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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 630681, 20237]*) (*NotebookOutlinePosition[ 631419, 20263]*) (* CellTagsIndexPosition[ 631375, 20259]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ The Application of Winding Numbers to the Interior Problem of Nonconvex \ Polygons and Polyhedra \ \>", "Title"], Cell["by Stefan Welke, Spwelke@aol.com", "Author"], Cell["In der Wehrhecke 29, D-53125 Bonn, Germany ", "Address"], Cell["\<\ Abstract: The winding number for closed curves and polygons in the Euclidean plane is \ generalized to polyhedra in three-space. The winding number allows to decide \ wether a point is in the interior or exterior of a closed polygon/polyhedron. \ Polygons and polyhedra need not be convex and self-intersections are allowed.\ \ \>", "Abstract"], Cell[CellGroupData[{ Cell["Introduction", "SectionFirst"], Cell[TextData[{ "Let ", Cell[BoxData[ \(TraditionalForm\`A\_1, A\_2, A\_3, ..., A\_n\)]], " be the vertices of a closed polygon \[ScriptCapitalP] in a plane. Given a \ point ", Cell[BoxData[ \(TraditionalForm\`\(\ P\ \[Epsilon]\ \[ScriptCapitalP]\)\)]], " then ", StyleBox["the interior problem", FontSlant->"Italic"], " for the polygon \[ScriptCapitalP] is to decide, if ", Cell[BoxData[ \(TraditionalForm\`P\)]], " is in the interior of the closed polygon or not. The corresponding \ problem for polyhedra in three-space is obvious. The purpose of this paper is \ to settle the nonconvex case.\nIf \[ScriptCapitalP] is convex and has \ anticlockwise orientation, then the interior problem is easily settled: A \ point ", Cell[BoxData[ \(TraditionalForm\`P\)]], " is in the interior of the polygon if and only if ", Cell[BoxData[ \(TraditionalForm\`P\)]], " is lying to the left of each straight line containing an edge of the \ polygon. This condition leads to a system of ", Cell[BoxData[ \(TraditionalForm\`n\)]], " linear inequalities, where ", Cell[BoxData[ \(TraditionalForm\`n\)]], " is the number of edges. A similar strategy with oriented planes \ containing the faces of a convex polyhedron applies to the three-dimensional \ case.\nWe now remind the topological concept of the ", StyleBox["winding number", FontSlant->"Italic"], " of a closed continuous curve ", Cell[BoxData[ \(TraditionalForm\`\[Gamma]\)]], " with respect to a point ", Cell[BoxData[ \(TraditionalForm\`P\)]], " in the plane. 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ImageRangeCache->{{{142.313, 429.313}, {256.063, 79.125}} -> {-5.21538, 0.523107, 0.0201719, 0.0510352}}], Cell[TextData[{ "The sum of the oriented angles ", Cell[BoxData[ \(TraditionalForm \`\(\(\[MeasuredAngle]z\_1\) p\[InvisibleComma] z\_2 + \(\[MeasuredAngle]z\_2\) p\[InvisibleComma] z\_3 + \(\[MeasuredAngle]z\_3\) p\[InvisibleComma] z\_4 + \(\[MeasuredAngle]z\_4\) p\[InvisibleComma] z\_5 + \(\[MeasuredAngle]z\_5\) p\[InvisibleComma] z\_6 + \(\[MeasuredAngle]z\_6\) p\[InvisibleComma] z\_1\)\)]], " equals ", Cell[BoxData[ \(TraditionalForm\`2 \[Pi]\)]], " if ", Cell[BoxData[ \(TraditionalForm\`P\)]], " is in the interior of the closed anticlockwise oriented polygon and ", Cell[BoxData[ \(TraditionalForm\`0\)]], " if ", Cell[BoxData[ \(TraditionalForm\`P\)]], " is in the exterior. If the polygon has clockwise orientation and ", Cell[BoxData[ \(TraditionalForm\`P\)]], " is in the interior, the this sum is ", Cell[BoxData[ \(TraditionalForm\`\(-2\) \[Pi]\)]], " and 0 otherwise. This sum equals ", Cell[BoxData[ \(TraditionalForm\`\(\[PlusMinus]2\)\ \[Pi]\)]], " for any interior point even if the polygon is not convex, because \ backward turns of the vector ", Cell[BoxData[ \(TraditionalForm\`\(p\[InvisibleComma] z\_k\)\&\[RightVector]\)]], " are counted with opposite sign. If a closed polygon surrounds a point ", Cell[BoxData[ \(TraditionalForm\`P\)]], " twice, then the sum of the oriented angles is ", Cell[BoxData[ \(TraditionalForm\`\(\[PlusMinus]4\) \[Pi]\)]], ", according to the orientation. The use of complex numbers allows a very \ elegant" }], "Text"], Cell[TextData[{ StyleBox["Definition:", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" ", FontWeight->"Bold"], "Let ", Cell[BoxData[ \(TraditionalForm\`\(\[ScriptCapitalP] := {z\_0, z\_1, ..., z\_n}\ \)\)]], "be a closed polygon in the complex plane with ", Cell[BoxData[ \(TraditionalForm\`z\_0 = z\_n\)]], ", and let ", Cell[BoxData[ \(TraditionalForm\`\(p\ \)\)]], "be a point not lying on any edge of the polygon, then the number" }], "Text"], Cell[BoxData[ StyleBox[ RowBox[{" ", RowBox[{ RowBox[{\(w \((p, \[ScriptCapitalP])\)\), ":=", RowBox[{ FractionBox[ StyleBox["1", "DisplayFormula"], \(2 \[Pi]\)], \(\[Sum]\+\(k = 1\)\%n\)}]}], "\[InvisibleComma]", \(Arg \((\(z\_k - p\)\/\(z\_\(k - 1\) - p\))\)\)}]}], "DisplayFormula"]], "NumberedEquation"], Cell[TextData[{ "is called winding number of ", Cell[BoxData[ \(TraditionalForm\`p\)]], " with respect to the polygon \[ScriptCapitalP]." }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(w(p, \[ScriptCapitalP])\ \)\)]], "is always an integer and counts how many times the closed polygon ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalP]\)]], " surrounds the point ", Cell[BoxData[ \(TraditionalForm\`p\)]], " in the positive or negative sense.\nThe winding number settles the \ two-dimensional interior problem: A closed polygon divides the complex plane \ into a finite set ", Cell[BoxData[ \(TraditionalForm\`\({U\_1, U\_2, ..., U\_m}\ \)\)]], "of bounded simply connected regions and one unbounded region ", Cell[BoxData[ \(TraditionalForm\`U\_\[Infinity]\)]], ". We call the union ", Cell[BoxData[ FormBox[ StyleBox[\(\(\[Union]\&m\)\+\(k = 1\)U\_k\), "DisplayFormula"], TraditionalForm]]], " the interior of the polygon and ", Cell[BoxData[ \(TraditionalForm\`U\_\[Infinity]\)]], " the exterior. Now a point ", Cell[BoxData[ \(TraditionalForm\`p\)]], " is in the interior of a polygon \[ScriptCapitalP] if and only if ", Cell[BoxData[ \(TraditionalForm\`w(p, \[ScriptCapitalP]) \[NotEqual] 0\)]], ". " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Computation of the two-dimensional winding number ", "Section"], Cell[TextData[{ "The following ", StyleBox["Mathematica", FontSlant->"Italic"], " code realizes the winding number. Note that the first vertex must \ coincide with the last one in the list ", StyleBox["poly", FontFamily->"Courier New", FontWeight->"Bold"], ". " }], "Text"], Cell[BoxData[ \(\(\n\t wn[a_, poly_] := \n\t\ \ \ \ \ \ \ 1\/\(2 \[Pi]\)* Sum[Arg[\((\((Part[poly, k + 1] - a)\).{1, \[ImaginaryI]})\)/ \((\((Part[poly, k] - a)\).