0012gSjm@]h0
04;N?[e2gP00@]hn_D;N0012gSjm001K6elKFa]O6e/K001K6elKFa]O6`00Ga]K6elKFa]O6`00Ga]K
6elKFa/002Yoo`03001oogoo019oo`03001oogoo045oo`00=Woo00<007ooOol05goo00<007ooOol0
9Goo010004;N?[dng@00@]hn_CkM0012gSjm?]d004;N?[d000EO6`03001O6elK009O6`03001O6elK
00=O6`03001O6elK009O6`03001oogoo02Qoo`03001oogoo019oo`03001oogoo041oo`00=goo00<0
07ooOol05goo00<007ooOol09Goo02H004;N?[e2gP00@]hn_D;N0012gSjm@]h004;N?[d005/KGa]K
6elKFa/005/KGa]K6elK001O6e/KGa]K6elK001O6e/KGa]K6`00:Woo00<007ooOol04Woo00<007oo
Ool0?goo000hOol00`00Oomoo`0GOol00`00Oomoo`0UOol04000@]hn_CkM0012gSjm?]d004;N?[dn
g@00@]hn_@001ElK00<005lKGa/00UlK00<005lKGa/00elK00<005lKGa/00UlK00<007ooOol0:7oo
00<007ooOol04Woo00<007ooOol0?Woo000iOol00`00Oomoo`0GOol00`00Oomoo`0UOol09P00@]hn
_D;N0012gSjm@]h004;N?[e2gP00@]hn_@00Fa]O6e/KGa]K6`00Fa]O6e/KGa/005lKFa]O6e/KGa/0
05lKFa]O6e/K000ZOol00`00Oomoo`0BOol00`00Oomoo`0mOol003Yoo`03001oogoo01Moo`03001o
ogoo02Eoo`0@0012gSjm?]d004;N?[dng@00@]hn_CkM0012gSjm0005Ga/00`00Ga]O6`02Ga/00`00
Ga]O6`03Ga/00`00Ga]O6`02Ga/00`00Oomoo`0XOol00`00Oomoo`0AOol00`00Oomoo`0mOol003]o
o`03001oogoo01Moo`03001oogoo02Eoo`0V0012gSjm@]h004;N?[e2gP00@]hn_D;N0012gSjm001K
6elKFa]O6e/K001K6elKFa]O6`00Ga]K6elKFa]O6`00Ga]K6elKFa/002Yoo`03001oogoo015oo`03
001oogoo03aoo`00?7oo00<007ooOol05goo00<007ooOol09Goo010004;N?[dng@00@]hn_@00?[e2
gSjm000n_D;N?[d000EO6`03001O6elK009O6`03001O6elK009O6`03001O6elK009O6`03001oogoo
02Qoo`03001oogoo019oo`03001oogoo03]oo`00?Goo00<007ooOol05goo00<007ooOol09Goo02D0
04;N?[e2gP00@]hn_@00?[e2gSjm000n_D;N?[d005/KGa]K6elKFa/005/KGa]K6elK001O6e/KGa]K
6`00Fa]O6e/KGa/0000ZOol00`00Oomoo`0BOol00`00Oomoo`0jOol003ioo`03001oogoo01Moo`03
001oogoo02Eoo`0@0012gSjm?]d004;N?[d003jm@]hn_@00?[e2gSjm0005Ga/01@00Ga]O6elK0000
1ElK00<005lKGa/00UlK00<007ooOol0:7oo00<007ooOol04Woo00<007ooOol0>Goo000oOol00`00
Oomoo`0GOol00`00Oomoo`0UOol09@00@]hn_D;N0012gSjm000n_D;N?[d003jm@]hn_@00Fa]O6e/K
Ga]K6`00Fa]O6e/K001K6elKFa]O6e/K001K6elKFa]O6`0002Yoo`03001oogoo019oo`03001oogoo
03Qoo`00@7oo00<007ooOol05goo00<007ooOol09Goo010004;N?[dng@00@]hn_@00?[e2gSjm000n
_D;N?[d000AO6`03001O6elK009O6`03001O6elK00=O6`03001O6elK009O6`03001oogoo02Qoo`03
001oogoo015oo`03001oogoo03Qoo`00@Goo00<007ooOol05goo00<007ooOol09Goo024004;N?[e2
gP00@]hn_@00?[e2gSjm000n_D;N?[d005/KGa]K6elK001O6e/KGa]K6`00Fa]O6e/KGa]K6`00Fa/0
0P0000=O6`00Ool0:Goo00<007ooOol04Goo00<007ooOol0=goo0012Ool00`00Oomoo`0GOol00`00
Oomoo`0UOol03`00@]hn_CkM0012gSjm000n_D;N?[d003jm@]h00005Ga/00`00Ga]O6`02Ga/00`00
Ga]O6`03Ga/00`00Ga]O6`02Ga/00`00Oomoo`0WOol00`00Oomoo`0BOol00`00Oomoo`0fOol004=o
o`03001oogoo01Moo`03001oogoo02Aoo`0M000n_D;N?[d003jm@]hn_@00?[e2gSjm000n_D;N001O
6e/KGa]K6elK001O6e/KGa]K6`00Fa]O6`0200001e/K001K6elKFa]O6`0002Uoo`03001oogoo019o
o`03001oogoo03Eoo`00A7oo00<007ooOol05Woo00<007ooOol09Goo010003jm@]hn_@00?[e2gSjm
000n_D;N?[d003jm@]h000EO6`03001O6elK009O6`8000AO6`03001O6elK009O6`03001oogoo02Mo
o`03001oogoo019oo`03001oogoo03Aoo`00AGoo00<007ooOol05Woo00<007ooOol09Goo01L003jm
@]hn_@00?[e2gP00@]hn_D;N0012gSjm@]h005lKFa]O6e/KGa/005lK0080009O6`0;Fa]O6e/KGa]K
6`00Fa]O6e/KGa/0000YOol00`00Oomoo`0BOol00`00Oomoo`0cOol004Ioo`03001oogoo01Ioo`03
001oogoo02Eoo`0@000n_D;N?[d003jm@]h003kM?[e2gP00?]dn_D;N0005Ga/00`00Ga]O6`07Ga/0
0`00Ga]O6`02Ga/00`00Oomoo`0XOol00`00Oomoo`0AOol00`00Oomoo`0cOol004Moo`03001oogoo
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The following shape is an umbilic torus.  It was formed by taking a slightly \
concave triangle and rotating it around an axis of revolution while rotating \
the triangle once.  The result is a torus that has a single crease that wraps \
around the torus three times.  (Helaman Ferguson has made a famous sculpture \
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Cell["3D shape", "Subsubsection"],

