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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[ 888059, 35661]*) (*NotebookOutlinePosition[ 933510, 37235]*) (* CellTagsIndexPosition[ 933340, 37226]*) (*WindowFrame->Normal*) Notebook[{ Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg"], "Graphics", ShowCellBracket->False, CellMargins->{{25, 24}, {5, 7}}, ImageSize->{81, 22}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "2D and 3D Measurements\n", StyleBox[ "Authors: Bill Davis, Horacio Porta and Jerry Uhl ", "Subtitle"], StyleBox["\[Copyright]1999", "Subtitle", FontSize->12], StyleBox["\nProducer: Bruce Carpenter\n", "Subtitle"], StyleBox["Publisher: ", "Subtitle", FontSize->12], StyleBox[ButtonBox["Math Everywhere, Inc.", ButtonData:>{ URL[ "http://www.matheverywhere.com"], None}, ButtonStyle->"MEIHyperlink"], FontSize->12], StyleBox[" Distributor: ", "Subtitle", FontSize->12], StyleBox[ButtonBox["Wolfram Research, Inc.", ButtonData:>{ URL[ "http://www.wolfram.com"], None}, ButtonStyle->"MEIHyperlink", ButtonNote->"Makers of Mathematica!"], FontSize->12] }], "PrefaceTitle", CellMargins->{{Inherited, Inherited}, {Inherited, 0}}], Cell[TextData[{ "VC.07 Transforming 2D Integrals\n", StyleBox["Samples", FontSize->16, FontSlant->"Italic"] }], "Title"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Initializations" }], "Special2"], Cell[BoxData[{ \(\(Off[General::spell];\)\), "\n", \(\(Off[General::spell1];\)\), "\n", \(\(Off[ParametricPlot3D::ppcom];\)\), "\n", \(\(Off[ParametricPlot::ppcom];\)\), "\n", \(\(Off[Plot::plnr];\)\), "\n", \(\(SetOptions[Limit, Analytic \[Rule] True];\)\)}], "Input", InitializationCell->True], Cell[BoxData[ \(\(CMView = {2.7, 1.6, 1.2};\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\($Pre\ = \ Chop; 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Cell[CellGroupData[{ Cell["\<\ B.3) Transforming 2D integrals: How you do it and why you do \ it\ \>", "Subsection", CellTags->"VC.07.B3"], Cell["This assumes familarity with B.1) and B.2)", "Special2"], Cell[TextData[{ "B.3.a) The instantanteous area conversion factor ", Cell[BoxData[ \(A\_xy[u, v]\)]] }], "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Knowing that\n ", Cell[BoxData[ RowBox[{ \(\[Integral]\_\(\ x[a]\)\%\(\ x[b]\)f[x] \[DifferentialD]x\), "=", RowBox[{\(\[Integral]\_\(\ a\)\%\(\ b\)\), RowBox[{\(f[x[u]]\), RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "u", "]"}], \(\[DifferentialD]u\)}]}]}]]], " \nis a mark of calculus literacy. \nBut without the fudge function \n \ ", Cell[BoxData[ RowBox[{\(fudge[u]\), "=", RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "u", "]"}]}]]], ",\nthe transformation fails.\nIn the two-variable case, when you use \ functions ", Cell[BoxData[ \(x[u, v]\)]], " and and ", Cell[BoxData[ \(y[u, v]\)]], " to go from integrating on ", Cell[BoxData[ \(xy\)]], "-paper to integrating on ", Cell[BoxData[ \(uv\)]], "-paper, you've got to come up with the fudge function, ", Cell[BoxData[ \(fudge[u, v]\)]], ", that makes\n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\ R\_xy\)f[x, y] \[DifferentialD]x \[DifferentialD]y\)\)]], " \n ", Cell[BoxData[ \(\( = \[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\ R\_uv\)f[x[u, v], y[u, v]] fudge[u, v] \[DifferentialD]u \[DifferentialD]v\)\)\)]], "\nwhere ", Cell[BoxData[ \(R\_uv\)]], " is the ", Cell[BoxData[ \(uv\)]], "-paper plot of the region ", Cell[BoxData[ \(\(\ R\_xy\)\)]], " orginally plotted on ", Cell[BoxData[ \(xy\)]], "-paper.\nWhat is the physical meaning of this function\n ", Cell[BoxData[ \(fudge[u, v]\)]], "?" }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "The function ", Cell[BoxData[ \(fudge[u, v]\)]], " has to satisfy\n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\ R\_xy\)f[x, y] \[DifferentialD]x \[DifferentialD]y\)\)]], " \n ", Cell[BoxData[ \(\( = \[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\ R\_uv\)f[x[u, v], y[u, v]] fudge[u, v] \[DifferentialD]u \[DifferentialD]v\)\)\)]], ".\nNo matter what ", Cell[BoxData[ \(f[x, y]\)]], " you have.\nYou get a pregnant clue to the physical meaning of ", Cell[BoxData[ \(fudge[u, v]\)]], " by taking \n ", Cell[BoxData[ \(f[x, y] = 1\)]], ". \nFor this particular ", Cell[BoxData[ \(f[x, y]\)]], ", you get\n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\ R\_xy\)\[DifferentialD]x \[DifferentialD]y\) = \[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\ R\_uv\)fudge[u, v] \[DifferentialD]u \[DifferentialD]v\)\)]], ".\nPut ", Cell[BoxData[ \(Area[R\_xy]\)]], " equal to the area of ", Cell[BoxData[ \(R\_xy\)]], " measured on ", Cell[BoxData[ \(xy\)]], "-paper and realize that this equation means that\n ", Cell[BoxData[ \(Area[R\_xy] = \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\ 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conversion factor because it converts ", Cell[BoxData[ \(uv\)]], "-paper area measurements into ", Cell[BoxData[ \(xy\)]], "-paper area measurements. " }], "SmallText"] }, Open ]], Cell[TextData[{ "B.3.b) The formula for the area conversion factor ", Cell[BoxData[ \(A\_xy[u, v]\)]] }], "Subsubsection"], Cell[TextData[{ "Now go for the throat.\nThe formula for the area conversion factor ", Cell[BoxData[ \(A\_xy[u, v]\)]], " that you integrate to convert ", Cell[BoxData[ \(uv\)]], "-paper area measurements to ", Cell[BoxData[ \(xy\)]], "-paper area measurements is: " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, y, u, v, gradx, grady, Axy];\)\), "\n", \(\(gradx[u_, v_] = {D[x[u, v], u], D[x[u, v], v]};\)\), "\n", \(\(grady[u_, v_] = {D[y[u, v], u], D[y[u, v], v]};\)\), "\n", \(Axy[u_, v_] = Abs[Det[{gradx[u, v], grady[u, v]}]]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{"Abs", "[", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["y", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(u, v\), "]"}], " ", RowBox[{ SuperscriptBox["x", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(u, v\), "]"}]}], "-", RowBox[{ RowBox[{ SuperscriptBox["x", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(u, v\), "]"}], " ", RowBox[{ SuperscriptBox["y", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(u, v\), "]"}]}]}], "]"}]], "Output"] }, Open ]], Cell["Some of the fancy folks call this the Jacobian determinant.", "Special2"], Cell[CellGroupData[{ Cell["Where does this beauty come from?", "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Ultimately, it comes from the cross product.\nStart with\n ", Cell[BoxData[ \(x = x[u, v]\)]], ", and \n ", Cell[BoxData[ \(y = y[u, v]\)]], ".\nAt a fixed ", Cell[BoxData[ \(uv\)]], "-paper point ", Cell[BoxData[ \({a, b}\)]], ", the linearized versions of ", Cell[BoxData[ \(x[u, v]\)]], " and ", Cell[BoxData[ \(y[u, v]\)]], " are calculated as follows:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, y, u, v, a, b, gradx, grady, linearx, lineary];\)\), "\n", \(\(gradx[u_, v_] = {D[x[u, v], u], D[x[u, v], v]};\)\), "\n", \(\(grady[u_, v_] = {D[y[u, v], u], D[y[u, v], v]};\)\), "\n", \(linearx[u_, v_] = x[a, b] + gradx[a, b] . {u - a, v - b}\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(x[a, b]\), "+", RowBox[{\((\(-b\) + v)\), " ", RowBox[{ SuperscriptBox["x", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}]}], "+", RowBox[{\((\(-a\) + u)\), " ", RowBox[{ SuperscriptBox["x", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}]}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(lineary[u_, v_] = y[a, b] + grady[a, b] . {u - a, v - b}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(y[a, b]\), "+", RowBox[{\((\(-b\) + v)\), " ", RowBox[{ SuperscriptBox["y", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}]}], "+", RowBox[{\((\(-a\) + u)\), " ", RowBox[{ SuperscriptBox["y", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}]}]}]], "Output"] }, Open ]], Cell[TextData[{ "Remember from B.2:\nOn ", Cell[BoxData[ \(uv\)]], "-paper, the closer you are to ", Cell[BoxData[ \({a, b}\)]], ", the more spectacularly the ", Cell[BoxData[ \(xy\)]], "-grid is approximated by the ", Cell[BoxData[ \(linearxlineary\)]], "-grid. As a result, at the point ", Cell[BoxData[ \({a, b}\)]], ", the two area conversion factors are the same; in other words \t \n \ ", Cell[BoxData[ \(A\_xy[a, b] = A\_linearxlineray[a, b]\)]], ".\nNow calculate ", Cell[BoxData[ \(A\_linearxlineray[a, b]\)]], ":\nTo do this, remember that the linear grid is a bunch of parallelograms. \ For this reason, the ", Cell[BoxData[ \(uv\)]], "-paper square with corners at \n ", Cell[BoxData[ \({a, b}\)]], ", ", Cell[BoxData[ \({a + \ h, b}\)]], ", ", Cell[BoxData[ \({a + \ h, b + \ h}\)]], " and ", Cell[BoxData[ \({a, b + \ h}\)]], " \nplots out on ", Cell[BoxData[ \(linearxlineary\)]], "-paper as the parallelogram with corners at:" }], "SmallText"], Cell["Activate the next cell.", "Special2"], Cell[BoxData[{ \(\(Clear[h];\)\), "\n", \(\(basepoint = {linearx[a, b], lineary[a, b]};\)\), "\n", \(\(corner1 = {linearx[a + h, b], lineary[a + h, b]};\)\), "\n", \(\(corner2 = {linearx[a + h, b + h], lineary[a + h, b + h]};\)\), "\n", \(\(corner3 = {linearx[a, b + h], lineary[a, b + h]};\)\)}], "Input", AspectRatioFixed->True], Cell["\<\ The area of this parallelogram is given by the absolute value of:\ \ \>", "SmallText"], Cell[CellGroupData[{ Cell["\<\ If you want to refresh yourself about where the ingredients of this \ calculation come from, double click the box.\ \>", "Special2"], Cell[TextData[{ "The area of the parallelogram determined by any vectors ", Cell[BoxData[ \(X\)]], " and ", Cell[BoxData[ \(Y\)]], " with their tails at a common base point is \n ", Cell[BoxData[ \(\[LeftDoubleBracketingBar]X\[RightDoubleBracketingBar] \[LeftDoubleBracketingBar]Y\[RightDoubleBracketingBar] \[LeftBracketingBar]Sin[angle\ between]\[RightBracketingBar]\)]], "\nwhich is the same as the length of the base times the perpendicular \ height. To understand the calculation, put everything in three dimensions as \ follows:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[X, x, Y, y];\)\), "\n", \(X = {x[1], x[2]}\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({x[1], x[2]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(X3D = {x[1], x[2], 0}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({x[1], x[2], 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Y = {y[1], y[2]}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({y[1], y[2]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Y3D = {y[1], y[2], 0}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({y[1], y[2], 0}\)], "Output"] }, Open ]], Cell["Note:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\@\(X . X\) == \@\(X3D . X3D\)\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\@\(Y . Y\) == \@\(Y3D . Y3D\)\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "This confirms that ", Cell[BoxData[ \(\[LeftDoubleBracketingBar]X\[RightDoubleBracketingBar] = \[LeftDoubleBracketingBar]X3D\[RightDoubleBracketingBar]\)]], " and ", Cell[BoxData[ \(\[LeftDoubleBracketingBar]Y\[RightDoubleBracketingBar] = \[LeftDoubleBracketingBar]Y3D\[RightDoubleBracketingBar]\)]], ". So\n ", Cell[BoxData[ \(area = \[LeftDoubleBracketingBar]X\[RightDoubleBracketingBar] \[LeftDoubleBracketingBar]Y\[RightDoubleBracketingBar] \[LeftBracketingBar]Sin[angle\ between]\[RightBracketingBar]\)]], " \n ", Cell[BoxData[ \(\( = \[LeftDoubleBracketingBar]X3D\[RightDoubleBracketingBar] \[LeftDoubleBracketingBar]Y3D\[RightDoubleBracketingBar] \[LeftBracketingBar]Sin[angle\ between]\[RightBracketingBar]\)\)]], ".\nNow look at:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(X3DcrossY3D = Cross[X3D, Y3D]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({0, 0, \(-x[2]\)\ y[1] + x[1]\ y[2]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\((X3D . Y3D)\)\^2 + X3DcrossY3D . X3DcrossY3D]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(x[1]\^2\ y[1]\^2 + x[2]\^2\ y[1]\^2 + x[1]\^2\ y[2]\^2 + x[2]\^2\ y[2]\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[X3D . X3D\ Y3D . Y3D]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(x[1]\^2\ y[1]\^2 + x[2]\^2\ y[1]\^2 + x[1]\^2\ y[2]\^2 + x[2]\^2\ y[2]\^2\)], "Output"] }, Open ]], Cell[TextData[{ "Remembering that \n ", Cell[BoxData[ \(\((X3D . Y3D)\)\^2 = \(\[LeftDoubleBracketingBar]X3D\[RightDoubleBracketingBar]\^2\) \(\[LeftDoubleBracketingBar]Y3D\[RightDoubleBracketingBar]\^2\) Cos[angle\ between]\^2\)]], ", \nyou can see from the calculations immediately above that\n ", Cell[BoxData[ \(\(\[LeftDoubleBracketingBar]X3D\[RightDoubleBracketingBar]\^2\) \(\[LeftDoubleBracketingBar]Y3D\[RightDoubleBracketingBar]\^2\) Cos[angle\ between]\^2 + \ \[LeftDoubleBracketingBar]X3D\[Cross]Y3D\[RightDoubleBracketingBar]\^2 \)]], "\n ", Cell[BoxData[ \(\( = \(\[LeftDoubleBracketingBar]X3D\[RightDoubleBracketingBar]\^2\) \[LeftDoubleBracketingBar]Y3D\[RightDoubleBracketingBar]\^2\)\)]], ".\nSo \n ", Cell[BoxData[ \(\[LeftDoubleBracketingBar]X3D\[Cross]Y3D \[RightDoubleBracketingBar]\^2 = \(\[LeftDoubleBracketingBar]X3D\[RightDoubleBracketingBar]\^2\) \(\[LeftDoubleBracketingBar]Y3D\[RightDoubleBracketingBar]\^2\) \((1 - \ Cos[angle\ between]\^2)\)\)]], "\n ", Cell[BoxData[ \(\( = \(\[LeftDoubleBracketingBar]X3D\[RightDoubleBracketingBar]\^2\) \(\[LeftDoubleBracketingBar]Y3D\[RightDoubleBracketingBar]\^2\) Sin[angle\ between]\^2\)\)]], "\nConsequently,\n ", Cell[BoxData[ \(\[LeftDoubleBracketingBar]X3D\[Cross]Y3D\[RightDoubleBracketingBar] = \[LeftDoubleBracketingBar]X3D\[RightDoubleBracketingBar] \[LeftDoubleBracketingBar]Y3D\[RightDoubleBracketingBar] \[LeftBracketingBar]Sin[angle\ between]\[RightBracketingBar]\)]], "\n ", Cell[BoxData[ \(\( = area\ of\ parallelogram\ determined\ by\ X\ and\ Y\)\)]], ".