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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[ 418346, 10484]*) (*NotebookOutlinePosition[ 419026, 10508]*) (* CellTagsIndexPosition[ 418982, 10504]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Celtic Knots", "Title"], Cell["As explained by Andrew Glassner in Computer Graphics.", "Subtitle"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell["\<\ This notebook is an implementation of the what Andrew Glassner explains in \ Computer Graphics.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Imported packages", "Section"], Cell[BoxData[ \(<< Graphics`Colors`\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Remove the speeling messages", "Section"], Cell[BoxData[ \(Off[\ General::spell1]\)], "Input"], Cell[BoxData[ \(Off[\ General::spell]\)], "Input"], Cell[BoxData[ \(\({delatv, deltah, NELowerBx, NELowerBy, NELowerCx, NELowerCy, NEUpperBx, \ NEUpperBy, \ NEUpperCx, \ NEUpperCy, BreakLines, BreakLine, \[IndentingNewLine]SALowerBx, SALowerBy, SALowerCx, SALowerCy, \[IndentingNewLine]SAUpperBx, \ SAUpperBy, \ SAUpperCx, \ SAUpperCy, \[IndentingNewLine]LALowerBx, LALowerBy, LALowerCx, LALowerCy, \[IndentingNewLine]LAUpperBx, \ LAUpperBy, \ LAUpperCx, \ LAUpperCy\[IndentingNewLine]};\)\)], "Input"], Cell[BoxData[ \(On[\ General::spell1]\)], "Input"], Cell[BoxData[ \(On[\ General::spell]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["First functions", "Section"], Cell[BoxData[ \(\(Grid\ = \ {};\)\)], "Input"], Cell[BoxData[ \(\({VisitedFlagIndex, \[IndentingNewLine]BandNumberIndex, \ \[IndentingNewLine]EdgeCodeIndex, \[IndentingNewLine]BreakLinesIndex, \ \[IndentingNewLine]SpecialCellIndex} = Range[5];\)\)], "Input"], Cell[BoxData[ \(\(emptyCell\ = \ {0, 0, {0, 0}, {0, 0}, 0};\)\)], "Input"], Cell[BoxData[ \(\(SetGridSize[h_, v_] := \[IndentingNewLine]\(Grid\ = \ Table[emptyCell, {2\ v}, {2\ h}]\);\)\)], "Input"], Cell[BoxData[ \(\(InitThread[] := Module[\[IndentingNewLine]{h, v}, \[IndentingNewLine]{v, h} = \(Dimensions[\ Grid]\)[\([{1, 2}]\)]; \[IndentingNewLine]Grid\ = \ MapIndexed[{#1[\([1]\)], #1[\([2]\)], If[\ OddQ[\ #2[\([1]\)] + #2[\([2]\)]\ ], \ {1, 5}, {3, 7}], #1[\([4]\)]\ , #[\([5]\)]} &, Grid, {2}];\[IndentingNewLine]];\)\)], "Input"], Cell[BoxData[ \(InitBreakLines[] := Module[\[IndentingNewLine]{h, v}, \[IndentingNewLine]{v, h} = \(Dimensions[\ Grid]\)[\([{1, 2}]\)]; \[IndentingNewLine]Grid\ = \ MapIndexed[{#1[\([1]\)], #1[\([2]\)], #1[\([3]\)], \ \[IndentingNewLine]Switch[#2, \[IndentingNewLine]{1, 1}, {2, 3}, \[IndentingNewLine]{1, h}, {1, 2}, \[IndentingNewLine]{v, 1}, {3, 4}, \[IndentingNewLine]{v, h}, {1, 4}, \[IndentingNewLine]{_, h}, {0, 1}, \[IndentingNewLine]{v, _}, {0, 4}, \[IndentingNewLine]{_, 1}, {0, 3}, \[IndentingNewLine]{1, _}, {0, 2}, \[IndentingNewLine]{_, _}, {0, 0}]\[IndentingNewLine]\[IndentingNewLine], #1[\([5]\)]\ \[IndentingNewLine]} &, Grid, {2}];\[IndentingNewLine]\[IndentingNewLine]]\)], "Input"], Cell[BoxData[{ \(\(EdgePosition[1] = {1, 1};\)\), "\[IndentingNewLine]", \(\(EdgePosition[2] = {1, .5};\)\), "\[IndentingNewLine]", \(\(EdgePosition[3] = {1, 0};\)\), "\[IndentingNewLine]", \(\(EdgePosition[4] = { .5, 