(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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In morphometric anlysis, i.e. quantifying \ shape and size changes, a kinematic\nformulation is adequate without any \ reference to stress or stress-strain like constitutive relations. Hence the \ algebraic expressions of shape functions can be used directly to calculate \ the non-mechanical growth strains --- the local changes in shape and size of \ the body --- in time. In a finite element region with ", StyleBox["n", "Input"], "-number of nodes there are ", StyleBox["n", "Input"], "-number of shape functions. Since the intent of the shape function \ interpolants is to reproduce an ", StyleBox["m-", "Input"], "number of pre-selected algebraic shapes, it is shown here that from ", StyleBox["n-m", "Input"], " given shape functions the remaining ", StyleBox["m", "Input"], " interpolants can be obtained. This new result substantially economizes \ the computing efforts required to deduce the complete set of interpolants." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Treil's Frame schematic", "Section"], Cell["\<\ Treil's maxillo-facial frame [Planche, 1998] and [Treil, 1997], \ according to the functional matrix hypothesis [Moss, 1969] is presented \ below.\ \>", "Text"], Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgE[FHaP00Ukg00AS65:D Kg]JeP9Nm`9S6009F]IS66F@9S6004F]IW>E[FG_L2HaP03e[FHaQJeV]JG_MJeV]JF]INmf@02G_L00e[FHaQW>@02F]H01V000010@Q000005kg0P@Q100000@48@0000048@H00003 124000@Q00h00005E[ES600000048@0>00000`Q2DYA[FP01F]H000ENm`0G HaQNmfF@03HaP00fLi F]IS6002HaP036LiHaQNmfFEkgHaQNmfmkG_M[FP02HaP2G_L00fLiHaQW>@02HaP025[FKg]S66EkgIcUS609W>@0;JeYW>EkgIcUNmfLiE[E[FV@02HaP016LiF]IW>Ekg0VLi00=S66mkG_L00Ukg00iS65kgIcUNmg>L DYAW>EJeIcU[FUJeG_MS66Li0ekg00]S65kgJeYNmf]JF]IW>FFEkgJeYF]E[F009S6003G_MW>Ekg009S6008JeYNme[FG_MS66]J 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anatomical symbols: O, A, B, C, D, E, F, H in the maxillo-facial frame \ are represented by nodes with indices 1 through 8. The coordinates of the \ i-th node is denoted by: ", StyleBox["{x[i],y[i],z[i]}.", "Input"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Maxillo-Facial Frames", "Section"], Cell[TextData[{ "There are eight nodes in the Treil maxillo-facial frame [Planche, 1998] \ and [Treil, 1997]. Only one shape function will be evaluated in", StyleBox[" ", "Input"], "the", StyleBox[" (x,y,z) ", "Input"], "coordinates", StyleBox[" ", "Input"], "system", StyleBox[" ", "Input"], "following", StyleBox[" ", "Input"], "a suitable method, such as Taig's isoparametric formulation [Taig, 1961] \ or Wachspress' rational polynomial interpolation [Wachspress, 1975] and \ [Laydi, 1985]. (", StyleBox["Mathematica", FontSlant->"Italic"], " coding of these two popular and effective procedures are included in the \ appendix.) This initial shape function in terms of ", StyleBox["(x,y,z)", "Input"], " will be indicated by the special symbol ", StyleBox["s[0]", "Input"], ", and the associated coordinates will be denoted by ", StyleBox["(x0, y0, z0).", "Input"], " Here the string ", StyleBox["\"0\"", "Input"], " reflects the starting choice for the proposed algorithm. For the \ eight-node maxillo-facial frame the remaining seven interpolants: ", StyleBox["Array[s,7]", "Input"], " will be calculated to reproduce the set of exact fields denoted by ", StyleBox["f ", "Input"], "as a list of functions in", StyleBox[" (x,y,z). ", "Input"], "In particular, here:", StyleBox[" f={1,x,y,z,x*y,y*z,z*x}. ", "Input"], "Its algebraic relations are crucial to this presentation as explained \ below.\n\nConsider a pre-assigned field ", StyleBox["u[x,y,z]", "Input"], " to be an element of the list ", StyleBox["f", "Input"], ". With the prescribed nodal values ", StyleBox["u[x[i],y[i], z[i]]", "Input"], ", the set of shape functions", StyleBox[" s[i] ", "Input"], "in terms of", StyleBox[" (x,y,z) ", "Input"], "should lead to", StyleBox[":", "Input"] }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(u[x, y, \ z]\), "\[Equal]", RowBox[{"Sum", "[", StyleBox[\(u[x[i], y[i], \ z[i]\ ]*s[i], {i, n}\), "Input"], "]"}]}], ";"}]], "Input"], Cell[TextData[{ "The aforementioned relation will be exactly satisfied in terms of the \ coordinate variables ", StyleBox["(x,y,z). ", "Input"], "Hence the name", StyleBox[" \"exact interpolated fields\" ", "Input"], "for", StyleBox[" f.", "Input"] }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Example" }], "Subsection"], Cell[BoxData[{ \(\(Clear[f, x, y, z, s];\)\), "\[IndentingNewLine]", \(\(f = {1, x, y, z, x*y, y*z, z*x};\)\)}], "Input"], Cell[TextData[{ "The following code generates the coefficient matrix ", StyleBox["mat", "Input"], " and the right hand side ", StyleBox["vec ", "Input"], "for", StyleBox[" LinearSolve:", "Input"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\[IndentingNewLine]\(shapeRelation // Clear;\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(shapeRelation::usage = \ "\"\ \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(\(shapeRelation[f_List, g_, {x_, y_, z_}, {x0_, y0_, z0_}]\)\(:=\)\(Module[{i}, \[IndentingNewLine]{\(-Table[ f /. {x \[Rule] x[i], y \[Rule] y[i], z \[Rule] z[i]}, {i, Length[f]}]\) // Thread, \(-f\) + \((g*f /. {x \[Rule] x0, y \[Rule] y0, z \[Rule] z0})\)}]\)\(\[IndentingNewLine]\)\)\)}], "Input"], Cell[BoxData[ \("shapeRelation[f,s[0],{x,y,z},{x0,y0,z0}]generates {mat,vec} for \ determining the shape functions as LinearSolve[mat, vec]with the initial \ shape function s[0] associated with the node at {x0, y0, z0} by exactly \ interpolating the elements of f in the coordinate system (x,y,z)."\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Execution", "Subsubsection"], Cell[BoxData[ \(Clear[mat, vec]; {mat, vec} = shapeRelation[f, s[0], {x, y, z}, {x0, y0, z0}];\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["A symmetric case", "Subsubsection"], Cell[TextData[{ "The anatomical points are symmetrically located on the left and right \ side of the face. The canonical maxillo-facial frame is then symmetric about \ ", StyleBox["x-", "Input"], " axis. Hence the following is a specialization for which a symmetric \ frame is generated according to the rule:" }], "Text"], Cell[BoxData[{ \(\(Clear[r];\)\), "\[IndentingNewLine]", \(\(r := {x[1] -> \(-x0\), y[1] -> y0, y0 -> 0, \[IndentingNewLine]z[1] -> z0, z0 -> 0, \[IndentingNewLine]x[2] -> \(-x[3]\), y[3] -> y[2], y[2] \[Rule] 0, z[3] -> z[2], x[4] -> \(-x[5]\), \ y[4] -> y[5], z[4] -> z[5], x[6] -> \(-x[7]\), y[6] -> y[7], z[6] -> z[7]};\)\[IndentingNewLine]\)}], "Input"], Cell["\<\ Now the coefficient matrix and the right hand side vector, matr, \ vecr, respectively, become:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[matr, \ vecr];\)\), "\[IndentingNewLine]", \(\({matr, \ vecr} = {mat, \ vec} /. r;\)\)}], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(matr\)\" is similar to \ existing symbol \"\!\(mat\)\"."\)], "Message"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(vecr\)\" is similar to \ existing symbol \"\!\(vec\)\"."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[matr]\)], "Input"], Cell[BoxData[ TagBox["mat", (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(vecr\)], "Input"], Cell[BoxData[ \(vec\)], "Output"] }, Open ]], Cell["\<\ Before attempting to solve for the unkown shape functions, the \ consistency of the assumption of the interpolated field should be \ established:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Det[matr]]\)], "Input"], Cell[BoxData[ \(Det[mat]\)], "Output"] }, Open ]], Cell["\<\ In general, the non-zero determinant signifies that the assumption:\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f\)], "Input"], Cell[BoxData[ \({1, x, y, z, x\ y, y\ z, x\ z}\)], "Output"] }, Open ]], Cell[TextData[{ "could be acceptable and the matrix ", StyleBox["matr ", "Input"], "is indeed invertable." }], "Text"], Cell["\<\ For given numerical data for the nodal coordinates, the shape \ functions will be obtained from the expression below after removing the \ comments (* and *).\ \>", "Text"], Cell[BoxData[ \( (*\[IndentingNewLine]\(Evaluate[Array[ss, 7]] = \ LinearSolve[matr, vecr];\)\[IndentingNewLine]*) \)], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Failure to interpolate: ", Cell[BoxData[ \(x\^2, y\^2, z\^2\)]], "--- \nan example of inconsistent data" }], "Subsection"], Cell[BoxData[ \(\(Clear[f, x, y, z, s];\)\)], "Input"], Cell[BoxData[ \(\(f = {1, x, y, z, x\^2, y\^2, z\^2};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\({mat, vec} = shapeRelation[f, s[0], {x, y, z}, {x0, y0, z0}];\)\)], "Input"], Cell[BoxData[ \(Set::"shape" \(\(:\)\(\ \)\) "Lists \!\({mat, vec}\) and \!\(shapeRelation[\(\(\[LeftSkeleton] 1 \ \[RightSkeleton]\)\)]\) are not the same shape."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell["The symmetric maxillo-facial frame", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\((mat //. r)\) // MatrixForm\)], "Input"], Cell[BoxData[ TagBox["mat", (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Det[\((mat //. r)\)]\)], "Input"], Cell[BoxData[ \(Det[mat]\)], "Output"] }, Open ]], Cell["\<\ Hence the following evaluation is not attempted since the \ coefficient matrix is not invertable.\ \>", "Text"], Cell[BoxData[ \( (*\ \ LinearSolve[mat, vec]\ \ *) \)], "Input"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Some 2-D results---Concave quadrilateral element", "Section"], Cell[TextData[{ "Standard literature in finite elements hardly addresses the convexity \ issue. There is no mention of it in the encyclopedic references in the two \ volumes of Zienkiewicz and Taylor [Zienkiewicz, 1977]. Wachpress adreeses \ the condition from projective geometry point of view. He develops concavity \ with curved sides. His general procedure [Gout, 1979] for n-gon is not \ applicable since the curve through the external intersection points \ intersects the element domain. It is then prudent to conclude that there has \ not been any consistent method available to generate the shape functions of a \ concave polygon. Recently the second author, Dasgupta, utilized the \ two-dimensionsal version of the aforementioned code in the form: ", StyleBox["shapeRelation[f,s,{x,y},{x0,y0}]", "Input"], " \nto generate the second, third and the fourth shape functions from the \ ", StyleBox["s", "Input"], " which pertained to the convex end." }], "Text"], Cell[CellGroupData[{ Cell["Contour plots of the shape functions", "Subsubsection"], Cell[TextData[{ "The following code calls the shape function generator, ", StyleBox["p", "Input"], " houses the nodal coordinates, and the physical domain is represented by \ ", StyleBox["x-y", "Input"], " frame. An example with the following nodes:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(p = {{0, 0}, {\(-1.5\), \(-0.5\)}, {1, 0}, {0.5, 0.5}}\)], "Input"], Cell[BoxData[ \({{0, 0}, {\(-1.5`\), \(-0.5`\)}, {1, 0}, {0.5`, 0.5`}}\)], "Output"] }, Open ]], Cell["is presented here.", "Text"], Cell[BoxData[ \( (*\ FourShapes[p, {x, y}]\ *) \)], "Input"], Cell["\<\ The linearity condition along the straight boundary segments of the \ calculated shape functions is verified:\ \>", "Text"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Shapes along sides:\"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(\(0.9999999999999999`\)\(\[InvisibleSpace]\)\) + 0.6666666666666676`\ x\), \(\(-0.6666666666666664`\)\ x\), "0", "0"}, { "0", \(\(\(0.4`\)\(\[InvisibleSpace]\)\) - 0.39999999999999997`\ x\), \(\(\(0.6000000000000001`\)\(\ \[InvisibleSpace]\)\) + 0.3999999999999996`\ x\), "0"}, {"0", "0", \(\(-1.`\) + 2.`\ x\), \ \(\(\(1.9999999999999998`\)\(\[InvisibleSpace]\)\) - 1.9999999999999996`\ x\)}, {\(\(\(1.`\)\(\[InvisibleSpace]\)\) - 2.`\ x\), "0", "0", \(2.`\ x\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]}], SequenceForm[ "Shapes along sides:", MatrixForm[ {{ Plus[ 0.99999999999999989, Times[ 0.66666666666666763, x]], Times[ -0.66666666666666641, x], 0, 0}, {0, Plus[ 0.40000000000000002, Times[ -0.39999999999999997, x]], Plus[ 0.60000000000000009, Times[ 0.39999999999999958, x]], 0}, {0, 0, Plus[ -1.0, Times[ 2.0, x]], Plus[ 1.9999999999999998, Times[ -1.9999999999999996, x]]}, { Plus[ 1.0, Times[ -2.0, x]], 0, 0, Times[ 2.0, x]}}]], Editable->False]], "Print"], Cell["\<\ In the above matrix form the row indicates the shape function and a \ column the corresponding straight line side of the quadrilateral.