Cell["\<\
Here is a more interesting example of a 3D mesh.  It consists of 8 cubes \
whose boundary quads are treated as crease surfaces.  Note the two edges \
curves on the bottom of the feet of the \"A\" are circles.\
\>", "Text"],

Cell[BoxData[
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                14}}, {{13, 14}, {15, 16}}, {{4, 14}, {6, 12}}, {6, 12}, {12, 
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                23}}, {{24, 23}, {26, 25}}, {{26, 25}, {28, 27}}, {{28, 
                27}, {30, 29}}, {{30, 29}, {32, 31}}, {{30, 20}, {28, 
                22}}, {22, 28}, {28, 26}, {26, 24}, {24, 22}, {15, 16}, {16, 
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                13}, {27, 29}}, {{13, 15}, {29, 31}}, {{15, 16}, {31, 
                32}}, {{16, 14}, {32, 30}}, {{14, 4}, {30, 20}}, {{4, 
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                22}}, {{{1, 2}, {3, 4}}, {{17, 18}, {19, 20}}}, {{{3, 4}, {5, 
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Cell["Mesh conversion function", "Subsubsection"],

Cell["\<\
This function converts a cubic mesh to polygonal meshes used in the rest of \
chapter 7.\
\>", "Text"],

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Cell["Creating surfaces of revolution by revolving cubic meshes", "Section"],

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Cell["\<\
Revolving a 1D cubic mesh to make a 2D cubic mesh for a surface of \
revolution.\
\>", "Subsubsection"],

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defined by revolving the profile curve around the ",
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Cell["Mathematical shapes", "Subsubsection"],

Cell["Here is the definition of the profile curve for a sphere", "Text"],

Cell[BoxData[
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Cell["Here are several rounds of subdivision for the profile curve", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
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