(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 215908, 10923]*) (*NotebookOutlinePosition[ 217097, 10961]*) (* CellTagsIndexPosition[ 217053, 10957]*) (*WindowFrame->Normal*) Notebook[{ Cell["Triangulations of a convex polygon", "Title", Evaluatable->False], Cell[TextData[{ "Keehong Song\nPusan National University,South Korea\nkhsong@m", StyleBox["athematica", FontSlant->"Italic"], ".co.kr" }], "Subsubtitle", Evaluatable->False], Cell[TextData[{ "This is a Mathematica Notebook which finds and visualizes the \ triangulations of a convex polygon. 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270.188}, {59.0625, 9.1875}} -> {-4, 0, 0, 0}, {{275.188, 325}, {59.0625, 9.1875}} -> {-5, 0, 0, 0}}], "\nIn general, n-polygon has the triangulations of the n-2 Catalan number, \ that is, ", Cell[BoxData[ \(1\/\(n - 1\)\)]], Cell[BoxData[ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(2 \((n - 2)\)\)}, {\(n - 2\)} }], "\[NegativeThinSpace]", ")"}]]], ".\nReferences:\nGrimaldi, R. ", StyleBox["Discrete and Combinatorial Mathematics", FontSlant->"Italic"], ", 3rd ed., Addison-Wesley,1994." }], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell["Counting binary tree", "Subsubsection"], Cell[BoxData[ RowBox[{ "The", " ", "number", " ", "of", " ", "binary", " ", "trees", " ", "with", " ", "n", " ", "nodes", " ", "is", " ", \(1\/\(n + 1\)\), RowBox[{"(", GridBox[{ {\(2 n\)}, {"n"} }], ")"}]}]], "Text"], Cell[BoxData[ \(catalanNumber[n_] := Binomial[2 n, n]/\((n + 1)\)\)], "Input", CellLabel->"In[1]:="] }, Closed]], Cell[CellGroupData[{ Cell["Generating binary tree", "Subsubsection"], Cell["\<\ Generating binary trees for traversal to find edges to be used in \ triangulations of a convex polygon, where the number of different binary \ trees is the Catalan number.\ \>", "Text"], Cell[BoxData[{ \(catalanTree[{}] := {{}}\), "\n", \(catalanTree[{a_}] := {{a}}\), "\n", \(catalanTree[l_List] := \n\t Module[{m = Rest[l], i, elem1st = First[l]}, \n\t\tJoin @@ Map[Insert[#, {elem1st}, {2}] &, \n\t\tTable[ Distribute[\n\t\t{catalanTree[Take[m, i]], \n\t\tcatalanTree[ Take[m, {i + 1, Length[m]}]]}, List], \n\t{i, 0, Length[m]}], {2}]]\)}], "Input", CellLabel->"In[2]:="], Cell["\<\ Finding all the binary trees made out of the nodes,{1,2,3,4}, when \ parenthesis seen as the tree structure.\ \>", "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(catalanTree[{1, 2, 3, 4}]\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\({}\), \({1}\), \({{}, {2}, {{}, {3}, {4}}}\)}, {\({}\), \({1}\), \({{}, {2}, {{4}, {3}, {}}}\)}, {\({}\), \({1}\), RowBox[{"(", "\[NoBreak]", GridBox[{ {"3"}, {"2"}, {"4"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}]}, {\({}\), \({1}\), \({{{}, {3}, {4}}, {2}, {}}\)}, {\({}\), \({1}\), \({{{4}, {3}, {}}, {2}, {}}\)}, {\({2}\), \({1}\), \({{}, {3}, {4}}\)}, {\({2}\), \({1}\), \({{4}, {3}, {}}\)}, {\({{}, {2}, {3}}\), \({1}\), \({4}\)}, {\({{3}, {2}, {}}\), \({1}\), \({4}\)}, {\({{}, {2}, {{}, {3}, {4}}}\), \({1}\), \({}\)}, {\({{}, {2}, {{4}, {3}, {}}}\), \({1}\), \({}\)}, { RowBox[{"(", "\[NoBreak]", GridBox[{ {"3"}, {"2"}, {"4"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], \({1}\), \({}\)}, {\({{{}, {3}, {4}}, {2}, {}}\), \({1}\), \({}\)}, {\({{{4}, {3}, {}}, {2}, {}}\), \({1}\), \({}\)} }], "\[NoBreak]", ")"}], TraditionalForm]], "Output", CellLabel->"Out[5]="] }, Closed]], Cell["\<\ By flattening the tree structure above, we get the order in which a group of \ cars can go in and out of the narrow-alley parking lot in an orderly \ fashion.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Flatten\ /@ \ %\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4"}, {"1", "2", "4", "3"}, {"1", "3", "2", "4"}, {"1", "3", "4", "2"}, {"1", "4", "3", "2"}, {"2", "1", "3", "4"}, {"2", "1", "4", "3"}, {"2", "3", "1", "4"}, {"3", "2", "1", "4"}, {"2", "3", "4", "1"}, {"2", "4", "3", "1"}, {"3", "2", "4", "1"}, {"3", "4", "2", "1"}, {"4", "3", "2", "1"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], TraditionalForm]], "Output", CellLabel->"Out[6]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Length[%] == catalanNumber[4]\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \(TraditionalForm\`True\)], "Output", CellLabel->"Out[7]="] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Finding edges involved in a triangulation of a convex polygon\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ Depth first traversal is used to find the edges which conduce to the \ triangulations of a convex polygon. There are three types of nodes in a given node of a tree: 1. both left and right children exist 2. only right child exists 3. only left child exists These three types determines the patterns of the edges of a triangulation\ \>", "Text"], Cell[BoxData[{ \(\(NodeQ = Function[c, c =!= {}];\)\), "\n", \(\(search[{lc_?NodeQ, {parent_}, rc_?NodeQ}, {l_, r_}]\n\t := \((\((AppendTo[storage, #] &)\) /@ \n\t{{l, l + Length[Flatten[lc]] + 1}, \n\t{l + Length[Flatten[lc]] + 1, r}}; \nsearch[lc, {l, l + Length[Flatten[lc]] + 1}]; \n search[rc, {l + Length[Flatten[lc]] + 1, r}])\);\)\), "\n", \(\(search[{lc_?NodeQ, {parent_}, {}}, {l_, r_}] := \n\t\((AppendTo[ storage, {l + 1, r}]; search[lc, {l + 1, r}])\);\)\), "\n", \(\(search[{{}, {parent_}, rc_?NodeQ}, {l_, r_}] := \n\t\((AppendTo[ storage, {l, v\[LeftDoubleBracket]r\[RightDoubleBracket]}]; search[rc, {l, v\[LeftDoubleBracket]r\[RightDoubleBracket]}])\);\)\n\), "\n", \(triangleEdges[n_] := \n Block[{storage = \(branches = {}\), v, m, trees}, \n\t\ v = RotateRight[Range[n + 2], 1]; \n\t\ trees = catalanTree[Range[n]]; \n\t\ Table[ search[trees\[LeftDoubleBracket]m\[RightDoubleBracket], {2, 1}]; \n\t\tAppendTo[branches, storage]; \n\t\tstorage = {}, \n\t\t{m, 1, catalanNumber[n]}]; \n\t\tbranches]\)}], "Input", CellLabel->"In[8]:=", PageWidth->Infinity, CellMargins->{{44, Inherited}, {Inherited, Inherited}}, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ We want to make sure that there is no duplicates in the set of edges for \ triangulations.