\ \>", "Text"], Cell["\<\ The original shape function is displayed with unit value at the \ concave end and with zero values at the remaining ends.\ \>", "Text"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .4 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.72623 .17279 L .77211 .18999 L .78371 .2 L .78171 .22721 L .77614 .25442 L .77211 .26949 L .76622 .28163 L .75147 .30884 L .73439 .33605 L .7159 .36327 L .70408 .3799 L .69591 .39048 L s .444 1 0 r .02381 .06641 m .05883 .09116 L .09184 .10528 L .15986 .09298 L .21303 .11837 L .22789 .13088 L .29592 .1187 L .35532 .14558 L .36395 .15367 L .42113 .14558 L .36395 .12307 L .29997 .11837 L .29592 .09337 L .22789 .09128 L .19953 .09116 L .22789 .06721 L .29592 .08224 L .33949 .09116 L .36395 .09464 L .43197 .10734 L .48105 .11837 L .5 .12294 L .56803 .13753 L .60432 .14558 L .63605 .15212 L .70408 .16913 L .71406 .17279 L .77189 .2 L .76962 .22721 L .76299 .25442 L .75252 .28163 L .73897 .30884 L .72322 .33605 L .70598 .36327 L .70408 .36614 L .68619 .39048 L .02381 .39048 L F 0 g .02381 .06641 m .05883 .09116 L .09184 .10528 L .15986 .09298 L .21303 .11837 L .22789 .13088 L .29592 .1187 L .35532 .14558 L .36395 .15367 L .42113 .14558 L .36395 .12307 L .29997 .11837 L .29592 .09337 L 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.11837 L .22789 .13099 L .29592 .11885 L .35501 .14558 L .36395 .15394 L .4221 .14558 L .36395 .12247 L .3019 .11837 L .29592 .09258 L .23044 .09116 L .29592 .08487 L .32745 .09116 L .36395 .09617 L .43197 .10966 L .47189 .11837 L .5 .12487 L .56803 .13973 L .59412 .14558 L .63605 .15449 L .70258 .17279 L .70408 .17326 L .75771 .2 L .75563 .22721 L .74955 .25442 L .73994 .28163 L .72746 .30884 L .71283 .33605 L .70408 .3511 L .69598 .36327 L .67686 .39048 L s .418 1 0 r .02381 .06648 m .05866 .09116 L .09184 .10539 L .15986 .09311 L .21276 .11837 L .22789 .1311 L .29592 .11899 L .3547 .14558 L .36395 .15421 L .42306 .14558 L .36395 .12187 L .30397 .11837 L .29592 .09179 L .23752 .09116 L .29592 .088 L .31278 .09116 L .36395 .09776 L .43197 .11217 L .4614 .11837 L .5 .12684 L .56803 .14198 L .58395 .14558 L .63605 .15698 L .69099 .17279 L .70408 .17722 L .74471 .2 L .7428 .22721 L .73721 .25442 L .72835 .28163 L .71679 .30884 L .70408 .33425 L .70302 .33605 L .68625 .36327 L .66791 .39048 L .02381 .39048 L F 0 g .02381 .06648 m .05866 .09116 L .09184 .10539 L .15986 .09311 L .21276 .11837 L .22789 .1311 L .29592 .11899 L .3547 .14558 L .36395 .15421 L .42306 .14558 L .36395 .12187 L .30397 .11837 L .29592 .09179 L .23752 .09116 L .29592 .088 L .31278 .09116 L .36395 .09776 L .43197 .11217 L .4614 .11837 L .5 .12684 L .56803 .14198 L .58395 .14558 L .63605 .15698 L .69099 .17279 L .70408 .17722 L .74471 .2 L .7428 .22721 L .73721 .25442 L .72835 .28163 L .71679 .30884 L .70408 .33425 L .70302 .33605 L .68625 .36327 L .66791 .39048 L s .405 1 0 r .02381 .06652 m .05857 .09116 L .09184 .10545 L .15986 .09318 L .21263 .11837 L .22789 .1312 L .29592 .11913 L .35439 .14558 L .36395 .15448 L .42403 .14558 L .36395 .12126 L .30621 .11837 L .36395 .09945 L .43197 .11487 L .44923 .11837 L .5 .12888 L .56803 .14427 L .57375 .14558 L .63605 .15963 L .67985 .17279 L .70408 .18165 L .73275 .2 L .73099 .22721 L .72583 .25442 L .71762 .28163 L .70682 .30884 L .70408 .31504 L .69299 .33605 L .67692 .36327 L .6593 .39048 L .02381 .39048 L F 0 g .02381 .06652 m .05857 .09116 L .09184 .10545 L .15986 .09318 L .21263 .11837 L .22789 .1312 L .29592 .11913 L .35439 .14558 L .36395 .15448 L .42403 .14558 L .36395 .12126 L .30621 .11837 L .36395 .09945 L .43197 .11487 L .44923 .11837 L .5 .12888 L .56803 .14427 L .57375 .14558 L .63605 .15963 L .67985 .17279 L .70408 .18165 L .73275 .2 L .73099 .22721 L .72583 .25442 L .71762 .28163 L .70682 .30884 L .70408 .31504 L .69299 .33605 L .67692 .36327 L .6593 .39048 L s .392 1 0 r .02381 .06656 m .05849 .09116 L .09184 .10551 L .15986 .09324 L .2125 .11837 L .22789 .13131 L .29592 .11927 L .35408 .14558 L .36395 .15475 L .42499 .14558 L .36395 .12064 L .30867 .11837 L .36395 .10126 L .43197 .11777 L .43505 .11837 L .5 .13098 L .5638 .14558 L .56803 .14666 L .63605 .16243 L .66909 .17279 L .70408 .1868 L .7217 .2 L .72007 .22721 L .71527 .25442 L .7076 .28163 L .70408 .29198 L .69684 .30884 L .68339 .33605 L .66797 .36327 L .65101 .39048 L .02381 .39048 L F 0 g .02381 .06656 m .05849 .09116 L .09184 .10551 L .15986 .09324 L .2125 .11837 L .22789 .13131 L .29592 .11927 L .35408 .14558 L .36395 .15475 L .42499 .14558 L .36395 .12064 L .30867 .11837 L .36395 .10126 L .43197 .11777 L .43505 .11837 L .5 .13098 L .5638 .14558 L .56803 .14666 L .63605 .16243 L .66909 .17279 L .70408 .1868 L .7217 .2 L .72007 .22721 L .71527 .25442 L .7076 .28163 L .70408 .29198 L .69684 .30884 L .68339 .33605 L .66797 .36327 L .65101 .39048 L s .379 1 0 r .02381 .0666 m .0584 .09116 L .09184 .10557 L .15986 .09331 L .21236 .11837 L .22789 .13142 L .29592 .11941 L .35377 .14558 L .36395 .15502 L .42596 .14558 L .36395 .12001 L .31141 .11837 L .36395 .10326 L .42344 .11837 L .43197 .13497 L .5 .13317 L .55411 .14558 L .56803 .14913 L .63605 .16541 L .65863 .17279 L .70408 .19322 L .71142 .2 L .70989 .22721 L .7054 .25442 L .70408 .26063 L .69764 .28163 L .68707 .30884 L .67418 .33605 L .65936 .36327 L .64299 .39048 L .02381 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.09116 L .09184 .10562 L .15986 .09338 L .21223 .11837 L .22789 .13152 L .29592 .11955 L .35347 .14558 L .36395 .15528 L .42692 .14558 L .36395 .11937 L .31453 .11837 L .36395 .10558 L .41148 .11837 L .43197 .13787 L .5 .13546 L .54416 .14558 L .56803 .15166 L .63605 .16859 L .64841 .17279 L .70157 .2 L .70001 .22721 L .69537 .25442 L .68784 .28163 L .67771 .30884 L .66533 .33605 L .65106 .36327 L .63605 .38908 L .6352 .39048 L s .354 1 0 r .02381 .06667 m .05823 .09116 L .09184 .10568 L .15986 .09344 L .2121 .11837 L .22789 .13163 L .29592 .11969 L .35316 .14558 L .36395 .15555 L .4279 .14558 L .36395 .11872 L .31824 .11837 L .36395 .10853 L .39596 .11837 L .43197 .13995 L .5 .13788 L .53375 .14558 L .56803 .15422 L .63605 .17198 L .63836 .17279 L .69161 .2 L .69011 .22721 L .68566 .25442 L .67845 .28163 L .66873 .30884 L .65682 .33605 L .64304 .36327 L .63605 .37586 L .62739 .39048 L .02381 .39048 L F 0 g .02381 .06667 m .05823 .09116 L .09184 .10568 L .15986 .09344 L .2121 .11837 L .22789 .13163 L .29592 .11969 L .35316 .14558 L .36395 .15555 L .4279 .14558 L .36395 .11872 L .31824 .11837 L .36395 .10853 L .39596 .11837 L .43197 .13995 L .5 .13788 L .53375 .14558 L .56803 .15422 L .63605 .17198 L .63836 .17279 L .69161 .2 L .69011 .22721 L .68566 .25442 L .67845 .28163 L .66873 .30884 L .65682 .33605 L .64304 .36327 L .63605 .37586 L .62739 .39048 L s .341 1 0 r .02381 .06671 m .05815 .09116 L .09184 .10574 L .15986 .09351 L .21196 .11837 L .22789 .13174 L .29592 .11983 L .35286 .14558 L .36395 .15581 L .42887 .14558 L .43197 .14163 L .5 .14047 L .52265 .14558 L .56803 .15683 L .62941 .17279 L .63605 .17527 L .68206 .2 L .68062 .22721 L .67636 .25442 L .66944 .28163 L .66009 .30884 L .64861 .33605 L .63605 .36178 L .63525 .36327 L .61979 .39048 L .02381 .39048 L F 0 g .02381 .06671 m .05815 .09116 L .09184 .10574 L .15986 .09351 L .21196 .11837 L .22789 .13174 L .29592 .11983 L .35286 .14558 L .36395 .15581 L .42887 .14558 L .43197 .14163 L .5 .14047 L .52265 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.61238 .39048 L s .315 1 0 r .02381 .06679 m .05798 .09116 L .09184 .10585 L .15986 .09364 L .2117 .11837 L .22789 .13195 L .29592 .1201 L .35225 .14558 L .36395 .15634 L .43084 .14558 L .43197 .1443 L .49813 .14558 L .5 .14922 L .56803 .16216 L .61173 .17279 L .63605 .18236 L .6641 .2 L .66278 .22721 L .65884 .25442 L .65242 .28163 L .64373 .30884 L .63605 .32882 L .63287 .33605 L .61984 .36327 L .60514 .39048 L .02381 .39048 L F 0 g .02381 .06679 m .05798 .09116 L .09184 .10585 L .15986 .09364 L .2117 .11837 L .22789 .13195 L .29592 .1201 L .35225 .14558 L .36395 .15634 L .43084 .14558 L .43197 .1443 L .49813 .14558 L .5 .14922 L .56803 .16216 L .61173 .17279 L .63605 .18236 L .6641 .2 L .66278 .22721 L .65884 .25442 L .65242 .28163 L .64373 .30884 L .63605 .32882 L .63287 .33605 L .61984 .36327 L .60514 .39048 L s .302 1 0 r .02381 .06682 m .0579 .09116 L .09184 .10591 L .15986 .0937 L .21157 .11837 L .22789 .13206 L .29592 .12024 L .35195 .14558 L .36395 .1566 L .43183 .14558 L .43197 .14543 L .49044 .14558 L .5 .16633 L .56803 .16488 L .6023 .17279 L .63605 .18642 L .65564 .2 L .65436 .22721 L .65056 .25442 L .64436 .28163 L .63605 .30852 L .63594 .30884 L .62512 .33605 L .61242 .36327 L .59808 .39048 L .02381 .39048 L F 0 g .02381 .06682 m .0579 .09116 L .09184 .10591 L .15986 .0937 L .21157 .11837 L .22789 .13206 L .29592 .12024 L .35195 .14558 L .36395 .1566 L .43183 .14558 L .43197 .14543 L .49044 .14558 L .5 .16633 L .56803 .16488 L .6023 .17279 L .63605 .18642 L .65564 .2 L .65436 .22721 L .65056 .25442 L .64436 .28163 L .63605 .30852 L .63594 .30884 L .62512 .33605 L .61242 .36327 L .59808 .39048 L s .289 1 0 r .02381 .06686 m .05781 .09116 L .09184 .10597 L .15986 .09377 L .21143 .11837 L .22789 .13216 L .29592 .12037 L .35165 .14558 L .36395 .15686 L .43197 .14595 L .5 .17165 L .56803 .16764 L .59207 .17279 L .63605 .19107 L .64747 .2 L .64623 .22721 L .64256 .25442 L .63656 .28163 L .63605 .28359 L .62811 .30884 L .61758 .33605 L .60519 .36327 L .59118 .39048 L .02381 .39048 L F 0 g .02381 .06686 m .05781 .09116 L .09184 .10597 L .15986 .09377 L .21143 .11837 L .22789 .13216 L .29592 .12037 L .35165 .14558 L .36395 .15686 L .43197 .14595 L .5 .17165 L .56803 .16764 L .59207 .17279 L .63605 .19107 L .64747 .2 L .64623 .22721 L .64256 .25442 L .63656 .28163 L .63605 .28359 L .62811 .30884 L .61758 .33605 L .60519 .36327 L .59118 .39048 L s .276 1 0 r .02381 .0669 m .05773 .09116 L .09184 .10603 L .15986 .09383 L .2113 .11837 L .22789 .13227 L .29592 .12051 L .35135 .14558 L .36395 .15712 L .43197 .14637 L .49915 .17279 L .5 .17375 L .5625 .17279 L .56803 .17043 L .58034 .17279 L .63605 .19675 L .63956 .2 L .63837 .22721 L .63605 .24669 L .63477 .25442 L .62873 .28163 L .62049 .30884 L .61022 .33605 L .59813 .36327 L .58443 .39048 L .02381 .39048 L F 0 g .02381 .0669 m .05773 .09116 L .09184 .10603 L .15986 .09383 L .2113 .11837 L .22789 .13227 L .29592 .12051 L .35135 .14558 L .36395 .15712 L .43197 .14637 L .49915 .17279 L .5 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.09184 .06433 L .15986 .06526 L .1771 .09116 L .22789 .09191 L .27198 .09116 L .22789 .07131 L .18811 .06395 L .15986 .05305 L .10882 .03673 L .09184 .03495 L .02381 .00952 L s .27 0 1 r .02381 .06486 m .06263 .09116 L .09184 .10293 L .15967 .09116 L .09184 .06602 L .05035 .06395 L .02381 .04109 L F 0 g .02381 .06486 m .06263 .09116 L .09184 .10293 L .15967 .09116 L .09184 .06602 L .05035 .06395 L .02381 .04109 L s .304 0 1 r .02381 .06474 m .06292 .09116 L .09184 .10275 L .15899 .09116 L .09184 .06635 L .04605 .06395 L .02381 .04162 L F 0 g .02381 .06474 m .06292 .09116 L .09184 .10275 L .15899 .09116 L .09184 .06635 L .04605 .06395 L .02381 .04162 L s .339 0 1 r .02381 .06461 m .06321 .09116 L .09184 .10258 L .15831 .09116 L .09184 .06668 L .04204 .06395 L .02381 .04216 L F 0 g .02381 .06461 m .06321 .09116 L .09184 .10258 L .15831 .09116 L .09184 .06668 L .04204 .06395 L .02381 .04216 L s .374 0 1 r .02381 .06449 m .0635 .09116 L .09184 .10241 L .15763 .09116 L .09184 .067 L .03829 .06395 L .02381 .0427 L F 0 g .02381 .06449 m .0635 .09116 L .09184 .10241 L .15763 .09116 L .09184 .067 L .03829 .06395 L .02381 .0427 L s .408 0 1 r .02381 .06437 m .0638 .09116 L .09184 .10223 L .15696 .09116 L .09184 .06733 L .03475 .06395 L .02381 .04326 L F 0 g .02381 .06437 m .0638 .09116 L .09184 .10223 L .15696 .09116 L .09184 .06733 L .03475 .06395 L .02381 .04326 L s .443 0 1 r .02381 .06425 m .0641 .09116 L .09184 .10206 L .15629 .09116 L .09184 .06765 L .0314 .06395 L .02381 .04382 L F 0 g .02381 .06425 m .0641 .09116 L .09184 .10206 L .15629 .09116 L .09184 .06765 L .0314 .06395 L .02381 .04382 L s .478 0 1 r .02381 .06413 m .06441 .09116 L .09184 .10188 L .15562 .09116 L .09184 .06797 L .02821 .06395 L .02381 .0444 L F 0 g .02381 .06413 m .06441 .09116 L .09184 .10188 L .15562 .09116 L .09184 .06797 L .02821 .06395 L .02381 .0444 L s .513 0 