0};\)\), "\[IndentingNewLine]", \(\(EdgePosition[5] = {0, 0};\)\), "\[IndentingNewLine]", \(\(EdgePosition[6] = {0, .5};\)\), "\[IndentingNewLine]", \(\(EdgePosition[7] = {0, 1};\)\), "\[IndentingNewLine]", \(\(EdgePosition[8] = { .5, 1};\)\[IndentingNewLine]\)}], "Input"], Cell[BoxData[ \(\(EdgeLine[offsetH_, offsetV_, EdgeCode_] := \[IndentingNewLine]If[\ EdgeCode \[Equal] {0, 0}, {}, \[IndentingNewLine]Line[{{offsetH, offsetV} + EdgePosition[EdgeCode[\([1]\)]\ ], {offsetH, offsetV} + EdgePosition[ EdgeCode[\([2]\)]\ ]}], \[IndentingNewLine]];\)\)], "Input"], Cell[BoxData[{ \(\(StartBreakLine[1] = {1, 1};\)\), "\[IndentingNewLine]", \(\(StartBreakLine[2] = {1, 0};\)\), "\[IndentingNewLine]", \(\(StartBreakLine[3] = {0, 0};\)\), "\[IndentingNewLine]", \(\(StartBreakLine[4] = {0, 1};\)\), "\[IndentingNewLine]", \(\(EndBreakLine[1]\ = \ {1, 0};\)\), "\[IndentingNewLine]", \(\(EndBreakLine[2]\ = \ {0, 0};\)\), "\[IndentingNewLine]", \(\(EndBreakLine[3]\ = \ {0, 1};\)\), "\[IndentingNewLine]", \(\(EndBreakLine[4]\ = \ {1, 1};\)\[IndentingNewLine]\)}], "Input"], Cell[BoxData[ \(\(BreakSegment[offsetH_, offsetV_, BreakLineValue_] := \[IndentingNewLine]Line[{{offsetH, offsetV} + StartBreakLine[BreakLineValue], {offsetH, offsetV} + EndBreakLine[BreakLineValue]}]\[IndentingNewLine];\)\)], "Input"], Cell[BoxData[ \(\(BreakLine[offsetH_, offsetV_, BreakLineValues_] := \[IndentingNewLine]\(BreakSegment[offsetH, offsetV, #] &\)\ /@ \ Complement[BreakLineValues, {0}]\[IndentingNewLine];\)\)], "Input"], Cell[BoxData[ \(\(\(FixEdgeCodes[] := \[IndentingNewLine]\(Grid\ = \ Grid /. {\[IndentingNewLine]{0, 0, {1, 5}, {0, 3}, 0} \[Rule] {0, 0, {1, 4}, {0, 3}, 0}, \[IndentingNewLine]{0, 0, {1, 5}, {0, 1}, 0} \[Rule] {0, 0, {5, 8}, {0, 1}, 0}, \[IndentingNewLine]{0, 0, {1, 5}, {0, 2}, 0} \[Rule] {0, 0, {1, 6}, {0, 2}, 0}, \[IndentingNewLine]{0, 0, {1, 5}, {0, 4}, 0} \[Rule] {0, 0, {2, 5}, {0, 4}, 0}, \[IndentingNewLine]{0, 0, {1, 5}, {1, 2}, 0} \[Rule] {0, 0, {6, 8}, {1, 2}, 0}, \[IndentingNewLine]{0, 0, {1, 5}, {3, 4}, 0} \[Rule] {0, 0, {2, 4}, {3, 4}, 0}, \[IndentingNewLine]{0, 0, {1, 5}, {1, 3}, 0} \[Rule] {0, 0, {4, 8}, {1, 3}, 0}, \[IndentingNewLine]{0, 0, {1, 5}, {2, 4}, 0} \[Rule] {0, 0, {2, 6}, {2, 4}, 0}, \[IndentingNewLine]\[IndentingNewLine]{0, 0, {3, 7}, {0, 3}, 0} \[Rule] {0, 0, {3, 8}, {0, 3}, 0}, \[IndentingNewLine]{0, 0, {3, 7}, {0, 1}, 0} \[Rule] {0, 0, {4, 7}, {0, 1}, 0}, \[IndentingNewLine]{0, 0, {3, 7}, {0, 2}, 0} \[Rule] {0, 0, {2, 7}, {0, 2}, 0}, \[IndentingNewLine]{0, 0, {3, 7}, {0, 4}, 0} \[Rule] {0, 0, {3, 6}, {0, 4}, 0}, \[IndentingNewLine]\[IndentingNewLine]{0, 0, {3, 7}, {1, 4}, 0} \[Rule] {0, 0, {4, 6}, {1, 4}, 0}, \[IndentingNewLine]{0, 0, {3, 7}, {2, 3}, 0} \[Rule] {0, 0, {2, 8}, {2, 3}, 0}, \[IndentingNewLine]\[IndentingNewLine]{0, 0, {3, 7}, {1, 3}, 0} \[Rule] {0, 0, {4, 8}, {1, 3}, 0}, \[IndentingNewLine]{0, 0, {3, 7}, {2, 4}, 0} \[Rule] {0, 0, {2, 6}, {2, 4}, 0}\[IndentingNewLine]\[IndentingNewLine]}\)\ \[IndentingNewLine];\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \(AddWallCell[x_, y_, wallNum_] := \[IndentingNewLine]Module[\[IndentingNewLine]{v, h}, \[IndentingNewLine]{v, h} = \(Dimensions[\ Grid]\)[\([{1, 