\ \>", "Text"], Cell[BoxData[ \(\(zigzag = triangleEdges[5];\)\)], "Input", CellLabel->"In[13]:="], Cell[CellGroupData[{ Cell[BoxData[ \(Length@Union[Sort /@ zigzag] == Length[zigzag]\)], "Input", CellLabel->"In[14]:="], Cell[BoxData[ \(TraditionalForm\`True\)], "Output", CellLabel->"Out[14]="] }, Closed]], Cell["\<\ The following routine 'triangulation' does the physical triangulation by \ placing the line segments to the necessary positions in an n-gon.\ \>", "Text"], Cell[BoxData[ \(triangulation[n_Integer, pairs_List] := \n\t Module[{x = N[2 Pi/n], poly, vertices, edges}, \n\t\tvertices = Chop@Table[N[{Cos[x\ i], Sin[x\ i]}], {i, n}]; \n\t\tpoly = Line /@ Partition[Append[vertices, First[vertices]], 2, 1]; \n\t\tedges = Line /@ \((\(vertices[\([#]\)] &\) /@ pairs)\); \n\t\tShow[ Graphics[{poly, edges}], \n\t\tDisplayFunction -> Identity, AspectRatio -> Automatic]\n\t]\)], "Input", CellLabel->"In[15]:="], Cell[CellGroupData[{ Cell["Making the visualization process as a single function", "Subsubsection"], Cell["\<\ In making a single function out of the process of visualizing triangulation, \ a minor trouble is met due to a limitation of the built-in 'Partition' \ function: it throws away the remainding parts after partitioning the given \ set to a size. Appending the remainder at the end of the partition result \ solves the problem.\ \>", "Text"] }, Closed]], Cell[BoxData[ \(showTriangulation[n_Integer, gons_] := \n\t Module[{pic}, \n\t (*If\ the\ size\ is\ too\ small, \ do\ not\ partition*) \n\t pic = If[n >= 5, \n\t Append[Partition[gons, 15], \[IndentingNewLine]Take[ gons, \(-Mod[catalanNumber[n - 2], 15]\)]], gons]; \n\t Show[GraphicsArray[pic], DisplayFunction -> $DisplayFunction]]\)], "Input", CellLabel->"In[16]:="] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["An example for the case of 9-polygon", "Subsubsection"], Cell[TextData[{ "The number of triangulations of a n-polygon is, ", Cell[BoxData[ \(1\/\(n - 1\)\)]], Cell[BoxData[ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(2 \((n - 2)\)\)}, {\(n - 2\)} }], "\[NegativeThinSpace]", ")"}]]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(size = 8;\)\), "\n", \(\(edges = triangleEdges[size - 2];\)\), "\n", \(\(tripic = \(triangulation[size, #] &\) /@ 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Ool3000?Ool3000?Ool3000@Ool3000?Ool3000?Ool00?moob5oo`00ogoo8Goo000>Ool3000?Ool3 000@Ool3000?Ool3000?Ool3000?Ool3000@Ool3000?Ool3000?Ool3000@Ool3000?Ool3000?Ool3 000?Ool3000@Ool3000?Ool3000?Ool000]oo`<000=oo`<000Yoo`<0009oo`<000Yoo`<000=oo`80 00Yoo`<000=oo`8000]oo`80009oo`@000Yoo`800003Ool0000000<000Yoo`P000Yoo`@0009oo`<0 00Yoo`8000=oo`<000Yoo`@00004Ool00000000:Ool30003Ool2000;Ool20002Ool4000:Ool30002 