1 r .02381 .064 m .06471 .09116 L .09184 .10171 L .15496 .09116 L .09184 .0683 L .02516 .06395 L .02381 .04499 L F 0 g .02381 .064 m .06471 .09116 L .09184 .10171 L .15496 .09116 L .09184 .0683 L .02516 .06395 L .02381 .04499 L s .131 0 1 r .80019 .39048 m .77228 .36327 L .77211 .3631 L .74433 .33605 L .71651 .30884 L .70408 .29664 L .68874 .28163 L .66101 .25442 L .63605 .22987 L .63335 .22721 L .60568 .2 L .56803 .17944 L .50558 .17279 L .5 .16852 L .46645 .14558 L .43197 .14293 L .37398 .11837 L .36395 .11273 L .29592 .11783 L .28892 .09116 L .22789 .06839 L .20425 .06395 L .15986 .04835 L .12472 .03673 L .09184 .03296 L .03575 .00952 L .97619 .00952 L .97619 .39048 L F 0 g .80019 .39048 m .77228 .36327 L .77211 .3631 L .74433 .33605 L .71651 .30884 L .70408 .29664 L .68874 .28163 L .66101 .25442 L .63605 .22987 L .63335 .22721 L .60568 .2 L .56803 .17944 L .50558 .17279 L .5 .16852 L .46645 .14558 L .43197 .14293 L .37398 .11837 L .36395 .11273 L .29592 .11783 L .28892 .09116 L .22789 .06839 L .20425 .06395 L .15986 .04835 L .12472 .03673 L .09184 .03296 L .03575 .00952 L s .166 0 1 r .81518 .39048 m .78638 .36327 L .77211 .34968 L .75775 .33605 L .72924 .30884 L .70408 .28469 L .70089 .28163 L .67259 .25442 L .64443 .22721 L .63605 .21883 L .61632 .2 L .56803 .17598 L .51591 .17279 L .5 .16106 L .48032 .14558 L .43197 .13967 L .39178 .11837 L .36395 .1069 L .30634 .09116 L .29592 .08822 L .22789 .06574 L .21847 .06395 L .15986 .0444 L .13772 .03673 L .09184 .03095 L .04737 .00952 L .97619 .00952 L .97619 .39048 L F 0 g .81518 .39048 m .78638 .36327 L .77211 .34968 L .75775 .33605 L .72924 .30884 L .70408 .28469 L .70089 .28163 L .67259 .25442 L .64443 .22721 L .63605 .21883 L .61632 .2 L .56803 .17598 L .51591 .17279 L .5 .16106 L .48032 .14558 L .43197 .13967 L .39178 .11837 L .36395 .1069 L .30634 .09116 L .29592 .08822 L .22789 .06574 L .21847 .06395 L .15986 .0444 L .13772 .03673 L .09184 .03095 L .04737 .00952 L s .2 0 1 r .83069 .39048 m .80092 .36327 L .77211 .33659 L .77153 .33605 L .74227 .30884 L .71327 .28163 L .70408 .27297 L .68441 .25442 L .6557 .22721 L .63605 .20813 L .62713 .2 L .56803 .17288 L .54001 .17279 L .5 .15196 L .49199 .14558 L .43197 .13533 L .40669 .11837 L .36395 .10266 L .32342 .09116 L .29592 .08339 L .23198 .06395 L .22789 .06236 L .15986 .04084 L .14849 .03673 L .09184 .02887 L .0587 .00952 L .97619 .00952 L .97619 .39048 L F 0 g .83069 .39048 m .80092 .36327 L .77211 .33659 L .77153 .33605 L .74227 .30884 L .71327 .28163 L .70408 .27297 L .68441 .25442 L .6557 .22721 L .63605 .20813 L .62713 .2 L .56803 .17288 L .54001 .17279 L .5 .15196 L .49199 .14558 L .43197 .13533 L .40669 .11837 L .36395 .10266 L .32342 .09116 L .29592 .08339 L .23198 .06395 L .22789 .06236 L .15986 .04084 L .14849 .03673 L .09184 .02887 L .0587 .00952 L s .547 0 1 r .09184 .06862 m .15429 .09116 L .09184 .10153 L .06502 .09116 L F 0 g .09184 .06862 m .15429 .09116 L .09184 .10153 L .06502 .09116 L .09184 .06862 L s .582 0 1 r .09184 .06894 m .15363 .09116 L .09184 .10135 L .06534 .09116 L F 0 g .09184 .06894 m .15363 .09116 L .09184 .10135 L .06534 .09116 L .09184 .06894 L s .617 0 1 r .09184 .06925 m .15297 .09116 L .09184 .10118 L .06565 .09116 L F 0 g .09184 .06925 m .15297 .09116 L .09184 .10118 L .06565 .09116 L .09184 .06925 L s .651 0 1 r .09184 .06957 m .15231 .09116 L .09184 .101 L .06597 .09116 L F 0 g .09184 .06957 m .15231 .09116 L .09184 .101 L .06597 .09116 L .09184 .06957 L s .686 0 1 r .09184 .06989 m .15165 .09116 L .09184 .10082 L .0663 .09116 L F 0 g .09184 .06989 m .15165 .09116 L .09184 .10082 L .0663 .09116 L .09184 .06989 L s .721 0 1 r .09184 .07021 m .15099 .09116 L .09184 .10064 L .06663 .09116 L F 0 g .09184 .07021 m .15099 .09116 L .09184 .10064 L .06663 .09116 L .09184 .07021 L s .756 0 1 r .09184 .07053 m .15032 .09116 L .09184 .10046 L .06696 .09116 L F 0 g .09184 .07053 m .15032 .09116 L .09184 .10046 L .06696 .09116 L .09184 .07053 L s .79 0 1 r .09184 .07085 m .14966 .09116 L .09184 .10028 L .06729 .09116 L F 0 g .09184 .07085 m .14966 .09116 L .09184 .10028 L .06729 .09116 L .09184 .07085 L s .825 0 1 r .09184 .07116 m .149 .09116 L .09184 .1001 L .06763 .09116 L F 0 g .09184 .07116 m .149 .09116 L .09184 .1001 L .06763 .09116 L .09184 .07116 L s .86 0 1 r .09184 .07148 m .14834 .09116 L .09184 .09992 L .06798 .09116 L F 0 g .09184 .07148 m .14834 .09116 L .09184 .09992 L .06798 .09116 L .09184 .07148 L s .894 0 1 r .09184 .0718 m .14768 .09116 L .09184 .09974 L .06833 .09116 L F 0 g .09184 .0718 m .14768 .09116 L .09184 .09974 L .06833 .09116 L .09184 .0718 L s .929 0 1 r .09184 .07212 m .14701 .09116 L .09184 .09956 L .06868 .09116 L F 0 g .09184 .07212 m .14701 .09116 L .09184 .09956 L .06868 .09116 L .09184 .07212 L s 1 0 0 r .09184 .07244 m .14635 .09116 L .09184 .09938 L .06904 .09116 L F 0 g .09184 .07244 m .14635 .09116 L .09184 .09938 L .06904 .09116 L .09184 .07244 L s .235 0 1 r .84671 .39048 m .84014 .38466 L .81594 .36327 L .78562 .33605 L .77211 .32379 L .75564 .30884 L .72591 .28163 L .70408 .26146 L .69647 .25442 L .66718 .22721 L .63812 .2 L .63605 .19845 L .58475 .17279 L .56803 .16827 L .5039 .14558 L .5 .14402 L .43197 .12862 L .42014 .11837 L .36395 .09903 L .33748 .09116 L .29592 .07917 L .24551 .06395 L .22789 .05732 L .15986 .03753 L .15774 .03673 L .09184 .02667 L .06974 .00952 L .97619 .00952 L .97619 .39048 L F 0 g .84671 .39048 m .84014 .38466 L .81594 .36327 L .78562 .33605 L .77211 .32379 L .75564 .30884 L .72591 .28163 L .70408 .26146 L .69647 .25442 L .66718 .22721 L .63812 .2 L .63605 .19845 L .58475 .17279 L .56803 .16827 L .5039 .14558 L .5 .14402 L .43197 .12862 L .42014 .11837 L .36395 .09903 L .33748 .09116 L .29592 .07917 L .24551 .06395 L .22789 .05732 L .15986 .03753 L .15774 .03673 L .09184 .02667 L .06974 .00952 L s .27 0 1 r .86327 .39048 m .84014 .37069 L .83148 .36327 L .80014 .33605 L .77211 .31128 L .76936 .30884 L .73886 .28163 L .70876 .25442 L .70408 .25016 L .67889 .22721 L .64928 .2 L .63605 .191 L .59914 .17279 L .56803 .16366 L .51813 .14558 L .5 .13851 L .43287 .11837 L .43197 .11807 L .36395 .09573 L .34924 .09116 L .29592 .07533 L .25815 .06395 L .22789 .05292 L .16871 .03673 L .15986 .03437 L .09184 .0242 L .08053 .00952 L .97619 .00952 L .97619 .39048 L F 0 g .86327 .39048 m .84014 .37069 L .83148 .36327 L .80014 .33605 L .77211 .31128 L .76936 .30884 L .73886 .28163 L .70876 .25442 L .70408 .25016 L .67889 .22721 L .64928 .2 L .63605 .191 L .59914 .17279 L .56803 .16366 L .51813 .14558 L .5 .13851 L .43287 .11837 L .43197 .11807 L .36395 .09573 L .34924 .09116 L .29592 .07533 L .25815 .06395 L .22789 .05292 L .16871 .03673 L .15986 .03437 L .09184 .0242 L .08053 .00952 L s .304 0 1 r .88057 .39048 m .84752 .36327 L .84014 .35708 L .81513 .33605 L .7834 .30884 L .77211 .29903 L .75214 .28163 L .7213 .25442 L .70408 .23905 L .69083 .22721 L .66064 .2 L .63605 .18476 L .61126 .17279 L .56803 .15911 L .53134 .14558 L .5 .13372 L .4462 .11837 L .43197 .11365 L .36395 .09261 L .35941 .09116 L .29592 .07174 L .27005 .06395 L .22789 .0491 L .18011 .03673 L .15986 .03131 L .09184 .02117 L .09108 .00952 L .97619 .00952 L .97619 .39048 L F 0 g .88057 .39048 m .84752 .36327 L .84014 .35708 L .81513 .33605 L .7834 .30884 L .77211 .29903 L .75214 .28163 L .7213 .25442 L .70408 .23905 L .69083 .22721 L .66064 .2 L .63605 .18476 L .61126 .17279 L .56803 .15911 L .53134 .14558 L .5 .13372 L .4462 .11837 L .43197 .11365 L .36395 .09261 L .35941 .09116 L .29592 .07174 L .27005 .06395 L .22789 .0491 L .18011 .03673 L .15986 .03131 L .09184 .02117 L .09108 .00952 L s .339 0 1 r .8987 .39048 m .86411 .36327 L .84014 .3438 L .83064 .33605 L .79784 .30884 L .77211 .28703 L .76577 .28163 L .73414 .25442 L .70408 .22814 L .70303 .22721 L .67221 .2 L .63605 .17947 L .62177 .17279 L .56803 .15463 L .54376 .14558 L .5 .12955 L .45897 .11837 L .43197 .10953 L .37036 .09116 L .36395 .08913 L .29592 .06833 L .28136 .06395 L .22789 .04576 L .19116 .03673 L .15986 .02831 L .10142 .00952 L .97619 .00952 L .97619 .39048 L F 0 g .8987 .39048 m .86411 .36327 L .84014 .3438 L .83064 .33605 L .79784 .30884 L .77211 .28703 L .76577 .28163 L .73414 .25442 L .70408 .22814 L .70303 .22721 L .67221 .2 L .63605 .17947 L .62177 .17279 L .56803 .15463 L .54376 .14558 L .5 .12955 L .45897 .11837 L .43197 .10953 L .37036 .09116 L .36395 .08913 L .29592 .06833 L .28136 .06395 L .22789 .04576 L .19116 .03673 L .15986 .02831 L .10142 .00952 L s .374 0 1 r .91776 .39048 m .90816 .3834 L .88145 .36327 L .84665 .33605 L .84014 .33084 L .81276 .30884 L .77973 .28163 L .77211 .27526 L .7473 .25442 L .71544 .22721 L .70408 .2171 L .68402 .2 L .63605 .17491 L .63135 .17279 L .56803 .15022 L .55562 .14558 L .5 .12587 L .4713 .11837 L .43197 .10566 L .38248 .09116 L .36395 .08537 L .29592 .06505 L .29225 .06395 L .22789 .04282 L .20191 .03673 L .15986 .02533 L .11157 .00952 L .97619 .00952 L .97619 .39048 L F 0 g .91776 .39048 m .90816 .3834 L .88145 .36327 L .84665 .33605 L .84014 .33084 L .81276 .30884 L .77973 .28163 L .77211 .27526 L .7473 .25442 L .71544 .22721 L .70408 .2171 L .68402 .2 L .63605 .17491 L .63135 .17279 L .56803 .15022 L .55562 .14558 L .5 .12587 L .4713 .11837 L .43197 .10566 L .38248 .09116 L .36395 .08537 L .29592 .06505 L .29225 .06395 L .22789 .04282 L .20191 .03673 L .15986 .02533 L .11157 .00952 L s .408 0 1 r .93784 .39048 m .90816 .36946 L .89963 .36327 L .86321 .33605 L .84014 .31818 L .82818 .30884 L .79406 .28163 L .77211 .26371 L .7608 .25442 L .72814 .22721 L .70408 .2065 L .69607 .2 L .64215 .17279 L .63605 .1703 L .56803 .1459 L .56716 .14558 L .5 .12259 L .48328 .11837 L .43197 .10201 L .39432 .09116 L .36395 .08176 L .30335 .06395 L .29592 .06168 L .22789 .04022 L .21241 .03673 L .15986 .02233 L .12155 .00952 L .97619 .00952 L .97619 .39048 L F 0 g .93784 .39048 m .90816 .36946 L .89963 .36327 L .86321 .33605 L .84014 .31818 L .82818 .30884 L .79406 .28163 L .77211 .26371 L .7608 .25442 L .72814 .22721 L .70408 .2065 L .69607 .2 L .64215 .17279 L .63605 .1703 L .56803 .1459 L .56716 .14558 L .5 .12259 L .48328 .11837 L .43197 .10201 L .39432 .09116 L .36395 .08176 L .30335 .06395 L .29592 .06168 L .22789 .04022 L .21241 .03673 L .15986 .02233 L .12155 .00952 L s .443 0 1 r .95903 .39048 m .91873 .36327 L .90816 .35588 L .88051 .33605 L .84413 .30884 L .84014 .30579 L .80885 .28163 L .77465 .25442 L .77211 .25237 L .74114 .22721 L .70836 .2 L .70408 .19708 L .65455 .17279 L .63605 .16542 L .57949 .14558 L .56803 .14176 L .5 .11959 L .49503 .11837 L .43197 .09855 L .40594 .09116 L .36395 .0783 L .31451 .06395 L .29592 .05832 L .22789 .03786 L .22272 .03673 L .15986 .01929 L .13139 .00952 L .97619 .00952 L .97619 .39048 L F 0 g .95903 .39048 m .91873 .36327 L .90816 .35588 L .88051 .33605 L .84413 .30884 L .84014 .30579 L .80885 .28163 L .77465 .25442 L .77211 .25237 L .74114 .22721 L .70836 .2 L .70408 .19708 L .65455 .17279 L .63605 .16542 L .57949 .14558 L .56803 .14176 L .5 .11959 L .49503 .11837 L .43197 .09855 L .40594 .09116 L .36395 .0783 L .31451 .06395 L .29592 .05832 L .22789 .03786 L .22272 .03673 L .15986 .01929 L .13139 .00952 L s .478 0 1 r .97619 .3872 m .93886 .36327 L .90816 .34264 L .89864 .33605 L .86058 .30884 L .84014 .29365 L .82414 .28163 L .78882 .25442 L .77211 .24123 L .75448 .22721 L .72089 .2 L .70408 .1895 L .66704 .17279 L .63605 .16075 L .59187 .14558 L .56803 .13771 L .50691 .11837 L .5 .11622 L .43197 .09525 L .41738 .09116 L .36395 .07498 L .32552 .06395 L .29592 .05507 L .23302 .03673 L .22789 .03567 L .15986 .01616 L .14109 .00952 L .97619 .00952 L F 0 g .97619 .3872 m .93886 .36327 L .90816 .34264 L .89864 .33605 L .86058 .30884 L .84014 .29365 L .82414 .28163 L .78882 .25442 L .77211 .24123 L .75448 .22721 L .72089 .2 L .70408 .1895 L .66704 .17279 L .63605 .16075 L .59187 .14558 L .56803 .13771 L .50691 .11837 L .5 .11622 L .43197 .09525 L .41738 .09116 L .36395 .07498 L .32552 .06395 L .29592 .05507 L .23302 .03673 L .22789 .03567 L .15986 .01616 L .14109 .00952 L s .513 0 1 r .97619 .37316 m .96012 .36327 L .91769 .33605 L .90816 .32971 L .87776 .30884 L 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.94331 .06395 L .90816 .07322 L .84014 .08895 L .83369 .09116 L .77211 .09793 L .70408 .10289 L .63605 .10141 L .56803 .09524 L .5305 .09116 L .5 .08605 L .43197 .07351 L .37373 .06395 L .36395 .06191 L .29592 .04253 L .24189 .03673 L .22789 .03398 L .15986 .02191 L .12568 .00952 L s .055 0 1 r .97619 .21673 m .90816 .20273 L .90174 .2 L .84014 .19232 L .77211 .18723 L .70408 .19 L .64353 .2 L .63605 .20223 L .56803 .21551 L .51602 .22721 L .5 .23201 L .43255 .25442 L .43197 .25463 L .36395 .28162 L .36391 .28163 L .3037 .30884 L .29592 .31259 L .24903 .33605 L .22789 .3472 L .19831 .36327 L .15986 .38507 L .15057 .39048 L .97619 .39048 L F 0 g .97619 .21673 m .90816 .20273 L .90174 .2 L .84014 .19232 L .77211 .18723 L .70408 .19 L .64353 .2 L .63605 .20223 L .56803 .21551 L .51602 .22721 L .5 .23201 L .43255 .25442 L .43197 .25463 L .36395 .28162 L .36391 .28163 L .3037 .30884 L .29592 .31259 L .24903 .33605 L .22789 .3472 L .19831 .36327 L .15986 .38507 L .15057 .39048 L s .129 0 1 r .97619 .04807 m .91184 .06395 L .90816 .06478 L .84014 .07806 L .77211 .08791 L .73234 .09116 L .70408 .09195 L .64116 .09116 L .63605 .09088 L .56803 .08502 L .5 .07606 L .43197 .06541 L .42163 .06395 L .36395 .05285 L .29592 .03759 L .27998 .03673 L .22789 .02636 L .15986 .01167 L .15604 .00952 L .97619 .00952 L F 0 g .97619 .04807 m .91184 .06395 L .90816 .06478 L .84014 .07806 L .77211 .08791 L .73234 .09116 L .70408 .09195 L .64116 .09116 L .63605 .09088 L .56803 .08502 L .5 .07606 L .43197 .06541 L .42163 .06395 L .36395 .05285 L .29592 .03759 L .27998 .03673 L .22789 .02636 L .15986 .01167 L .15604 .00952 L s .092 0 1 r .97619 .22424 m .90816 .21172 L .84014 .20282 L .80303 .2 L .77211 .19915 L .73377 .2 L .70408 .20251 L .63605 .21493 L .56803 .22285 L .54638 .22721 L .5 .23999 L .45445 .25442 L .43197 .26242 L .38199 .28163 L .36395 .28918 L .31941 .30884 L .29592 .31992 L .26306 .33605 L .22789 .35428 L .21109 .36327 L .16233 .39048 L .97619 .39048 L F 0 g .97619 .22424 m .90816 .21172 L .84014 .20282 L .80303 .2 L .77211 .19915 L .73377 .2 L .70408 .20251 L .63605 .21493 L .56803 .22285 L .54638 .22721 L .5 .23999 L .45445 .25442 L .43197 .26242 L .38199 .28163 L .36395 .28918 L .31941 .30884 L .29592 .31992 L .26306 .33605 L .22789 .35428 L .21109 .36327 L .16233 .39048 L s .129 0 1 r .97619 .23196 m .92378 .22721 L .90816 .21959 L .84014 .21197 L .77211 .20929 L .70408 .21272 L .63605 .22224 L .58084 .22721 L .56803 .23028 L .5 .2479 L .47832 .25442 L .43197 .27014 L .40106 .28163 L .36395 .29669 L .33573 .30884 L .29592 .3272 L .27753 .33605 L .22789 .36131 L .22417 .36327 L .1743 .39048 L .97619 .39048 L F 0 g .97619 .23196 m .92378 .22721 L .90816 .21959 L .84014 .21197 L .77211 .20929 L .70408 .21272 L .63605 .22224 L .58084 .22721 L .56803 .23028 L .5 .2479 L .47832 .25442 L .43197 .27014 L .40106 .28163 L .36395 .29669 L .33573 .30884 L .29592 .3272 L .27753 .33605 L .22789 .36131 L .22417 .36327 L .1743 .39048 L s .165 0 1 r .97619 .04115 m .90816 .0566 L .87289 .06395 L .84014 .069 L .77211 .07751 L .70408 .08145 L .63605 .0805 L .56803 .07547 L .5 .06754 L .47264 .06395 L .43197 .05738 L .36395 .04516 L .31618 .03673 L .29592 .03625 L .22789 .01931 L .18613 .00952 L .97619 .00952 L F 0 g .97619 .04115 m .90816 .0566 L .87289 .06395 L .84014 .069 L .77211 .07751 L .70408 .08145 L .63605 .0805 L .56803 .07547 L .5 .06754 L .47264 .06395 L .43197 .05738 L .36395 .04516 L .31618 .03673 L .29592 .03625 L .22789 .01931 L .18613 .00952 L s .165 0 1 r .97619 .23994 m .90881 .22721 L .90816 .22685 L .84014 .21978 L .77211 .21751 L .70408 .22032 L .63605 .22653 L .62514 .22721 L .56803 .23827 L .5044 .25442 L .5 .25574 L .43197 .2778 L .42129 .28163 L .36395 .30414 L .35274 .30884 L .29592 .33442 L .29245 .33605 L .23754 .36327 L .22789 .36829 L .18653 .39048 L .97619 .39048 L F 0 g .97619 .23994 m .90881 .22721 L .90816 .22685 L .84014 .21978 L .77211 .21751 L .70408 .22032 L .63605 .22653 L .62514 .22721 L .56803 .23827 L .5044 .25442 L .5 .25574 L .43197 .2778 L .42129 .28163 L .36395 .30414 L .35274 .30884 L .29592 .33442 L .29245 .33605 L .23754 .36327 L .22789 .36829 L .18653 .39048 L s .202 0 1 r .97619 .24786 m .90816 .23482 L .84746 .22721 L .84014 .22686 L .77211 .22452 L .70408 .2265 L .69171 .22721 L .63605 .2337 L .56803 .2462 L .53276 .25442 L .5 .26351 L .44261 .28163 L .43197 .28539 L .37042 .30884 L .36395 .31153 L .30778 .33605 L .29592 .34159 L .25125 .36327 L .22789 .37523 L .19903 .39048 L .97619 .39048 L F 0 g .97619 .24786 m .90816 .23482 L .84746 .22721 L .84014 .22686 L .77211 .22452 L .70408 .2265 L .69171 .22721 L .63605 .2337 L .56803 .2462 L .53276 .25442 L .5 .26351 L .44261 .28163 L .43197 .28539 L .37042 .30884 L .36395 .31153 L .30778 .33605 L .29592 .34159 L .25125 .36327 L .22789 .37523 L .19903 .39048 L s .239 0 1 r .97619 .2557 m .95581 .25442 L .90816 .24278 L .84014 .23483 L .77211 .23197 L .70408 .23427 L .63605 .24167 L .56803 .25405 L .56629 .25442 L .5 .27123 L .46537 .28163 L .43197 .29293 L .38885 .30884 L .36395 .31886 L .32364 .33605 L .29592 .3487 L .26535 .36327 L .22789 .38212 L .21183 .39048 L .97619 .39048 L F 0 g .97619 .2557 m .95581 .25442 L .90816 .24278 L .84014 .23483 L .77211 .23197 L .70408 .23427 L .63605 .24167 L .56803 .25405 L .56629 .25442 L .5 .27123 L .46537 .28163 L .43197 .29293 L .38885 .30884 L .36395 .31886 L .32364 .33605 L .29592 .3487 L .26535 .36327 L .22789 .38212 L .21183 .39048 L s .202 0 1 r .97619 .03473 m .96536 .03673 L .90816 .04907 L .84014 .06093 L .82423 .06395 L .77211 .06869 L .70408 .07223 L .63605 .07153 L .56803 .06712 L .53261 .06395 L .5 .05968 L .43197 .04979 L .36395 .03849 L .35317 .03673 L .29592 .03535 L .22789 .01247 L .21544 .00952 L .97619 .00952 L F 0 g .97619 .03473 m .96536 .03673 L .90816 .04907 L .84014 .06093 L .82423 .06395 L .77211 .06869 L .70408 .07223 L .63605 .07153 L .56803 .06712 L .53261 .06395 L .5 .05968 L .43197 .04979 L .36395 .03849 L .35317 .03673 L .29592 .03535 L .22789 .01247 L .21544 .00952 L s .497 0 1 r .22789 .09763 m .28608 .11837 L .22789 .12881 L .21604 .11837 L F 0 g .22789 .09763 m .28608 .11837 L .22789 .12881 L .21604 .11837 L .22789 .09763 L s .534 0 1 r .22789 .0982 m .28493 .11837 L .22789 .12846 L .21646 .11837 L F 0 g .22789 .0982 m .28493 .11837 L .22789 .12846 L .21646 .11837 L .22789 .0982 L s .571 0 1 r .22789 .09876 m .28378 .11837 L .22789 .12812 L .21689 .11837 L F 0 g .22789 .09876 m .28378 .11837 L .22789 .12812 L .21689 .11837 L .22789 .09876 L s .608 0 1 r .22789 .09933 m .28263 .11837 L .22789 .12777 L .21731 .11837 L F 0 g .22789 .09933 m .28263 .11837 L .22789 .12777 L .21731 .11837 L .22789 .09933 L s .645 0 1 r .22789 .0999 m .28147 .11837 L .22789 .12742 L .21774 .11837 L F 0 g .22789 .0999 m .28147 .11837 L .22789 .12742 L .21774 .11837 L .22789 .0999 L s .682 0 1 r .22789 .10047 m .28031 .11837 L .22789 .12706 L .21817 .11837 L F 0 g .22789 .10047 m .28031 .11837 L .22789 .12706 L .21817 .11837 L .22789 .10047 L s .719 0 1 r .22789 .10105 m .27914 .11837 L .22789 .12671 L .2186 .11837 L F 0 g .22789 .10105 m .27914 .11837 L .22789 .12671 L .2186 .11837 L .22789 .10105 L s .755 0 1 r .22789 .10163 m .27796 .11837 L .22789 .12635 L .21903 .11837 L F 0 g .22789 .10163 m .27796 .11837 L .22789 .12635 L .21903 .11837 L .22789 .10163 L s .792 0 1 r .22789 .10222 m .27677 .11837 L .22789 .12599 L .21947 .11837 L F 0 g .22789 .10222 m .27677 .11837 L .22789 .12599 L .21947 .11837 L .22789 .10222 L s .829 0 1 r .22789 .10281 m .27557 .11837 L .22789 .12562 L .21991 .11837 L F 0 g .22789 .10281 m .27557 .11837 L .22789 .12562 L .21991 .11837 L .22789 .10281 L s .866 0 1 r .22789 .10341 m .27435 .11837 L .22789 .12525 L .22035 .11837 L F 0 g .22789 .10341 m .27435 .11837 L .22789 .12525 L .22035 .11837 L .22789 .10341 L s .903 0 1 r .22789 .10402 m .27311 .11837 L .22789 .12487 L .2208 .11837 L F 0 g .22789 .10402 m .27311 .11837 L .22789 .12487 L .2208 .11837 L .22789 .10402 L s 1 0 0 r .22789 .10465 m .27186 .11837 L .22789 .12449 L .22124 .11837 L F 0 g .22789 .10465 m .27186 .11837 L .22789 .12449 L .22124 .11837 L .22789 .10465 L s .276 0 1 r .97619 .26347 m .91153 .25442 L .90816 .25066 L .84014 .24278 L .77211 .23995 L .70408 .24223 L .63605 .24956 L .60386 .25442 L .56803 .26184 L .5 .27887 L .49032 .28163 L .43197 .30041 L .40832 .30884 L .36395 .32615 L .34014 .33605 L .29592 .35577 L .27989 .36327 L .22789 .38896 L .22493 .39048 L .97619 .39048 L F 0 g .97619 .26347 m .91153 .25442 L .90816 .25066 L .84014 .24278 L .77211 .23995 L .70408 .24223 L .63605 .24956 L .60386 .25442 L .56803 .26184 L .5 .27887 L .49032 .28163 L .43197 .30041 L .40832 .30884 L .36395 .32615 L .34014 .33605 L .29592 .35577 L .27989 .36327 L .22789 .38896 L .22493 .39048 L s .313 0 1 r .97619 .27119 m .90816 .25848 L .88972 .25442 L .84014 .25067 L .77211 .24786 L .70408 .25012 L .65866 .25442 L .63605 .25739 L .56803 .26957 L .51717 .28163 L .5 .28646 L .43197 .30783 L .42902 .30884 L .36395 .33337 L .35734 .33605 L .29592 .36279 L .29489 .36327 L .23832 .39048 L .97619 .39048 L F 0 g .97619 .27119 m .90816 .25848 L .88972 .25442 L .84014 .25067 L .77211 .24786 L .70408 .25012 L .65866 .25442 L .63605 .25739 L .56803 .26957 L .51717 .28163 L .5 .28646 L .43197 .30783 L .42902 .30884 L .36395 .33337 L .35734 .33605 L .29592 .36279 L .29489 .36327 L .23832 .39048 L s .239 0 1 r .97619 .02869 m .93459 .03673 L .90816 .04215 L .84014 .05315 L .77211 .06073 L .7148 .06395 L .70408 .0642 L .64677 .06395 L .63605 .06359 L .56803 .05934 L .5 .05219 L .43197 .04287 L .39164 .03673 L .36395 .03251 L .29592 .0317 L .24449 .00952 L .97619 .00952 L F 0 g .97619 .02869 m .93459 .03673 L .90816 .04215 L .84014 .05315 L .77211 .06073 L .7148 .06395 L .70408 .0642 L .64677 .06395 L .63605 .06359 L .56803 .05934 L .5 .05219 L .43197 .04287 L .39164 .03673 L .36395 .03251 L .29592 .0317 L .24449 .00952 L s .35 0 1 r .97619 .27884 m .90816 .26624 L .84014 .25849 L .77211 .25571 L .70408 .25794 L .63605 .26515 L .56803 .27723 L .54773 .28163 L .5 .29399 L .45072 .30884 L .43197 .31519 L .37518 .33605 L .36395 .34055 L .31028 .36327 L .29592 .36976 L .25205 .39048 L .97619 .39048 L F 0 g .97619 .27884 m .90816 .26624 L .84014 .25849 L .77211 .25571 L .70408 .25794 L .63605 .26515 L .56803 .27723 L .54773 .28163 L .5 .29399 L .45072 .30884 L .43197 .31519 L .37518 .33605 L .36395 .34055 L .31028 .36327 L .29592 .36976 L .25205 .39048 L s .387 0 1 r .97619 .28642 m .92202 .28163 L .90816 .27393 L .84014 .26624 L .77211 .26348 L .70408 .2657 L .63605 .27285 L .58232 .28163 L .56803 .28483 L .5 .30146 L .47423 .30884 L .43197 .3225 L .39386 .33605 L .36395 .34768 L .32624 .36327 L .29592 .37669 L .26618 .39048 L .97619 .39048 L F 0 g .97619 .28642 m .92202 .28163 L .90816 .27393 L .84014 .26624 L .77211 .26348 L .70408 .2657 L .63605 .27285 L .58232 .28163 L .56803 .28483 L .5 .30146 L .47423 .30884 L .43197 .3225 L .39386 .33605 L .36395 .34768 L .32624 .36327 L .29592 .37669 L .26618 .39048 L s .276 0 1 r .97619 .02288 m .90816 .03573 L .90263 .03673 L .84014 .04601 L .77211 .05309 L .70408 .0564 L .63605 .0559 L .56803 .05198 L .5 .04527 L .43371 .03673 L .43197 .03648 L .36395 .02686 L .29592 .02213 L .27392 .00952 L .97619 .00952 L F 0 g .97619 .02288 m .90816 .03573 L .90263 .03673 L .84014 .04601 L .77211 .05309 L .70408 .0564 L .63605 .0559 L .56803 .05198 L .5 .04527 L .43371 .03673 L .43197 .03648 L .36395 .02686 L .29592 .02213 L .27392 .00952 L s .424 0 1 r .97619 .29395 m .90831 .28163 L .90816 .28155 L .84014 .27393 L .77211 .27119 L .70408 .27339 L .63605 .28049 L .62739 .28163 L .56803 .29237 L .50009 .30884 L .5 .30887 L .43197 .32975 L .41364 .33605 L .36395 .35475 L .34284 .36327 L .29592 .38357 L .28074 .39048 L .97619 .39048 L F 0 g .97619 .29395 m .90831 .28163 L .90816 .28155 L .84014 .27393 L .77211 .27119 L .70408 .27339 L .63605 .28049 L .62739 .28163 L .56803 .29237 L .50009 .30884 L .5 .30887 L .43197 .32975 L .41364 .33605 L .36395 .35475 L .34284 .36327 L .29592 .38357 L .28074 .39048 L s .46 0 1 r .97619 .30142 m .90816 .28912 L .84164 .28163 L .84014 .28156 L .77211 .27884 L .70408 .28102 L .69564 .28163 L .63605 .28806 L .56803 .29985 L .52777 .30884 L .5 .31623 L .43462 .33605 L .43197 .33695 L .36395 .36178 L .36017 .36327 L .29592 .39041 L .29576 .39048 L .97619 .39048 L F 0 g .97619 .30142 m .90816 .28912 L .84164 .28163 L .84014 .28156 L .77211 .27884 L .70408 .28102 L .69564 .28163 L .63605 .28806 L .56803 .29985 L .52777 .30884 L .5 .31623 L .43462 .33605 L .43197 .33695 L .36395 .36178 L .36017 .36327 L .29592 .39041 L .29576 .39048 L s .313 0 1 r .97619 .01718 m .90816 .02969 L .86184 .03673 L .84014 .03939 L .77211 .04604 L .70408 .04919 L .63605 .04878 L .56803 .04513 L .5 .03881 L .48117 .03673 L .43197 .03049 L .36395 .02121 L .30405 .00952 L .97619 .00952 L F 0 g .97619 .01718 m .90816 .02969 L .86184 .03673 L .84014 .03939 L .77211 .04604 L .70408 .04919 L .63605 .04878 L .56803 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presented at IAS in 1998, Murmansk, Russia." }], "Text"], Cell[BoxData[ \({{\((\(-1.