\nEnd of explanation." }], "SmallText"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(X = corner1 - basepoint;\)\), "\n", \(\(Y = corner3 - basepoint;\)\), "\n", \(\(X3D = Append[X, 0];\)\), "\n", \(\(Y3D = Append[Y, 0];\)\), "\n", \(\(X3DcrossY3D = Cross[X3D, Y3D];\)\), "\n", \(AreaofParallelogram = \[Sqrt]Collect[Expand[X3DcrossY3D . X3DcrossY3D], h\^4]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{\(h\^4\), " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["y", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}], "2"], " ", SuperscriptBox[ RowBox[{ SuperscriptBox["x", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}], "2"]}], "-", RowBox[{"2", " ", RowBox[{ SuperscriptBox["x", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}], " ", RowBox[{ SuperscriptBox["y", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}], " ", RowBox[{ SuperscriptBox["x", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}], " ", RowBox[{ SuperscriptBox["y", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}]}], "+", RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["x", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}], "2"], " ", SuperscriptBox[ RowBox[{ SuperscriptBox["y", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}], "2"]}]}], ")"}]}], ")"}]}]], "Output"] }, Open ]], Cell[TextData[{ "The original ", Cell[BoxData[ \(uv\)]], "-paper square with corners at \n ", Cell[BoxData[ \({a, b}\)]], ", ", Cell[BoxData[ \({a + \ h, b}\)]], ", ", Cell[BoxData[ \({a + \ h, b + \ h}\)]], ", ", Cell[BoxData[ \({a, b + \ h}\)]], " \nhas area measuring out to ", Cell[BoxData[ \(h\^2\)]], ". \nSo the area conversion factor\n ", Cell[BoxData[ \(A\_xy[a, b] = A\_linearxlineary[a, b]\)]], "\nis given by the absolute value of: " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[AreaofParallelogram\/h\^2]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FractionBox[ SqrtBox[ RowBox[{\(h\^4\), " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["y", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}], " ", RowBox[{ SuperscriptBox["x", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}]}], "-", RowBox[{ RowBox[{ SuperscriptBox["x", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}], " ", RowBox[{ SuperscriptBox["y", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}]}]}], ")"}], "2"]}]], \(h\^2\)]], "Output"] }, Open ]], Cell["\<\ Not so bad because this is the same as the absolute value of: \ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Det[{gradx[u, v], grady[u, v]}] /. {u \[Rule] a, v \[Rule] b}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["y", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}], " ", RowBox[{ SuperscriptBox["x", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(a, b\), "]"}]}], "-", RowBox[{ RowBox[{ SuperscriptBox["x", TagBox[\((0, 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Don't worry about the stray ends.\nCalculate\n \ ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\ R\_xy\)\%\[Null] x\ \(y\^2\) \[DifferentialD]x \[DifferentialD]y\)\)]], "\nwithout a lot of weeping, wailing, and gnashing of teeth by transforming \ the integral to new variables ", Cell[BoxData[ \(u\)]], " and ", Cell[BoxData[ \(v\)]], "." }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "This is a real bastard to calculate without leaving ", Cell[BoxData[ \(xy\)]], "-paper because this integral will have to be broken into three integrals \ with lots of weeping, wailing, and gnashing of teeth. \nBut this is a great \ set-up for transforming to an integral easily set up on ", Cell[BoxData[ \(uv\)]], "-paper.\nThe ", Cell[BoxData[ \(R\_xy\)]], " region has its boundary formed by the curves \n ", Cell[BoxData[ \(y = 0.8 x\)]], ", ", Cell[BoxData[ \(y = 0.8 x + \ 0.5\)]], " , \n ", Cell[BoxData[ \(x\ y = 0.2\)]], ", and ", Cell[BoxData[ \(x\ y = 0.6\)]], ".\nUse \n 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\(y\^2\) \[DifferentialD]x \[DifferentialD]y\)\)]], "\n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\ R\_uv\)\%\[Null] x[u, v]\ \(y[u, v]\^2\) A\_xy[u, v] \[DifferentialD]u \[DifferentialD]v\)\)]], "\n ", Cell[BoxData[ \(\[Integral]\_0.2\%0.6\(\[Integral]\_0\%0.5\ x[u, v]\ \(y[u, v]\^2\) A\_xy[u, v] \[DifferentialD]u \[DifferentialD]v\)\)]], ".\nYou can let ", StyleBox["Mathematica", FontSlant->"Italic"], " mop this up as soon as you find out what ", Cell[BoxData[ \(x[u, v]\)]], ", ", Cell[BoxData[ \(y[u, v]\)]], ", and ", Cell[BoxData[ \(A\_xy[u, v]\)]], " are:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{u == u[x, y], v == v[x, y]}, {x, y}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({{x \[Rule] 0.25`\ \((\(-2.5`\)\ u - 0.5`\ \@\(25.`\ u\^2 + 80.`\ v\))\), y \[Rule] 0.1`\ \((5.`\ u - 1.`\ \@\(25.`\ u\^2 + 80.`\ v\))\)}, {x \[Rule] 0.25`\ \((\(-2.5`\)\ u + 0.5`\ \@\(25.`\ u\^2 + 80.`\ v\))\), y \[Rule] 0.1`\ \((5.`\ u + \@\(25.`\ u\^2 + 80.`\ v\))\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "The original region ", Cell[BoxData[ \(R\_xy\)]], " consists of ", Cell[BoxData[ \({x, y}\)]], "'s with positive coordinates, so you use:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[Axy, gradx, grady];\)\), "\n", \(\(x[u_, v_] = 0.25\ \((\(-2.5\)\ u + 0.5\ \@\(25\ u\^2 + 80\ v\))\);\)\), "\n", \(\(y[u_, v_] = 0.1\ \((5\ u + \@\(25\ u\^2 + 80\ v\))\);\)\), "\n", \(\(gradx[u_, v_] = {D[x[u, v], u], D[x[u, v], v]};\)\), "\n", \(\(grady[u_, v_] = {D[y[u, v], u], D[y[u, v], v]};\)\), "\n", \(Axy[u_, v_] = Det[{gradx[u, v], grady[u, v]}]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-\(5.`\/\@\(25\ u\^2 + 80\ v\)\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "This is negative; so throw in an extra minus sign to arrive at\n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\ R\_xy\)\%\[Null] x\ y\^2\ \[DifferentialD]x \[DifferentialD]y\)\)]], "\n ", Cell[BoxData[ \(\( = 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The choice of ", Cell[BoxData[ \(uv\)]], "-paper was more or less dictated by the set-up of the problem. \nIt saved \ a lot of miserable, irritating algebra. " }], "SmallText"] }, Open ]], Cell["B.3.c.iii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "In calculating the last two integrals, what was the decisive advantage in \ switching from ", Cell[BoxData[ \(xy\)]], "-paper to ", Cell[BoxData[ \(uv\)]], "-paper?" }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Setting up the integrals on ", Cell[BoxData[ \(xy\)]], "-paper would have been frustrating.\nSetting up the integrals by \ transforming them to ", Cell[BoxData[ \(uv\)]], "-paper was a breeze; it's the modern, up-to-date way of doing it. " }], "SmallText"] }, Open ]] }, Closed]], Cell["Tutorial Problem", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "T.1) Transforming ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \)\(R\_xy\)\)\%\ \[Null]\ f[x, y]\ \[DifferentialD]x \[DifferentialD]y\)\)]], " when the boundary of ", Cell[BoxData[ \(R\_xy\)]], " is given by parametric formulas" }], "Subsection", CellTags->"VC.07.T1"], Cell[TextData[{ "When the boundary of a region ", Cell[BoxData[ \(R\_xy\)]], " is plotted with parametric formulas, you often have a good shot at using \ the parametric formulas to come up with ", Cell[BoxData[ \(uv\)]], "-paper on which ", Cell[BoxData[ \(R\_uv\)]], " is a rectangle. 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Cos[t], \(-2\) + \@5\ Sin[t]}\)], "Output"] }, Open ]], Cell["Put", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, y, u, t];\)\), "\n", \({x[u_, t_], y[u_, t_]} = {1, \(-2\)} + u\ {Cos[t], Sin[t]}\)}], "Input",\ AspectRatioFixed->True], Cell[BoxData[ \({1 + u\ Cos[t], \(-2\) + u\ Sin[t]}\)], "Output"] }, Open ]], Cell[TextData[{ "Realize this:\nWhen you run ", Cell[BoxData[ \(u\)]], " from ", Cell[BoxData[ \(0\)]], " to ", Cell[BoxData[ \(\@5\)]], " , and you run ", Cell[BoxData[ \(t\)]], " from ", Cell[BoxData[ \(0\)]], " to ", Cell[BoxData[ \(2 \[Pi]\)]], ", then ", Cell[BoxData[ \({x[u, t], y[u, t]}\)]], " runs through all of ", Cell[BoxData[ \(R\_xy\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \({{ulow = 0, uhigh = \@5}, {tlow = 0, thigh = 2\ \[Pi]}}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({{0, \@5}, {0, 2\ \[Pi]}}\)], "Output"] }, Open ]], Cell[TextData[{ "The upshot:\n", Cell[BoxData[ \(R\_uv\)]], " is the rectangle \n ", Cell[BoxData[ \(ulow \[LessEqual] u \[LessEqual] uhigh\)]], " and ", Cell[BoxData[ \(tlow \[LessEqual] t \[LessEqual] thigh\)]], "\non ", Cell[BoxData[ \(ut\)]], "-paper.\nThis, and the fact that \n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \)\(R\_xy\)\)\%\ \[Null]\ f[x, y] \[DifferentialD]x \[DifferentialD]y\)\)]], "\n \n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \ \)\(R\_ut\)\)\%\[Null]\ f[x[u, t], y[u, t]] A\_xy[u, t] \[DifferentialD]u \[DifferentialD]t\)\)\)\)]], "\n \n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_tlow\%thigh\(\[Integral]\_ulow\%uhigh\ f[x[u, t], y[u, t]] A\_xy[u, t] \[DifferentialD]u \[DifferentialD]t\)\)\)\)]], ",\ntell you that you can turn everything over to the machine,\nright now:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[gradx, grady, Axy];\)\), "\n", \(\(gradx[u_, t_] = {D[x[u, t], u], D[x[u, t], t]};\)\), "\n", \(\(grady[u_, t_] = {D[y[u, t], u], D[y[u, t], t]};\)\), "\n", \(Axy[u_, t_] = TrigExpand[Det[{gradx[u, t], grady[u, t]}]]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(u\)], "Output"] }, Open ]], Cell[TextData[{ "Here comes the calculation of ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \)\(R\_xy\)\)\%\ \[Null]\ \((3 x\^2 + \ 5 y\^4)\)\ \[DifferentialD]x \[DifferentialD]y\)\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(calculation = Simplify[\[Integral]\_tlow\%thigh\(\[Integral]\_ulow\%uhigh\(\((3\ x[u, \ t]\^2 + 5\ y[u, t]\^4)\)\ Axy[u, t]\) \[DifferentialD]u \[DifferentialD]t\)]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(10095\ \[Pi]\)\/8\)], "Output"] }, Open ]], Cell["Nasty answer, but it wasn't hard to get.", "SmallText"] }, Open ]], Cell["T.1.a.ii)", "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ Could you have used the Gauss-Green formula to calculate the \ integral in part i)?\ \>", "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Yes. \nHere is how it goes:\nYou want 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\[DifferentialD]t\)\)\)]], ",\nwhere ", Cell[BoxData[ \(a = 0\)]], " and ", Cell[BoxData[ \(b = 2 \[Pi]\)]], ". \nTo calculate \n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \)\(R\_xy\)\)\ f[x, y] \[DifferentialD]x \[DifferentialD]y\)\)]], ",\nYou just say \n ", Cell[BoxData[ \(m[x, y] = 0\)]], ", and \n ", Cell[BoxData[ \(n[x, y] = \[Integral]\_0\%x f[s, y] \[DifferentialD]s\)]], ": " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[n, m, s, y];\)\), "\n", \(\(m[x_, y_] = 0;\)\), "\n", \(n[x_, y_] = \[Integral]\_0\%x f[s, y] \[DifferentialD]s\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(x\^3 + 5\ x\ y\^4\)], "Output"] }, Open ]], Cell[TextData[{ "This gives you ", Cell[BoxData[ \(f[x, y] = D[n[x, y], x] - \ D[m[x, y], y]\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \({D[n[x, y], x] - D[m[x, y], y], f[x, y]}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({3\ x\^2 + 5\ y\^4, 3\ x\^2 + 5\ y\^4}\)], "Output"] }, Open ]], Cell[TextData[{ "Now you know that\n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \)\(R\_xy\)\)\ f[x, y] \[DifferentialD]x \[DifferentialD]y\)\)]], "\n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \)\(R\_xy\)\)\ \((D[ n[x, y], x] - \ D[m[x, y], y])\) \[DifferentialD]x \[DifferentialD]y\)\)]], "\n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_a\%b\ m[x[t], y[t]] \(x'\)[t] + \ n[x[t], y[t]]\ \(y'\)[t] \[DifferentialD]t\)\)\)]], ": " }], "SmallText"], Cell[TextData[{ "The reason you re-enter ", Cell[BoxData[ \({x[t], y[t]}\)]], " is that ", Cell[BoxData[ \(x\)]], " and ", Cell[BoxData[ \(y\)]], " were cleared \nwhen you calculated ", Cell[BoxData[ \(m[x, y]\)]], " and ", Cell[BoxData[ \(n[x, y]\)]], "." }], "Special2"], Cell[CellGroupData[{ Cell[BoxData[{\({x[t_], y[t_]} = {1, \(-2\)} + \@5\ {Cos[t], Sin[t]};\), "\n", RowBox[{"GGcalculation", "=", RowBox[{\(\[Integral]\_a\%b\), RowBox[{ RowBox[{"(", RowBox[{ RowBox[{\(m[x[t], y[t]]\), " ", RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", RowBox[{\(n[x[t], y[t]]\), " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]}], ")"}], \(\[DifferentialD]t\)}]}]}]}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(10095\ \[Pi]\)\/8\)], "Output"] }, Open ]], Cell["\<\ As expected, this is the same nasty answer you got in part \ i).\ \>", "SmallText"] }, Open ]], Cell["T.1.a.iii)", "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ How do you decide whether to go with transformations as in part i), \ or with the Gauss-Green approach as in part ii)?\ \>", "Text"], Cell["Answer:", "Special1"], Cell["\<\ From the scientific point of view, it's a toss-up. Both methods \ work, so the decision is really a matter of personal choice. Most folks prefer the approach using transformations in part i). But when \ you're putting on the ritz, you might want to go with the Gauss-Green \ approach. Because you're already familiar with the Gauss-Green formula, this lesson \ will concentrate on the approach using transformations. You should do that \ too, because the approach using transformations has a clear extension to \ three dimensions. 