Ool3000:Ool30003Ool2000:Ool400000goo00000002000Ool3000?Ool3000@Ool3000?Ool3000?Ool3000?Ool3000@Ool3000?Ool3000?Ool3000@ Ool3000?Ool3000?Ool3000?Ool3000@Ool3000?Ool3000?Ool00?moob5oo`00ogoo8Goo003oOolQ Ool00?moob5oo`00\ \>"], ImageRangeCache->{{{0, 287}, {171.063, 0}} -> {-0.395206, -0.224132, \ 0.0550188, 0.0550188}, {{7.125, 23.75}, {166.938, 150.313}} -> {-1.95002, \ -1.57107, 0.126317, 0.126317}, {{25.4375, 42.0625}, {166.938, 150.313}} -> \ {-4.2632, -1.57107, 0.126317, 0.126317}, {{43.75, 60.375}, {166.938, \ 150.313}} -> {-6.57639, -1.57107, 0.126317, 0.126317}, {{62, 78.625}, \ {166.938, 150.313}} -> {-8.88168, -1.57107, 0.126317, 0.126317}, {{80.3125, \ 96.9375}, {166.938, 150.313}} -> {-11.1949, -1.57107, 0.126317, 0.126317}, \ {{98.5625, 115.188}, {166.938, 150.313}} -> {-13.5001, -1.57107, 0.126317, \ 0.126317}, {{116.875, 133.5}, {166.938, 150.313}} -> {-15.8133, -1.57107, \ 0.126317, 0.126317}, {{135.188, 151.813}, {166.938, 150.313}} -> {-18.1265, \ -1.57107, 0.126317, 0.126317}, {{153.438, 170.063}, {166.938, 150.313}} -> \ {-20.4318, -1.57107, 0.126317, 0.126317}, {{171.75, 188.375}, {166.938, \ 150.313}} -> {-22.745, -1.57107, 0.126317, 0.126317}, {{190, 206.625}, \ {166.938, 150.313}} -> {-25.0503, -1.57107, 0.126317, 0.126317}, {{208.313, \ 224.938}, {166.938, 150.313}} -> {-27.3635, -1.57107, 0.126317, 0.126317}, \ {{7.125, 23.75}, {148.688, 132.063}} -> {-1.95002, -3.87636, 0.126317, \ 0.126317}, {{25.4375, 42.0625}, {148.688, 132.063}} -> {-4.2632, -3.87636, \ 0.126317, 0.126317}, {{43.75, 60.375}, {148.688, 132.063}} -> {-6.57639, \ -3.87636, 0.126317, 0.126317}, {{62, 78.625}, {148.688, 132.063}} -> \ {-8.88168, -3.87636, 0.126317, 0.126317}, {{80.3125, 96.9375}, {148.688, \ 132.063}} -> {-11.1949, -3.87636, 0.126317, 0.126317}, {{98.5625, 115.188}, \ {148.688, 132.063}} -> {-13.5001, -3.87636, 0.126317, 0.126317}, {{116.875, \ 133.5}, {148.688, 132.063}} -> {-15.8133, -3.87636, 0.126317, 0.126317}, \ {{135.188, 151.813}, {148.688, 132.063}} -> {-18.1265, -3.87636, 0.126317, \ 0.126317}, {{153.438, 170.063}, {148.688, 132.063}} -> {-20.4318, -3.87636, \ 0.126317, 0.126317}, {{171.75, 188.375}, {148.688, 132.063}} -> {-22.745, \ -3.87636, 0.126317, 0.126317}, {{190, 206.625}, {148.688, 132.063}} -> \ {-25.0503, -3.87636, 0.126317, 0.126317}, {{208.313, 224.938}, {148.688, \ 132.063}} -> {-27.3635, -3.87636, 0.126317, 0.126317}, {{226.563, 243.188}, \ {148.688, 132.063}} -> {-29.6687, -3.87636, 0.126317, 0.126317}, {{244.875, \ 261.5}, {148.688, 132.063}} -> {-31.9819, -3.87636, 0.126317, 0.126317}, \ {{263.188, 279.813}, {148.688, 132.063}} -> {-34.2951, -3.87636, 0.126317, \ 0.126317}, {{7.125, 23.75}, {130.375, 113.75}} -> {-1.95002, -6.18954, \ 0.126317, 0.126317}, {{25.4375, 42.0625}, {130.375, 113.75}} -> {-4.2632, \ -6.18954, 0.126317, 0.126317}, {{43.75, 60.375}, {130.375, 113.75}} -> \ {-6.57639, -6.18954, 