`\) + x)\)\^2 - 1.`\ y\^2, \(-0.3333333333333333`\)\ \((\(-1.`\) + x)\)\ x - 0.3333333333333333`\ y + 0.3333333333333333`\ y\^2, \(-0.3333333333333333`\)\ \((\(-4.`\) + x)\)\ x - 1.3333333333333333`\ y + 0.3333333333333333`\ y\^2, \(-0.3333333333333333`\)\ \ \((\(-1.0000000000000004`\) + x)\)\ x + 1.6666666666666665`\ y + 0.3333333333333333`\ y\^2}, {\(-9.`\)\ \((\(-0.5555555555555556`\) \ + y)\)\ \((\(\(0.19999999999999998`\)\(\[InvisibleSpace]\)\) + y)\) + x\ \((\(-1.3599999999999999`\) + x + \(0.1280000000000001`\ x\)\/\(\(-0.2`\)\ x + y\))\), y\ \((\(-1.4`\) + 3.`\ y)\) + \(x\ \((0.06666666666666665`\ \((\(-1.`\) + x)\)\ \ x - 0.3333333333333333`\ \((\(-0.36000000000000015`\) + x)\)\ \ y)\)\)\/\(\(-0.2`\)\ x + y\), y\ \((\(-2.4000000000000004`\) + 3.`\ y)\) + \(x\ \((0.06666666666666665`\ \((\(-4.`\) + x)\)\ \ x - 0.3333333333333333`\ \((\(-3.359999999999999`\) + x)\)\ y)\)\)\/\(\(-0.2`\ \)\ x + 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The three other shape functions, ", StyleBox["n2[x,y], n3[x,y]", "Input"], " and ", StyleBox["n4[x,y]", "Input"], " were obtained from:" }], "Text"], Cell[BoxData[ \(\(Shape1ToAll[ n1xy_, \[IndentingNewLine]{{x1_, y1_}, {x2_, y2_}, {x3_, y3_}, {x4_, y4_}}] := Module[{n1}, {n1, \(({\(-\((x3\ y - x4\ y - n1\ x3\ y1 + n1\ x4\ y1 - x\ y3 + n1\ x1\ y3 + x4\ y3 - n1\ x4\ y3 + x\ y4 - n1\ x1\ y4 - x3\ y4 + n1\ x3\ y4)\)\), \[IndentingNewLine]x2\ y - x4\ y - n1\ x2\ y1 + n1\ x4\ y1 - x\ y2 + n1\ x1\ y2 + x4\ y2 - n1\ x4\ y2 + x\ y4 - n1\ x1\ y4 - x2\ y4 + n1\ x2\ y4, \(-x2\)\ y + x3\ y + n1\ x2\ y1 - n1\ x3\ y1 + x\ y2 - n1\ x1\ y2 - x3\ y2 + n1\ x3\ y2 - x\ y3 + n1\ x1\ y3 + x2\ y3 - n1\ x2\ y3}/ Det[{{1, x2, y2}, {1, x3, y3}, {1, x4, y4}}])\)\[IndentingNewLine]} /. n1 \[Rule] n1xy];\)\)], "Input"], Cell[TextData[{ "A remarkable result is that if ", StyleBox["n1[x,y]", "Input"], " satisfies the elliptic condition:\n", StyleBox["D[n1,{x,2}]+\[Alpha][x,y] D[n1,{y,2}]==0", "Input"], ";\nthen the same partial differential equation, for arbitrary ", StyleBox["\[Alpha][x,y]>0", "Input"], ", is satisfied by the three compute shape functions as verified below:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"Simplify", "[", RowBox[{\((\[Alpha]\ \ \ D[#, {x, 2}] + \ \ D[#, {y, 2}])\), "/.", RowBox[{ RowBox[{ SuperscriptBox["n1", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}], "\[Rule]", RowBox[{\(-\ \[Alpha]\), " ", RowBox[{ SuperscriptBox["n1", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}]}]}]}], "]"}], ")"}], "&"}], "/@", \({n2, n3, n4}\)}]], "Input"], Cell[BoxData[ \({0, 0, 0}\)], "Output"] }, Open ]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Conclusions", "Section"], Cell[TextData[{ "The biomedical application of maxillo-facial morphometric analysis can be \ cited to be an important application of the present methodology. The idea is \ now being utilized for general three-dimensionsal cases. For unconventional \ concave quadrilateral finite element, the proposed formulation made it \ possible to generate all shape functions starting from the one which pertains \ to the concave node.\n\nTo our knowledge there is neither any published work \ nor any hint that the shape functions for elements can be generated from a \ single one by satisfying (desirable) interpolation conditions. Perhaps no \ investigator other than Wachspress could concieve the shape functions in the \ ", StyleBox["x-y", "Input"], " frame. In fact, except for Taig's isoparametric formulation in the \ computational domain of a unit square, there exists no other original idea, \ to our knowledge, of generating shape functions. For the quadrilateral \ example presented here, it has been shown that if the first shape function is \ generated by isoparametric formulation then the three remainder can be \ constructed according to the proposed scheme automatically satisfying the \ patch test." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ "\n[1] Gout, J. L.,\"Construction of an Hermite rational interpolation \ Wachspress type finite element,\" Comput. Math. Appl., 5, 1979, 337--347.\n\n\ [2] Irons, B. and Ahmad, S., Techniques of finite elements, Ellis Horwood, \ John Wiley and sons, 1980.\n\n[3] Laydi, M. and Lesaint, P., \"Sur les \ elements finis rationnels de Serendip de degre quelconque, Numerische \ Mathematik, Math. 46, 1985, Springer Verlag, 175--187.\n\n", StyleBox["[4] Moss, M. L., Salentijn, L., \"The primary role of the \ functional matrix in facial growth,\" Angle Orthod. 1969, 55, 566-574.\n\n[5] \ Planche P., Treil J., \"Cephalometrics for the next millennium,\" \ Proceedings: The future of orthodontics, Leuven University Press, France, \ 1998, 181-192.\n\n[6] Treil, J., Madrid C., Jaeger M., Casteigt, J., \ Borianne Ph., Biometrie tridimensionnelle maxillo-faciale, Biom. Hum et \ Anthropol, France, 1997,15, 1.2, 65-73.\n\n", FontFamily->"Times New Roman"], "[6] Taig, I., C., Structural analysis by the matrix displacement method,\n\ Engl. Electric Aviation, Report No. S017, 1961.\n\n[7] Wachspress, E. L. , A \ rational finite element basis,\nAcademic Press, New York, 1975.\n\n[8] \ Zienkiewicz, O. C. and Taylor, R. L.,The finite element method, third \ edition,\nMcGraw-Hill, 1977.", StyleBox["\n", FontFamily->"Times New Roman"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Acknowledgement", "Section"], Cell["\<\ Gautam Dasgupta was supported by a grant from the National Science \ Foundation, cms-9820353.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Appendices", "Section"], Cell[CellGroupData[{ Cell["Appendix-I: Isoparametric Shape Functions:", "Subsection"], Cell[CellGroupData[{ Cell["Analytical Isoparametric case", "Subsubsection"], Cell[TextData[{ "The ", StyleBox["nodes", "Input"], " of the quadrilateral elements are indexed as:" }], "Text"], Cell[BoxData[ \(nodeNumber[p_, color_] := Graphics[{{\[IndentingNewLine]Text["\<1\>", p[\([1]\)], {1, 0}], \[IndentingNewLine]Text["\<2\>", p[\([2]\)], {\(-1\), 1}], \[IndentingNewLine]Text["\<3\>", p[\([3]\)], {\(-1\), \(-1\)}], \[IndentingNewLine]Text["\<4\>", p[\([4]\)], {1, \(-1\)}]}, {color, Polygon[p]}}]\)], "Input", 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4\ x\ y[3] + 4\ x[4]\ y[3] + 4\ x\ y[4] - 4\ x[3]\ y[4])\)^2 - 4\ \((\(-4\)\ x[2]\ y[1] + 4\ x[3]\ y[1] + 4\ x[1]\ y[2] - 4\ x[4]\ y[2] - 4\ x[1]\ y[3] + 4\ x[4]\ y[3] + 4\ x[2]\ y[4] - 4\ x[3]\ y[4])\)\ \((\(-2\)\ y\ x[1] + 2\ y\ x[2] + 2\ y\ x[3] - 2\ y\ x[4] + 2\ x\ y[1] - x[2]\ y[1] - x[3]\ y[1] - 2\ x\ y[2] + x[1]\ y[2] + x[4]\ y[2] - 2\ x\ y[3] + x[1]\ y[3] + x[4]\ y[3] + 2\ x\ y[4] - x[2]\ y[4] - x[3]\ y[4])\))\))\))\)/\((\(-4\)\ x[ 2]\ y[1] + 4\ x[3]\ y[1] + 4\ x[1]\ y[2] - 4\ x[4]\ y[2] - 4\ x[1]\ y[3] + 4\ x[4]\ y[3] + 4\ x[2]\ y[4] - 4\ x[3]\ y[4])\) + \((x[4]\ y[ 3]\ \((\(-4\)\ y\ x[1] + 4\ y\ x[2] - 4\ y\ x[3] + 4\ y\ x[4] + 4\ x\ y[1] - 4\ x[2]\ y[1] - 4\ x\ y[2] + 4\ x[1]\ y[2] + 4\ x\ y[3] - 4\ x[4]\ y[3] - 4\ x\ y[4] + 4\ x[3]\ y[ 4] - \[Sqrt]\((\((4\ y\ x[1] - 4\ y\ x[2] + 4\ y\ x[3] - 4\ y\ x[4] - 4\ x\ y[1] + 4\ x[2]\ y[1] + 4\ x\ y[2] - 4\ x[1]\ y[2] - 4\ x\ y[3] + 4\ x[4]\ y[3] + 4\ x\ y[4] - 4\ x[3]\ y[4])\)^2 - 4\ \((\(-4\)\ x[2]\ y[1] + 4\ x[3]\ y[1] + 4\ x[1]\ y[2] - 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x[2]\ y[4] - x[3]\ y[4])\))\))\))\)/\((\(-4\)\ x[ 2]\ y[1] + 4\ x[3]\ y[1] + 4\ x[1]\ y[2] - 4\ x[4]\ y[2] - 4\ x[1]\ y[3] + 4\ x[4]\ y[3] + 4\ x[2]\ y[4] - 4\ x[3]\ y[4])\))\)/\((\(-2\)\ x[3]\ y[ 1] + 2\ x[4]\ y[1] + 2\ x[3]\ y[2] - 2\ x[4]\ y[2] + 2\ x[1]\ y[3] - 2\ x[2]\ y[3] - 2\ x[1]\ y[4] + 2\ x[2]\ y[4])\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \( (*location = {"\"}; \ \[IndentingNewLine]$Path = Union[$Path, location]\[IndentingNewLine]*) \)], "Input"], Cell[BoxData[ \({"Users:Gautam:Research:ConcaveElement"}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[""], "Input"], Cell[BoxData[ \({":", "Applications:Mathematica 4.0 Files:AddOns:Applications", "Applications:Mathematica 4.0 Files:AddOns:Autoload", "Applications:Mathematica 4.0 Files:AddOns:ExtraPackages", "Applications:Mathematica 4.0 Files:AddOns:StandardPackages", "Applications:Mathematica 4.0 Files:AddOns:StandardPackages:StartUp", "Applications:Mathematica 4.0 Files:Configuration:Kernel", "Applications:Mathematica 4.0 Files:SystemFiles:Graphics:Packages", "System:System Folder:Preferences:Mathematica:4.0:AddOns:Applications", "System:System Folder:Preferences:Mathematica:4.0:AddOns:Autoload", "System:System Folder:Preferences:Mathematica:4.0:Kernel", "Users:Gautam:Research:ConcaveElement"}\)], "Output"] }, Open ]], Cell[BoxData[ \( (*\ << Shape1ToAll . m\ \ *) \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?Shape1ToAll\)\)], "Input"], Cell[BoxData[ \("Shape1ToAll[p,n1,{x,y}] yields shape functins for a quadrilateral with \ nodes p in x-y for a given shape n1 for p[[1]]."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Sxy - 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1.0503933615080605`\ u + 0.688782532136433`\ v)\), \((\(\(0.530605598401636`\)\ \(\[InvisibleSpace]\)\) + 0.2635341862087222`\ u - 0.07748803486534875`\ u\^2 - 0.1467406264116749`\ v - 0.10331737982046499`\ u\ \ v)\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)}, \ {\((\(\(1.0827904974169502`\)\(\[InvisibleSpace]\)\) - 0.4154893769158065`\ u - 0.4523048913260678`\ v + 0.17355885364837487`\ u\ \ v)\)/\((\(\(4.010637059632144`\)\(\[InvisibleSpace]\)\) - 0.12397060974883906`\ u - 0.2975294633972142`\ v)\), \((u\ \ \((\(\(0.44516718955264994`\)\(\[InvisibleSpace]\)\) - 0.18595591462325883`\ v)\) + 0.07438236584930352`\ \((\(-5.87878787878788`\) + v)\)\ \((\(-2.3939393939393936`\) + v)\))\)/\((\(\(4.010637059632144`\)\(\ \[InvisibleSpace]\)\) - 0.12397060974883906`\ u - 0.2975294633972142`\ v)\), \ \((\(-0.030992652437209806`\)\ u\^2 + u\ \((\(\(0.30654550774258427`\)\(\[InvisibleSpace]\)\ \) + 0.19835297559814272`\ v)\) - 0.07438236584930352`\ \((\(-5.87878787878788`\) + v)\)\ \((\(\(2.0404040404040407`\)\(\ \[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.010637059632144`\)\(\ \[InvisibleSpace]\)\) - 0.12397060974883906`\ u - 0.2975294633972142`\ v)\), \ \((\(\(0.988805065453092`\)\(\[InvisibleSpace]\)\) - 0.4601939301282667`\ u + 0.0309926524372098`\ u\^2 + 0.4846123835636441`\ v - 0.18595591462325878`\ u\ \ v)\)/\((\(\(4.010637059632144`\)\(\[InvisibleSpace]\)\) - 0.12397060974883906`\ u - 0.2975294633972142`\ v)\)}}, LocalStrainDisplacementMatrices \[Rule] \ {{{\(-\(\((0.3315815789626931`\ \((\(-0.033158157896269275`\)\ u\^2 + u\ \((\(-0.14736959065008567`\) + 0.17684350878010283`\ v)\) + 0.13263263158507713`\ \ \((\(-5.833333333333334`\) + v)\)\ \((\(-1.3888888888888886`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\ \(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2\)\) + \ \((\(-0.14736959065008567`\) - 0.06631631579253855`\ u + 0.17684350878010283`\ \ v)\)/\((\(\(4.052663742877359`\)\(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), 0, \(-\(\((0.3315815789626931`\ \((0.03315815789626929`\ \ u\^2 + u\ \((\(\(0.41447697370336617`\)\(\[InvisibleSpace]\)\) - 0.23210710527388506`\ v)\) + 0.19894894737761573`\ \ \((\(-4.166666666666667`\) + v)\)\ \((\(-1.388888888888889`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\ \(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2\)\) + \ \((\(\(0.41447697370336617`\)\(\[InvisibleSpace]\)\) + 0.06631631579253858`\ u - 0.23210710527388506`\ \ v)\)/\((\(\(4.052663742877359`\)\(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), 0, \((\(\(0.2578967836376502`\)\(\[InvisibleSpace]\)\) + 0.22105438597512875`\ \ v)\)/\((\(\(4.052663742877359`\)\(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\) - \ \((0.3315815789626931`\ \((u\ \((\(\(0.2578967836376502`\)\(\[InvisibleSpace]\ \)\) + 0.22105438597512875`\ v)\) - 0.2210543859751287`\ \((\(-4.166666666666667`\ \) + v)\)\ \((\(\(1.1666666666666665`\)\(\[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2, 