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1.00409 L p .831 .763 .788 r F P 0 g s .55297 .91215 m .54933 .90047 L .61534 .79739 L p .925 .916 .789 r F P 0 g s .22165 .48169 m .2366 .51506 L .20597 .56298 L p .329 .337 .717 r F P 0 g s .22249 .59062 m .20597 .56298 L .2366 .51506 L p .335 .348 .726 r F P 0 g s .38708 1.09492 m .39479 1.06243 L .47426 1.03031 L p .742 .636 .742 r F P 0 g s .47636 1.00409 m .47426 1.03031 L .39479 1.06243 L p .741 .641 .749 r F P 0 g s .31495 1.03174 m .32733 1.00704 L .39479 1.06243 L p .683 .571 .722 r F P 0 g s .39479 1.06243 m .38708 1.09492 L .31495 1.03174 L p .683 .571 .722 r F P 0 g s .29487 .87861 m .27929 .88623 L .25501 .74717 L p .602 .512 .725 r F P 0 g s .27929 .88623 m .29487 .87861 L .33972 .98371 L p .645 .539 .72 r F P 0 g s .33972 .98371 m .32733 1.00704 L .27929 .88623 L p .645 .539 .72 r F P 0 g s .7257 .64166 m .74204 .61585 L .69655 .67075 L p 0 0 0 r F P 0 g s .39479 1.06243 m .4025 1.03185 L .47636 1.00409 L p .741 .641 .749 r F P 0 g s .47847 .97944 m .47636 1.00409 L .4025 1.03185 L p .74 .647 .756 r F P 0 g s .47636 1.00409 m .47847 .97944 L .54933 .90047 L p .831 .763 .788 r F P 0 g s .32733 1.00704 m .33972 .98371 L .4025 1.03185 L p .683 .575 .728 r F P 0 g s .4025 1.03185 m .39479 1.06243 L .32733 1.00704 L p .683 .575 .728 r F P 0 g s .54568 .88937 m .54933 .90047 L .47847 .97944 L p .827 .769 .798 r F P 0 g s .54933 .90047 m .54568 .88937 L .60642 .80156 L p .915 .918 .804 r F P 0 g s .60642 .80156 m .61534 .79739 L .54933 .90047 L p .915 .918 .804 r F P 0 g s .23917 .61671 m .25602 .64124 L .25501 .74717 L p .521 .464 .733 r F P 0 g s .27209 .7572 m .25501 .74717 L .25602 .64124 L p .524 .472 .741 r F P 0 g s .25501 .74717 m .27209 .7572 L .29487 .87861 L p .602 .512 .725 r F P 0 g s .80587 .49542 m .79016 .52804 L .81981 .56718 L p .297 .05 .36 r F P 0 g s .81981 .56718 m .83787 .54028 L .80587 .49542 L p .297 .05 .36 r F P 0 g s .3105 .87128 m .29487 .87861 L .27209 .7572 L p .603 .518 .732 r F P 0 g s .29487 .87861 m .3105 .87128 L .35212 .96176 L p .645 .544 .726 r F P 0 g s .35212 .96176 m .33972 .98371 L .29487 .87861 L p .645 .544 .726 r F P 0 g s .33972 .98371 m .35212 .96176 L .4102 1.00317 L p .683 .581 .734 r F P 0 g s .4102 1.00317 m .4025 1.03185 L .33972 .98371 L p .683 .581 .734 r F P 0 g s .4025 1.03185 m .4102 1.00317 L .47847 .97944 L p .74 .647 .756 r F P 0 g s .48057 .95635 m .47847 .97944 L .4102 1.00317 L p .739 .654 .765 r F P 0 g s .47847 .97944 m .48057 .95635 L .54568 .88937 L p .827 .769 .798 r F P 0 g s .2366 .51506 m .25168 .54677 L .22249 .59062 L p .335 .348 .726 r F P 0 g s .23917 .61671 m .22249 .59062 L .25168 .54677 L p .342 .36 .735 r F P 0 g s .54202 .87884 m .54568 .88937 L .48057 .95635 L p .822 .775 .808 r F P 0 g s .54568 .88937 m .54202 .87884 L .59743 .80511 L p .904 .919 .819 r F P 0 g s .59743 .80511 m .60642 .80156 L .54568 .88937 L p .904 .919 .819 r F P 0 g s .4179 .97638 m .4102 1.00317 L .35212 .96176 L p .683 .587 .742 r F P 0 g s .4102 1.00317 m .4179 .97638 L .48057 .95635 L p .739 .654 .765 r F P 0 g s .70922 .66569 m .7257 .64166 L .68348 .6905 L p 0 0 0 r F P 0 g s .36455 .94119 m .35212 .96176 L .3105 .87128 L p .645 .55 .733 r F P 0 g s .35212 .96176 m .36455 .94119 L .4179 .97638 L p .683 .587 .742 r F P 0 g s .25602 .64124 m .27302 .66417 L .27209 .7572 L p .524 .472 .741 r F P 0 g s .2893 .76638 m .27209 .7572 L .27302 .66417 L p .527 .481 .749 r F P 0 g s .27209 .7572 m .2893 .76638 L .3105 .87128 L p .603 .518 .732 r F P 0 g s .32621 .86423 m .3105 .87128 L .2893 .76638 L p .604 .525 .739 r F P 0 g s .3105 .87128 m .32621 .86423 L .36455 .94119 L p .645 .55 .733 r F P 0 g s .48268 .93483 m .48057 .95635 L .4179 .97638 L p .738 .661 .774 r F P 0 g s .48057 .95635 m .48268 .93483 L .54202 .87884 L p .822 .775 .808 r F P 0 g s .79016 .52804 m .77428 .55901 L .80165 .59272 L p .288 .033 .342 r F P 0 g s .80165 .59272 m .81981 .56718 L .79016 .52804 L p .288 .033 .342 r F P 0 g s .42562 .95148 m .4179 .97638 L .36455 .94119 L p .683 .594 .75 r F P 0 g s .4179 .97638 m .42562 .95148 L .48268 .93483 L p .738 .661 .774 r F P 0 g s .377 .92199 m .36455 .94119 L .32621 .86423 L p .645 .557 .741 r F P 0 g s .36455 .94119 m .377 .92199 L .42562 .95148 L p .683 .594 .75 r F P 0 g s .25168 .54677 m .26689 .57677 L .23917 .61671 L p .342 .36 .735 r F P 0 g s .25602 .64124 m .23917 .61671 L .26689 .57677 L p .35 .373 .745 r F P 0 g s .53834 .86889 m .54202 .87884 L .48268 .93483 L p .816 .781 .82 r F P 0 g s .54202 .87884 m .53834 .86889 L .58838 .80806 L p .891 .92 .835 r F P 0 g s .58838 .80806 m .59743 .80511 L .54202 .87884 L p .891 .92 .835 r F P 0 g s .4848 .91488 m .48268 .93483 L .42562 .95148 L p .736 .669 .785 r F P 0 g s .48268 .93483 m .4848 .91488 L .53834 .86889 L p .816 .781 .82 r F P 0 g s .2893 .76638 m .30663 .77468 L .32621 .86423 L p .604 .525 .739 r F P 0 g s .342 .85747 m .32621 .86423 L .30663 .77468 L p .605 .533 .747 r F P 0 g s .32621 .86423 m .342 .85747 L .377 .92199 L p .645 .557 .741 r F P 0 g s .43334 .92846 m .42562 .95148 L .377 .92199 L p .682 .602 .76 r F P 0 g s .42562 .95148 m .43334 .92846 L .4848 .91488 L p .736 .669 .785 r F P 0 g s .27302 .66417 m .29018 .68547 L .2893 .76638 L p .527 .481 .749 r F P 0 g s .30663 .77468 m .2893 .76638 L .29018 .68547 L p .53 .491 .758 r F P 0 g s .6926 .68789 m .70922 .66569 L .67027 .70878 L p 0 0 0 r F P 0 g s .38948 .90417 m .377 .92199 L .342 .85747 L p .645 .565 .75 r F P 0 g s .377 .92199 m .38948 .90417 L .43334 .92846 L p .682 .602 .76 r F P 0 g s .48692 .89648 m .4848 .91488 L .43334 .92846 L p .733 .679 .797 r F P 0 g s .4848 .91488 m .48692 .89648 L .53463 .85953 L p .809 .788 .834 r F P 0 g s .53463 .85953 m .53834 .86889 L .4848 .91488 L p .809 .788 .834 r F P 0 g s .77428 .55901 m .75824 .58829 L .78339 .61688 L p .278 .014 .322 r F P 0 g s .78339 .61688 m .80165 .59272 L .77428 .55901 L p .278 .014 .322 r F P 0 g s .53834 .86889 m .53463 .85953 L .57926 .81038 L p .876 .92 .852 r F P 0 g s .57926 .81038 m .58838 .80806 L .53834 .86889 L p .876 .92 .852 r F P 0 g s .44109 .90732 m .43334 .92846 L .38948 .90417 L p .681 .611 .771 r F P 0 g s .43334 .92846 m .44109 .90732 L .48692 .89648 L p .733 .679 .797 r F P 0 g s .30663 .77468 m .32408 .78211 L .342 .85747 L p .605 .533 .747 r F P 0 g s .35785 .851 m .342 .85747 L .32408 .78211 L p .606 .542 .757 r F P 0 g s .342 .85747 m .35785 .851 L .38948 .90417 L p .645 .565 .75 r F P 0 g s .26689 .57677 m .28222 .60503 L .25602 .64124 L p .35 .373 .745 r F P 0 g s .27302 .66417 m .25602 .64124 L .28222 .60503 L p .358 .387 .755 r F P 0 g s .402 .88771 m .38948 .90417 L .35785 .851 L p .645 .575 .761 r F P 0 g s .38948 .90417 m .402 .88771 L .44109 .90732 L p .681 .611 .771 r F P 0 g s .29018 .68547 m .30747 .70511 L .30663 .77468 L p .53 .491 .758 r F P 0 g s .32408 .78211 m .30663 .77468 L .30747 .70511 L p .533 .503 .768 r F P 0 g s .67587 .70826 m .6926 .68789 L .65695 .72558 L p 0 0 0 r F P 0 g s .48905 .87965 m .48692 .89648 L .44109 .90732 L p .73 .69 .812 r F P 0 g s .48692 .89648 m .48905 .87965 L .53092 .85075 L p .8 .796 .848 r F P 0 g s .53092 .85075 m .53463 .85953 L .48692 .89648 L p .8 .796 .848 r F P 0 g s .44885 .88804 m .44109 .90732 L .402 .88771 L p .68 .622 .784 r F P 0 g s .44109 .90732 m .44885 .88804 L .48905 .87965 L p .73 .69 .812 r F P 0 g s .53463 .85953 m .53092 .85075 L .57009 .81209 L p .859 .919 .869 r F P 0 g s .57009 .81209 m .57926 .81038 L .53463 .85953 L p .859 .919 .869 r F P 0 g s .32408 .78211 m .34165 .78864 L .35785 .851 L p .606 .542 .757 r F P 0 g s .37379 .84482 m .35785 .851 L .34165 .78864 L p .607 .553 .768 r F P 0 g s .35785 .851 m .37379 .84482 L .402 .88771 L p .645 .575 .761 r F P 0 g s .75824 .58829 m .74204 .61585 L .76505 .63965 L p .266 0 .298 r F P 0 g s .76505 .63965 m .78339 .61688 L .75824 .58829 L p .266 0 .298 r F P 0 g s .41456 .87264 m .402 .88771 L .37379 .84482 L p .645 .585 .774 r F P 0 g s .402 .88771 m .41456 .87264 L .44885 .88804 L p .68 .622 .784 r F P 0 g s .28222 .60503 m .29767 .63153 L .27302 .66417 L p .358 .387 .755 r F P 0 g s .29018 .68547 m .27302 .66417 L .29767 .63153 L p .366 .403 .767 r F P 0 g s .30747 .70511 m .3249 .72308 L .32408 .78211 L p .533 .503 .768 r F P 0 g s .34165 .78864 m .32408 .78211 L .3249 .72308 L p .537 .515 .779 r F P 0 g s .49119 .86438 m .48905 .87965 L .44885 .88804 L p .725 .702 .828 r F P 0 g s .48905 .87965 m .49119 .86438 L .52718 .84257 L p .789 .804 .865 r F P 0 g s .52718 .84257 m .53092 .85075 L .48905 .87965 L p .789 .804 .865 r F P 0 g s .22165 .48169 m .20684 .44668 L .27144 .53327 L p 0 .092 .628 r F P 0 g s .45664 .87065 m .44885 .88804 L .41456 .87264 L p .678 .635 .799 r F P 0 g s .44885 .88804 m .45664 .87065 L .49119 .86438 L p .725 .702 .828 r F P 0 g s .65904 .72676 m .67587 .70826 L .64352 .74086 L p 0 0 0 r F P 0 g s .57926 .81038 m .57009 .81209 L .60258 .77739 L p .829 .999 .83 r F P 0 g s .34165 .78864 m .35933 .79428 L .37379 .84482 L p .607 .553 .768 r F P 0 g s .3898 .83894 m .37379 .84482 L .35933 .79428 L p .608 .566 .781 r F P 0 g s .37379 .84482 m .3898 .83894 L .41456 .87264 L p .645 .585 .774 r F P 0 g s .53092 .85075 m .52718 .84257 L .56085 .81316 L p .839 .916 .887 r F P 0 g s .56085 .81316 m .57009 .81209 L .53092 .85075 L p .839 .916 .887 r F P 0 g s .42718 .85894 m .41456 .87264 L .3898 .83894 L p .644 .599 .788 r F P 0 g s .41456 .87264 m .42718 .85894 L .45664 .87065 L p .678 .635 .799 r F P 0 g s .74204 .61585 m .7257 .64166 L .74664 .66101 L p .251 0 .27 r F P 0 g s .74664 .66101 m .76505 .63965 L .74204 .61585 L p .251 0 .27 r F P 0 g s .29767 .63153 m .31321 .65624 L .29018 .68547 L p .366 .403 .767 r F P 0 g s .30747 .70511 m .29018 .68547 L .31321 .65624 L p .375 .419 .779 r F P 0 g s .3249 .72308 m .34245 .73934 L .34165 .78864 L p .537 .515 .779 r F P 0 g s .35933 .79428 m .34165 .78864 L .34245 .73934 L p .541 .53 .792 r F P 0 g s .49334 .85068 m .49119 .86438 L .45664 .87065 L p .719 .717 .847 r F P 0 g s .49119 .86438 m .49334 .85068 L .52342 .83498 L p .775 .812 .883 r F P 0 g s .52342 .83498 m .52718 .84257 L .49119 .86438 L p .775 .812 .883 r F P 0 g s .46447 .85513 m .45664 .87065 L .42718 .85894 L p .675 .65 .818 r F P 0 g s .45664 .87065 m .46447 .85513 L .49334 .85068 L p .719 .717 .847 r F P 0 g s .27144 .53327 m .2836 .55876 L .22165 .48169 L p 0 .092 .628 r F P 0 g s .2366 .51506 m .22165 .48169 L .2836 .55876 L p 0 .113 .644 r F P 0 g s .35933 .79428 m .37712 .79901 L .3898 .83894 L p .608 .566 .781 r F P 0 g s .40589 .83334 m .3898 .83894 L .37712 .79901 L p .609 .58 .796 r F P 0 g s .3898 .83894 m .40589 .83334 L .42718 .85894 L p .644 .599 .788 r F P 0 g s .57009 .81209 m .56085 .81316 L .58875 .78639 L p .813 .994 .866 r F P 0 g s .58875 .78639 m .60258 .77739 L .57009 .81209 L p .813 .994 .866 r F P 0 g s .52718 .84257 m .52342 .83498 L .55156 .8136 L p .815 .912 .906 r F P 0 g s .55156 .8136 m .56085 .81316 L .52718 .84257 L p .815 .912 .906 r F P 0 g s .43985 .84663 m .42718 .85894 L .40589 .83334 L p .643 .614 .806 r F P 0 g s .42718 .85894 m .43985 .84663 L .46447 .85513 L p .675 .65 .818 r F P 0 g s .7257 .64166 m .70922 .66569 L .72815 .68094 L p .234 0 .236 r F P 0 g s .72815 .68094 m .74664 .66101 L .7257 .64166 L p .234 0 .236 r F P 0 g s .31321 .65624 m .32885 .67913 L .30747 .70511 L p .375 .419 .779 r F P 0 g s .3249 .72308 m .30747 .70511 L .32885 .67913 L p .385 .438 .792 r F P 0 g s .34245 .73934 m .36011 .75388 L .35933 .79428 L p .541 .53 .792 r F P 0 g s .37712 .79901 m .35933 .79428 L .36011 .75388 L p .545 .546 .806 r F P 0 g s .47234 .8415 m .46447 .85513 L .43985 .84663 L p .67 .669 .841 r F P 0 g s .46447 .85513 m .47234 .8415 L .4955 .83856 L p .71 .733 .869 r F P 0 g s .4955 .83856 m .49334 .85068 L .46447 .85513 L p .71 .733 .869 r F P 0 g s .49334 .85068 m .4955 .83856 L .51964 .82798 L p .756 .82 .902 r F P 0 g s .51964 .82798 m .52342 .83498 L .49334 .85068 L p .756 .82 .902 r F P 0 g s .2836 .55876 m .27144 .53327 L .27322 .53697 L p 0 0 .319 r F P 0 g s .2836 .55876 m .2958 .58304 L .2944 .58144 L p 0 .113 .644 r F P 0 g s .25168 .54677 m .2366 .51506 L .2958 .58304 L p 0 .135 .661 r F P 0 g s .2958 .58304 m .2836 .55876 L .28547 .56238 L p 0 0 .35 r F P 0 g s .2958 .58304 m .30803 .60612 L .30656 .60457 L p 0 .135 .661 r F P 0 g s .26689 .57677 m .25168 .54677 L .30803 .60612 L p 0 .159 .678 r F P 0 g s .30803 .60612 m .2958 .58304 L .29776 .58659 L p 0 0 .383 r F P 0 g s .30803 .60612 m .32029 .62797 L .31875 .6265 L p 0 .159 .678 r F P 0 g s .28222 .60503 m .26689 .57677 L .32029 .62797 L p .01 .184 .697 r F P 0 g s .32029 .62797 m .33257 .64861 L .32843 .64502 L p .01 .184 .697 r F P 0 g s .32843 .64502 m .28222 .60503 L p .31918 .62731 L .01 .184 .697 r F P 0 g s .31918 .62731 m .32029 .62797 L .32843 .64502 L .01 .184 .697 r F .29776 .58659 m .30088 .59225 L p .30803 .60612 L 0 0 .383 r F P 0 g s .29767 .63153 m .28222 .60503 L .33257 .64861 L p .032 .211 .716 r F P 0 g s .33257 .64861 m .34487 .668 L .29767 .63153 L p .032 .211 .716 r F P 0 g s .31321 .65624 m .29767 .63153 L .34487 .668 L p .054 .24 .736 r F P 0 g s .34487 .668 m .35719 .68616 L .31321 .65624 L p .054 .24 .736 r F P 0 g s .32885 .67913 m .31321 .65624 L .35719 .68616 L p .079 .271 .757 r F P 0 g s .35719 .68616 m .36952 .70307 L .32885 .67913 L p .079 .271 .757 r F P 0 g s .34457 .70018 m .32885 .67913 L .36952 .70307 L p .104 .304 .78 r F P 0 g s .36952 .70307 m .38184 .71872 L .34457 .70018 L p .104 .304 .78 r F P 0 g s .36037 .71937 m .34457 .70018 L .38184 .71872 L p .132 .339 .803 r F P 0 g s .38184 .71872 m .39417 .73312 L .36037 .71937 L p .132 .339 .803 r F P 0 g s .37623 .73669 m .36037 .71937 L .39417 .73312 L p .162 .378 .827 r F P 0 g s .39417 .73312 m .40649 .74626 L .37623 .73669 L p .162 .378 .827 r F P 0 g s .39214 .75211 m .37623 .73669 L .40649 .74626 L p .194 .419 .852 r F P 0 g s .40649 .74626 m .41881 .75814 L .39214 .75211 L p .194 .419 .852 r F P 0 g s .40811 .76561 m .39214 .75211 L .41881 .75814 L p .229 .464 .877 r F P 0 g s .41881 .75814 m .4311 .76876 L .40811 .76561 L p .229 .464 .877 r F P 0 g s .4311 .76876 m .41881 .75814 L .45542 .77922 L p 0 .356 .823 r F P 0 g s .44827 .77529 m .45542 .77922 L .41881 .75814 L p 0 .283 .776 r F P 0 g s .41881 .75814 m .40649 .74626 L .44827 .77529 L p 0 .283 .776 r F P 0 g s .44107 .77133 m .44827 .77529 L .40649 .74626 L p 0 .213 .727 r F P 0 g s .40649 .74626 m .39417 .73312 L .44107 .77133 L p 0 .213 .727 r F P 0 g s .43383 .76735 m .44107 .77133 L .39417 .73312 L p 0 .147 .679 r F P 0 g s .39417 .73312 m .38184 .71872 L .43383 .76735 L p 0 .147 .679 r F P 0 g s .42654 .76335 m .43383 .76735 L .38184 .71872 L p 0 .085 .631 r F P 0 g s .38184 .71872 m .36952 .70307 L .42654 .76335 L p 0 .085 .631 r F P 0 g s .4192 .75931 m .42654 .76335 L .36952 .70307 L p 0 .027 .584 r F P 0 g s .36952 .70307 m .35719 .68616 L .4192 .75931 L p 0 .027 .584 r F P 0 g s .41182 .75526 m .4192 .75931 L .35719 .68616 L p 0 0 .539 r F P 0 g s .35719 .68616 m .34487 .668 L .41182 .75526 L p 0 0 .539 r F P 0 g s .40439 .75118 m .41182 .75526 L .34487 .668 L p 0 0 .497 r F P 0 g s .34487 .668 m .33257 .64861 L .40439 .75118 L p 0 0 .497 r F P 0 g s .39692 .74707 m .40439 .75118 L .33257 .64861 L p 0 0 .457 r F P 0 g s .33257 .64861 m .32029 .62797 L .39692 .74707 L p 0 0 .457 r F P 0 g s .3894 .74294 m .39692 .74707 L .32029 .62797 L p 0 0 .419 r F P 0 g s .32029 .62797 m .30803 .60612 L .3894 .74294 L p 0 0 .419 r F P 0 g s .38182 .73878 m .3894 .74294 L .30803 .60612 L p 0 0 .383 r F P 0 g s .30088 .59225 m .38182 .73878 L p .30954 .60884 L 0 0 .383 r F P 0 g s .30803 .60612 m .30088 .59225 L .30954 .60884 L 0 0 .383 r F .31875 .6265 m .31633 .62418 L p .30803 .60612 L 0 .159 .678 r F P 0 g s .31633 .62418 m .26689 .57677 L p .30696 .60535 L 0 .159 .678 r F P 0 g s .30696 .60535 m .30803 .60612 L .31633 .62418 L 0 .159 .678 r F .28547 .56238 m .28847 .5682 L p .2958 .58304 L 0 0 .35 r F P 0 g s .36037 .71937 m .37623 .73669 L .36011 .75388 L p .407 .48 .823 r F P 0 g s .3742 .73459 m .38182 .73878 L .2958 .58304 L p 0 0 .35 r F P 0 g s .28847 .5682 m .3742 .73459 L p .29725 .58586 L 0 0 .35 r F P 0 g s .2958 .58304 m .28847 .5682 L .29725 .58586 L 0 0 .35 r F .30656 .60457 m .30424 .60213 L p .2958 .58304 L 0 .135 .661 r F P 0 g s .30424 .60213 m .25168 .54677 L p .29476 .58219 L 0 .135 .661 r F P 0 g s .29476 .58219 m .2958 .58304 L .30424 .60213 L 0 .135 .661 r F .27322 .53697 m .2761 .54294 L p .2836 .55876 L 0 0 .319 r F P 0 g s .36654 .73037 m .3742 .73459 L .2836 .55876 L p 0 0 .319 r F P 0 g s .2761 .54294 m .36654 .73037 L p .285 .56166 L 0 0 .319 r F P 0 g s .2836 .55876 m .2761 .54294 L .285 .56166 L 0 0 .319 r F .2944 .58144 m .2366 .51506 L p .2836 .55876 L 0 .113 .644 r F P 0 g s .37712 .79901 m .39502 .80283 L .40589 .83334 L p .609 .58 .796 r F P 0 g s .42207 .82805 m .40589 .83334 L .39502 .80283 L p .61 .598 .814 r F P 0 g s .40589 .83334 m .42207 .82805 L .43985 .84663 L p .643 .614 .806 r F P 0 g s .56085 .81316 m .55156 .8136 L .57484 .79379 L p .791 .984 .899 r F P 0 g s .57484 .79379 m .58875 .78639 L .56085 .81316 L p .791 .984 .899 r F P 0 g s .45258 .8357 m .43985 .84663 L .42207 .82805 L p .64 .634 .828 r F P 0 g s .43985 .84663 m .45258 .8357 L .47234 .8415 L p .67 .669 .841 r F P 0 g s .52342 .83498 m .51964 .82798 L .54221 .8134 L p .786 .906 .924 r F P 0 g s .54221 .8134 m .55156 .8136 L .52342 .83498 L p .786 .906 .924 r F P 0 g s .70922 .66569 m .6926 .68789 L .70961 .69944 L p .214 0 .196 r F P 0 g s .70961 .69944 m .72815 .68094 L .70922 .66569 L p .214 0 .196 r F P 0 g s .32885 .67913 m .34457 .70018 L .3249 .72308 L p .385 .438 .792 r F P 0 g s .34245 .73934 m .3249 .72308 L .34457 .70018 L p .396 .458 .807 r F P 0 g s .36011 .75388 m .37787 .76667 L .37712 .79901 L p .545 .546 .806 r F P 0 g s .39502 .80283 m .37712 .79901 L .37787 .76667 L p .55 .565 .822 r F P 0 g s .48024 .82977 m .47234 .8415 L .45258 .8357 L p .662 .692 .868 r F P 0 g s .47234 .8415 m .48024 .82977 L .49767 .82803 L p .696 .753 .896 r F P 0 g s .49767 .82803 m .4955 .83856 L .47234 .8415 L p .696 .753 .896 r F P 0 g s .4955 .83856 m .49767 .82803 L .51583 .82159 L p .733 .828 .924 r F P 0 g s .51583 .82159 m .51964 .82798 L .4955 .83856 L p .733 .828 .924 r F P 0 g s .39502 .80283 m .41301 .80572 L .42207 .82805 L p .61 .598 .814 r F P 0 g s .43833 .82305 m .42207 .82805 L .41301 .80572 L p .61 .619 .835 r F P 0 g s .42207 .82805 m .43833 .82305 L .45258 .8357 L p .64 .634 .828 r F P 0 g s .60258 .77739 m .58875 .78639 L .60796 .77081 L p .554 .902 .682 r F P 0 g s .6926 .68789 m .67587 .70826 L .69102 .7165 L p .189 0 .147 r F P 0 g s .69102 .7165 m .70961 .69944 L .6926 .68789 L p .189 0 .147 r F P 0 g s .55156 .8136 m .54221 .8134 L .56085 .79955 L p .764 .967 .928 r F P 0 g s .56085 .79955 m .57484 .79379 L .55156 .8136 L p .764 .967 .928 r F P 0 g s .46538 .82617 m .45258 .8357 L .43833 .82305 L p .636 .658 .854 r F P 0 g s .45258 .8357 m .46538 .82617 L .48024 .82977 L p .662 .692 .868 r F P 0 g s .34457 .70018 m .36037 .71937 L .34245 .73934 L p .396 .458 .807 r F P 0 g s .36011 .75388 m .34245 .73934 L .36037 .71937 L p .407 .48 .823 r F P 0 g s .51964 .82798 m .51583 .82159 L .53281 .81255 L p .753 .896 .942 r F P 0 g s .53281 .81255 m .54221 .8134 L .51964 .82798 L p .753 .896 .942 r F P 0 g s .37787 .76667 m .39573 .7777 L .39502 .80283 L p .55 .565 .822 r F P 0 g s .41301 .80572 m .39502 .80283 L .39573 .7777 L p .555 .587 .841 r F P 0 g s .58875 .78639 m .57484 .79379 L .59079 .78163 L p .585 .937 .777 r F P 0 g s .59079 .78163 m .60796 .77081 L .58875 .78639 L p .585 .937 .777 r F P 0 g s .41301 .80572 m .43109 .80768 L .43833 .82305 L p .61 .619 .835 r F P 0 g s .45468 .81835 m .43833 .82305 L .43109 .80768 L p .609 .645 .861 r F P 0 g s .43833 .82305 m .45468 .81835 L .46538 .82617 L p .636 .658 .854 r F P 0 g s .67587 .70826 m .65904 .72676 L .67239 .7321 L p .157 0 .086 r F P 0 g s .67239 .7321 m .69102 .7165 L .67587 .70826 L p .157 0 .086 r F P 0 g s .4882 .81994 m .48024 .82977 L .46538 .82617 L p .648 .72 .901 r F P 0 g s .48024 .82977 m .4882 .81994 L .49986 .8191 L p .675 .775 .925 r F P 0 g s .49986 .8191 m .49767 .82803 L .48024 .82977 L p .675 .775 .925 r F P 0 g s .49767 .82803 m .49986 .8191 L .51201 .8158 L p .701 .835 .946 r F P 0 g s .51201 .8158 m .51583 .82159 L .49767 .82803 L p .701 .835 .946 r F P 0 g s .37787 .76667 m .36011 .75388 L .37623 .73669 L p .42 .505 .841 r F P 0 g s .54221 .8134 m .53281 .81255 L .54679 .80367 L p .73 .944 .952 r F P 0 g s .54679 .80367 m .56085 .79955 L .54221 .8134 L p .73 .944 .952 r F P 0 g s .39573 .7777 m .41367 .78694 L .41301 .80572 L p .555 .587 .841 r F P 0 g s .43109 .80768 m .41301 .80572 L .41367 .78694 L p .56 .614 .863 r F P 0 g s .47826 .81805 m .46538 .82617 L .45468 .81835 L p .627 .688 .887 r F P 0 g s .46538 .82617 m .47826 .81805 L .4882 .81994 L p .648 .72 .901 r F P 0 g s .51583 .82159 m .51201 .8158 L .52335 .81104 L p .713 .882 .959 r F P 0 g s .52335 .81104 m .53281 .81255 L .51583 .82159 L p .713 .882 .959 r F P 0 g s .65904 .72676 m .6421 .74336 L .65372 .74623 L p .118 0 .009 r F P 0 g s .65372 .74623 m .67239 .7321 L .65904 .72676 L p .118 0 .009 r F P 0 g s .57484 .79379 m .56085 .79955 L .57356 .79048 L p .605 .954 .86 r F P 0 g s .56021 .79476 m .56108 .79444 L p .57356 .79048 L .296 .784 .723 r F P 0 g s .57356 .79048 m .59079 .78163 L .58319 .78742 L p .605 .954 .86 r F P 0 g s .59079 .78163 m .5901 .78198 L p .57891 .78802 L 0 0 0 r F P 0 g s .5901 .78198 m .57356 .79048 L .57891 .78802 L p 0 0 0 r F P 0 g s .56108 .79444 m .57891 .78802 L .57356 .79048 L p .296 .784 .723 r F P 0 g s .58319 .78742 m .57484 .79379 L p .57356 .79048 L .605 .954 .86 r F P 0 g s .55628 .79735 m .57356 .79048 L .56085 .79955 L p .612 .949 .925 r F P 0 g s .57356 .79048 m .55628 .79735 L .56021 .79476 L p .296 .784 .723 r F P 0 g s .56021 .79476 m .55628 .79735 L p .54993 .7985 L .395 .852 .873 r F P 0 g s .54153 .80001 m .56021 .79476 L p .54993 .7985 L .395 .852 .873 r F P 0 g s .56021 .79476 m .54153 .80001 L .54027 .79944 L p 0 0 0 r F P 0 g s .54027 .79944 m .54006 .79934 L p .54331 .7986 L 0 0 0 r F P 0 g s .54331 .7986 m .56021 .79476 L .54027 .79944 L 0 0 0 r F .55808 .79546 m .57049 .79256 L p .57592 .78963 L .065 0 0 r F P 0 g s .57891 .78802 m .56021 .79476 L p .56894 .79158 L .065 0 0 r F P 0 g s .5693 .79145 m .57592 .78963 L .57891 .78802 L .065 0 0 r F .54006 .79934 m .55808 .79546 L .56021 .79476 L p 0 0 0 r F P 0 g s .56021 .79476 m .55808 .79546 L p .56894 .79158 L .065 0 0 r F P 0 g s .55808 .79546 m .57592 .78963 L .5693 .79145 L .065 0 0 r F .57049 .79256 m .57618 .79122 L .57891 .78802 L p F P 0 g s .37623 .73669 m .39214 .75211 L .37787 .76667 L p .42 .505 .841 r F P 0 g s .39573 .7777 m .37787 .76667 L .39214 .75211 L p .433 .533 .86 r F P 0 g s .43109 .80768 m .44927 .8087 L .45468 .81835 L p .609 .645 .861 r F P 0 g s .47112 .81395 m .45468 .81835 L .44927 .8087 L p .606 .677 .892 r F P 0 g s .45468 .81835 m .47112 .81395 L .47826 .81805 L p .627 .688 .887 r F P 0 g s .41367 .78694 m .43169 .7944 L .43109 .80768 L p .56 .614 .863 r F P 0 g s .44927 .8087 m .43109 .80768 L .43169 .7944 L p .565 .645 .888 r F P 0 g s .53281 .81255 m .52335 .81104 L .53267 .80614 L p .691 .915 .971 r F P 0 g s .53267 .80614 m .54679 .80367 L .53281 .81255 L p .691 .915 .971 r F P 0 g s .49621 .81203 m .4882 .81994 L .47826 .81805 L p .622 .754 .941 r F P 0 g s .4882 .81994 m .49621 .81203 L .50206 .81177 L p .64 .797 .958 r F P 0 g s .50206 .81177 m .49986 .8191 L .4882 .81994 L p .64 .797 .958 r F P 0 g s .49986 .8191 m .50206 .81177 L .50816 .81062 L p .659 .838 .969 r F P 0 g s .50816 .81062 m .51201 .8158 L .49986 .8191 L p .659 .838 .969 r F P 0 g s .63503 .7589 m .65372 .74623 L .6421 .74336 L p .067 0 0 r F P 0 g s .39214 .75211 m .40811 .76561 L .39573 .7777 L p .433 .533 .86 r F P 0 g s .41367 .78694 m .39573 .7777 L .40811 .76561 L p .448 .565 .88 r F P 0 g s .49123 .81134 m .47826 .81805 L .47112 .81395 L p .61 .725 .927 r F P 0 g s .47826 .81805 m .49123 .81134 L .49621 .81203 L p .622 .754 .941 r F P 0 g s .56085 .79955 m .54679 .80367 L .55628 .79735 L p .612 .949 .925 r F P 0 g s .51201 .8158 m .50816 .81062 L .51384 .80888 L p .666 .864 .975 r F P 0 g s .51384 .80888 m .52335 .81104 L .51201 .8158 L p .666 .864 .975 r F P 0 g s .43169 .7944 m .44976 .80005 L .44927 .8087 L p .565 .645 .888 r F P 0 g s .46753 .80877 m .44927 .8087 L .44976 .80005 L p .569 .682 .916 r F P 0 g s .44927 .8087 m .46753 .80877 L .47112 .81395 L p .606 .677 .892 r F P 0 g s .48765 .80985 m .47112 .81395 L .46753 .80877 L p .597 .717 .928 r F P 0 g s .47112 .81395 m .48765 .80985 L .49123 .81134 L p .61 .725 .927 r F P 0 g s .61633 .77009 m .63503 .7589 L .62507 .75805 L p 0 0 0 r F P 0 g s .40811 .76561 m .4241 .7772 L .41367 .78694 L p .448 .565 .88 r F P 0 g s .43169 .7944 m .41367 .78694 L .4241 .7772 L p .464 .601 .903 r F P 0 g s .52335 .81104 m .51384 .80888 L .5185 .80694 L p .647 .879 .983 r F P 0 g s .5185 .80694 m .53267 .80614 L .52335 .81104 L p .647 .879 .983 r F P 0 g s .54679 .80367 m .53267 .80614 L .53897 .80225 L p .605 .926 .968 r F P 0 g s .53897 .80225 m .55628 .79735 L .54679 .80367 L p .605 .926 .968 r F P 0 g s .85583 .51204 m .83787 .54028 L .83752 .69411 L p .494 .25 .472 r F P 0 g s .83752 .69411 m .85669 .68441 L .85583 .51204 L p .494 .25 .472 r F P 0 g s .60796 .77081 m .59079 .78163 L .59762 .77979 L p 0 0 0 r F P 0 g s .59762 .77979 m .61633 .77009 L .60796 .77081 L p 0 0 0 r F P 0 g s .4241 .7772 m .44013 .78686 L .43169 .7944 L p .464 .601 .903 r F P 0 g s .44976 .80005 m .43169 .7944 L .44013 .78686 L p .481 .641 .926 r F P 0 g s .44976 .80005 m .4679 .80388 L .46753 .80877 L p .569 .682 .916 r F P 0 g s .48587 .8079 m .46753 .80877 L .4679 .80388 L p .57 .725 .947 r F P 0 g s .46753 .80877 m .48587 .8079 L .48765 .80985 L p .597 .717 .928 r F P 0 g s .50428 .80606 m .49621 .81203 L .49123 .81134 L closepath p .573 .789 .981 r F P 0 g s .50428 .80606 m .50206 .81177 L .49621 .81203 L closepath p .583 .816 .988 r F P 0 g s .50428 .80606 m .50816 .81062 L .50206 .81177 L closepath p .6 .834 .988 r F P 0 g s .50428 .80606 m .49123 .81134 L .48765 .80985 L closepath p .574 .769 .971 r F P 0 g s .50428 .80606 m .51384 .80888 L .50816 .81062 L closepath p .61 .839 .987 r F P 0 g s .53267 .80614 m .5185 .80694 L .52163 .80515 L p .587 .887 .991 r F P 0 g s .52163 .80515 m .53897 .80225 L .53267 .80614 L p .587 .887 .991 r F P 0 g s .57891 .78802 m .59762 .77979 L .59079 .78163 L p 0 0 0 r F P 0 g s .50428 .80606 m .48765 .80985 L .48587 .8079 L closepath p .576 .764 .968 r F P 0 g s .83787 .54028 m .81981 .56718 L .81841 .70344 L p .491 .241 .463 r F P 0 g s .81841 .70344 m .83752 .69411 L .83787 .54028 L p .491 .241 .463 r F P 0 g s .44013 .78686 m .45616 .79457 L .44976 .80005 L p .481 .641 .926 r F P 0 g s .4679 .80388 m .44976 .80005 L .45616 .79457 L p .499 .685 .95 r F P 0 g s .50428 .80606 m .5185 .80694 L .51384 .80888 L closepath p .598 .837 .99 r F P 0 g s .4679 .80388 m .48607 .80589 L .48587 .8079 L p .57 .725 .947 r F P 0 g s .50428 .80606 m .48587 .8079 L .48607 .80589 L closepath p .566 .775 .977 r F P 0 g s .45616 .79457 m .47221 .80035 L .4679 .80388 L p .499 .685 .95 r F P 0 g s .48607 .80589 m .4679 .80388 L .47221 .80035 L p .516 .735 .973 r F P 0 g s .50428 .80606 m .52163 .80515 L .5185 .80694 L closepath p .562 .839 .997 r F P 0 g s .81981 .56718 m .80165 .59272 L .79936 .71239 L p .487 .232 .453 r F P 0 g s .79936 .71239 m .81841 .70344 L .81981 .56718 L p .487 .232 .453 r F P 0 g s .55628 .79735 m .53897 .80225 L .54153 .80001 L p .395 .852 .873 r F P 0 g s .47221 .80035 m .48825 .80418 L .48607 .80589 L p .516 .735 .973 r F P 0 g s .50428 .80606 m .48607 .80589 L .48825 .80418 L closepath p .532 .787 .991 r F P 0 g s .53897 .80225 m .52163 .80515 L .52289 .80378 L p .466 .867 .963 r F P 0 g s .52289 .80378 m .54153 .80001 L .53897 .80225 L p .466 .867 .963 r F P 0 g s .50428 .80606 m .52289 .80378 L .52163 .80515 L closepath p .505 .841 .998 r F P 0 g s .80165 .59272 m .78339 .61688 L .78037 .72097 L p .482 .221 .441 r F P 0 g s .78037 .72097 m .79936 .71239 L .80165 .59272 L p .482 .221 .441 r F P 0 g s .50428 .80606 m .49218 .80298 L .49743 .8023 L closepath p .445 .777 .998 r F P 0 g s .50428 .80606 m .48825 .80418 L .49218 .80298 L closepath p .481 .787 .998 r F P 0 g s .50428 .80606 m .52213 .80288 L .52289 .80378 L closepath p .453 .839 .985 r F P 0 g s .50428 .80606 m .51942 .80237 L .52213 .80288 L closepath p .442 .838 .98 r F P 0 g s .50428 .80606 m .51507 .80212 L .51942 .80237 L closepath p .464 .835 .991 r F P 0 g s .50428 .80606 m .50954 .80201 L .51507 .80212 L closepath p .476 .817 .999 r F P 0 g s .50428 .80606 m .50343 .80203 L .50954 .80201 L closepath p .465 .793 .999 r F P 0 g s .50428 .80606 m .49743 .8023 L .50343 .80203 L closepath p .445 .777 .998 r F P 0 g s .48003 .79864 m .49218 .80298 L .48825 .80418 L p .436 .73 .988 r F P 0 g s .48825 .80418 m .47221 .80035 L .48003 .79864 L p .436 .73 .988 r F P 0 g s .4241 .7772 m .40811 .76561 L .4311 .76876 L p .266 .512 .903 r F P 0 g s .46785 .79306 m .48003 .79864 L .47221 .80035 L p .391 .673 .972 r F P 0 g s .47221 .80035 m .45616 .79457 L .46785 .79306 L p .391 .673 .972 r F P 0 g s .4311 .76876 m .44338 .77812 L .4241 .7772 L p .266 .512 .903 r F P 0 g s .44013 .78686 m .4241 .7772 L .44338 .77812 L p .305 .563 .928 r F P 0 g s .78339 .61688 m .76505 .63965 L .76146 .72919 L p .477 .208 .427 r F P 0 g s .76146 .72919 m .78037 .72097 L .78339 .61688 L p .477 .208 .427 r F P 0 g s .54153 .80001 m .52289 .80378 L .52213 .80288 L p .312 .8 .878 r F P 0 g s .45563 .78622 m .46785 .79306 L .45616 .79457 L p .347 .617 .951 r F P 0 g s .45616 .79457 m .44013 .78686 L .45563 .78622 L p .347 .617 .951 r F P 0 g s .44338 .77812 m .45563 .78622 L .44013 .78686 L p .305 .563 .928 r F P 0 g s .49053 .79851 m .49743 .8023 L .49218 .80298 L p .366 .718 .988 r F P 0 g s .49218 .80298 m .48003 .79864 L .49053 .79851 L p .366 .718 .988 r F P 0 g s .76505 .63965 m .74664 .66101 L .7426 .73703 L p .471 .193 .41 r F P 0 g s .7426 .73703 m .76146 .72919 L .76505 .63965 L p .471 .193 .41 r F P 0 g s .52213 .80288 m .54006 .79934 L .54153 .80001 L p .312 .8 .878 r F P 0 g s .54006 .79934 m .52213 .80288 L .51942 .80237 L p .224 .745 .822 r F P 0 g s .74664 .66101 m .72815 .68094 L .72382 .7445 L p .463 .176 .391 r F P 0 g s .72382 .7445 m .7426 .73703 L .74664 .66101 L p .463 .176 .391 r F P 0 g s .50258 .7992 m .50343 .80203 L .49743 .8023 L p .335 .724 .983 r F P 0 g s .49743 .8023 m .49053 .79851 L .50258 .7992 L p .335 .724 .983 r F P 0 g s .51942 .80237 m .53468 .79961 L .54006 .79934 L p .224 .745 .822 r F P 0 g s .53468 .79961 m .51942 .80237 L .51507 .80212 L p .237 .75 .861 r F P 0 g s .4836 .7947 m .49053 .79851 L .48003 .79864 L p .283 .653 .969 r F P 0 g s .48003 .79864 m .46785 .79306 L .4836 .7947 L p .283 .653 .969 r F P 0 g s .72815 .68094 m .70961 .69944 L .7051 .75161 L p .454 .155 .367 r F P 0 g s .7051 .75161 m .72382 .7445 L .72815 .68094 L p .454 .155 .367 r F P 0 g s .59762 .77979 m .57891 .78802 L .57618 .79122 L p .198 0 0 r F P 0 g s .51484 .79987 m .50954 .80201 L .50343 .80203 L p .323 .748 .974 r F P 0 g s .50343 .80203 m .50258 .7992 L .51484 .79987 L p .323 .748 .974 r F P 0 g s .52595 .79999 m .51484 .79987 L p .52236 .79981 L .027 .599 .733 r F P 0 g s .52595 .79999 m .53468 .79961 L p .53012 .80019 L .237 .75 .861 r F P 0 g s .55808 .79546 m .54006 .79934 L .53468 .79961 L p .042 0 0 r F P 0 g s .53468 .79961 m .55004 .7978 L .55808 .79546 L p .042 0 0 r F P 0 g s .55004 .7978 m .53468 .79961 L .52817 .79989 L p .056 0 0 r F P 0 g s .52817 .79989 m .52595 .79999 L p .5272 .79987 L .056 0 0 r F P 0 g s .5272 .79987 m .55004 .7978 L .52817 .79989 L .056 0 0 r F .53693 .7997 m .55004 .7978 L p .53963 .79874 L F P 0 g s .52595 .79999 m .53693 .7997 L p .53963 .79874 L .056 0 0 r F P 0 g s .52019 .79968 m .53693 .7997 L p .5234 .7998 L .027 .599 .733 r F P 0 g s .51484 .79987 m .51907 .79972 L p .5234 .7998 L .027 .599 .733 r F P 0 g s .53693 .7997 m .52595 .79999 L p .52236 .79981 L .027 .599 .733 r F P 0 g s .51507 .80212 m .52595 .79999 L p .53012 .80019 L .237 .75 .861 r F P 0 g s .52595 .79999 m .51507 .80212 L .50954 .80201 L p .291 .764 .935 r F P 0 g s .50954 .80201 m .51484 .79987 L .52595 .79999 L p .291 .764 .935 r F P 0 g s .53693 .7997 m .52019 .79968 L p .53465 .8005 L .227 0 0 r F P 0 g s .54802 .80127 m .53693 .7997 L p .53465 .8005 L .227 0 0 r F P 0 g s .51907 .79972 m .52019 .79968 L p .5234 .7998 L .027 .599 .733 r F P 0 g s .70961 .69944 m .69102 .7165 L .68646 .75835 L p .442 .129 .338 r F P 0 g s .68646 .75835 m .7051 .75161 L .70961 .69944 L p .442 .129 .338 r F P 0 g s .61633 .77009 m .59762 .77979 L .59437 .78664 L p .285 0 0 r F P 0 g s .57618 .79122 m .59437 .78664 L .59762 .77979 L p .198 0 0 r F P 0 g s .69102 .7165 m .67239 .7321 L .66789 .76473 L p .427 .098 .302 r F P 0 g s .66789 .76473 m .68646 .75835 L .69102 .7165 L p .427 .098 .302 r F P 0 g s .63503 .7589 m .61633 .77009 L .61263 .7817 L p .342 0 .113 r F P 0 g s .59437 .78664 m .61263 .7817 L .61633 .77009 L p .285 0 0 r F P 0 g s .67239 .7321 m .65372 .74623 L .64939 .77074 L p .407 .057 .256 r F P 0 g s .64939 .77074 m .66789 .76473 L .67239 .7321 L p .407 .057 .256 r F P 0 g s .65372 .74623 m .63503 .7589 L .63097 .7764 L p .38 .005 .195 r F P 0 g s .61263 .7817 m .63097 .7764 L .63503 .7589 L p .342 0 .113 r F P 0 g s .63097 .7764 m .64939 .77074 L .65372 .74623 L p .38 .005 .195 r F P 0 g s .47662 .79086 m .4836 .7947 L .46785 .79306 L p .198 .582 .942 r F P 0 g s .46785 .79306 m .45563 .78622 L .47662 .79086 L p .198 .582 .942 r F P 0 g s .50172 .7976 m .50258 .7992 L .49053 .79851 L p .206 .649 .949 r F P 0 g s .49053 .79851 m .4836 .7947 L .50172 .7976 L p .206 .649 .949 r F P 0 g s .4696 .78701 m .47662 .79086 L .45563 .78622 L p .113 .507 .908 r F P 0 g s .45563 .78622 m .44338 .77812 L .4696 .78701 L p .113 .507 .908 r F P 0 g s .57618 .79122 m .55808 .79546 L .55004 .7978 L p .242 0 0 r F P 0 g s .50258 .7992 m .50172 .7976 L .52019 .79968 L p .131 .65 .887 r F P 0 g s .52019 .79968 m .51484 .79987 L .51318 .79978 L p .131 .65 .887 r F P 0 g s .51199 .79946 m .52019 .79968 L .51318 .79978 L .131 .65 .887 r F .5256 .80147 m .52019 .79968 L .51236 .7988 L p 0 .495 .729 r F P 0 g s .51236 .7988 m .50172 .7976 L p .51276 .79938 L 0 .495 .729 r F P 0 g s .51276 .79938 m .5256 .80147 L .51236 .7988 L 0 .495 .729 r F .50172 .7976 m .50085 .79723 L .5087 .79857 L p F P 0 g s .50172 .7976 m .5087 .79857 L .51016 .79896 L 0 .495 .729 r F .51318 .79978 m .50258 .7992 L p .51199 .79946 L .131 .65 .887 r F P 0 g s .50085 .79723 m .50172 .7976 L .4836 .7947 L p .062 .555 .89 r F P 0 g s .50085 .79723 m .49997 .79811 L .51408 .80135 L p .292 0 0 r F P 0 g s .51408 .80135 m .53107 .80525 L p .51358 .80061 L .292 0 0 r F P 0 g s .50085 .79723 m .51408 .80135 L .51358 .80061 L .292 0 0 r F .5087 .79857 m .5256 .80147 L p .51016 .79896 L 0 .495 .729 r F P 0 g s .46253 .78312 m .4696 .78701 L .44338 .77812 L p .029 .432 .868 r F P 0 g s .44338 .77812 m .4311 .76876 L .46253 .78312 L p .029 .432 .868 r F P 0 g s .4836 .7947 m .47662 .79086 L .50085 .79723 L p .062 .555 .89 r F P 0 g s .59437 .78664 m .57618 .79122 L .56553 .79693 L p .361 0 0 r F P 0 g s .55004 .7978 m .56553 .79693 L .57618 .79122 L p .242 0 0 r F P 0 g s .45542 .77922 m .46253 .78312 L .4311 .76876 L p 0 .356 .823 r F P 0 g s .56553 .79693 m .55004 .7978 L .53693 .7997 L p .282 0 0 r F P 0 g s .61263 .7817 m .59437 .78664 L .58114 .79703 L p .43 .008 .135 r F P 0 g s .56553 .79693 m .58114 .79703 L .59437 .78664 L p .361 0 0 r F P 0 g s .49997 .79811 m .50085 .79723 L .47662 .79086 L p 0 .443 .807 r F P 0 g s .47662 .79086 m .4696 .78701 L .49997 .79811 L p 0 .443 .807 r F P 0 g s .53693 .7997 m .54802 .80127 L .56553 .79693 L p .282 0 0 r F P 0 g s .52019 .79968 m .5256 .80147 L .54802 .80127 L p .227 0 0 r F P 0 g s .63097 .7764 m .61263 .7817 L .59688 .79811 L p .472 .087 .227 r F P 0 g s .58114 .79703 m .59688 .79811 L .61263 .7817 L p .43 .008 .135 r F P 0 g s .58114 .79703 m .56553 .79693 L .54802 .80127 L p .412 0 0 r F P 0 g s .49909 .80026 m .49997 .79811 L .4696 .78701 L p 0 .323 .706 r F P 0 g s .4696 .78701 m .46253 .78312 L .49909 .80026 L p 0 .323 .706 r F P 0 g s .53107 .80525 m .5256 .80147 L .50085 .79723 L p .292 0 0 r F P 0 g s .64939 .77074 m .63097 .7764 L .61276 .80017 L p .499 .142 .292 r F P 0 g s .59688 .79811 m .61276 .80017 L .63097 .7764 L p .472 .087 .227 r F P 0 g s .54802 .80127 m .55923 .80473 L .58114 .79703 L p .412 0 0 r F P 0 g s .55923 .80473 m .54802 .80127 L .5256 .80147 L p .399 0 0 r F P 0 g s .5256 .80147 m .53107 .80525 L .55923 .80473 L p .399 0 0 r F P 0 g s .59688 .79811 m .58114 .79703 L .55923 .80473 L p .485 .047 .132 r F P 0 g s .4982 .80369 m .49909 .80026 L .46253 .78312 L p .368 0 0 r F P 0 g s .46253 .78312 m .45542 .77922 L .4982 .80369 L p .368 0 0 r F P 0 g s .66789 .76473 m .64939 .77074 L .62879 .80323 L p .518 .183 .339 r F P 0 g s .61276 .80017 m .62879 .80323 L .64939 .77074 L p .499 .142 .292 r F P 0 g s .5366 .81108 m .53107 .80525 L .49997 .79811 L p .447 0 0 r F P 0 g s .49997 .79811 m .49909 .80026 L .5366 .81108 L p .447 0 0 r F P 0 g s .68646 .75835 m .66789 .76473 L .64497 .8073 L p .532 .214 .374 r F P 0 g s .62879 .80323 m .64497 .8073 L .66789 .76473 L p .518 .183 .339 r F P 0 g s .49729 .80844 m .4982 .80369 L .45542 .77922 L p .481 0 0 r F P 0 g s .45542 .77922 m .44827 .77529 L .49729 .80844 L p .481 0 0 r F P 0 g s .61276 .80017 m .59688 .79811 L .57056 .8101 L p .528 .129 .228 r F P 0 g s .55923 .80473 m .57056 .8101 L .59688 .79811 L p .485 .047 .132 r F P 0 g s .57056 .8101 m .55923 .80473 L .53107 .80525 L p .5 0 0 r F P 0 g s .7051 .75161 m .68646 .75835 L .6613 .81241 L p .542 .238 .402 r F P 0 g s .64497 .8073 m .6613 .81241 L .68646 .75835 L p .532 .214 .374 r F P 0 g s .53107 .80525 m .5366 .81108 L .57056 .8101 L p .5 0 0 r F P 0 g s .49638 .81451 m .49729 .80844 L .44827 .77529 L p .573 .011 0 r F P 0 g s .44827 .77529 m .44107 .77133 L .49638 .81451 L p .573 .011 0 r F P 0 g s .72382 .7445 m .7051 .75161 L .67781 .81855 L p .55 .258 .424 r F P 0 g s .6613 .81241 m .67781 .81855 L .7051 .75161 L p .542 .238 .402 r F P 0 g s .5422 .81898 m .5366 .81108 L .49909 .80026 L p .553 0 0 r F P 0 g s .49909 .80026 m .4982 .80369 L .5422 .81898 L p .553 0 0 r F P 0 g s .62879 .80323 m .61276 .80017 L .58202 .81744 L p .555 .185 .295 r F P 0 g s .57056 .8101 m .58202 .81744 L .61276 .80017 L p .528 .129 .228 r F P 0 g s .7426 .73703 m .72382 .7445 L .69448 .82576 L p .556 .273 .442 r F P 0 g s .67781 .81855 m .69448 .82576 L .72382 .7445 L p .55 .258 .424 r F P 0 g s .58202 .81744 m .57056 .8101 L .5366 .81108 L p .56 .102 .122 r F P 0 g s .49546 .82193 m .49638 .81451 L .44107 .77133 L p .647 .098 0 r F P 0 g s .44107 .77133 m .43383 .76735 L .49546 .82193 L p .647 .098 0 r F P 0 g s .5366 .81108 m .5422 .81898 L .58202 .81744 L p .56 .102 .122 r F P 0 g s .76146 .72919 m .7426 .73703 L .71134 .83404 L p .561 .287 .457 r F P 0 g s .69448 .82576 m .71134 .83404 L .7426 .73703 L p .556 .273 .442 r F P 0 g s .54787 .82899 m .5422 .81898 L .4982 .80369 L p .623 .096 0 r F P 0 g s .4982 .80369 m .49729 .80844 L .54787 .82899 L p .623 .096 0 r F P 0 g s .64497 .8073 m .62879 .80323 L .59362 .82676 L p .573 .226 .342 r F P 0 g s .58202 .81744 m .59362 .82676 L .62879 .80323 L p .555 .185 .295 r F P 0 g s .49453 .83072 m .49546 .82193 L .43383 .76735 L p .705 .171 0 r F P 0 g s .43383 .76735 m .42654 .76335 L .49453 .83072 L p .705 .171 0 r F P 0 g s .78037 .72097 m .76146 .72919 L .72839 .84342 L p .565 .298 .469 r F P 0 g s .71134 .83404 m .72839 .84342 L .76146 .72919 L p .561 .287 .457 r F P 0 g s .59362 .82676 m .58202 .81744 L .5422 .81898 L p .597 .174 .21 r F P 0 g s .79936 .71239 m .78037 .72097 L .74563 .85391 L p .569 .307 .48 r F P 0 g s .72839 .84342 m .74563 .85391 L .78037 .72097 L p .565 .298 .469 r F P 0 g s .49358 .84092 m .49453 .83072 L .42654 .76335 L p .75 .233 0 r F P 0 g s .42654 .76335 m .4192 .75931 L .49358 .84092 L p .75 .233 0 r F P 0 g s .5422 .81898 m .54787 .82899 L .59362 .82676 L p .597 .174 .21 r F P 0 g s .55362 .84116 m .54787 .82899 L .49729 .80844 L p .67 .175 .063 r F P 0 g s .49729 .80844 m .49638 .81451 L .55362 .84116 L p .67 .175 .063 r F P 0 g s .6613 .81241 m .64497 .8073 L .60538 .8381 L p .586 .256 .378 r F P 0 g s .59362 .82676 m .60538 .8381 L .64497 .8073 L p .573 .226 .342 r F P 0 g s .81841 .70344 m .79936 .71239 L .76309 .86554 L p .572 .315 .489 r F P 0 g s .74563 .85391 m .76309 .86554 L .79936 .71239 L p .569 .307 .48 r F P 0 g s .83752 .69411 m .81841 .70344 L .78076 .87833 L p .574 .322 .497 r F P 0 g s .76309 .86554 m .78076 .87833 L .81841 .70344 L p .572 .315 .489 r F P 0 g s .49262 .85254 m .49358 .84092 L .4192 .75931 L p .785 .285 0 r F P 0 g s .4192 .75931 m .41182 .75526 L .49262 .85254 L p .785 .285 0 r F P 0 g s .60538 .8381 m .59362 .82676 L .54787 .82899 L p .62 .225 .273 r F P 0 g s .85669 .68441 m .83752 .69411 L .79865 .8923 L p .576 .329 .504 r F P 0 g s .78076 .87833 m .79865 .8923 L .83752 .69411 L p .574 .322 .497 r F P 0 g s .67781 .81855 m .6613 .81241 L .61729 .85152 L p .595 .28 .406 r F P 0 g s .60538 .8381 m .61729 .85152 L .6613 .81241 L p .586 .256 .378 r F P 0 g s .55945 .85553 m .55362 .84116 L .49638 .81451 L p .701 .234 .139 r F P 0 g s .49638 .81451 m .49546 .82193 L .55945 .85553 L p .701 .234 .139 r F P 0 g s .54787 .82899 m .55362 .84116 L .60538 .8381 L p .62 .225 .273 r F P 0 g s .49166 .86562 m .49262 .85254 L .41182 .75526 L p .813 .328 0 r F P 0 g s .41182 .75526 m .40439 .75118 L .49166 .86562 L p .813 .328 0 r F P 0 g s .79865 .8923 m .81678 .90748 L .85669 .68441 L p .576 .329 .504 r F P 0 g s .49067 .8802 m .49166 .86562 L .40439 .75118 L p .836 .365 .033 r F P 0 g s .40439 .75118 m .39692 .74707 L .49067 .8802 L p .836 .365 .033 r F P 0 g s .69448 .82576 m .67781 .81855 L .62938 .86706 L p .602 .299 .428 r F P 0 g s .61729 .85152 m .62938 .86706 L .67781 .81855 L p .595 .28 .406 r F P 0 g s .61729 .85152 m .60538 .8381 L .55362 .84116 L p .637 .263 .32 r F P 0 g s .56537 .87216 m .55945 .85553 L .49546 .82193 L p .723 .28 .198 r F P 0 g s .49546 .82193 m .49453 .83072 L .56537 .87216 L p .723 .28 .198 r F P 0 g s .55362 .84116 m .55945 .85553 L .61729 .85152 L p .637 .263 .32 r F P 0 g s .48968 .89629 m .49067 .8802 L .39692 .74707 L p .853 .396 .073 r F P 0 g s .39692 .74707 m .3894 .74294 L .48968 .89629 L p .853 .396 .073 r F P 0 g s .71134 .83404 m .69448 .82576 L .64165 .88476 L p .608 .315 .446 r F P 0 g s .62938 .86706 m .64165 .88476 L .69448 .82576 L p .602 .299 .428 r F P 0 g s .48866 .91395 m .48968 .89629 L .3894 .74294 L p .868 .423 .109 r F P 0 g s .3894 .74294 m .38182 .73878 L .48866 .91395 L p .868 .423 .109 r F P 0 g s .62938 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Cell[TextData[{ "Measure the volume under the plotted surface and above its base in the ", Cell[BoxData[ \(xy\)]], "-plane." }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "It's duck soup.\nThe volume is measured by \n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \)\(R\_xy\)\)\%\ \[Null]\((x\ y + \ 5)\)\ \[DifferentialD]x \[DifferentialD]y\)\)]], ", \nwhere ", Cell[BoxData[ \(R\_xy\)]], " is everything inside and on the ellipse \n ", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", StyleBox[\(x\/2\), FontSize->16], ")"}], "2"], "+", " ", SuperscriptBox[ RowBox[{"(", StyleBox[\(y\/3\), FontSize->16], ")"}], "2"]}], "=", "1"}]]], " \nplotted in the ", Cell[BoxData[ \(xy\)]], "-plane.\nGo with ", Cell[BoxData[ \(rt\)]], "-paper coming from:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \({x[r, t], y[r, t]}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({2\ r\ Cos[t], 3\ r\ Sin[t]}\)], "Output"] }, Open ]], Cell[TextData[{ "As you run ", Cell[BoxData[ \(r\)]], " from ", Cell[BoxData[ \(rlow\)]], " to ", Cell[BoxData[ \(rhigh\)]], ", and as you run ", Cell[BoxData[ \(t\)]], " from ", Cell[BoxData[ \(tlow\)]], " to ", Cell[BoxData[ \(thigh\)]], ", ", Cell[BoxData[ \({x[r, t], y[r, t]}\)]], " sweeps out everything inside and on ", Cell[BoxData[ \(R\_xy\)]], ". \nSo ", Cell[BoxData[ \(R\_rt\)]], " is the rectangle ", Cell[BoxData[ \(rlow \[LessEqual] r \[LessEqual] rhigh\)]], " and ", Cell[BoxData[ \(tlow \[LessEqual] t \[LessEqual] thigh\)]], ", and\n ", Cell[BoxData[ \(volume = \[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \ \)\(R\_xy\)\)\%\[Null]\((x\ y + \ 5)\)\ \[DifferentialD]x \[DifferentialD]y\)\)]], " \n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \ \)\(R\_\(\(\ \)\(r\ t\)\)\)\)\%\[Null]\((x[r, t]\ y[r, t] + \ 5)\)\ A\_xy[r, t] \[DifferentialD]r \[DifferentialD]t\)\)\)\)]], "\n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_tlow\%thigh\(\[Integral]\_rlow\%rhigh\ \((x[r, t]\ y[r, t] + \ 5)\)\ A\_xy[r, t] \[DifferentialD]r \[DifferentialD]t\)\)\)\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[gradx, grady, Axy];\)\), "\n", \(\(gradx[r_, t_] = {D[x[r, t], r], D[x[r, t], t]};\)\), "\n", \(\(grady[r_, t_] = {D[y[r, t], r], D[y[r, t], t]};\)\), "\n", \(Axy[r_, t_] = TrigExpand[Det[{gradx[r, t], grady[r, t]}]]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(6\ r\)], "Output"] }, Open ]], Cell["Measure the volume:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_tlow\%thigh\(\[Integral]\_rlow\%rhigh\(\((x[r, t]\ y[r, t] + 5)\)\ Axy[r, t]\) \[DifferentialD]r \[DifferentialD]t\)\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(30\ \[Pi]\)], "Output"] }, Open ]], Cell[TextData[{ "Nice answer. \nBut if you hadn't gone to ", Cell[BoxData[ \(rt\)]], "-paper to calculate the integral, getting this nice answer wouldn't have \ been so simple." }], "SmallText"] }, Open ]], Cell["T.1.b.ii)", "Subsubsection"], 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The integral \n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \)\(R\_xy\)\)\%\ \[Null]\ \[DifferentialD]x \[DifferentialD]y\)\)]], "\nmeasures the area of the ribbon. \nGo to ", Cell[BoxData[ \(rt\)]], "-paper with:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\({rlow, rhigh} = {\(-0.5\), 0.5};\)\), "\n", \(\(Clear[x, y, r, t];\)\), "\n", \({x[r_, t_], y[r_, t_]} = {4\ Cos[t], 3\ Sin[t]} + r\ unitnormal[t]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({4\ Cos[ t] - \(6\ r\ Cos[t]\ \@\(9\ Cos[t]\^2 + 16\ \ Sin[t]\^2\)\)\/\(\(-25\) + 7\ Cos[t]\^2 - 7\ Sin[t]\^2\), 3\ Sin[t] - \(8\ r\ Sin[t]\ \@\(9\ Cos[t]\^2 + 16\ \ Sin[t]\^2\)\)\/\(\(-25\) + 7\ Cos[t]\^2 - 7\ Sin[t]\^2\)}\)], "Output"] }, Open ]], Cell[TextData[{ "When you look at the plotting instructions above, then you see that when \ you run ", Cell[BoxData[ \(r\)]], " from ", Cell[BoxData[ \(rlow\)]], " to ", Cell[BoxData[ \(rhigh\)]], ", and you run ", Cell[BoxData[ \(t\)]], " from ", Cell[BoxData[ \(tlow\)]], " to ", Cell[BoxData[ \(thigh\)]], ", then ", Cell[BoxData[ \({x[r, t], y[r, t]}\)]], " describes the whole ribbon ", Cell[BoxData[ \(R\_xy\)]], ". 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The integral \n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \)\(R\_xy\)\)\%\ \[Null]\ \[DifferentialD]x \[DifferentialD]y\)\)]], "\nmeasures the area of the ribbon. \nGo to ", Cell[BoxData[ \(rt\)]], "-paper with:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, y, r, t];\)\), "\n", \({x[r_, t_], y[r_, t_]} = {4\ Cos[t], 3\ Sin[t]} + r\ halfwidth[t]\ unitnormal[t]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({4\ Cos[ t] - \(1.2000000000000002`\ r\ t\ Cos[t]\ \@\(9\ Cos[t]\^2 + 16\ \ Sin[t]\^2\)\)\/\(\(-25\) + 7\ Cos[t]\^2 - 7\ Sin[t]\^2\), 3\ Sin[t] - \(1.6`\ r\ t\ Sin[t]\ \@\(9\ Cos[t]\^2 + 16\ \ Sin[t]\^2\)\)\/\(\(-25\) + 7\ Cos[t]\^2 - 7\ Sin[t]\^2\)}\)], "Output"] }, Open ]], Cell[TextData[{ "When you look at the plotting instructions above, then you see that when \ you run ", Cell[BoxData[ \(r\)]], " from ", Cell[BoxData[ \(\(-1\)\)]], " to ", Cell[BoxData[ \(1\)]], ", and you run ", Cell[BoxData[ \(t\)]], " from ", Cell[BoxData[ \(tlow\)]], " to ", Cell[BoxData[ \(thigh\)]], ", then ", Cell[BoxData[ \({x[r, t], y[r, t]}\)]], " describes ", Cell[BoxData[ \(R\_xy\)]], ". This tells you that on ", Cell[BoxData[ \(rt\)]], "-paper, ", Cell[BoxData[ \(R\_rt\)]], " is the rectangle \n ", Cell[BoxData[ \(\(-1\) \[LessEqual] r \[LessEqual] 1\)]], " and ", Cell[BoxData[ \(tlow \[LessEqual] t \[LessEqual] thigh\)]], ",\nso\n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \)\(R\_xy\)\)\%\ \[Null]\ \[DifferentialD]x \[DifferentialD]y\) = \ \[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \)\(R\_\(\(\ \)\(r\ t\)\)\)\ \)\%\[Null] A\_xy[r, t]\ \[DifferentialD]r \[DifferentialD]t\)\)]], "\n ", Cell[BoxData[ \(\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(=\)\(\[Integral]\_tlow\%thigh\(\ \[Integral]\_\(-1\)\%1\ A\_xy[r, t] \[DifferentialD]r \[DifferentialD]t\)\)\)\)\)]], "." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[gradx, grady, Axy];\)\), "\n", \(\(gradx[r_, t_] = {D[x[r, t], r], D[x[r, t], t]};\)\), "\n", \(\(grady[r_, t_] = {D[y[r, t], r], D[y[r, t], t]};\)\), "\n", \(Axy[r_, t_] = Det[{gradx[r, t], grady[r, t]}]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(17.280000000000005`\ r\ t\^2\ Cos[t]\^4\)\/\((\(-25\) + 7\ Cos[t]\^2 \ - 7\ Sin[t]\^2)\)\^2 + \(48.000000000000014`\ r\ t\^2\ Cos[t]\^2\ Sin[t]\^2\)\ \/\((\(-25\) + 7\ Cos[t]\^2 - 7\ Sin[t]\^2)\)\^2 + \(30.720000000000006`\ r\ \ t\^2\ Sin[t]\^4\)\/\((\(-25\) + 7\ Cos[t]\^2 - 7\ Sin[t]\^2)\)\^2 - \ \(3.6000000000000005`\ t\ Cos[t]\^2\ \@\(9\ Cos[t]\^2 + 16\ Sin[t]\^2\)\)\/\(\ \(-25\) + 7\ Cos[t]\^2 - 7\ Sin[t]\^2\) - \(6.4`\ t\ Sin[t]\^2\ \@\(9\ Cos[t]\ \^2 + 16\ Sin[t]\^2\)\)\/\(\(-25\) + 7\ Cos[t]\^2 - 7\ Sin[t]\^2\)\)], \ "Output"] }, Open ]], Cell[TextData[{ "Analyzing this mess see whether it can ever go negative is a scary \ prospect. Take the easy way out and integrate its absolute value,using ", Cell[BoxData[ \(NIntegrate\)]], " to calculate\n ", Cell[BoxData[ \(\[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \)\(R\_xy\)\)\%\ \[Null]\ \[DifferentialD]x \[DifferentialD]y\) = \ \[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\(\ \)\(R\_\(\(\ \)\(r\ t\)\)\)\ \)\%\[Null] A\_xy[r, t]\ \[DifferentialD]r \[DifferentialD]t\)\)]], "\n ", Cell[BoxData[ \(\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(=\)\(\[Integral]\_tlow\%thigh\(\ \[Integral]\_\(-1\)\%1\ A\_xy[r, t] \[DifferentialD]r \[DifferentialD]t\)\)\)\)\)]], "." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[Evaluate[Abs[Axy[r, t]]], {t, tlow, thigh}, {r, \(-1\), 1}, AccuracyGoal \[Rule] 2]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(17.25940067170122`\)], "Output"] }, Open ]], Cell[TextData[{ "About ", Cell[BoxData[ \(17.3\)]], " square units.\nPity those poor souls in the traditional calculus course. \ Most of them couldn't even dream of making 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.68607 .41907 L p .19 0 0 r F P 0 g s .55266 .40131 m .55201 .40751 L .60967 .41767 L p .643 .13 .009 r F P 0 g s .60417 .42517 m .60967 .41767 L .55201 .40751 L p .696 .226 .131 r F P 0 g s .60967 .41767 m .60417 .42517 L .65046 .42624 L p .538 .047 .037 r F P 0 g s .48345 .37559 m .48785 .38122 L .55266 .40131 L p .64 .085 0 r F P 0 g s .55201 .40751 m .55266 .40131 L .48785 .38122 L p .728 .221 .02 r F P 0 g s .76791 .37898 m .75301 .38205 L .76024 .35401 L p .755 .968 .93 r F P 0 g s .76024 .35401 m .77601 .34883 L .76791 .37898 L p .755 .968 .93 r F P 0 g s .70758 .40416 m .72285 .39456 L .69917 .41096 L p .429 .832 .632 r F P 0 g s .64059 .4357 m .65046 .42624 L .60417 .42517 L p .582 .125 .133 r F P 0 g s .65046 .42624 m .64059 .4357 L .67282 .42925 L p .264 0 0 r F P 0 g s .67282 .42925 m .68607 .41907 L .65046 .42624 L p .264 0 0 r F P 0 g s .75301 .38205 m .73799 .3872 L .7444 .36131 L p .783 .986 .897 r F P 0 g s .7444 .36131 m .76024 .35401 L .75301 .38205 L p .783 .986 .897 r F P 0 g s .55201 .40751 m .55136 .41593 L .60417 .42517 L p .696 .226 .131 r F P 0 g s .5986 .43474 m .60417 .42517 L .55136 .41593 L p .725 .287 .209 r F P 0 g s .60417 .42517 m .5986 .43474 L .64059 .4357 L p .582 .125 .133 r F P 0 g s .48785 .38122 m .49229 .38908 L .55201 .40751 L p .728 .221 .02 r F P 0 g s .55136 .41593 m .55201 .40751 L .49229 .38908 L p .776 .309 .135 r F P 0 g s .85396 .34584 m .83849 .34428 L .81977 .29639 L p .651 .851 .977 r F P 0 g s .81977 .29639 m .83479 .29539 L .85396 .34584 L p .651 .851 .977 r F P 0 g s .83849 .34428 m .82296 .34354 L .80475 .29823 L p .678 .866 .97 r F P 0 g s .80475 .29823 m .81977 .29639 L .83849 .34428 L p .678 .866 .97 r F P 0 g s .48785 .38122 m .48345 .37559 L .41729 .34561 L p .651 .093 0 r F P 0 g s .73799 .3872 m .72285 .39456 L .72849 .37087 L p .806 .997 .857 r F P 0 g s .72849 .37087 m .7444 .36131 L .73799 .3872 L p .806 .997 .857 r F P 0 g s .68607 .41907 m .67282 .42925 L .69216 .41586 L p 0 0 0 r F P 0 g s .67658 .4293 m .69216 .41586 L .67282 .42925 L p 0 0 0 r F P 0 g s .69216 .41586 m .67658 .4293 L .67695 .42752 L p .836 .993 .741 r F P 0 g s .7059 .40533 m .68607 .41907 L p .69216 .41586 L 0 0 0 r F P 0 g s .67695 .42752 m .68012 .41229 L p .69216 .41586 L .836 .993 .741 r F P 0 g s .69216 .41586 m .70758 .40416 L .7059 .40533 L p 0 0 0 r F P 0 g s .82296 .34354 m .80737 .34387 L .78973 .30119 L p .711 .884 .961 r F P 0 g s .78973 .30119 m .80475 .29823 L .82296 .34354 L p .711 .884 .961 r F P 0 g s .63058 .4469 m .64059 .4357 L .5986 .43474 L p .606 .173 .193 r F P 0 g s .64059 .4357 m .63058 .4469 L .6594 .44117 L p .308 0 0 r F P 0 g s .6594 .44117 m .67282 .42925 L .64059 .4357 L p .308 0 0 r F P 0 g s .49229 .38908 m .49676 .39919 L .55136 .41593 L p .776 .309 .135 r F P 0 g s .5507 .42642 m .55136 .41593 L .49676 .39919 L p .802 .365 .209 r F P 0 g s .55136 .41593 m .5507 .42642 L .5986 .43474 L p .725 .287 .209 r F P 0 g s .59296 .44604 m .5986 .43474 L .5507 .42642 L p .741 .324 .257 r F P 0 g s .5986 .43474 m .59296 .44604 L .63058 .4469 L p .606 .173 .193 r F P 0 g s .41729 .34561 m .42666 .35357 L .48785 .38122 L p .651 .093 0 r F P 0 g s .49229 .38908 m .48785 .38122 L .42666 .35357 L p .752 .235 0 r F P 0 g s .37088 .33272 m .36168 .33369 L .27859 .28116 L p .429 .747 .993 r F P 0 g s .80737 .34387 m .79172 .34554 L .77471 .30552 L p .747 .901 .947 r F P 0 g s .77471 .30552 m .78973 .30119 L .80737 .34387 L p .747 .901 .947 r F P 0 g s .72285 .39456 m .70758 .40416 L .71248 .38269 L p .822 1 .814 r F P 0 g s .71248 .38269 m .72849 .37087 L .72285 .39456 L p .822 1 .814 r F P 0 g s .67282 .42925 m .6594 .44117 L .67658 .4293 L p 0 0 0 r F P 0 g s .27859 .28116 m .29245 .28291 L .37088 .33272 L p .429 .747 .993 r F P 0 g s .3801 .33259 m .37088 .33272 L .29245 .28291 L p .351 .709 .985 r F P 0 g s .79172 .34554 m .77601 .34883 L .75968 .31151 L p .785 .918 .927 r F P 0 g s .75968 .31151 m .77471 .30552 L .79172 .34554 L p .785 .918 .927 r F P 0 g s .49676 .39919 m .50127 .4114 L .5507 .42642 L p .802 .365 .209 r F P 0 g s .55004 .43866 m .5507 .42642 L .50127 .4114 L p .816 .399 .254 r F P 0 g s .5507 .42642 m .55004 .43866 L .59296 .44604 L p .741 .324 .257 r F P 0 g s .42666 .35357 m .43607 .3638 L .49229 .38908 L p .752 .235 0 r F P 0 g s .49676 .39919 m .49229 .38908 L .43607 .3638 L p .809 .33 .064 r F P 0 g s .29245 .28291 m .30631 .28552 L .3801 .33259 L p .351 .709 .985 r F P 0 g s .62041 .45929 m .63058 .4469 L .59296 .44604 L p .618 .198 .224 r F P 0 g s .63058 .4469 m .62041 .45929 L .64579 .45428 L p .331 0 .034 r F P 0 g s .64579 .45428 m .6594 .44117 L .63058 .4469 L p .331 0 .034 r F P 0 g s .58722 .45854 m .59296 .44604 L .55004 .43866 L p .749 .343 .283 r F P 0 g s .59296 .44604 m .58722 .45854 L .62041 .45929 L p .618 .198 .224 r F P 0 g s .70758 .40416 m .69216 .41586 L .69636 .39661 L p .831 .997 .774 r F P 0 g s .69636 .39661 m .71248 .38269 L .70758 .40416 L p .831 .997 .774 r F P 0 g s .77601 .34883 m .76024 .35401 L .74465 .31943 L p .823 .932 .903 r F P 0 g s .74465 .31943 m .75968 .31151 L .77601 .34883 L p .823 .932 .903 r F P 0 g s .6594 .44117 m .64579 .45428 L .6608 .44393 L p 0 0 0 r F P 0 g s .6608 .44393 m .67658 .4293 L .6594 .44117 L p 0 0 0 r F P 0 g s .43607 .3638 m .44553 .37629 L .49676 .39919 L p .809 .33 .064 r F P 0 g s .50127 .4114 m .49676 .39919 L .44553 .37629 L p .84 .39 .144 r F P 0 g s .76024 .35401 m .7444 .36131 L .72961 .3295 L p .856 .941 .876 r F P 0 g s .72961 .3295 m .74465 .31943 L .76024 .35401 L p .856 .941 .876 r F P 0 g s .50127 .4114 m .50582 .42535 L .55004 .43866 L p .816 .399 .254 r F P 0 g s .54936 .4521 m .55004 .43866 L .50582 .42535 L p .823 .417 .278 r F P 0 g s .55004 .43866 m .54936 .4521 L .58722 .45854 L p .749 .343 .283 r F P 0 g s .68012 .41229 m .69636 .39661 L .69216 .41586 L p .836 .993 .741 r F P 0 g s .61009 .47212 m .62041 .45929 L .58722 .45854 L p .621 .204 .23 r F P 0 g s .62041 .45929 m .61009 .47212 L .63197 .46784 L p .336 0 .044 r F P 0 g s .63197 .46784 m .64579 .45428 L .62041 .45929 L p .336 0 .044 r F P 0 g s .5814 .47148 m .58722 .45854 L .54936 .4521 L p .751 .348 .288 r F P 0 g s .58722 .45854 m .5814 .47148 L .61009 .47212 L p .621 .204 .23 r F P 0 g s .7444 .36131 m .72849 .37087 L .71455 .34184 L p .884 .946 .846 r F P 0 g s .71455 .34184 m .72961 .3295 L .7444 .36131 L p .884 .946 .846 r F P 0 g s .44553 .37629 m .45504 .39089 L .50127 .4114 L p .84 .39 .144 r F P 0 g s .50582 .42535 m .50127 .4114 L .45504 .39089 L p .857 .426 .194 r F P 0 g s .64579 .45428 m .63197 .46784 L .64483 .45901 L p 0 0 0 r F P 0 g s .64483 .45901 m .6608 .44393 L .64579 .45428 L p 0 0 0 r F P 0 g s .72849 .37087 m .71248 .38269 L .69947 .35647 L p .906 .948 .818 r F P 0 g s .69947 .35647 m .71455 .34184 L .72849 .37087 L p .906 .948 .818 r F P 0 g s .67658 .4293 m .6608 .44393 L .66374 .42915 L p .838 .988 .72 r F P 0 g s .66374 .42915 m .68012 .41229 L .67658 .4293 L p .838 .988 .72 r F P 0 g s .50582 .42535 m .51043 .4405 L .54936 .4521 L p .823 .417 .278 r F P 0 g s .54868 .46597 m .54936 .4521 L .51043 .4405 L p .824 .42 .283 r F P 0 g s .54936 .4521 m .54868 .46597 L .5814 .47148 L p .751 .348 .288 r F P 0 g s .59958 .48446 m .61009 .47212 L .5814 .47148 L p .613 .188 .211 r F P 0 g s .61009 .47212 m .59958 .48446 L .61793 .4809 L p .322 0 .016 r F P 0 g s .61793 .4809 m .63197 .46784 L .61009 .47212 L p .322 0 .016 r F P 0 g s .45504 .39089 m .46462 .40724 L .50582 .42535 L p .857 .426 .194 r F P 0 g s .51043 .4405 m .50582 .42535 L .46462 .40724 L p .865 .445 .22 r F P 0 g s .57548 .48393 m .5814 .47148 L .54868 .46597 L p .746 .335 .272 r F P 0 g s .5814 .47148 m .57548 .48393 L .59958 .48446 L p .613 .188 .211 r F P 0 g s .71248 .38269 m .69636 .39661 L .68434 .37319 L p .922 .947 .794 r F P 0 g s .68434 .37319 m .69947 .35647 L .71248 .38269 L p .922 .947 .794 r F P 0 g s .63197 .46784 m .61793 .4809 L .62863 .47357 L p 0 0 0 r F P 0 g s .62863 .47357 m .64483 .45901 L .63197 .46784 L p 0 0 0 r F P 0 g s .6608 .44393 m .64483 .45901 L .64719 .44642 L p .838 .987 .716 r F P 0 g s .64719 .44642 m .66374 .42915 L .6608 .44393 L p .838 .987 .716 r F P 0 g s .51043 .4405 m .51509 .45608 L .54868 .46597 L p .824 .42 .283 r F P 0 g s .54798 .47935 m .54868 .46597 L .51509 .45608 L p .82 .409 .268 r F P 0 g s .54868 .46597 m .54798 .47935 L .57548 .48393 L p .746 .335 .272 r F P 0 g s .69636 .39661 m .68012 .41229 L .66914 .39166 L p .933 .945 .776 r F P 0 g s .66914 .39166 m .68434 .37319 L .69636 .39661 L p .933 .945 .776 r F P 0 g s .83479 .29539 m .81977 .29639 L .76913 .24275 L p .666 .843 .968 r F P 0 g s .76913 .24275 m .78187 .23868 L .83479 .29539 L p .666 .843 .968 r F P 0 g s .46462 .40724 m .47427 .42477 L .51043 .4405 L p .865 .445 .22 r F P 0 g s .51509 .45608 m .51043 .4405 L .47427 .42477 L p .867 .449 .226 r F P 0 g s .81977 .29639 m .80475 .29823 L .75643 .2477 L p .696 .854 .958 r F P 0 g s .75643 .2477 m .76913 .24275 L .81977 .29639 L p .696 .854 .958 r F P 0 g s .80475 .29823 m .78973 .30119 L .74379 .25376 L p .73 .865 .944 r F P 0 g s .74379 .25376 m .75643 .2477 L .80475 .29823 L p .73 .865 .944 r F P 0 g s .58889 .49528 m .59958 .48446 L .57548 .48393 L p .591 .142 .155 r F P 0 g s .59958 .48446 m .58889 .49528 L .60365 .49243 L p .281 0 0 r F P 0 g s .60365 .49243 m .61793 .4809 L .59958 .48446 L p .281 0 0 r F P 0 g s .56945 .49485 m .57548 .48393 L .54798 .47935 L p .731 .3 .226 r F P 0 g s .57548 .48393 m .56945 .49485 L .58889 .49528 L p .591 .142 .155 r F P 0 g s .78973 .30119 m .77471 .30552 L .73119 .26122 L p .767 .874 .926 r F P 0 g s .73119 .26122 m .74379 .25376 L .78973 .30119 L p .767 .874 .926 r F P 0 g s .46462 .40724 m .45504 .39089 L .41689 .36674 L p .854 .396 .05 r F P 0 g s .77471 .30552 m .75968 .31151 L .71865 .27036 L p .802 .881 .903 r F P 0 g s .71865 .27036 m .73119 .26122 L .77471 .30552 L p .802 .881 .903 r F P 0 g s .64483 .45901 m .62863 .47357 L .63048 .46315 L p .837 .99 .729 r F P 0 g s .63048 .46315 m .64719 .44642 L .64483 .45901 L p .837 .99 .729 r F P 0 g s .68012 .41229 m .66374 .42915 L .65388 .41127 L p .939 .944 .764 r F P 0 g s .65388 .41127 m .66914 .39166 L .68012 .41229 L p .939 .944 .764 r F P 0 g s .61793 .4809 m .60365 .49243 L .6122 .48659 L p 0 0 0 r F P 0 g s .6122 .48659 m .62863 .47357 L .61793 .4809 L p 0 0 0 r F P 0 g s .75968 .31151 m .74465 .31943 L .70615 .28144 L p .835 .885 .878 r F P 0 g s .70615 .28144 m .71865 .27036 L .75968 .31151 L p .835 .885 .878 r F P 0 g s .51509 .45608 m .51982 .47115 L .54798 .47935 L p .82 .409 .268 r F P 0 g s .54728 .49119 m .54798 .47935 L .51982 .47115 L p .807 .377 .225 r F P 0 g s .54798 .47935 m .54728 .49119 L .56945 .49485 L p .731 .3 .226 r F P 0 g s .47427 .42477 m .48401 .4427 L .51509 .45608 L p .867 .449 .226 r F P 0 g s .51982 .47115 m .51509 .45608 L .48401 .4427 L p .862 .437 .209 r F P 0 g s .74465 .31943 m .72961 .3295 L .6937 .29469 L p .863 .886 .851 r F P 0 g s .6937 .29469 m .70615 .28144 L .74465 .31943 L p .863 .886 .851 r F P 0 g s .41689 .36674 m .43078 .38599 L .46462 .40724 L p .854 .396 .05 r F P 0 g s .47427 .42477 m .46462 .40724 L .43078 .38599 L p .867 .421 .083 r F P 0 g s .66374 .42915 m .64719 .44642 L .63852 .43126 L p .94 .943 .762 r F P 0 g s .63852 .43126 m .65388 .41127 L .66374 .42915 L p .94 .943 .762 r F P 0 g s .29245 .28291 m .27859 .28116 L .21872 .22372 L p .456 .747 .991 r F P 0 g s .72961 .3295 m .71455 .34184 L .68128 .31021 L p .886 .884 .825 r F P 0 g s .68128 .31021 m .6937 .29469 L .72961 .3295 L p .886 .884 .825 r F P 0 g s .21872 .22372 m .23635 .22864 L .29245 .28291 L p .456 .747 .991 r F P 0 g s .30631 .28552 m .29245 .28291 L .23635 .22864 L p .398 .715 .987 r F P 0 g s .57802 .50347 m .58889 .49528 L .56945 .49485 L p .537 .044 .034 r F P 0 g s .58889 .49528 m .57802 .50347 L .58915 .50134 L p .192 0 0 r F P 0 g s .58915 .50134 m .60365 .49243 L .58889 .49528 L p .192 0 0 r F P 0 g s .56332 .50315 m .56945 .49485 L .54728 .49119 L p .694 .223 .128 r F P 0 g s .56945 .49485 m .56332 .50315 L .57802 .50347 L p .537 .044 .034 r F P 0 g s .23635 .22864 m .25389 .23443 L .30631 .28552 L p .398 .715 .987 r F P 0 g s .32015 .28923 m .30631 .28552 L .25389 .23443 L p .315 .666 .974 r F P 0 g s .62863 .47357 m .6122 .48659 L .61358 .4783 L p .832 .996 .763 r F P 0 g s .61358 .4783 m .63048 .46315 L .62863 .47357 L p .832 .996 .763 r F P 0 g s .25389 .23443 m .27135 .24134 L .32015 .28923 L p .315 .666 .974 r F P 0 g s .33398 .29433 m .32015 .28923 L .27135 .24134 L p .2 .592 .945 r F P 0 g s .71455 .34184 m .69947 .35647 L .66889 .328 L p .903 .88 .802 r F P 0 g s .66889 .328 m .68128 .31021 L .71455 .34184 L p .903 .88 .802 r F P 0 g s .27135 .24134 m .28872 .24965 L .33398 .29433 L p .2 .592 .945 r F P 0 g s .3478 .3011 m .33398 .29433 L .28872 .24965 L p .051 .486 .888 r F P 0 g s .60365 .49243 m .58915 .50134 L .59555 .49696 L p .422 .828 .631 r F P 0 g s .59555 .49696 m .6122 .48659 L .60365 .49243 L p .422 .828 .631 r F P 0 g s .43078 .38599 m .44473 .40637 L .47427 .42477 L p .867 .421 .083 r F P 0 g s .48401 .4427 m .47427 .42477 L .44473 .40637 L p .87 .426 .09 r F P 0 g s .48401 .4427 m .49384 .46008 L .51982 .47115 L p .862 .437 .209 r F P 0 g s .52461 .48465 m .51982 .47115 L .49384 .46008 L p .846 .402 .162 r F P 0 g s .51982 .47115 m .52461 .48465 L .54728 .49119 L p .807 .377 .225 r F P 0 g s .28872 .24965 m .306 .25964 L .3478 .3011 L p .051 .486 .888 r F P 0 g s .36161 .30981 m .3478 .3011 L .306 .25964 L p 0 .348 .796 r F P 0 g s .54656 .50041 m .54728 .49119 L .52461 .48465 L p .774 .306 .131 r F P 0 g s .54728 .49119 m .54656 .50041 L .56332 .50315 L p .694 .223 .128 r F P 0 g s .64719 .44642 m .63048 .46315 L .62306 .45065 L p .936 .945 .769 r F P 0 g s .62306 .45065 m .63852 .43126 L .64719 .44642 L p .936 .945 .769 r F P 0 g s .69947 .35647 m .68434 .37319 L .65651 .34787 L p .916 .877 .783 r F P 0 g s .65651 .34787 m .66889 .328 L .69947 .35647 L p .916 .877 .783 r F P 0 g s .306 .25964 m .3232 .27158 L .36161 .30981 L p 0 .348 .796 r F P 0 g s .68434 .37319 m .66914 .39166 L .64414 .36944 L p .924 .873 .769 r F P 0 g s .64414 .36944 m .65651 .34787 L .68434 .37319 L p .924 .873 .769 r F P 0 g s .44473 .40637 m .45874 .42711 L .48401 .4427 L p .87 .426 .09 r F P 0 g s .49384 .46008 m .48401 .4427 L .45874 .42711 L p .862 .411 .069 r F P 0 g s .6122 .48659 m .59555 .49696 L .5965 .49077 L p .817 1 .815 r F P 0 g s .5965 .49077 m .61358 .4783 L .6122 .48659 L p .817 1 .815 r F P 0 g s .56696 .50803 m .57802 .50347 L .56332 .50315 L p .398 0 0 r F P 0 g s .57802 .50347 m .56696 .50803 L .57442 .5066 L p .014 0 0 r F P 0 g s .57442 .5066 m .58915 .50134 L .57802 .50347 L p .014 0 0 r F P 0 g s .55709 .50782 m .56332 .50315 L .54656 .50041 L p .59 .048 0 r F P 0 g s .56332 .50315 m .55709 .50782 L .56696 .50803 L p .398 0 0 r F P 0 g s .49384 .46008 m .50377 .47586 L .52461 .48465 L p .846 .402 .162 r F P 0 g s .52945 .49551 m .52461 .48465 L .50377 .47586 L p .806 .326 .061 r F P 0 g s .52461 .48465 m .52945 .49551 L .54656 .50041 L p .774 .306 .131 r F P 0 g s .63048 .46315 m .61358 .4783 L .60749 .46838 L p .926 .947 .788 r F P 0 g s .60749 .46838 m .62306 .45065 L .63048 .46315 L p .926 .947 .788 r F P 0 g s .58915 .50134 m .57442 .5066 L .57868 .50368 L p .495 .901 .795 r F P 0 g s .57868 .50368 m .59555 .49696 L .58915 .50134 L p .495 .901 .795 r F P 0 g s .66914 .39166 m .65388 .41127 L .63175 .3921 L p .928 .871 .761 r F P 0 g s .63175 .3921 m .64414 .36944 L .66914 .39166 L p .928 .871 .761 r F P 0 g s .54583 .50598 m .54656 .50041 L .52945 .49551 L p .678 .143 0 r F P 0 g s .54656 .50041 m .54583 .50598 L .55709 .50782 L p .59 .048 0 r F P 0 g s .45874 .42711 m .47283 .44724 L .49384 .46008 L p .862 .411 .069 r F P 0 g s .50377 .47586 m .49384 .46008 L .47283 .44724 L p .836 .366 .011 r F P 0 g s .65388 .41127 m .63852 .43126 L .61934 .41505 L p .929 .871 .759 r F P 0 g s .61934 .41505 m .63175 .3921 L .65388 .41127 L p .929 .871 .759 r F P 0 g s .59555 .49696 m .57868 .50368 L .57926 .49956 L p .782 .992 .883 r F P 0 g s .57926 .49956 m .5965 .49077 L .59555 .49696 L p .782 .992 .883 r F P 0 g s .61358 .4783 m .5965 .49077 L .59183 .48339 L p .904 .95 .821 r F P 0 g s .59183 .48339 m .60749 .46838 L .61358 .4783 L p .904 .95 .821 r F P 0 g s .50377 .47586 m .51379 .48895 L .52945 .49551 L p .806 .326 .061 r F P 0 g s .53436 .50271 m .52945 .49551 L .51379 .48895 L p .695 .153 0 r F P 0 g s .52945 .49551 m .53436 .50271 