0.126317, 0.126317}, {{62, 78.625}, {130.375, 113.75}} -> \ {-8.88168, -6.18954, 0.126317, 0.126317}, {{80.3125, 96.9375}, {130.375, \ 113.75}} -> {-11.1949, -6.18954, 0.126317, 0.126317}, {{98.5625, 115.188}, \ {130.375, 113.75}} -> {-13.5001, -6.18954, 0.126317, 0.126317}, {{116.875, \ 133.5}, {130.375, 113.75}} -> {-15.8133, -6.18954, 0.126317, 0.126317}, \ {{135.188, 151.813}, {130.375, 113.75}} -> {-18.1265, -6.18954, 0.126317, \ 0.126317}, {{153.438, 170.063}, {130.375, 113.75}} -> {-20.4318, -6.18954, \ 0.126317, 0.126317}, {{171.75, 188.375}, {130.375, 113.75}} -> {-22.745, \ -6.18954, 0.126317, 0.126317}, {{190, 206.625}, {130.375, 113.75}} -> \ {-25.0503, -6.18954, 0.126317, 0.126317}, {{208.313, 224.938}, {130.375, \ 113.75}} -> {-27.3635, -6.18954, 0.126317, 0.126317}, {{226.563, 243.188}, \ {130.375, 113.75}} -> {-29.6687, -6.18954, 0.126317, 0.126317}, {{244.875, \ 261.5}, {130.375, 113.75}} -> {-31.9819, -6.18954, 0.126317, 0.126317}, \ {{263.188, 279.813}, {130.375, 113.75}} -> {-34.2951, -6.18954, 0.126317, \ 0.126317}, {{7.125, 23.75}, {112.125, 95.5}} -> {-1.95002, -8.49483, \ 0.126317, 0.126317}, {{25.4375, 42.0625}, {112.125, 95.5}} -> {-4.2632, \ -8.49483, 0.126317, 0.126317}, {{43.75, 60.375}, {112.125, 95.5}} -> \ {-6.57639, -8.49483, 0.126317, 0.126317}, {{62, 78.625}, {112.125, 95.5}} -> \ {-8.88168, -8.49483, 0.126317, 0.126317}, {{80.3125, 96.9375}, {112.125, \ 95.5}} -> {-11.1949, -8.49483, 0.126317, 0.126317}, {{98.5625, 115.188}, \ {112.125, 95.5}} -> {-13.5001, -8.49483, 0.126317, 0.126317}, {{116.875, \ 133.5}, {112.125, 95.5}} -> {-15.8133, -8.49483, 0.126317, 0.126317}, \ {{135.188, 151.813}, {112.125, 95.5}} -> {-18.1265, -8.49483, 0.126317, \ 0.126317}, {{153.438, 170.063}, {112.125, 95.5}} -> {-20.4318, -8.49483, \ 0.126317, 0.126317}, {{171.75, 188.375}, {112.125, 95.5}} -> {-22.745, \ -8.49483, 0.126317, 0.126317}, {{190, 206.625}, {112.125, 95.5}} -> \ {-25.0503, -8.49483, 0.126317, 0.126317}, {{208.313, 224.938}, {112.125, \ 95.5}} -> {-27.3635, -8.49483, 0.126317, 0.126317}, {{226.563, 243.188}, \ {112.125, 95.5}} -> {-29.6687, -8.49483, 0.126317, 0.126317}, {{244.875, \ 261.5}, {112.125, 95.5}} -> {-31.9819, -8.49483, 0.126317, 0.126317}, \ {{263.188, 279.813}, {112.125, 95.5}} -> {-34.2951, -8.49483, 0.126317, \ 0.126317}, {{7.125, 23.75}, {93.8125, 77.1875}} -> {-1.95002, -10.808, \ 0.126317, 0.126317}, {{25.4375, 42.0625}, {93.8125, 77.1875}} -> {-4.2632, \ -10.808, 0.126317, 0.126317}, {{43.75, 60.375}, {93.8125, 77.1875}} -> \ {-6.57639, -10.808, 0.126317, 0.126317}, {{62, 78.625}, {93.8125, 77.1875}} -> \ {-8.88168, -10.808, 0.126317, 0.126317}, {{80.3125, 96.9375}, {93.8125, \ 77.1875}} -> {-11.1949, -10.808, 0.126317, 0.126317}, {{98.5625, 115.188}, \ {93.8125, 77.1875}} -> {-13.5001, -10.808, 0.126317, 0.126317}, {{116.875, \ 133.5}, {93.8125, 77.1875}} -> {-15.8133, -10.808, 0.126317, 0.126317}, \ {{135.188, 151.813}, {93.8125, 77.1875}} -> {-18.1265, -10.808, 0.126317, \ 0.126317}, {{153.438, 170.063}, {93.8125, 77.1875}} -> {-20.4318, -10.808, \ 0.126317, 0.126317}, {{171.75, 188.375}, {93.8125, 77.1875}} -> {-22.745, \ -10.808, 0.126317, 0.126317}, {{190, 206.625}, {93.8125, 77.1875}} -> \ {-25.0503, -10.808, 0.126317, 0.126317}, {{208.313, 224.938}, {93.8125, \ 77.1875}} -> {-27.3635, -10.808, 0.126317, 0.126317}, {{226.563, 243.188}, \ {93.8125, 77.1875}} -> {-29.6687, -10.808, 0.126317, 0.126317}, {{244.875, \ 261.5}, {93.8125, 77.1875}} -> {-31.9819, -10.808, 0.126317, 0.126317}, \ {{263.188, 279.813}, {93.8125, 77.1875}} -> {-34.2951, -10.808, 0.126317, \ 0.126317}, {{7.125, 23.75}, {75.5, 58.875}} -> {-1.95002, -13.1212, 0.126317, \ 0.126317}, {{25.4375, 42.0625}, {75.5, 58.875}} -> {-4.2632, -13.1212, \ 0.126317, 0.126317}, {{43.75, 60.375}, {75.5, 58.875}} -> {-6.57639, \ -13.1212, 0.126317, 0.126317}, {{62, 78.625}, {75.5, 58.875}} -> {-8.88168, \ -13.1212, 0.126317, 0.126317}, {{80.3125, 96.9375}, {75.5, 58.875}} -> \ {-11.1949, -13.1212, 0.126317, 0.126317}, {{98.5625, 115.188}, {75.5, \ 58.875}} -> {-13.5001, -13.1212, 0.126317, 0.126317}, {{116.875, 133.5}, \ {75.5, 58.875}} -> {-15.8133, -13.1212, 0.126317, 0.126317}, {{135.188, \ 151.813}, {75.5, 58.875}} -> {-18.1265, -13.1212, 0.126317, 0.126317}, \ {{153.438, 170.063}, {75.5, 58.875}} -> {-20.4318, -13.1212, 0.126317, \ 0.126317}, {{171.75, 188.375}, {75.5, 58.875}} -> {-22.745, -13.1212, \ 0.126317, 0.126317}, {{190, 206.625}, {75.5, 58.875}} -> {-25.0503, -13.1212, \ 0.126317, 0.126317}, {{208.313, 224.938}, {75.5, 58.875}} -> {-27.3635, \ -13.1212, 0.126317, 0.126317}, {{226.563, 243.188}, {75.5, 58.875}} -> \ {-29.6687, -13.1212, 0.126317, 0.126317}, {{244.875, 261.5}, {75.5, 58.875}} -> \ {-31.9819, -13.1212, 0.126317, 0.126317}, {{263.188, 279.813}, {75.5, \ 58.875}} -> {-34.2951, -13.1212, 0.126317, 0.126317}, {{7.125, 23.75}, \ {57.25, 40.625}} -> {-1.95002, -15.4265, 0.126317, 0.126317}, {{25.4375, \ 42.0625}, {57.25, 40.625}} -> {-4.2632, -15.4265, 0.126317, 0.126317}, \ {{43.75, 60.375}, {57.25, 40.625}} -> {-6.57639, -15.4265, 0.126317, \ 0.126317}, {{62, 78.625}, {57.25, 40.625}} -> {-8.88168, -15.4265, 0.126317, \ 0.126317}, {{80.3125, 96.9375}, {57.25, 40.625}} -> {-11.1949, -15.4265, \ 0.126317, 0.126317}, {{98.5625, 115.188}, {57.25, 40.625}} -> {-13.5001, \ -15.4265, 0.126317, 0.126317}, {{116.875, 133.5}, {57.25, 40.625}} -> \ {-15.8133, -15.4265, 0.126317, 0.126317}, {{135.188, 151.813}, {57.25, \ 40.625}} -> {-18.1265, -15.4265, 0.126317, 0.126317}, {{153.438, 170.063}, \ {57.25, 40.625}} -> {-20.4318, -15.4265, 0.126317, 0.126317}, {{171.75, \ 188.375}, {57.25, 40.625}} -> {-22.745, -15.4265, 0.126317, 0.126317}, {{190, \ 206.625}, {57.25, 40.625}} -> {-25.0503, -15.4265, 0.126317, 0.126317}, \ {{208.313, 224.938}, {57.25, 40.625}} -> {-27.3635, -15.4265, 