2}]\)]; \[IndentingNewLine]\[IndentingNewLine]If[\ x > 0\ && \ y > 0\ && \ x \[LessEqual] h\ && \ y \[LessEqual] v, \[IndentingNewLine]\(If[\ MemberQ[\ Grid[\([y, x, BreakLinesIndex]\)], 0], \[IndentingNewLine]\(Grid[\([y, x, BreakLinesIndex]\)]\ = \ Rest[Sort[\ Append[Grid[\([y, x, BreakLinesIndex]\)], wallNum\ ]]];\)\[IndentingNewLine], \(Print[\ "\", {x, y}];\)\[IndentingNewLine]];\)\[IndentingNewLine]]\ \[IndentingNewLine]]\)], "Input"], Cell[BoxData[ \(AddBreakLines[{{x1_, y1_}, {x2_, y2_}}] := Module[\[IndentingNewLine]{deltah, deltav}, \[IndentingNewLine]\[IndentingNewLine]{deltah\ , deltav} = {x2, y2} - {x1, y1}; \[IndentingNewLine]If[\ deltah \[Equal] 0, \[IndentingNewLine]\(\(\((AddWallCell[\ x2, #, 1]; AddWallCell[\ x2 + 1, #, 3];)\) &\)\ /@ \ \((Range @@ \ \((Sort[{y1, y2}] + {1, 0})\))\);\)\[IndentingNewLine], \[IndentingNewLine]If[ deltav \[Equal] 0, \[IndentingNewLine]\(\(\((AddWallCell[#, y1, 4]; AddWallCell[\ #, y1 + 1, 2];)\) &\)\ /@ \ \((Range @@ \ \((Sort[{x1, x2}] + {1, 0})\))\);\)\[IndentingNewLine]]\[IndentingNewLine]]\ \[IndentingNewLine]]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Normal Elbows", "Section"], Cell[BoxData[ \(\(BezierSegment[{p0_, p1_, p2_, p3_, steps_}] := \[IndentingNewLine]Table[\ p0\ \((u^3)\)\ + \ 3\ p1\ \((u^2)\) \((1 - u)\)\ + \ 3\ p2\ u\ \((\((1 - u)\)^2)\)\ + \ p3\ \((\((1 - u)\)^3)\), {u, 0, 1, 1/steps}];\)\)], "Input"], Cell[BoxData[ \(\(\(NormalElbows = {\[IndentingNewLine]{{24, 0}, {30, 0}}, \[IndentingNewLine]{{24, 3}, {26, 5}}, \[IndentingNewLine]{{17, 4}, {22, 8}}, \[IndentingNewLine]{{15, 8}, {26, 10}}, \[IndentingNewLine]{{11, 10}, {23, 13}}, \[IndentingNewLine]{{9, 10}, {21, 14}}, \[IndentingNewLine]{{11, 19}, {18, 20}}, \[IndentingNewLine]{{10, 23}, {23, 25}}, \[IndentingNewLine]{{3, 23}, {10, 31}}, \[IndentingNewLine]{{3, 25}, {10, 31}}\[IndentingNewLine]};\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \(\(bandSizes = {32/Sqrt[2. ], 15, 9, 5, 2, 0}/32;\)\)], "Input"], Cell[BoxData[{ \(\(Clear[NELowerBx, NELowerBy, NELowerCx, NELowerCy];\)\), "\n", \(\({\ NELowerBx, NELowerBy, NELowerCx, NELowerCy}\ = \[IndentingNewLine]Interpolation\ /@ \ \n\ \((\(Transpose[{bandSizes, #}] &\)\ /@ \ Transpose[\ Flatten\ /@ \ \(Take[\ NormalElbows, 5]\)[\([{1, 2, 3, 4, 5, 5}]\)]/ 32\ ])\);\)\)}], "Input"], Cell[BoxData[{ \(\(Clear[NEUpperBx, \ NEUpperBy, \ NEUpperCx, \ NEUpperCy];\)\), "\n", \(\({\ NEUpperBx, \ NEUpperBy, \ NEUpperCx, \ NEUpperCy}\ = \[IndentingNewLine]Interpolation\ /@ \ \n\ \((\(Transpose[{bandSizes, #}] &\)\ /@ \ Transpose[\ Flatten\ /@ \((\ \(Reverse[ Take[\ NormalElbows, \(-5\)]]\)\ [\([{1, 2, 3, 4, 5, 5}]\)])\)/32\ ])\);\)\)}], "Input"], Cell[BoxData[ \(\(\(NEedge\ := \ Join[\ \ BezierSegment[{{w/Sqrt[2. ] + underW, 0 + underW}, {NELowerBx[w/2], NELowerBy[w/2]}, {NELowerCx[w/2], NELowerCy[w/2]}, {1, .5 - w/2}, 10}], \[IndentingNewLine]\ BezierSegment[{{1, .5 + w/2}, \ {NEUpperCx[w/2], NEUpperCy[w/2]}, {NEUpperBx[w/2], NEUpperBy[w/2]}\ , \[IndentingNewLine]{0 + underW, w/Sqrt[2. ] + underW}, 10}]\ ];\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \(\(\(NEedgeWithFoot\ := \ Join[\ \ BezierSegment[{{\(\(+w\)/2\)/Sqrt[2. ], \(\(-w\)/2\)/ Sqrt[2. ]}, {NELowerBx[w/2], NELowerBy[w/2]}, {NELowerCx[w/2], NELowerCy[w/2]}, {1, .5 - w/2}, 10}], \[IndentingNewLine]\ BezierSegment[{{1, .5 + w/2}, \ {NEUpperCx[w/2], NEUpperCy[w/2]}, {NEUpperBx[w/2], NEUpperBy[w/2]}\ , \[IndentingNewLine]{\(\(-w\)/2\)/ Sqrt[2. ], \(w/2\)/Sqrt[2. ]}, 10}]\ ];\)\(\[IndentingNewLine]\)\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Short Arcs", "Section"], Cell[BoxData[ \(\(ShortArcCurves\ = \ {\[IndentingNewLine]{{\ 24, 0}, {29, 0}}, \[IndentingNewLine]{{22, 1}, {29, 1}}, \[IndentingNewLine]{{19, 6}, {21, 7}}, \[IndentingNewLine]{{13, 7}, {18, 11}}, \[IndentingNewLine]{{8, 6}, {16, 15}}, \[IndentingNewLine]{{7, 8}, {15, 17}}, \[IndentingNewLine]{{9, 17}, {16, 21}}, \[IndentingNewLine]{{9, 22}, {14, 25}}, \[IndentingNewLine]{{4, 25}, {11, 31}}, \[IndentingNewLine]{{4, 26}, {11, 32}}\[IndentingNewLine]};\)\)], "Input"], Cell[BoxData[{ \(\(Clear[SALowerBx, SALowerBy, SALowerCx, SALowerCy];\)\), "\n", \(\({\ SALowerBx, SALowerBy, SALowerCx, SALowerCy}\ = \[IndentingNewLine]Interpolation\ /@ \ \n\ \((\(Transpose[{bandSizes, #}] &\)\ /@ \ Transpose[\ Flatten\ /@ \ \(Take[\ ShortArcCurves, 5]\)[\([{1, 2, 3, 4, 5, 5}]\)]/ 32\ ])\);\)\)}], "Input"], Cell[BoxData[{ \(\(Clear[SAUpperBx, \ SAUpperBy, \ SAUpperCx, \ SAUpperCy];\)\), "\n", \(\({\ SAUpperBx, \ SAUpperBy, \ SAUpperCx, \ SAUpperCy}\ = \[IndentingNewLine]Interpolation\ /@ \ \n\ \((\(Transpose[{bandSizes, #}] &\)\ /@ \ Transpose[\ Flatten\ /@ \((\ \(Reverse[ Take[\ ShortArcCurves, \(-5\)]]\)\ [\([{1, 2, 3, 4, 5, 5}]\)])\)/32\ ])\);\)\)}], "Input"], Cell[BoxData[ \(\(\(SAedge\ := \ Join[\ \ BezierSegment[{{w/Sqrt[2. ] + underW, 0 + underW}, {SALowerBx[w/2], SALowerBy[w/2]}, {SALowerCx[w/2], SALowerCy[w/2]}, {1, .5 - w/2}, 10}], \[IndentingNewLine]\ BezierSegment[{{1, .5 + w/2}, \ {SAUpperCx[w/2], SAUpperCy[w/2]}, {SAUpperBx[w/2], SAUpperBy[w/2]}\ , \[IndentingNewLine]{0 + underW, w/Sqrt[2. ] + underW}, 10}]\ ];\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \(\(\(SAedgeWithFoot\ := \ Join[\ \ BezierSegment[{{\(\(+w\)/2\)/Sqrt[2. ], \(\(-w\)/2\)/ Sqrt[2. ]}, {SALowerBx[w/2], SALowerBy[w/2]}, {SALowerCx[w/2], SALowerCy[w/2]}, {1, .5 - w/2}, 10}], \[IndentingNewLine]\ BezierSegment[{{1, .5 + w/2}, \ {SAUpperCx[w/2], SAUpperCy[w/2]}, {SAUpperBx[w/2], SAUpperBy[w/2]}\ , \[IndentingNewLine]{\(\(-w\)/2\)/ Sqrt[2. ], \(w/2\)/Sqrt[2. ]}, 10}]\ ];\)\(\[IndentingNewLine]\)\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Detects short arcs", "Section"], Cell[BoxData[ \(\(\(CheckOneCell[\ Cell_, CurrentCell_] := Module[\[IndentingNewLine]{}, \[IndentingNewLine]ThisEdgeCode = Cell[\([\ EdgeCodeIndex]\)]; \ \[IndentingNewLine]\[IndentingNewLine]MatchedArc\ = \ Select[\ {\[IndentingNewLine]{{2, 5}, {0, 1}, {3, 6}}, \[IndentingNewLine]{{4, 7}, {\(-1\), 0}, {5, 8}}, \[IndentingNewLine]{{1, 6}, {0, \(-1\)}, {2, 7}}, \[IndentingNewLine]{{3, 8}, {1, 0}, {1, 4}}\[IndentingNewLine]}, \ #[\([1]\)]\ \[Equal] \ ThisEdgeCode &]; \[IndentingNewLine]\[IndentingNewLine]If[ MatchedArc \[NotEqual] {}, MatchedArc\ = \ First[MatchedArc]; \[IndentingNewLine]If[\ Extract[Grid, Join[\ CurrentCell + \ MatchedArc[\([2]\)], {EdgeCodeIndex}]] \[Equal] \ Last[MatchedArc], \[IndentingNewLine]\ Grid\ = \ ReplacePart[Grid, 1, \ Join[CurrentCell, {SpecialCellIndex}]]; \[IndentingNewLine]\ \ Grid\ = \ ReplacePart[Grid, 1, \ Join[CurrentCell + \ MatchedArc[\([2]\)], {SpecialCellIndex}]\ ];\ \[IndentingNewLine]];\[IndentingNewLine]];\[IndentingNewLine]];\)\(\ \[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \(\(\(FixShortArcs[] := MapIndexed[\ CheckOneCell[#1, #2] &, \ Grid, \ {2}];\)\(\n\)\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Long Arcs", "Section"], Cell[BoxData[ \(\(LongArcCurves\ = \ {\[IndentingNewLine]{{\ 24, 0}, {60, 0}}, \[IndentingNewLine]{{\ 22, 1}, {61, 1}}, \[IndentingNewLine]{{21, 8}, {56, 7}}, \[IndentingNewLine]{{18, 11}, {36, 11}}, \[IndentingNewLine]{{17, 5}, {32, 15}}, \[IndentingNewLine]{{17, 19}, {30, 17}}, \[IndentingNewLine]{{16, 23}, {40, 21}}, \[IndentingNewLine]{{13, 26}, {46, 25}}, \[IndentingNewLine]{{11, 32}, {43, 31}}, \[IndentingNewLine]{{11, 34}, {43, 32}}\[IndentingNewLine]};\)\)], "Input"], Cell[BoxData[{ \(\(Clear[LALowerBx, LALowerBy, LALowerCx, LALowerCy];\)\), "\n", \(\({\ LALowerBx, LALowerBy, LALowerCx, LALowerCy}\ = \[IndentingNewLine]Interpolation\ /@ \ \n\ \((\(Transpose[{bandSizes, #}] &\)\ /@ \ Transpose[\ Flatten\ /@ \ \(Take[\ LongArcCurves, 5]\)[\([{1, 2, 3, 4, 5, 5}]\)]/ 32\ ])\);\)\)}], "Input"], Cell[BoxData[{ \(\(Clear[LAUpperBx, \ LAUpperBy, \ LAUpperCx, \ LAUpperCy];\)\), "\n", \(\({\ LAUpperBx, \ LAUpperBy, \ LAUpperCx, \ LAUpperCy}\ = \[IndentingNewLine]Interpolation\ /@ \ \n\ \((\(Transpose[{bandSizes, #}] &\)\ /@ \ Transpose[\ Flatten\ /@ \((\ \(Reverse[ Take[\ LongArcCurves, \(-5\)]]\)\ [\([{1, 2, 3, 4, 5, 5}]\)])\)/32\ ])\);\)\)}], "Input"], Cell[BoxData[ \(\(\(LAedge\ := \ Join[\ \ BezierSegment[{{w/Sqrt[2. ] + underW, 0 + underW}, {LALowerBx[w/2], LALowerBy[w/2]}, {LALowerCx[w/2], LALowerCy[w/2]}, {2, .5 - w/2}, 10}], \[IndentingNewLine]\ BezierSegment[{{2, .5 + w/2}, \ {LAUpperCx[w/2], LAUpperCy[w/2]}, {LAUpperBx[w/2], LAUpperBy[w/2]}\ , \[IndentingNewLine]{0 + underW, w/Sqrt[2. ] + underW}, 10}]\ ];\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \(\(\(LAedgeWithFoot\ := \ Join[\ \ BezierSegment[{{\(\(+w\)/2\)/Sqrt[2. ], \(\(-w\)/2\)/ Sqrt[2. ]}, {LALowerBx[w/2], LALowerBy[w/2]}, {LALowerCx[w/2], LALowerCy[w/2]}, {2, .5 - w/2}, 10}], \[IndentingNewLine]\ BezierSegment[{{2, .5 + w/2}, \ {LAUpperCx[w/2], LAUpperCy[w/2]}, {LAUpperBx[w/2], LAUpperBy[w/2]}\ , \[IndentingNewLine]{\(\(-w\)/2\)/ Sqrt[2. ], \(w/2\)/Sqrt[2. ]}, 10}]\ ];\)\(\[IndentingNewLine]\)\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Detects long arcs", "Section"], Cell[BoxData[ \(\(\(CheckOneCellLA[\ Cell_, CurrentCell_] := Module[\[IndentingNewLine]{}, \[IndentingNewLine]ThisEdgeCode = Cell[\([\ EdgeCodeIndex]\)]; \ \[IndentingNewLine]\[IndentingNewLine]MatchedArc\ = \ Select[\ {\[IndentingNewLine]{{2, 5}, {0, 1}, {2, 6}}, \[IndentingNewLine]{{4, 7}, {\(-1\), 0}, {4, 8}}, \[IndentingNewLine]{{1, 6}, {0, \(-1\)}, {2, 6}}, \[IndentingNewLine]{{3, 8}, {1, 0}, {4, 8}}, \[IndentingNewLine]{{3, 6}, {0, \(-1\)}, {2, 6}}, \[IndentingNewLine]{{5, 8}, {1, 0}, {4, 8}}, \[IndentingNewLine]{{2, 7}, {0, 1}, {2, 6}}, \[IndentingNewLine]{{1, 4}, {\(-1\), 0}, {4, 8}}\[IndentingNewLine]}, \ #[\([1]\)]\ \[Equal] \ ThisEdgeCode &]; \[IndentingNewLine]\[IndentingNewLine]If[ MatchedArc \[NotEqual] {}, MatchedArc\ = \ First[MatchedArc]; \[IndentingNewLine]If[\ Extract[Grid, Join[\ CurrentCell + \ MatchedArc[\([2]\)], {EdgeCodeIndex}]] \[Equal] \ Last[MatchedArc], \[IndentingNewLine]\ Grid\ = \ ReplacePart[Grid, 2, \ Join[CurrentCell, {SpecialCellIndex}]]; \[IndentingNewLine]\ \ Grid\ = \ ReplacePart[Grid, 2, \ Join[CurrentCell + \ MatchedArc[\([2]\)], {SpecialCellIndex}]\ ];\ \[IndentingNewLine]];\[IndentingNewLine]];\[IndentingNewLine]];\)\(\ \[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \(\(\(FixLongArcs[] := MapIndexed[\ CheckOneCellLA[#1, #2] &, \ Grid, \ {2}];\)\(\n\)\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Building the bands.", "Section"], Cell[BoxData[{ \(\(VertMirror[{x_, y_}] := {1 - x, y};\)\), "\[IndentingNewLine]", \(\(HorzMirror[{x_, y_}] := {x, 1 - y};\)\)}], "Input"], Cell[BoxData[ \(\(Rot90[{x_, y_}] := \ {1 - y, x};\)\)], "Input"], Cell[BoxData[{ \(\(OppositeIndex\ = \ {5, 6, 7, 8, 1, 2, 3, 4};\)\), "\[IndentingNewLine]", \(\(moveDelta\ = \ {{1, 1}, {0, 1}, {\(-1\), 1}, {\(-1\), 0}, {\(-1\), \(-1\)}, {0, \(-1\)}, {1, \(-1\)}, {1, 0}};\)\), "\[IndentingNewLine]", \(\(colorBands\ = \ Symbol\ /@ \ AllColors;\)\)}], "Input"], Cell[BoxData[ \(\(\(Colorize[] := Module[\[IndentingNewLine]{}, \[IndentingNewLine]{v, h} = \(Dimensions[\ Grid]\)[\([{1, 2}]\)]; \[IndentingNewLine]BandNumber = 1; \[IndentingNewLine]cellsLeft\ = Flatten[\ MapIndexed[#2 &, \ Grid, {2}], 1]; \[IndentingNewLine]ThisIndex\ = 0; pass = 0; \[IndentingNewLine]While[\ Length[cellsLeft] > 0\ && \ \ \((\(pass++\))\) < 3, \[IndentingNewLine]Current\ = \ First[cellsLeft]; \[IndentingNewLine]\[IndentingNewLine]\ \[IndentingNewLine]While\ [\ Current[\([1]\)] <= v\ && \ Current[\([1]\)] > 0\ && \ Current[\([2]\)] <= h\ && \ Current[\([2]\)] > 0\ && \ Extract[Grid, \ Join[Current, {BandNumberIndex}]] \[Equal] 0, \[IndentingNewLine]\[IndentingNewLine]Grid\ = \ ReplacePart[Grid, BandNumber, Join[Current, {BandNumberIndex}]]; \ \[IndentingNewLine]cellsLeft\ = \ Complement[ cellsLeft, {Current}\ ]; \[IndentingNewLine]nextOne\ = \ Extract[Grid, \ Join[Current, {EdgeCodeIndex}]]; \[IndentingNewLine]\ \[IndentingNewLine]nextOne\ = First[Complement[ nextOne, {ThisIndex}]]; \[IndentingNewLine]ThisIndex\ = \ \ OppositeIndex[\([nextOne]\)]; \[IndentingNewLine]Current += \ moveDelta[\([\ nextOne\ ]\)];\[IndentingNewLine]]; \ \[IndentingNewLine]\(BandNumber++\);\[IndentingNewLine]];\[IndentingNewLine]];\ \)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \(MakeBand[{y_, x_}, EdgeCode_, SpecialCell_: 0] := \[IndentingNewLine]Module[\[IndentingNewLine]{parity, pA, pB, pC, pD}, \[IndentingNewLine]parity\ = \ EvenQ[x]; \[IndentingNewLine]theEdge\ = \ Switch[\ SpecialCell, \[IndentingNewLine]0, \ If[parity, NEedgeWithFoot, NEedge], \[IndentingNewLine]1, \ If[parity, SAedgeWithFoot, SAedge], \[IndentingNewLine]2, \ If[parity, LAedgeWithFoot, LAedge]\[IndentingNewLine]]; \[IndentingNewLine]theOtherEdge\ \ = Switch[\ SpecialCell, \[IndentingNewLine]0, \ If[\(! parity\), NEedgeWithFoot, NEedge], \[IndentingNewLine]1, If[\(! parity\), SAedgeWithFoot, SAedge], \[IndentingNewLine]2, If[\(! parity\), LAedgeWithFoot, LAedge]\[IndentingNewLine]]; \[IndentingNewLine]\ \[IndentingNewLine]Switch[EdgeCode, \[IndentingNewLine]{2, 6}, If[SpecialCell \[NotEqual] 2, \ Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ {{0, .5 - w/2}, {0, .5 + w/2}, {1, .5 + w/2}, {1, .5 - w/2}, {0, .5 - w/2}\ }]], \[IndentingNewLine]{4, 8}, If[SpecialCell \[NotEqual] 2, \ \ Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ {{ .5 - w/2, 0}, { .5 + w/2, 0}, { .5 + w/2, 1}, { .5 - w/2, 1}, { .5 - w/2, 1}\ }]], \[IndentingNewLine]{4, 6}, \ Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ {{ .5 - w/2, 0}, { .5 + w/2, 0}, { .5 + w/2, .5 + w/2}, {0, .5 + w/2}, {0, .5 - w/2}, { .5 - w/2, .5 - w/2}, { .5 - w/2, 0}\ }], \[IndentingNewLine]{6, 8}, \ Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ {{0, .5 - w/2}, {0, .5 + w/2}, { .5 - w/2, .5 + w/2}, { .5 - w/2, 1}, { .5 + w/2, 1}, { .5 + w/2, .5 - w/2}, {0, .5 - w/2}\ }], \[IndentingNewLine]{2, 8}, \ Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ {{ .5 - w/2, 1}, { .5 + w/2, 1}, { .5 + w/2, .5 + w/2}, {1, .5 + w/2}, {1, .5 - w/2}, { .5 - w/2, .5 - w/2}, { .5 - w/2, 1}\ }], \[IndentingNewLine]{2, 4}, \ Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ {{1, .5 - w/2}, {1, .5 + w/2}, { .5 - w/2, .5 + w/2}, { .5 - w/2, 0}, { .5 + w/2, 0}, { .5 + w/2, .5 - w/2}, {1, .5 - w/2}\ }], \[IndentingNewLine]{1, 5}, \ \[IndentingNewLine]If[\ parity, \[IndentingNewLine]\t pA\ = \ {\(w/2\)/Sqrt[2. ], \(\(-w\)/2\)/ Sqrt[2. ]}; \[IndentingNewLine]\t pB\ = \ {1 - underW, 1 - w/Sqrt[2. ] - underW}; \[IndentingNewLine]\t pC\ = \ {1 - w/Sqrt[2. ] - underW, 1 - underW}; \[IndentingNewLine]\t pD\ = \ {\(\(-w\)/2\)/Sqrt[2. ], \(\(+w\)/2\)/ Sqrt[2. ]};\[IndentingNewLine], \[IndentingNewLine]\t pA\ = \ {w/Sqrt[2. ] + underW, 0 + underW}; \[IndentingNewLine]\t pB\ = \ {1 + \(w/2\)/Sqrt[2. ], 1 - \(w/2\)/Sqrt[2. ]}; \[IndentingNewLine]\t pC\ = \ {1 - \(w/2\)/Sqrt[2. ], 1 + \(w/2\)/ Sqrt[2. ]}; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ pD\ = \ \ {0 + underW, \(+w\)/Sqrt[2. ] + underW};\[IndentingNewLine]]; \[IndentingNewLine]Polygon[\ \ \(Plus[#, {x - 1, y - 1}] &\)\ /@ {pA, pB, pC\ , pD, pA}], \[IndentingNewLine]{3, 7}, \ \[IndentingNewLine]If[\ parity, \[IndentingNewLine]\t pA\ = \ {1 + \(w/2\)/Sqrt[2. ], \(\(+w\)/2\)/ Sqrt[2. ]}; \[IndentingNewLine]\t pB\ = \ {w/Sqrt[2. ] + underW, 1 - underW}; \[IndentingNewLine]\t pC\ = \ {0 + underW, 1 - w/Sqrt[2. ] - underW}; \[IndentingNewLine]\t pD\ = {1 - \(w/2\)/Sqrt[2. ], \(\(-w\)/2\)/ Sqrt[2. ]};\[IndentingNewLine], \[IndentingNewLine]\t pA\ = \ {1 - underW, \(+w\)/Sqrt[2. ] + underW}; \[IndentingNewLine]\t pB\ = {\(w/2\)/Sqrt[2. ], 1 + \(w/2\)/Sqrt[2. ]}; \[IndentingNewLine]\t pC\ = \ {\(\(-w\)/2\)/Sqrt[2. ], 1 - \(w/2\)/ Sqrt[2. ]}; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ pD\ = \ {1 - w/Sqrt[2. ] - underW, 0 + underW};\[IndentingNewLine]]; \[IndentingNewLine]Polygon[\ \ \(Plus[#, {x - 1, y - 1}] &\)\ /@ {pA, pB, pC\ , pD, pA}\ ], \[IndentingNewLine]{2, 5}, \[IndentingNewLine]Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ \ theEdge\ ]\[IndentingNewLine], \[IndentingNewLine]{3, 