0, \((\(-0.1934225877282376`\) - 0.1657907894813465`\ \ v)\)/\((\(\(4.052663742877359`\)\(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\) - \ \((0.3315815789626931`\ \((u\ \((\(-0.1934225877282376`\) - 0.1657907894813465`\ v)\) - 0.11052719298756437`\ \ \((\(-5.833333333333333`\) + v)\)\ \((\(\(1.1666666666666665`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2, 0}, {0, \((0.8842175439005134`\ \ \((\(-0.033158157896269275`\)\ u\^2 + u\ \((\(-0.14736959065008567`\) + 0.17684350878010283`\ v)\) + 0.13263263158507713`\ \ \((\(-5.833333333333334`\) + v)\)\ \((\(-1.3888888888888886`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2 + \ \((0.17684350878010283`\ u + 0.13263263158507713`\ \((\(-5.833333333333334`\) \ + v)\) + 0.13263263158507713`\ \((\(-1.3888888888888886`\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), 0, \((0.8842175439005134`\ \((0.03315815789626929`\ u\^2 \ + u\ \((\(\(0.41447697370336617`\)\(\[InvisibleSpace]\)\) - 0.23210710527388506`\ v)\) + 0.19894894737761573`\ \ \((\(-4.166666666666667`\) + v)\)\ \((\(-1.388888888888889`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2 + \ \((\(-0.23210710527388506`\)\ u + 0.19894894737761573`\ \((\(-4.166666666666667`\) \ + v)\) + 0.19894894737761573`\ \((\(-1.388888888888889`\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), 0, \((0.22105438597512875`\ u - 0.2210543859751287`\ \((\(-4.166666666666667`\) + v)\) - 0.2210543859751287`\ \((\(\(1.1666666666666665`\)\ \(\[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\) + \ \((0.8842175439005134`\ \((u\ \((\(\(0.2578967836376502`\)\(\[InvisibleSpace]\ \)\) + 0.22105438597512875`\ v)\) - 0.2210543859751287`\ \((\(-4.166666666666667`\ \) + v)\)\ \((\(\(1.1666666666666665`\)\(\[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2, 0, \((\(-0.1657907894813465`\)\ u - 0.11052719298756437`\ \((\(-5.833333333333333`\) \ + v)\) - 0.11052719298756437`\ \ \((\(\(1.1666666666666665`\)\(\[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\) + \ \((0.8842175439005134`\ \((u\ \((\(-0.1934225877282376`\) - 0.1657907894813465`\ v)\) - 0.11052719298756437`\ \ \((\(-5.833333333333333`\) + v)\)\ \((\(\(1.1666666666666665`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2}, \ {\((0.8842175439005134`\ \((\(-0.033158157896269275`\)\ u\^2 + u\ \((\(-0.14736959065008567`\) + 0.17684350878010283`\ v)\) + 0.13263263158507713`\ \ \((\(-5.833333333333334`\) + v)\)\ \((\(-1.3888888888888886`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2 + \ \((0.17684350878010283`\ u + 0.13263263158507713`\ \((\(-5.833333333333334`\) \ + v)\) + 0.13263263158507713`\ \((\(-1.3888888888888886`\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), \ \(-\(\((0.3315815789626931`\ \((\(-0.033158157896269275`\)\ u\^2 + u\ \((\(-0.14736959065008567`\) + 0.17684350878010283`\ v)\) + 0.13263263158507713`\ \ \((\(-5.833333333333334`\) + v)\)\ \((\(-1.3888888888888886`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\ \(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2\)\) + \ \((\(-0.14736959065008567`\) - 0.06631631579253855`\ u + 0.17684350878010283`\ \ v)\)/\((\(\(4.052663742877359`\)\(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), \((0.8842175439005134`\ \ \((0.03315815789626929`\ u\^2 + u\ \((\(\(0.41447697370336617`\)\(\ \[InvisibleSpace]\)\) - 0.23210710527388506`\ v)\) + 0.19894894737761573`\ \ \((\(-4.166666666666667`\) + v)\)\ \((\(-1.388888888888889`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2 + \ \((\(-0.23210710527388506`\)\ u + 0.19894894737761573`\ \((\(-4.166666666666667`\) \ + v)\) + 0.19894894737761573`\ \((\(-1.388888888888889`\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), \ \(-\(\((0.3315815789626931`\ \((0.03315815789626929`\ u\^2 + u\ \((\(\(0.41447697370336617`\)\(\ \[InvisibleSpace]\)\) - 0.23210710527388506`\ v)\) + 0.19894894737761573`\ \ \((\(-4.166666666666667`\) + v)\)\ \((\(-1.388888888888889`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\ \(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2\)\) + \ \((\(\(0.41447697370336617`\)\(\[InvisibleSpace]\)\) + 0.06631631579253858`\ u - 0.23210710527388506`\ \ v)\)/\((\(\(4.052663742877359`\)\(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), \ \((0.22105438597512875`\ u - 0.2210543859751287`\ \((\(-4.166666666666667`\) + v)\) - 0.2210543859751287`\ \((\(\(1.1666666666666665`\)\ \(\[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\) + \ \((0.8842175439005134`\ \((u\ \((\(\(0.2578967836376502`\)\(\[InvisibleSpace]\ \)\) + 0.22105438597512875`\ v)\) - 0.2210543859751287`\ \((\(-4.166666666666667`\ \) + v)\)\ \((\(\(1.1666666666666665`\)\(\[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2, \ \((\(\(0.2578967836376502`\)\(\[InvisibleSpace]\)\) + 0.22105438597512875`\ \ v)\)/\((\(\(4.052663742877359`\)\(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\) - \ \((0.3315815789626931`\ \((u\ \((\(\(0.2578967836376502`\)\(\[InvisibleSpace]\ \)\) + 0.22105438597512875`\ v)\) - 0.2210543859751287`\ \((\(-4.166666666666667`\ \) + v)\)\ \((\(\(1.1666666666666665`\)\(\[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2, \ \((\(-0.1657907894813465`\)\ u - 0.11052719298756437`\ \((\(-5.833333333333333`\) \ + v)\) - 0.11052719298756437`\ \ \((\(\(1.1666666666666665`\)\(\[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\) + \ \((0.8842175439005134`\ \((u\ \((\(-0.1934225877282376`\) - 0.1657907894813465`\ v)\) - 0.11052719298756437`\ \ \((\(-5.833333333333333`\) + v)\)\ \((\(\(1.1666666666666665`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2, \ \((\(-0.1934225877282376`\) - 0.1657907894813465`\ \ v)\)/\((\(\(4.052663742877359`\)\(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\) - \ \((0.3315815789626931`\ \((u\ \((\(-0.1934225877282376`\) - 0.1657907894813465`\ v)\) - 0.11052719298756437`\ \ \((\(-5.833333333333333`\) + v)\)\ \((\(\(1.1666666666666665`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)^2}}, \ {{\((\(-0.6606219426304372`\) - 0.816062399719952`\ u + 0.2720207999066507`\ \ v)\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ u)\) - \ \((1.6321247994399033`\ \((\(\(0.8660254037844387`\)\(\[InvisibleSpace]\)\) - 0.6606219426304372`\ u - 0.408031199859976`\ u\^2 - 0.233160685634272`\ v + 0.2720207999066507`\ u\ \ v)\))\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u)\)\^2, 0, \((\(\(1.4766843423503888`\)\(\[InvisibleSpace]\)\) + 0.8160623997199519`\ u - 0.2720207999066506`\ \ v)\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ u)\) - \ \((1.6321247994399033`\ \((\(\(1.1547005383792515`\)\(\[InvisibleSpace]\)\) + 1.4766843423503888`\ u + 0.40803119985997593`\ u\^2 - 0.3108809141790293`\ v - 0.2720207999066506`\ u\ \ v)\))\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u)\)\^2, 0, \((\(\(1.4766843423503888`\)\(\[InvisibleSpace]\)\) + 0.8160623997199518`\ u + 0.2720207999066507`\ \ v)\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ u)\) - \ \((1.6321247994399033`\ \((\(\(1.1547005383792515`\)\(\[InvisibleSpace]\)\) + 1.4766843423503888`\ u + 0.4080311998599759`\ u\^2 + 0.3108809141790293`\ v + 0.2720207999066506`\ u\ \ v)\))\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u)\)\^2, 0, \((\(-0.6606219426304372`\) - 0.816062399719952`\ u - 0.2720207999066506`\ \ v)\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ u)\) - \ \((1.6321247994399033`\ \((\(\(0.8660254037844387`\)\(\[InvisibleSpace]\)\) - 0.6606219426304372`\ u - 0.408031199859976`\ u\^2 + 0.233160685634272`\ v - 0.2720207999066507`\ u\ \ v)\))\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u)\)\^2, 0}, {0, \(\(-0.233160685634272`\) + 0.2720207999066507`\ \ u\)\/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ \ u\), 0, \(\(-0.3108809141790293`\) - 0.2720207999066506`\ \ u\)\/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ \ u\), 0, \(\(\(0.3108809141790293`\)\(\[InvisibleSpace]\)\) + \ 0.2720207999066507`\ u\)\/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u\), 0, \(\(\(0.233160685634272`\)\(\[InvisibleSpace]\)\) - \ 0.2720207999066506`\ u\)\/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u\)}, {\(\(-0.233160685634272`\) + 0.2720207999066507`\ \ u\)\/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ \ u\), \((\(-0.6606219426304372`\) - 0.816062399719952`\ u + 0.2720207999066507`\ \ v)\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ u)\) - \ \((1.6321247994399033`\ \((\(\(0.8660254037844387`\)\(\[InvisibleSpace]\)\) - 0.6606219426304372`\ u - 0.408031199859976`\ u\^2 - 0.233160685634272`\ v + 0.2720207999066507`\ u\ \ v)\))\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u)\)\^2, \(\(-0.3108809141790293`\) - \ 0.2720207999066506`\ u\)\/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u\), \((\(\(1.4766843423503888`\)\(\[InvisibleSpace]\)\) \ + 0.8160623997199519`\ u - 0.2720207999066506`\ \ v)\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ u)\) - \ \((1.6321247994399033`\ \((\(\(1.1547005383792515`\)\(\[InvisibleSpace]\)\) + 1.4766843423503888`\ u + 0.40803119985997593`\ u\^2 - 0.3108809141790293`\ v - 0.2720207999066506`\ u\ \ v)\))\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u)\)\^2, \ \(\(\(0.3108809141790293`\)\(\[InvisibleSpace]\)\) + 0.2720207999066507`\ u\)\ \/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ \ u\), \((\(\(1.4766843423503888`\)\(\[InvisibleSpace]\)\) + 0.8160623997199518`\ u + 0.2720207999066507`\ \ v)\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ u)\) - \ \((1.6321247994399033`\ \((\(\(1.1547005383792515`\)\(\[InvisibleSpace]\)\) + 1.4766843423503888`\ u + 0.4080311998599759`\ u\^2 + 0.3108809141790293`\ v + 0.2720207999066506`\ u\ \ v)\))\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u)\)\^2, \(\(\(0.233160685634272`\)\(\[InvisibleSpace]\)\ \) - 0.2720207999066506`\ \ u\)\/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ \ u\), \((\(-0.6606219426304372`\) - 0.816062399719952`\ u - 0.2720207999066506`\ \ v)\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ u)\) - \ \((1.6321247994399033`\ \((\(\(0.8660254037844387`\)\(\[InvisibleSpace]\)\) - 0.6606219426304372`\ u - 0.408031199859976`\ u\^2 + 0.233160685634272`\ v - 0.2720207999066507`\ u\ \ v)\))\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u)\)\^2}}, {{\(\(-0.23316068563427186`\) + \ 0.18134719993776696`\ v\)\/\(\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + \ 1.088083199626602`\ v\), 0, \(\(\(0.2331606856342719`\)\(\[InvisibleSpace]\)\) - \ 0.181347199937767`\ v\)\/\(\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + \ 1.088083199626602`\ v\), 0, \(\(\(0.3108809141790292`\)\(\[InvisibleSpace]\)\) + \ 0.18134719993776702`\ v\)\/\(\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + \ 1.088083199626602`\ v\), 0, \(\(-0.3108809141790292`\) - 0.18134719993776702`\ v\)\ \/\(\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + 1.088083199626602`\ v\), 0}, {0, \((0.18134719993776696`\ u - 0.12089813329184464`\ \((\(-1.285714285714286`\) \ + v)\) - 0.12089813329184464`\ \((\(\(5.571428571428572`\)\(\[InvisibleSpace]\ \)\) + v)\))\)/\((\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + 1.088083199626602`\ v)\) - \((1.088083199626602`\ \ \((u\ \((\(-0.23316068563427186`\) + 0.18134719993776696`\ v)\) - 0.12089813329184464`\ \ \((\(-1.285714285714286`\) + v)\)\ \((\(\(5.571428571428572`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\)\^2, 0, \((\(-0.181347199937767`\)\ u - 0.24179626658368936`\ \((\(-1.2857142857142858`\) \ + v)\) - 0.24179626658368936`\ \ \((\(\(2.7857142857142856`\)\(\[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\) - \((1.088083199626602`\ \ \((u\ \((\(\(0.2331606856342719`\)\(\[InvisibleSpace]\)\) - 0.181347199937767`\ v)\) - 0.24179626658368936`\ \ \((\(-1.2857142857142858`\) + v)\)\ \((\(\(2.7857142857142856`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\)\^2, 0, \((0.18134719993776702`\ u + 0.24179626658368933`\ \ \((\(\(1.7142857142857153`\)\(\[InvisibleSpace]\)\) + v)\) + 0.24179626658368933`\ \((\(\(2.785714285714284`\)\ \(\[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\) - \((1.088083199626602`\ \ \((u\ \((\(\(0.3108809141790292`\)\(\[InvisibleSpace]\)\) + 0.18134719993776702`\ v)\) + 0.24179626658368933`\ \ \((\(\(1.7142857142857153`\)\(\[InvisibleSpace]\)\) + v)\)\ \((\(\(2.785714285714284`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\)\^2, 0, \((\(-0.181347199937767`\)\ u + 0.12089813329184464`\ \ \((\(\(1.7142857142857146`\)\(\[InvisibleSpace]\)\) + v)\) + 0.12089813329184464`\ \((\(\(5.571428571428573`\)\ \(\[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\) - \((1.088083199626602`\ \ \((u\ \((\(-0.3108809141790292`\) - 0.18134719993776702`\ v)\) + 0.12089813329184464`\ \ \((\(\(1.7142857142857146`\)\(\[InvisibleSpace]\)\) + v)\)\ \((\(\(5.571428571428573`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\)\^2}, \ {\((0.18134719993776696`\ u - 0.12089813329184464`\ \((\(-1.285714285714286`\) \ + v)\) - 0.12089813329184464`\ \((\(\(5.571428571428572`\)\(\[InvisibleSpace]\ \)\) + 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\[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\)\^2, \ \(\(\(0.2331606856342719`\)\(\[InvisibleSpace]\)\) - 0.181347199937767`\ \ v\)\/\(\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + 1.088083199626602`\ v\ \), \((0.18134719993776702`\ u + 0.24179626658368933`\ \ \((\(\(1.7142857142857153`\)\(\[InvisibleSpace]\)\) + v)\) + 0.24179626658368933`\ \((\(\(2.785714285714284`\)\ \(\[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\) - \((1.088083199626602`\ \ \((u\ \((\(\(0.3108809141790292`\)\(\[InvisibleSpace]\)\) + 0.18134719993776702`\ v)\) + 0.24179626658368933`\ \ \((\(\(1.7142857142857153`\)\(\[InvisibleSpace]\)\) + v)\)\ \((\(\(2.785714285714284`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\)\^2, \ \(\(\(0.3108809141790292`\)\(\[InvisibleSpace]\)\) + 0.18134719993776702`\ \ v\)\/\(\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + 1.088083199626602`\ v\ \), \((\(-0.181347199937767`\)\ u + 0.12089813329184464`\ \ \((\(\(1.7142857142857146`\)\(\[InvisibleSpace]\)\) + v)\) + 0.12089813329184464`\ \((\(\(5.571428571428573`\)\ \(\[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\) - \((1.088083199626602`\ \ \((u\ \((\(-0.3108809141790292`\) - 0.18134719993776702`\ v)\) + 0.12089813329184464`\ \ \((\(\(1.7142857142857146`\)\(\[InvisibleSpace]\)\) + v)\)\ \((\(\(5.571428571428573`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\)\^2, \(\(-0.3108809141790292`\ \) - 0.18134719993776702`\ \ v\)\/\(\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + 1.088083199626602`\ v\ \)}}, {{\((\(\(0.5547694090359752`\)\(\[InvisibleSpace]\)\) - 0.4477086458886815`\ u + 0.2238543229443408`\ \ v)\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\) + \((1.0503933615080605`\ \ \((\(\(1.239497513433882`\)\(\[InvisibleSpace]\)\) + 0.5547694090359752`\ u - 0.22385432294434074`\ u\^2 + 0.3179380238919623`\ v + 0.2238543229443408`\ u\ \ v)\))\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)^2, 0, \((\(-0.999732907151649`\) + 0.3443912660682165`\ u - 0.24107388624775158`\ \ v)\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\) + \((1.0503933615080605`\ \ \((0.17219563303410826`\ u\^2 + u\ \((\(-0.999732907151649`\) - 0.24107388624775158`\ v)\) + 0.06887825321364331`\ \ \((\(\(3.8985507246376856`\)\(\[InvisibleSpace]\)\) + v)\)\ \((\(\(4.76811594202898`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.238009043746706`\)\(\ \[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)^2, 0, \((\(-0.8689640496011088`\) + 0.2582934495511625`\ u + 0.12053694312387578`\ \ v)\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\) + \((1.0503933615080605`\ \ \((0.12914672477558126`\ u\^2 + u\ \((\(-0.8689640496011088`\) + 0.12053694312387578`\ v)\) - 0.06887825321364333`\ \ \((\(-3.615942028985507`\) + v)\)\ \((\(\(4.768115942028985`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.238009043746706`\)\(\ \[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)^2, 0, \((\(\(0.2635341862087222`\)\(\[InvisibleSpace]\)\) - 0.1549760697306975`\ u - 0.10331737982046499`\ \ v)\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\) + \((1.0503933615080605`\ \ \((\(\(0.530605598401636`\)\(\[InvisibleSpace]\)\) + 0.2635341862087222`\ u - 0.07748803486534875`\ u\^2 - 0.1467406264116749`\ v - 0.10331737982046499`\ u\ \ v)\))\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)^2, 0}, {0, \((\(\(0.3179380238919623`\)\(\[InvisibleSpace]\)\ \) + 0.2238543229443408`\ u)\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\ \)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\) - \((0.688782532136433`\ \ \((\(\(1.239497513433882`\)\(\[InvisibleSpace]\)\) + 0.5547694090359752`\ u - 0.22385432294434074`\ u\^2 + 0.3179380238919623`\ v + 0.2238543229443408`\ u\ \ v)\))\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)^2, 0, \((\(-0.24107388624775158`\)\ u + 0.06887825321364331`\ \ \((\(\(3.8985507246376856`\)\(\[InvisibleSpace]\)\) + v)\) + 0.06887825321364331`\ \ \((\(\(4.76811594202898`\)\(\[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.238009043746706`\)\(\ \[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\) - \((0.688782532136433`\ \ \((0.17219563303410826`\ u\^2 + u\ \((\(-0.999732907151649`\) - 0.24107388624775158`\ v)\) + 0.06887825321364331`\ \ \((\(\(3.8985507246376856`\)\(\[InvisibleSpace]\)\) + v)\)\ \((\(\(4.76811594202898`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.238009043746706`\)\(\ \[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)^2, 0, \((0.12053694312387578`\ u - 0.06887825321364333`\ \((\(-3.615942028985507`\) \ + v)\) - 0.06887825321364333`\ \((\(\(4.768115942028985`\)\(\[InvisibleSpace]\ \)\) + v)\))\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\) - \((0.688782532136433`\ \ \((0.12914672477558126`\ u\^2 + u\ \((\(-0.8689640496011088`\) + 0.12053694312387578`\ v)\) - 0.06887825321364333`\ \ \((\(-3.615942028985507`\) + v)\)\ \((\(\(4.768115942028985`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.238009043746706`\)\(\ \[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)^2, 0, \((\(-0.1467406264116749`\) - 0.10331737982046499`\ \ u)\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\) - \((0.688782532136433`\ \ \((\(\(0.530605598401636`\)\(\[InvisibleSpace]\)\) + 0.2635341862087222`\ u - 0.07748803486534875`\ u\^2 - 0.1467406264116749`\ v - 0.10331737982046499`\ u\ \ v)\))\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)^2}, \ {\((\(\(0.3179380238919623`\)\(\[InvisibleSpace]\)\) + 0.2238543229443408`\ \ u)\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\) - \((0.688782532136433`\ \ \((\(\(1.239497513433882`\)\(\[InvisibleSpace]\)\) + 0.5547694090359752`\ u - 0.22385432294434074`\ u\^2 + 0.3179380238919623`\ v + 0.2238543229443408`\ u\ \ v)\))\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)^2, \ \((\(\(0.5547694090359752`\)\(\[InvisibleSpace]\)\) - 0.4477086458886815`\ u + 0.2238543229443408`\ \ v)\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\) + \((1.0503933615080605`\ \ \((\(\(1.239497513433882`\)\(\[InvisibleSpace]\)\) + 0.5547694090359752`\ u - 0.22385432294434074`\ u\^2 + 0.3179380238919623`\ v + 0.2238543229443408`\ u\ \ v)\))\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)^2, \ \((\(-0.24107388624775158`\)\ u + 0.06887825321364331`\ \ \((\(\(3.8985507246376856`\)\(\[InvisibleSpace]\)\) + v)\) + 0.06887825321364331`\ \ \((\(\(4.76811594202898`\)\(\[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.238009043746706`\)\(\ \[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\) - \((0.688782532136433`\ \ \((0.17219563303410826`\ u\^2 + u\ \((\(-0.999732907151649`\) - 0.24107388624775158`\ v)\) + 0.06887825321364331`\ \ \((\(\(3.8985507246376856`\)\(\[InvisibleSpace]\)\) + v)\)\ \((\(\(4.76811594202898`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.238009043746706`\)\(\ \[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)^2, \ \((\(-0.999732907151649`\) + 0.3443912660682165`\ u - 0.24107388624775158`\ \ v)\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\) + \((1.0503933615080605`\ \ \((0.17219563303410826`\ u\^2 + u\ \((\(-0.999732907151649`\) - 0.24107388624775158`\ v)\) + 0.06887825321364331`\ \ \((\(\(3.8985507246376856`\)\(\[InvisibleSpace]\)\) + v)\)\ \((\(\(4.76811594202898`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.238009043746706`\)\(\ \[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)^2, \ \((0.12053694312387578`\ u - 0.06887825321364333`\ \((\(-3.615942028985507`\) \ + v)\) - 0.06887825321364333`\ \((\(\(4.768115942028985`\)\(\[InvisibleSpace]\ \)\) + v)\))\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\) - \((0.688782532136433`\ \ \((0.12914672477558126`\ u\^2 + u\ \((\(-0.8689640496011088`\) + 0.12053694312387578`\ v)\) - 0.06887825321364333`\ \ \((\(-3.615942028985507`\) + v)\)\ 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{0.26666666666666666`, 1.0666666666666667`}, {0, 0, 0.4`}}, {{1.0666666666666667`}, {0.26666666666666666`, 1.0666666666666667`}, {0, 0, 0.4`}}, {{1.0666666666666667`}, {0.26666666666666666`, 1.0666666666666667`}, {0, 0, 0.4`}}, {{1.0666666666666667`}, {0.26666666666666666`, 1.0666666666666667`}, {0, 0, 0.4`}}}, LocalInterpolants \[Rule] {{\((\(-0.033158157896269275`\)\ u\^2 + u\ \((\(-0.14736959065008567`\) + 0.17684350878010283`\ v)\) + 0.13263263158507713`\ \((\(-5.833333333333334`\) + v)\)\ \((\(-1.3888888888888886`\) + v)\))\)/\((\(\(4.052663742877359`\)\(\[InvisibleSpace]\ \)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), \((0.03315815789626929`\ u\^2 + u\ \((\(\(0.41447697370336617`\)\(\[InvisibleSpace]\)\) - 0.23210710527388506`\ v)\) + 0.19894894737761573`\ \((\(-4.166666666666667`\) + v)\)\ \((\(-1.388888888888889`\) + v)\))\)/\((\(\(4.052663742877359`\)\(\[InvisibleSpace]\ \)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), \((u\ \((\(\(0.2578967836376502`\ 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v)\)/\((\(\(4.041451884327381`\)\(\ \[InvisibleSpace]\)\) + 1.6321247994399033`\ u)\), \((\(\(1.1547005383792515`\)\(\ \[InvisibleSpace]\)\) + 1.4766843423503888`\ u + 0.4080311998599759`\ u\^2 + 0.3108809141790293`\ v + 0.2720207999066506`\ u\ v)\)/\((\(\(4.041451884327381`\)\(\ \[InvisibleSpace]\)\) + 1.6321247994399033`\ u)\), \((\(\(0.8660254037844387`\)\(\ \[InvisibleSpace]\)\) - 0.6606219426304372`\ u - 0.408031199859976`\ u\^2 + 0.233160685634272`\ v - 0.2720207999066507`\ u\ v)\)/\((\(\(4.041451884327381`\)\(\ \[InvisibleSpace]\)\) + 1.6321247994399033`\ u)\)}, {\((u\ \ \((\(-0.23316068563427186`\) + 0.18134719993776696`\ v)\) - 0.12089813329184464`\ \((\(-1.285714285714286`\) + v)\)\ \((\(\(5.571428571428572`\)\(\[InvisibleSpace]\)\ \) + v)\))\)/\((\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + 1.088083199626602`\ v)\), \((u\ \ \((\(\(0.2331606856342719`\)\(\[InvisibleSpace]\)\) - 0.181347199937767`\ v)\) - 0.24179626658368936`\ \((\(-1.2857142857142858`\) + v)\)\ \((\(\(2.7857142857142856`\)\(\[InvisibleSpace]\ \)\) + v)\))\)/\((\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + 1.088083199626602`\ v)\), \((u\ \ \((\(\(0.3108809141790292`\)\(\[InvisibleSpace]\)\) + 0.18134719993776702`\ v)\) + 0.24179626658368933`\ \((\(\(1.7142857142857153`\)\(\ \[InvisibleSpace]\)\) + v)\)\ \((\(\(2.785714285714284`\)\(\[InvisibleSpace]\)\ \) + v)\))\)/\((\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + 1.088083199626602`\ v)\), \((u\ \((\(-0.3108809141790292`\) \ - 0.18134719993776702`\ v)\) + 0.12089813329184464`\ \((\(\(1.7142857142857146`\)\(\ \[InvisibleSpace]\)\) + v)\)\ \((\(\(5.571428571428573`\)\(\[InvisibleSpace]\)\ \) + v)\))\)/\((\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + 1.088083199626602`\ v)\)}, {\((\(\(1.239497513433882`\)\(\ \[InvisibleSpace]\)\) + 0.5547694090359752`\ u - 0.22385432294434074`\ u\^2 + 0.3179380238919623`\ v + 0.2238543229443408`\ u\ v)\)/\((\(\(4.238009043746706`\)\(\ \[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\), \((0.17219563303410826`\ u\^2 + u\ \((\(-0.999732907151649`\) - 0.24107388624775158`\ v)\) + 0.06887825321364331`\ \((\(\(3.8985507246376856`\)\(\ \[InvisibleSpace]\)\) + v)\)\ \((\(\(4.76811594202898`\)\(\[InvisibleSpace]\)\ \) + v)\))\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\), \((0.12914672477558126`\ u\^2 + u\ \((\(-0.8689640496011088`\) + 0.12053694312387578`\ v)\) - 0.06887825321364333`\ \((\(-3.615942028985507`\) + v)\)\ \((\(\(4.768115942028985`\)\(\[InvisibleSpace]\)\ \) + v)\))\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\), \((\(\(0.530605598401636`\)\(\ \[InvisibleSpace]\)\) + 0.2635341862087222`\ u - 0.07748803486534875`\ u\^2 - 0.1467406264116749`\ v - 0.10331737982046499`\ u\ v)\)/\((\(\(4.238009043746706`\)\(\ \[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v)\)}, {\(\(\(1.0827904974169502`\)\(\ \[InvisibleSpace]\)\) - 0.4154893769158065`\ u - 0.4523048913260678`\ v + \ 0.17355885364837487`\ u\ v\)\/\(\(\(4.010637059632144`\)\(\[InvisibleSpace]\)\ \) - 0.12397060974883906`\ u - 0.2975294633972142`\ v\), \((u\ \ \((\(\(0.44516718955264994`\)\(\[InvisibleSpace]\)\) - 0.18595591462325883`\ v)\) + 0.07438236584930352`\ \((\(-5.87878787878788`\) + v)\)\ \((\(-2.3939393939393936`\) + v)\))\)/\((\(\(4.010637059632144`\)\(\[InvisibleSpace]\ \)\) - 0.12397060974883906`\ u - 0.2975294633972142`\ v)\), \((\(-0.030992652437209806`\)\ u\ \^2 + u\ \((\(\(0.30654550774258427`\)\(\[InvisibleSpace]\)\) + 0.19835297559814272`\ v)\) - 0.07438236584930352`\ \((\(-5.87878787878788`\) + v)\)\ \((\(\(2.0404040404040407`\)\(\[InvisibleSpace]\ \)\) + v)\))\)/\((\(\(4.010637059632144`\)\(\[InvisibleSpace]\)\) - 0.12397060974883906`\ u - 0.2975294633972142`\ v)\), \((\(\(0.988805065453092`\)\(\ \[InvisibleSpace]\)\) - 0.4601939301282667`\ u + 0.0309926524372098`\ u\^2 + 0.4846123835636441`\ v - 0.18595591462325878`\ u\ v)\)/\((\(\(4.010637059632144`\)\(\ \[InvisibleSpace]\)\) - 0.12397060974883906`\ u - 0.2975294633972142`\ v)\)}}, LocalStrainDisplacementMatrices \[Rule] {{{\(-\(\((0.3315815789626931`\ \ \((\(-0.033158157896269275`\)\ u\^2 + u\ \((\(-0.14736959065008567`\) + 0.17684350878010283`\ v)\) + 0.13263263158507713`\ \((\(-5.833333333333334`\) \ + v)\)\ \((\(-1.3888888888888886`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)\^2\ \)\) + \(\(-0.14736959065008567`\) - 0.06631631579253855`\ u + \ 0.17684350878010283`\ v\)\/\(\(\(4.052663742877359`\)\(\[InvisibleSpace]\)\) \ + 0.3315815789626931`\ u - 0.8842175439005134`\ v\), 0, \(-\(\((0.3315815789626931`\ \((0.03315815789626929`\ u\^2 + u\ \((\(\(0.41447697370336617`\)\(\ \[InvisibleSpace]\)\) - 0.23210710527388506`\ v)\) + 0.19894894737761573`\ \((\(-4.166666666666667`\) \ + v)\)\ \((\(-1.388888888888889`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)\^2\ \)\) + \(\(\(0.41447697370336617`\)\(\[InvisibleSpace]\)\) + \ 0.06631631579253858`\ u - 0.23210710527388506`\ v\)\/\(\(\(4.052663742877359`\ \)\(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ \ v\), 0, \(\(\(0.2578967836376502`\)\(\[InvisibleSpace]\)\) + \ 0.22105438597512875`\ v\)\/\(\(\(4.052663742877359`\)\(\[InvisibleSpace]\)\) \ + 0.3315815789626931`\ u - 0.8842175439005134`\ v\) - \((0.3315815789626931`\ \ \((u\ \((\(\(0.2578967836376502`\)\(\[InvisibleSpace]\)\) + 0.22105438597512875`\ v)\) - 0.2210543859751287`\ \((\(-4.166666666666667`\) + v)\)\ \((\(\(1.1666666666666665`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ \ v)\)\^2, 0, \(\(-0.1934225877282376`\) - 0.1657907894813465`\ \ v\)\/\(\(\(4.052663742877359`\)\(\[InvisibleSpace]\)\) + 0.3315815789626931`\ \ u - 0.8842175439005134`\ v\) - \((0.3315815789626931`\ \((u\ \ \((\(-0.1934225877282376`\) - 0.1657907894813465`\ v)\) - 0.11052719298756437`\ \((\(-5.833333333333333`\) + v)\)\ \((\(\(1.1666666666666665`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ \ v)\)\^2, 0}, {0, \((0.8842175439005134`\ \((\(-0.033158157896269275`\)\ u\^2 \ + u\ \((\(-0.14736959065008567`\) + 0.17684350878010283`\ v)\) + 0.13263263158507713`\ \((\(-5.833333333333334`\) + v)\)\ \((\(-1.3888888888888886`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)\^2 \ + \((0.17684350878010283`\ u + 0.13263263158507713`\ \((\(-5.833333333333334`\) + v)\) + 0.13263263158507713`\ \((\(-1.3888888888888886`\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), 0, \((0.8842175439005134`\ \((0.03315815789626929`\ u\^2 + u\ \((\(\(0.41447697370336617`\)\(\[InvisibleSpace]\ \)\) - 0.23210710527388506`\ v)\) + 0.19894894737761573`\ \((\(-4.166666666666667`\) + v)\)\ \((\(-1.388888888888889`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)\^2 \ + \((\(-0.23210710527388506`\)\ u + 0.19894894737761573`\ \((\(-4.166666666666667`\) + v)\) + 0.19894894737761573`\ \((\(-1.388888888888889`\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), 0, \((0.22105438597512875`\ u - 0.2210543859751287`\ \((\(-4.166666666666667`\) + v)\) - 0.2210543859751287`\ \((\(\(1.1666666666666665`\)\(\ \[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\) + \((0.8842175439005134`\ \ \((u\ \((\(\(0.2578967836376502`\)\(\[InvisibleSpace]\)\) + 0.22105438597512875`\ v)\) - 0.2210543859751287`\ \((\(-4.166666666666667`\) + v)\)\ \((\(\(1.1666666666666665`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ \ v)\)\^2, 0, \((\(-0.1657907894813465`\)\ u - 0.11052719298756437`\ \((\(-5.833333333333333`\) + v)\) - 0.11052719298756437`\ \((\(\(1.1666666666666665`\)\(\ \[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\) + \((0.8842175439005134`\ \ \((u\ \((\(-0.1934225877282376`\) - 0.1657907894813465`\ v)\) - 0.11052719298756437`\ \((\(-5.833333333333333`\) + v)\)\ \((\(\(1.1666666666666665`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ \ v)\)\^2}, {\((0.8842175439005134`\ \((\(-0.033158157896269275`\)\ u\^2 + u\ \((\(-0.14736959065008567`\) + 0.17684350878010283`\ v)\) + 0.13263263158507713`\ \((\(-5.833333333333334`\) + v)\)\ \((\(-1.3888888888888886`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)\^2 \ + \((0.17684350878010283`\ u + 0.13263263158507713`\ \((\(-5.833333333333334`\) + v)\) + 0.13263263158507713`\ \((\(-1.3888888888888886`\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), \(-\(\((0.3315815789626931`\ \ \((\(-0.033158157896269275`\)\ u\^2 + u\ \((\(-0.14736959065008567`\) + 0.17684350878010283`\ v)\) + 0.13263263158507713`\ \((\(-5.833333333333334`\) \ + v)\)\ \((\(-1.3888888888888886`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)\^2\ \)\) + \(\(-0.14736959065008567`\) - 0.06631631579253855`\ u + \ 0.17684350878010283`\ v\)\/\(\(\(4.052663742877359`\)\(\[InvisibleSpace]\)\) \ + 0.3315815789626931`\ u - 0.8842175439005134`\ v\), \((0.8842175439005134`\ \ \((0.03315815789626929`\ u\^2 + u\ \((\(\(0.41447697370336617`\)\(\[InvisibleSpace]\ \)\) - 0.23210710527388506`\ v)\) + 0.19894894737761573`\ \((\(-4.166666666666667`\) + v)\)\ \((\(-1.388888888888889`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)\^2 \ + \((\(-0.23210710527388506`\)\ u + 0.19894894737761573`\ \((\(-4.166666666666667`\) + v)\) + 0.19894894737761573`\ \((\(-1.388888888888889`\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\), \(-\(\((0.3315815789626931`\ \ \((0.03315815789626929`\ u\^2 + u\ \((\(\(0.41447697370336617`\)\(\ \[InvisibleSpace]\)\) - 0.23210710527388506`\ v)\) + 0.19894894737761573`\ \((\(-4.166666666666667`\) \ + v)\)\ \((\(-1.388888888888889`\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\)\^2\ \)\) + \(\(\(0.41447697370336617`\)\(\[InvisibleSpace]\)\) + \ 0.06631631579253858`\ u - 0.23210710527388506`\ v\)\/\(\(\(4.052663742877359`\ \)\(\[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ \ v\), \((0.22105438597512875`\ u - 0.2210543859751287`\ \((\(-4.166666666666667`\) + v)\) - 0.2210543859751287`\ \((\(\(1.1666666666666665`\)\(\ \[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ v)\) + \((0.8842175439005134`\ \ \((u\ \((\(\(0.2578967836376502`\)\(\[InvisibleSpace]\)\) + 0.22105438597512875`\ v)\) - 0.2210543859751287`\ \((\(-4.166666666666667`\) + v)\)\ \((\(\(1.1666666666666665`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.052663742877359`\)\(\ \[InvisibleSpace]\)\) + 0.3315815789626931`\ u - 0.8842175439005134`\ \ v)\)\^2, \(\(\(0.2578967836376502`\)\(\[InvisibleSpace]\)\) + \ 0.22105438597512875`\ v\)\/\(\(\(4.052663742877359`\)\(\[InvisibleSpace]\)\) \ + 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{\(\(-0.233160685634272`\) + 0.2720207999066507`\ \ u\)\/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ \ u\), \(\(-0.6606219426304372`\) - 0.816062399719952`\ u + 0.2720207999066507`\ \ v\)\/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u\) - \((1.6321247994399033`\ \((\(\(0.8660254037844387`\ \)\(\[InvisibleSpace]\)\) - 0.6606219426304372`\ u - 0.408031199859976`\ u\^2 - 0.233160685634272`\ v + 0.2720207999066507`\ u\ \ v)\))\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u)\)\^2, \(\(-0.3108809141790293`\) - \ 0.2720207999066506`\ u\)\/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u\), \(\(\(1.4766843423503888`\)\(\[InvisibleSpace]\)\) \ + 0.8160623997199519`\ u - 0.2720207999066506`\ v\)\/\(\(\(4.041451884327381`\ \)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ u\) - \((1.6321247994399033`\ \ \((\(\(1.1547005383792515`\)\(\[InvisibleSpace]\)\) + 1.4766843423503888`\ u + 0.40803119985997593`\ u\^2 - 0.3108809141790293`\ v - 0.2720207999066506`\ u\ \ v)\))\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u)\)\^2, \ \(\(\(0.3108809141790293`\)\(\[InvisibleSpace]\)\) + 0.2720207999066507`\ u\)\ \/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ \ u\), \(\(\(1.4766843423503888`\)\(\[InvisibleSpace]\)\) + 0.8160623997199518`\ \ u + 0.2720207999066507`\ v\)\/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\ \)\) + 1.6321247994399033`\ u\) - \((1.6321247994399033`\ \ \((\(\(1.1547005383792515`\)\(\[InvisibleSpace]\)\) + 1.4766843423503888`\ u + 0.4080311998599759`\ u\^2 + 0.3108809141790293`\ v + 0.2720207999066506`\ u\ \ v)\))\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u)\)\^2, \(\(\(0.233160685634272`\)\(\[InvisibleSpace]\)\ \) - 0.2720207999066506`\ \ u\)\/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + 1.6321247994399033`\ \ u\), \(\(-0.6606219426304372`\) - 0.816062399719952`\ u - 0.2720207999066506`\ \ v\)\/\(\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u\) - \((1.6321247994399033`\ \((\(\(0.8660254037844387`\ \)\(\[InvisibleSpace]\)\) - 0.6606219426304372`\ u - 0.408031199859976`\ u\^2 + 0.233160685634272`\ v - 0.2720207999066507`\ u\ \ v)\))\)/\((\(\(4.041451884327381`\)\(\[InvisibleSpace]\)\) + \ 1.6321247994399033`\ u)\)\^2}}, {{\(\(-0.23316068563427186`\) + \ 0.18134719993776696`\ v\)\/\(\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + \ 1.088083199626602`\ v\), 0, \(\(\(0.2331606856342719`\)\(\[InvisibleSpace]\)\) - \ 0.181347199937767`\ v\)\/\(\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + \ 1.088083199626602`\ v\), 0, \(\(\(0.3108809141790292`\)\(\[InvisibleSpace]\)\) + \ 0.18134719993776702`\ v\)\/\(\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + \ 1.088083199626602`\ v\), 0, \(\(-0.3108809141790292`\) - 0.18134719993776702`\ v\)\/\(\(\ \(4.04145188432738`\)\(\[InvisibleSpace]\)\) + 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0.24179626658368933`\ \((\(\(2.785714285714284`\)\(\ \[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\) - \((1.088083199626602`\ \((u\ \ \((\(\(0.3108809141790292`\)\(\[InvisibleSpace]\)\) + 0.18134719993776702`\ v)\) + 0.24179626658368933`\ \ \((\(\(1.7142857142857153`\)\(\[InvisibleSpace]\)\) + v)\)\ \((\(\(2.785714285714284`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\)\^2, \ \(\(\(0.3108809141790292`\)\(\[InvisibleSpace]\)\) + 0.18134719993776702`\ \ v\)\/\(\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + 1.088083199626602`\ v\ \), \((\(-0.181347199937767`\)\ u + 0.12089813329184464`\ \((\(\(1.7142857142857146`\)\(\ \[InvisibleSpace]\)\) + v)\) + 0.12089813329184464`\ \((\(\(5.571428571428573`\)\(\ \[InvisibleSpace]\)\) + v)\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\) - \((1.088083199626602`\ \((u\ \ \((\(-0.3108809141790292`\) - 0.18134719993776702`\ v)\) + 0.12089813329184464`\ \ \((\(\(1.7142857142857146`\)\(\[InvisibleSpace]\)\) + v)\)\ \((\(\(5.571428571428573`\)\(\ \[InvisibleSpace]\)\) + v)\))\))\)/\((\(\(4.04145188432738`\)\(\ \[InvisibleSpace]\)\) + 1.088083199626602`\ v)\)\^2, \(\(-0.3108809141790292`\ \) - 0.18134719993776702`\ \ v\)\/\(\(\(4.04145188432738`\)\(\[InvisibleSpace]\)\) + 1.088083199626602`\ v\ \)}}, {{\(\(\(0.5547694090359752`\)\(\[InvisibleSpace]\)\) - \ 0.4477086458886815`\ u + 0.2238543229443408`\ v\)\/\(\(\(4.238009043746706`\)\ \(\[InvisibleSpace]\)\) - 1.0503933615080605`\ u + 0.688782532136433`\ v\) + \ \((1.0503933615080605`\ \((\(\(1.239497513433882`\)\(\[InvisibleSpace]\)\) + 0.5547694090359752`\ u - 0.22385432294434074`\ u\^2 + 0.3179380238919623`\ v + 0.2238543229443408`\ u\ \ v)\))\)/\((\(\(4.238009043746706`\)\(\[InvisibleSpace]\)\) - \ 1.0503933615080605`\ u + 0.688782532136433`\ v)\)\^2, 0, \(\(-0.999732907151649`\) + 0.3443912660682165`\ u 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