0.126317, \ 0.126317}, {{226.563, 243.188}, {57.25, 40.625}} -> {-29.6687, -15.4265, \ 0.126317, 0.126317}, {{244.875, 261.5}, {57.25, 40.625}} -> {-31.9819, \ -15.4265, 0.126317, 0.126317}, {{263.188, 279.813}, {57.25, 40.625}} -> \ {-34.2951, -15.4265, 0.126317, 0.126317}, {{7.125, 23.75}, {38.9375, \ 22.3125}} -> {-1.95002, -17.7397, 0.126317, 0.126317}, {{25.4375, 42.0625}, \ {38.9375, 22.3125}} -> {-4.2632, -17.7397, 0.126317, 0.126317}, {{43.75, \ 60.375}, {38.9375, 22.3125}} -> {-6.57639, -17.7397, 0.126317, 0.126317}, \ {{62, 78.625}, {38.9375, 22.3125}} -> {-8.88168, -17.7397, 0.126317, \ 0.126317}, {{80.3125, 96.9375}, {38.9375, 22.3125}} -> {-11.1949, -17.7397, \ 0.126317, 0.126317}, {{98.5625, 115.188}, {38.9375, 22.3125}} -> {-13.5001, \ -17.7397, 0.126317, 0.126317}, {{116.875, 133.5}, {38.9375, 22.3125}} -> \ {-15.8133, -17.7397, 0.126317, 0.126317}, {{135.188, 151.813}, {38.9375, \ 22.3125}} -> {-18.1265, -17.7397, 0.126317, 0.126317}, {{153.438, 170.063}, \ {38.9375, 22.3125}} -> {-20.4318, -17.7397, 0.126317, 0.126317}, {{171.75, \ 188.375}, {38.9375, 22.3125}} -> {-22.745, -17.7397, 0.126317, 0.126317}, \ {{190, 206.625}, {38.9375, 22.3125}} -> {-25.0503, -17.7397, 0.126317, \ 0.126317}, {{208.313, 224.938}, {38.9375, 22.3125}} -> {-27.3635, -17.7397, \ 0.126317, 0.126317}, {{226.563, 243.188}, {38.9375, 22.3125}} -> {-29.6687, \ -17.7397, 0.126317, 0.126317}, {{244.875, 261.5}, {38.9375, 22.3125}} -> \ {-31.9819, -17.7397, 0.126317, 0.126317}, {{263.188, 279.813}, {38.9375, \ 22.3125}} -> {-34.2951, -17.7397, 0.126317, 0.126317}, {{7.125, 23.75}, \ {20.6875, 4.0625}} -> {-1.95002, -20.045, 0.126317, 0.126317}, {{25.4375, \ 42.0625}, {20.6875, 4.0625}} -> {-4.2632, -20.045, 0.126317, 0.126317}, \ {{43.75, 60.375}, {20.6875, 4.0625}} -> {-6.57639, -20.045, 0.126317, \ 0.126317}, {{62, 78.625}, {20.6875, 4.0625}} -> {-8.88168, -20.045, 0.126317, \ 0.126317}, {{80.3125, 96.9375}, {20.6875, 4.0625}} -> {-11.1949, -20.045, \ 0.126317, 0.126317}, {{98.5625, 115.188}, {20.6875, 4.0625}} -> {-13.5001, \ -20.045, 0.126317, 0.126317}, {{116.875, 133.5}, {20.6875, 4.0625}} -> \ {-15.8133, -20.045, 0.126317, 0.126317}, {{135.188, 151.813}, {20.6875, \ 4.0625}} -> {-18.1265, -20.045, 0.126317, 0.126317}, {{153.438, 170.063}, \ {20.6875, 4.0625}} -> {-20.4318, -20.045, 0.126317, 0.126317}, {{171.75, \ 188.375}, {20.6875, 4.0625}} -> {-22.745, -20.045, 0.126317, 0.126317}, \ {{190, 206.625}, {20.6875, 4.0625}} -> {-25.0503, -20.045, 0.126317, \ 0.126317}, {{208.313, 224.938}, {20.6875, 4.0625}} -> {-27.3635, -20.045, \ 0.126317, 0.126317}, {{226.563, 243.188}, {20.6875, 4.0625}} -> {-29.6687, \ -20.045, 0.126317, 0.126317}, {{244.875, 261.5}, {20.6875, 4.0625}} -> \ {-31.9819, -20.045, 0.126317, 0.126317}, {{263.188, 279.813}, {20.6875, \ 4.0625}} -> {-34.2951, -20.045, 0.126317, 0.126317}}], Cell[BoxData[ 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