L .54583 .50598 L p .678 .143 0 r F P 0 g s .55573 .50809 m .56696 .50803 L .55709 .50782 L p .046 0 0 r F P 0 g s .56696 .50803 m .55573 .50809 L .55948 .50737 L p .26 .769 .836 r F P 0 g s .55948 .50737 m .57442 .5066 L .56696 .50803 L p .26 .769 .836 r F P 0 g s .55076 .50798 m .55709 .50782 L .54583 .50598 L p .257 0 0 r F P 0 g s .55709 .50782 m .55076 .50798 L .55573 .50809 L p .046 0 0 r F P 0 g s .47283 .44724 m .487 .46569 L .50377 .47586 L p .836 .366 .011 r F P 0 g s .51379 .48895 m .50377 .47586 L .487 .46569 L p .774 .271 0 r F P 0 g s .63852 .43126 m .62306 .45065 L .6069 .43734 L p .926 .872 .764 r F P 0 g s .6069 .43734 m .61934 .41505 L .63852 .43126 L p .926 .872 .764 r F P 0 g s .57442 .5066 m .55948 .50737 L .5616 .5059 L p .545 .92 .936 r F P 0 g s .5616 .5059 m .57868 .50368 L .57442 .5066 L p .545 .92 .936 r F P 0 g s .54509 .50706 m .54583 .50598 L .53436 .50271 L p .354 0 0 r F P 0 g s .54583 .50598 m .54509 .50706 L .55076 .50798 L p .257 0 0 r F P 0 g s .5965 .49077 m .57926 .49956 L .57606 .49466 L p .864 .948 .868 r F P 0 g s .57606 .49466 m .59183 .48339 L .5965 .49077 L p .864 .948 .868 r F P 0 g s .62306 .45065 m .60749 .46838 L .59443 .45788 L p .918 .876 .779 r F P 0 g s .59443 .45788 m .6069 .43734 L .62306 .45065 L p .918 .876 .779 r F P 0 g s .57868 .50368 m .5616 .5059 L .56186 .50384 L p .715 .955 .95 r F P 0 g s .56186 .50384 m .57926 .49956 L .57868 .50368 L p .715 .955 .95 r F P 0 g s .47283 .44724 m .45874 .42711 L .44222 .41092 L p .735 .267 0 r F P 0 g s .487 .46569 m .50125 .48139 L .51379 .48895 L p .774 .271 0 r F P 0 g s .5239 .49835 m .51379 .48895 L .50125 .48139 L p .618 .07 0 r F P 0 g s .51379 .48895 m .5239 .49835 L .53436 .50271 L p .695 .153 0 r F P 0 g s .53933 .50542 m .53436 .50271 L .5239 .49835 L p .348 0 0 r F P 0 g s .53436 .50271 m .53933 .50542 L .54509 .50706 L p .354 0 0 r F P 0 g s .44222 .41092 m .45921 .43395 L .47283 .44724 L p .735 .267 0 r F P 0 g s .487 .46569 m .47283 .44724 L .45921 .43395 L p .686 .206 0 r F P 0 g s .60749 .46838 m .59183 .48339 L .58192 .4756 L p .902 .883 .804 r F P 0 g s .58192 .4756 m .59443 .45788 L .60749 .46838 L p .902 .883 .804 r F P 0 g s .57926 .49956 m .56186 .50384 L .56023 .50139 L p .787 .931 .928 r F P 0 g s .56023 .50139 m .57606 .49466 L .57926 .49956 L p .787 .931 .928 r F P 0 g s .54434 .50306 m .55573 .50809 L .55076 .50798 L closepath p .406 .801 .986 r F P 0 g s .54434 .50306 m .55948 .50737 L .55573 .50809 L closepath p .479 .837 .994 r F P 0 g s .54434 .50306 m .55076 .50798 L .54509 .50706 L closepath p .345 .764 .977 r F P 0 g s .54434 .50306 m .5616 .5059 L .55948 .50737 L closepath p .548 .862 .996 r F P 0 g s .54434 .50306 m .54509 .50706 L .53933 .50542 L closepath p .309 .734 .973 r F P 0 g s .50125 .48139 m .51557 .49334 L .5239 .49835 L p .618 .07 0 r F P 0 g s .5239 .49835 m .53409 .50323 L .53933 .50542 L p .348 0 0 r F P 0 g s .53409 .50323 m .5239 .49835 L .51557 .49334 L p 0 .314 .711 r F P 0 g s .45921 .43395 m .47622 .45521 L .487 .46569 L p .686 .206 0 r F P 0 g s .50125 .48139 m .487 .46569 L .47622 .45521 L p .582 .085 0 r F P 0 g s .70615 .28144 m .6937 .29469 L .63918 .26061 L p .846 .838 .837 r F P 0 g s .6937 .29469 m .68128 .31021 L .63112 .27937 L p .864 .832 .813 r F P 0 g s .63112 .27937 m .63918 .26061 L .6937 .29469 L p .864 .832 .813 r F P 0 g s .71865 .27036 m .70615 .28144 L .64729 .24411 L p .824 .843 .862 r F P 0 g s .63918 .26061 m .64729 .24411 L .70615 .28144 L p .846 .838 .837 r F P 0 g s .59183 .48339 m .57606 .49466 L .5694 .48952 L p .871 .89 .845 r F P 0 g s .5694 .48952 m .58192 .4756 L .59183 .48339 L p .871 .89 .845 r F P 0 g s .54434 .50306 m .56186 .50384 L .5616 .5059 L closepath p .604 .873 .992 r F P 0 g s .68128 .31021 m .66889 .328 L .62311 .30036 L p .878 .826 .793 r F P 0 g s .62311 .30036 m .63112 .27937 L .68128 .31021 L p .878 .826 .793 r F P 0 g s .73119 .26122 m .71865 .27036 L .65547 .22975 L p .797 .846 .888 r F P 0 g s .64729 .24411 m .65547 .22975 L .71865 .27036 L p .824 .843 .862 r F P 0 g s .54434 .50306 m .53933 .50542 L .53409 .50323 L closepath p .304 .718 .975 r F P 0 g s .66889 .328 m .65651 .34787 L .61515 .32338 L p .887 .821 .777 r F P 0 g s .61515 .32338 m .62311 .30036 L .66889 .328 L p .887 .821 .777 r F P 0 g s .74379 .25376 m .73119 .26122 L .6637 .21732 L p .765 .846 .912 r F P 0 g s .65547 .22975 m .6637 .21732 L .73119 .26122 L p .797 .846 .888 r F P 0 g s .65651 .34787 m .64414 .36944 L .60721 .34803 L p .893 .816 .765 r F P 0 g s .60721 .34803 m .61515 .32338 L .65651 .34787 L p .893 .816 .765 r F P 0 g s .75643 .2477 m .74379 .25376 L .67201 .20657 L p .733 .843 .932 r F P 0 g s .6637 .21732 m .67201 .20657 L .74379 .25376 L p .765 .846 .912 r F P 0 g s .47622 .45521 m .49325 .47363 L .50125 .48139 L p .582 .085 0 r F P 0 g s .51557 .49334 m .50125 .48139 L .49325 .47363 L p 0 .128 .612 r F P 0 g s .54434 .50306 m .56023 .50139 L .56186 .50384 L closepath p .643 .872 .984 r F P 0 g s .57606 .49466 m .56023 .50139 L .55686 .49884 L p .807 .892 .905 r F P 0 g s .55686 .49884 m .5694 .48952 L .57606 .49466 L p .807 .892 .905 r F P 0 g s .64414 .36944 m .63175 .3921 L .5993 .3737 L p .897 .814 .758 r F P 0 g s .5993 .3737 m .60721 .34803 L .64414 .36944 L p .897 .814 .758 r F P 0 g s .76913 .24275 m .75643 .2477 L .68038 .19721 L p .7 .838 .949 r F P 0 g s .67201 .20657 m .68038 .19721 L .75643 .2477 L p .733 .843 .932 r F P 0 g s .51557 .49334 m .52994 .50073 L .53409 .50323 L p 0 .314 .711 r F P 0 g s .54434 .50306 m .53409 .50323 L .52994 .50073 L closepath p .328 .716 .982 r F P 0 g s .63175 .3921 m .61934 .41505 L .5914 .39957 L p .897 .813 .757 r F P 0 g s .5914 .39957 m .5993 .3737 L .63175 .3921 L p .897 .813 .757 r F P 0 g s .78187 .23868 m .76913 .24275 L .68882 .18895 L p .67 .832 .961 r F P 0 g s .68038 .19721 m .68882 .18895 L .76913 .24275 L p .7 .838 .949 r F P 0 g s .49325 .47363 m .5103 .48822 L .51557 .49334 L p 0 .128 .612 r F P 0 g s .52994 .50073 m .51557 .49334 L .5103 .48822 L p 0 .438 .85 r F P 0 g s .54434 .50306 m .55686 .49884 L .56023 .50139 L closepath p .665 .864 .975 r F P 0 g s .61934 .41505 m .6069 .43734 L .58352 .42467 L p .895 .815 .761 r F P 0 g s .58352 .42467 m .5914 .39957 L .61934 .41505 L p .895 .815 .761 r F P 0 g s .68882 .18895 m .69733 .18156 L .78187 .23868 L p .67 .832 .961 r F P 0 g s .5103 .48822 m .52734 .49819 L .52994 .50073 L p 0 .438 .85 r F P 0 g s .54434 .50306 m .52994 .50073 L .52734 .49819 L closepath p .374 .725 .989 r F P 0 g s .54434 .50306 m .52774 .49406 L .53072 .49293 L closepath p .543 .772 .983 r F P 0 g s .54434 .50306 m .52657 .49588 L .52774 .49406 L closepath p .489 .756 .989 r F P 0 g s .54434 .50306 m .52734 .49819 L .52657 .49588 L closepath p .431 .739 .992 r F P 0 g s .54434 .50306 m .55212 .49644 L .55686 .49884 L closepath p .672 .851 .968 r F P 0 g s .54434 .50306 m .54651 .49447 L .55212 .49644 L closepath p .666 .835 .965 r F P 0 g s .54434 .50306 m .54066 .49314 L .54651 .49447 L closepath p .65 .82 .965 r F P 0 g s .54434 .50306 m .53519 .49261 L .54066 .49314 L closepath p .624 .804 .969 r F P 0 g s .54434 .50306 m .53072 .49293 L .53519 .49261 L closepath p .588 .788 .975 r F P 0 g s .6069 .43734 m .59443 .45788 L .57565 .44792 L p .889 .82 .773 r F P 0 g s .57565 .44792 m .58352 .42467 L .6069 .43734 L p .889 .82 .773 r F P 0 g s .59443 .45788 m .58192 .4756 L .56779 .46824 L p .877 .828 .795 r F P 0 g s .56779 .46824 m .57565 .44792 L .59443 .45788 L p .877 .828 .795 r F P 0 g s .5694 .48952 m .55686 .49884 L .55212 .49644 L p .801 .855 .89 r F P 0 g s .32664 .25245 m .34523 .272 L .35739 .30203 L p 0 .319 .806 r F P 0 g s .35739 .30203 m .34033 .28567 L .32664 .25245 L p 0 .319 .806 r F P 0 g s .36369 .29377 m .38204 .31755 L .39136 .34136 L p 0 .152 .689 r F P 0 g s .30791 .23514 m .32664 .25245 L .34033 .28567 L p .057 .414 .863 r F P 0 g s .34033 .28567 m .3232 .27158 L .30791 .23514 L p .057 .414 .863 r F P 0 g s .58192 .4756 m .5694 .48952 L .55994 .48468 L p .853 .84 .832 r F P 0 g s .55994 .48468 m .56779 .46824 L .58192 .4756 L p .853 .84 .832 r F P 0 g s .55212 .49644 m .55994 .48468 L .5694 .48952 L p .801 .855 .89 r F P 0 g s .38204 .31755 m .40029 .34295 L .40831 .36374 L p 0 .094 .645 r F P 0 g s .5087 .48355 m .52657 .49588 L .52734 .49819 L p .206 .558 .934 r F P 0 g s .52734 .49819 m .5103 .48822 L .5087 .48355 L p .206 .558 .934 r F P 0 g s .28902 .21997 m .30791 .23514 L .3232 .27158 L p .162 .504 .91 r F P 0 g s .3232 .27158 m .306 .25964 L .28902 .21997 L p .162 .504 .91 r F P 0 g s .40029 .34295 m .41848 .36935 L .42526 .38719 L p 0 .06 .618 r F P 0 g s .41848 .36935 m .43661 .39592 L .44222 .41092 L p 0 .053 .612 r F P 0 g s .45921 .43395 m .44222 .41092 L .43661 .39592 L p 0 .075 .63 r F P 0 g s .49076 .46652 m .5087 .48355 L .5103 .48822 L p .003 .376 .84 r F P 0 g s .5103 .48822 m .49325 .47363 L .49076 .46652 L p .003 .376 .84 r F P 0 g s .26996 .20673 m .28902 .21997 L .306 .25964 L p .259 .582 .944 r F P 0 g s .306 .25964 m .28872 .24965 L .26996 .20673 L p .259 .582 .944 r F P 0 g s .43661 .39592 m .4547 .42168 L .45921 .43395 L p 0 .075 .63 r F P 0 g s .47622 .45521 m .45921 .43395 L .4547 .42168 L p 0 .132 .674 r F P 0 g s .55994 .48468 m .55212 .49644 L .54651 .49447 L p .781 .821 .884 r F P 0 g s .47275 .44558 m .49076 .46652 L .49325 .47363 L p 0 .231 .746 r F P 0 g s .49325 .47363 m .47622 .45521 L .47275 .44558 L p 0 .231 .746 r F P 0 g s .4547 .42168 m .47275 .44558 L .47622 .45521 L p 0 .132 .674 r F P 0 g s .511 .47985 m .52774 .49406 L .52657 .49588 L p .402 .642 .954 r F P 0 g s .52657 .49588 m .5087 .48355 L .511 .47985 L p .402 .642 .954 r F P 0 g s .25074 .19516 m .26996 .20673 L .28872 .24965 L p .341 .645 .966 r F P 0 g s .28872 .24965 m .27135 .24134 L .25074 .19516 L p .341 .645 .966 r F P 0 g s .54651 .49447 m .54871 .48068 L .55994 .48468 L p .781 .821 .884 r F P 0 g s .54871 .48068 m .54651 .49447 L .54066 .49314 L p .747 .79 .887 r F P 0 g s .51696 .47754 m .53072 .49293 L .52774 .49406 L p .541 .695 .939 r F P 0 g s .52774 .49406 m .511 .47985 L .51696 .47754 L p .541 .695 .939 r F P 0 g s .23133 .18496 m .25074 .19516 L .27135 .24134 L p .406 .692 .978 r F P 0 g s .27135 .24134 m .25389 .23443 L .23133 .18496 L p .406 .692 .978 r F P 0 g s .56779 .46824 m .55994 .48468 L .54871 .48068 L p .823 .799 .829 r F P 0 g s .54066 .49314 m .53694 .47798 L .54871 .48068 L p .747 .79 .887 r F P 0 g s .53694 .47798 m .54066 .49314 L .53519 .49261 L p .701 .762 .899 r F P 0 g s .52594 .4769 m .53519 .49261 L .53072 .49293 L p .635 .731 .918 r F P 0 g s .53072 .49293 m .51696 .47754 L .52594 .4769 L p .635 .731 .918 r F P 0 g s .53519 .49261 m .52594 .4769 L .53694 .47798 L p .701 .762 .899 r F P 0 g s .21176 .17587 m .23133 .18496 L .25389 .23443 L p .456 .727 .984 r F P 0 g s .25389 .23443 m .23635 .22864 L .21176 .17587 L p .456 .727 .984 r F P 0 g s .49413 .46087 m .511 .47985 L .5087 .48355 L p .33 .547 .911 r F P 0 g s .5087 .48355 m .49076 .46652 L .49413 .46087 L p .33 .547 .911 r F P 0 g s .57565 .44792 m .56779 .46824 L .55092 .46215 L p .842 .784 .795 r F P 0 g s .54871 .48068 m .55092 .46215 L .56779 .46824 L p .823 .799 .829 r F P 0 g s .19201 .16763 m .21176 .17587 L .23635 .22864 L p .493 .752 .987 r F P 0 g s .23635 .22864 m .21872 .22372 L .19201 .16763 L p .493 .752 .987 r F P 0 g s .55092 .46215 m .54871 .48068 L .53694 .47798 L p .783 .763 .835 r F P 0 g s .47716 .4379 m .49413 .46087 L .49076 .46652 L p .277 .476 .874 r F P 0 g s .49076 .46652 m .47275 .44558 L .47716 .4379 L p .277 .476 .874 r F P 0 g s .58352 .42467 m .57565 .44792 L .55314 .43963 L p .852 .774 .775 r F P 0 g s .55092 .46215 m .55314 .43963 L .57565 .44792 L p .842 .784 .795 r F P 0 g s .50307 .45735 m .51696 .47754 L .511 .47985 L p .532 .637 .9 r F P 0 g s .511 .47985 m .49413 .46087 L .50307 .45735 L p .532 .637 .9 r F P 0 g s .5914 .39957 m .58352 .42467 L .55536 .41409 L p .857 .769 .764 r F P 0 g s .55314 .43963 m .55536 .41409 L .58352 .42467 L p .852 .774 .775 r F P 0 g s .46009 .41187 m .47716 .4379 L .47275 .44558 L p .241 .429 .847 r F P 0 g s .47275 .44558 m .4547 .42168 L .46009 .41187 L p .241 .429 .847 r F P 0 g s .53694 .47798 m .53318 .45803 L .55092 .46215 L p .783 .763 .835 r F P 0 g s .53318 .45803 m .53694 .47798 L .52594 .4769 L p .729 .728 .85 r F P 0 g s .5993 .3737 m .5914 .39957 L .5576 .3866 L p .858 .767 .76 r F P 0 g s .55536 .41409 m .5576 .3866 L .5914 .39957 L p .857 .769 .764 r F P 0 g s .5166 .45636 m .52594 .4769 L .51696 .47754 L p .652 .69 .873 r F P 0 g s .51696 .47754 m .50307 .45735 L .5166 .45636 L p .652 .69 .873 r F P 0 g s .52594 .4769 m .5166 .45636 L .53318 .45803 L p .729 .728 .85 r F P 0 g s .60721 .34803 m .5993 .3737 L .55985 .35824 L p .858 .768 .762 r F P 0 g s .5576 .3866 m .55985 .35824 L .5993 .3737 L p .858 .767 .76 r F P 0 g s .55314 .43963 m .55092 .46215 L .53318 .45803 L p .799 .745 .803 r F P 0 g s .44294 .38387 m .46009 .41187 L .4547 .42168 L p .219 .401 .83 r F P 0 g s .4547 .42168 m .43661 .39592 L .44294 .38387 L p .219 .401 .83 r F P 0 g s .61515 .32338 m .60721 .34803 L .56212 .32997 L p .855 .77 .768 r F P 0 g s .55985 .35824 m .56212 .32997 L .60721 .34803 L p .858 .768 .762 r F P 0 g s .48907 .43308 m .50307 .45735 L .49413 .46087 L p .523 .598 .871 r F P 0 g s .49413 .46087 m .47716 .4379 L .48907 .43308 L p .523 .598 .871 r F P 0 g s .42569 .35497 m .44294 .38387 L .43661 .39592 L p .211 .39 .823 r F P 0 g s .43661 .39592 m .41848 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