8}, \[IndentingNewLine]Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ \ \((Rot90\ /@ \ \ theEdge)\)]\ \[IndentingNewLine], \[IndentingNewLine]{2, 7}, \[IndentingNewLine]Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ \ \((HorzMirror\ /@ \ theOtherEdge)\)]\[IndentingNewLine], \[IndentingNewLine]{5, 8}, \[IndentingNewLine]Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ \ \((Rot90\ /@ \((HorzMirror\ /@ \ theEdge)\))\)]\[IndentingNewLine], \ \[IndentingNewLine]{3, 6}, \[IndentingNewLine]Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ \((VertMirror\ /@ \ \ theEdge)\)], \ \[IndentingNewLine]{1, 4}, \[IndentingNewLine]Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ \((Rot90\ /@ \ \((VertMirror\ /@ \ \ \ theOtherEdge)\))\)], \[IndentingNewLine]{1, 6}, \[IndentingNewLine]Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ \((HorzMirror /@ \((VertMirror\ /@ \ theOtherEdge)\))\)], \[IndentingNewLine]{4, 7}, \[IndentingNewLine]Polygon[\ \(Plus[#, {x - 1, y - 1}] &\)\ /@ \((Rot90 /@ \((HorzMirror\ /@ \ \ \((VertMirror\ /@ \ \ theOtherEdge)\))\))\)], \[IndentingNewLine]\ \[IndentingNewLine]_, {}\[IndentingNewLine]\[IndentingNewLine]]\ \[IndentingNewLine]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(DrawGrid[DoPrimary_: True, \ DoSecondary_: True, \ DoTertiary_: True, \ DoBreakLines_: True, \ DoThread_: True] := Module[\[IndentingNewLine]{v, h, i, j}, \[IndentingNewLine]{v, h} = \(Dimensions[\ Grid]\)[\([{1, 2}]\)]; \[IndentingNewLine]\[IndentingNewLine]\ \[IndentingNewLine]PrimaryGrid\ = \ {Red, PointSize[0.02], Table[Point[{j, i}], {i, 0, v, 2}, {j, 0, h, 2}]}; \[IndentingNewLine]SecondaryGrid\ = \ {Blue, PointSize[0.02], Table[Point[{j, i}], {i, 1, v, 2}, {j, 1, h, 2}]}; \[IndentingNewLine]TertiaryGrid\ = \ {\ Blue, \[IndentingNewLine]Table[\ Line[{{i, 0}, {i, v}}], {i, 0, h}], \[IndentingNewLine]Table[ Line[{{0, i}, {h, i}}], {i, 0, v}]}; \[IndentingNewLine]ThreadLine\ = {Red, \ Table[\ EdgeLine[i - 1, j - 1, Grid[\([j, i, EdgeCodeIndex]\)]], {i, 1, h}, {j, 1, v}]}; \[IndentingNewLine]BreakLines\ = \ {Black, Thickness[0.02], Table[\ BreakLine[i - 1, j - 1, Grid[\([j, i, BreakLinesIndex]\)]], {i, 1, h}, {j, 1, v}]}; \[IndentingNewLine]Band\ = \ {MapIndexed[\ \ {colorBands[\([1 + #1[\([\ BandNumberIndex\ ]\)]\ ]\)], MakeBand[#2, #1[\([\ EdgeCodeIndex]\)], \ #1[\([\ SpecialCellIndex]\)]\ ]} &\ , \ Grid, {2}]}; \ \[IndentingNewLine]\[IndentingNewLine]allGraphics\ = \ \ {\[IndentingNewLine]If[\ DoTertiary, TertiaryGrid, {}], \[IndentingNewLine]If[\ DoPrimary, PrimaryGrid, {}], \[IndentingNewLine]If[\ DoSecondary, SecondaryGrid, {}], \[IndentingNewLine]If[\ \ DoBreakLines, BreakLines, {}], \[IndentingNewLine]If[\ DoThread, ThreadLine, {}], \ \[IndentingNewLine]Band\[IndentingNewLine]}\ //. \ {{before___, {}, after___} \[Rule] {before, after}, \[IndentingNewLine]{before___, Null, after___} \[Rule] {before, after}\[IndentingNewLine]}; \[IndentingNewLine]\ \[IndentingNewLine]\[IndentingNewLine]\(Show[ Graphics[\[IndentingNewLine]allGraphics\[IndentingNewLine]], AspectRatio \[Rule] Automatic];\)\[IndentingNewLine]\[IndentingNewLine]];\)\), "\ \[IndentingNewLine]", \(\(w = .4;\)\), "\n", \(\(underW\ = 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