{1, \[ImaginaryI]})\)], {k, 1, Length[poly] - 1}] // Round\)\)], "Input", CellLabel->"In[1]:="], Cell["\<\ Points in the plane are represented here as pairs of real numbers. An example \ demonstrates the use of the winding number:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(a\_0 = {0, 4}; a\_1 = {7, 4}; a\_2 = {7, \(-3\)}; a\_3 = {\(-2\), \(-3\)}; a\_4 = {0, 2}; a\_5 = {5, 2}; a\_6 = {3, \(-2\)}; a\_7 = {\(-4\), 3}; a\_8 = {\(-5\), 0}; a\_9 = {\(-3\), \(-3\)}; a\_10 = {0, 4}\ ; \np\_1 = {\(-3\), \(-1\)}; \ p\_2 = {5, 3}\ ; \ p\_3 = {2, 1}; \ p\_4 = {\(-2\), 4}\ ; \n wn[p\_1, {a\_0, a\_1, a\_2, a\_3, a\_4, a\_5, a\_6, a\_7, a\_8, a\_9, a\_10}]\)], "Input", CellLabel->"In[2]:="], Cell[BoxData[ \(1\)], "Output", CellLabel->"Out[2]="] }, Open ]], Cell["\<\ A change of the order of vertices produces the expected result:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(wn[p\_1, Reverse[{a\_0, a\_1, a\_2, a\_3, a\_4, a\_5, a\_6, a\_7, a\_8, a\_9, a\_10}]]\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ \(\(-1\)\)], "Output", CellLabel->"Out[3]="] }, Open ]], Cell[TextData[{ "The next figure shows a polygon and the positions of points ", Cell[BoxData[ \(TraditionalForm\`p\_1, p\_2, p\_3\)]], " in the interior and ", Cell[BoxData[ \(TraditionalForm\`p\_4\)]], " in the exterior of a polygon. 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0000;Oooo`030000oooooooo0;3oool00`000?ooooooo`06oooo001?oooo00D0003oooooooooool0 00000oooo`80000^oooo00<0003oooooool0[oooo`030000oooooooo00Koool004ooool01@000?oo ooooooooo`000004oooo00<0003oooooool0;?oook<00008oooo001?oooo0P0000;oool00`000?oo ooooo`3/oooo001>oooo0P000003oooo000000000>ooool00?oooom3oooo003ooooo@oooo`00\ \>"], ImageRangeCache->{{{142.313, 399.625}, {605.188, 446.688}} -> {-12.313, 17.5389, 0.0491309, 0.0463724}}], Cell[CellGroupData[{ Cell[BoxData[ \(\(wn[#, {a\_0, a\_1, a\_2, a\_3, a\_4, a\_5, a\_6, a\_7, a\_8, a\_9, a\_10}]&\)\ /@\ {p\_1, p\_2, p\_3, p\_4}\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \({1, \(-1\), \(-2\), 0}\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell[TextData[{ "There is still a problem left: The winding number is not defined for \ points ", Cell[BoxData[ \(TraditionalForm\`p\)]], " on the boundary of a polygon. Thus we must provide a test for this \ situation. The following function tests, wether a point ", Cell[BoxData[ \(TraditionalForm\`p\)]], " lies on the edge ", Cell[BoxData[ \(TraditionalForm\`{v, w}\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ptInEdge[p_, {v_, w_}] := \n\t\ \ \ \ \ \ \ \ \ \ Module[{t, u}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u = t /. Flatten@Solve[p == t*v + \((1 - t)\)*w, t]; \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ If[And[u \[GreaterEqual] 0, u \[LessEqual] 1] == True, True, False, False]]\ ; \n \(ptInEdge[#, {a\_2, a\_3}]&\)\ /@ \ {a\_1, a\_2, \((2 a\_2 + 3 a\_3)\)\/5}\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ \({False, True, True}\)], "Output", CellLabel->"Out[5]="] }, Open ]], Cell[TextData[{ "The final function ", StyleBox["WindingNumber2D", FontFamily->"Courier New", FontWeight->"Bold"], StyleBox[" ", FontFamily->"Courier New"], "at first performs this test for each edge. If this test gives ", StyleBox["True", FontFamily->"Courier New", FontWeight->"Bold"], " for one edge, nothing will be computed but the unevaluated expression is \ returned. Otherwise the function wn[..] will be evaluated. Note that ", StyleBox["WindingNumber2D", FontFamily->"Courier New", FontWeight->"Bold"], " now does not require the coincidence of the first and last point in the \ polygon list!" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(WindingNumber2D::notdefined = "\"; \nWindingNumber2D[p_, poly_] := \n\t Module[{v = Append[poly, First[poly]], w}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ w = Table[ptInEdge[p, Take[v, {k - 1, k}]], {k, 2, Length[v]}]; \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ If[MemberQ[w, True], \((Message[WindingNumber2D::notdefined, p, poly]; \ Short@HoldForm[WindingNumber2D[p, poly]])\), wn[p, v]]\n \t\t\ \ \ \ \ \ \ \ \ \ \ \ ]; \n WindingNumber2D[ p\_3, {a\_0, a\_1, a\_2, a\_3, a\_4, a\_5, a\_6, a\_7, a\_8, a\_9}]\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \(\(-2\)\)], "Output", CellLabel->"Out[6]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(WindingNumber2D[ \(a\_2 + a\_3\)\/2, {a\_0, a\_1, a\_2, a\_3, a\_4, a\_5, a\_6, a\_7, a\_8, a\_9}]\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \(WindingNumber2D::"notdefined" \( : \ \) "point \!\({5\/2, \(-3\)}\) is on boundary of \!\(\\[LeftSkeleton] 1 \ \\[RightSkeleton]\) ."\)], "Message"], Cell[BoxData[ TagBox[ \(WindingNumber2D[{5\/2, \(-3\)}, {{0, 4}, {7, 4}, {7, \(-3\)}, \[LeftSkeleton]5\[RightSkeleton], {\(-5\), 0}, {\(-3\), \(-3\)}}]\), Short]], "Output", CellLabel->"Out[7]//Short="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The interior problem for polyhedra", "Section"], Cell[TextData[{ "Let us first examine the two-dimensional winding number from a different \ point of view: ", Cell[BoxData[ \(TraditionalForm\`Arg(z/w)\)]], " is the length of the radial projection of the line ", Cell[BoxData[ \(TraditionalForm\`{z, w}\)]], " onto the unit circle. Then the winding number of a closed polygon counts, \ how many times the unit circle is completely covered by the totality of all \ projections of the polygon's edges.\nThis geometric interpretation of the \ winding number allows for a generalization to higher dimensions, especially \ to three dimensions. The next figure suggests what we shall do! 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.01 w .28414 .68262 Mdot .38578 .49535 Mdot .61977 .72745 Mdot .5 Mabswid .27458 .64702 m .28469 .62172 L s .01 w .28107 .64476 Mdot .2839 .62172 Mdot .3931 .49074 Mdot .5 Mabswid .33595 .52393 m .35109 .50107 L s .01 w .31539 .76964 Mdot .57094 .80074 Mdot .5 Mabswid .26568 .67238 m .27458 .64702 L s .01 w .57091 .80145 Mdot .52565 .48509 Mdot .58369 .52144 Mdot .2731 .64401 Mdot .5 Mabswid .5657 .83127 m .56473 .80785 L s .01 w .45798 .47017 Mdot .32219 .52861 Mdot .5646 .50275 Mdot .34856 .50416 Mdot .37795 .48663 Mdot .5 Mabswid .49516 .47217 m .48653 .4493 L s .35109 .50107 m .367 .47913 L s .01 w .26948 .66311 Mdot .5 Mabswid .63824 .56064 m .61448 .55124 L s .34984 .50105 m .32412 .50107 L s .01 w .55286 .49226 Mdot .26722 .66253 Mdot .5 Mabswid .25807 .69763 m .26568 .67238 L s .01 w .3477 .49568 Mdot .26466 .69638 Mdot .28532 .7526 Mdot .5 Mabswid .367 .47913 m .38359 .45823 L s .01 w .27108 .72739 Mdot .37363 .83761 Mdot .63727 .56666 Mdot .60166 .51588 Mdot .65275 .70924 Mdot .32057 .50765 Mdot .29964 .78168 Mdot .53912 .47047 Mdot .55269 .47611 Mdot .63292 .76018 Mdot .5 Mabswid .66089 .57068 m .63824 .56064 L s .25177 .72263 m .25807 .69763 L s .32412 .50107 m .29917 .50208 L s .01 w .5795 .49127 Mdot .42448 .45312 Mdot .47213 .45069 Mdot .5 Mabswid .56594 .85358 m .5657 .83127 L s .01 w .38834 .46155 Mdot .62628 .53465 Mdot .42488 .4505 Mdot .5 Mabswid .48653 .4493 m .47765 .4276 L s .01 w .664 .68897 Mdot .49868 .4508 Mdot .40355 .45361 Mdot .24853 .63006 Mdot .5 Mabswid .38359 .45823 m .40075 .43851 L s .01 w .29697 .7928 Mdot .64086 .54805 Mdot .30981 .50288 Mdot .59976 .49624 Mdot .67115 .64362 Mdot .42255 .44165 Mdot .5 Mabswid .24685 .74723 m .25177 .72263 L s .01 w .34672 .46895 Mdot .46417 .43838 Mdot .37315 .45517 Mdot .64592 .54155 Mdot .5 Mabswid .68229 .58128 m .66089 .57068 L s .29917 .50208 m .27514 .50407 L s .01 w .24353 .71086 Mdot .32744 .83439 Mdot .5107 .43987 Mdot .47112 .43228 Mdot .43015 .43341 Mdot .5 Mabswid .40075 .43851 m .41839 .42011 L s .01 w .55027 .45022 Mdot .68221 .63239 Mdot .63161 .79911 Mdot .5 Mabswid .56544 .87463 m .56594 .85358 L s .01 w .65629 .76357 Mdot .43501 .42945 Mdot .23599 .60622 Mdot .50124 .43102 Mdot .5 Mabswid .47765 .4276 m .46856 .40722 L s .01 w .42988 .42759 Mdot .22865 .65776 Mdot .68407 .69471 Mdot .28042 .80106 Mdot .26225 .53438 Mdot .23251 .60844 Mdot .5 Mabswid .24335 .77127 m .24685 .74723 L s .01 w .39531 .43218 Mdot .43003 .42494 Mdot .61204 .48142 Mdot .23933 .73388 Mdot .58943 .46193 Mdot .33334 .45643 Mdot .61494 .48065 Mdot .36376 .4398 Mdot .35616 .44324 Mdot .56133 .86673 Mdot .5 Mabswid .70229 .5924 m .68229 .58128 L s .01 w .52246 .42739 Mdot .67224 .75468 Mdot .5 Mabswid .27514 .50407 m .2522 .50705 L s .01 w .31197 .46774 Mdot .64597 .79946 Mdot .5 Mabswid .41839 .42011 m .43641 .40313 L s .01 w .68712 .59106 Mdot .41283 .42064 Mdot .69427 .62379 Mdot .33276 .45037 Mdot .24756 .53874 Mdot .42799 .41612 Mdot .52633 .42284 Mdot .39796 .42041 Mdot .44242 .41331 Mdot .56998 .86892 Mdot .41677 .41599 Mdot .53681 .42486 Mdot .50394 .41523 Mdot .48341 .41211 Mdot .5 Mabswid .24129 .7946 m .24335 .77127 L s .01 w .54974 .88137 Mdot .5 Mabswid .56419 .8943 m .56544 .87463 L s .01 w .35619 .87788 Mdot .22111 .72176 Mdot .40299 .41432 Mdot .55776 .42874 Mdot .24683 .78267 Mdot .43968 .40774 Mdot .5 Mabswid .46856 .40722 m .45931 .3883 L s .01 w .2557 .51094 Mdot .21243 .61423 Mdot .28812 .46975 Mdot .65346 .49662 Mdot .5054 .40719 Mdot .65174 .49345 Mdot .5 Mabswid .43641 .40313 m .45469 .38771 L s .72076 .60398 m .70229 .5924 L s .01 w .41943 .40475 Mdot .5 Mabswid .2522 .50705 m .2305 .51101 L s .01 w .51904 .40887 Mdot .51472 .40778 Mdot .53458 .89589 Mdot .58005 .87739 Mdot .32125 .86948 Mdot .55523 .41795 Mdot .29213 .45952 Mdot .32224 .87121 Mdot .45478 .3973 Mdot .21443 .57201 Mdot .30698 .4455 Mdot .39053 .40544 Mdot .5 Mabswid .24072 .81706 m .24129 .7946 L s .01 w .45265 .39565 Mdot .62379 .85192 Mdot .4598 .39509 Mdot .20175 .69327 Mdot .71672 .6668 Mdot .2924 .85372 Mdot .69544 .75953 Mdot .20491 .71593 Mdot .53968 .40595 Mdot .6276 .852 Mdot .72056 .65415 Mdot .56782 .41496 Mdot .40609 .39584 Mdot .5 Mabswid .56219 .91242 m .56419 .8943 L s .01 w .4783 .39021 Mdot .19928 .59585 Mdot .45381 .38851 Mdot .62206 .44542 Mdot .62708 .44939 Mdot .5 Mabswid .45931 .3883 m .44996 .37099 L s .01 w .51388 .39418 Mdot .70772 .74292 Mdot .5 Mabswid .45469 .38771 m .47313 .37396 L s .01 w .20593 .56767 Mdot .44745 .38754 Mdot .56014 .40773 Mdot .20019 .72129 Mdot .29214 .44475 Mdot .49548 .3895 Mdot .42885 .38827 Mdot .29954 .86835 Mdot .53779 .39836 Mdot .5 Mabswid .73756 .61593 m .72076 .60398 L s .01 w .43082 .38717 Mdot .43419 .38684 Mdot .51382 .3917 Mdot .45256 .38564 Mdot .59904 .42568 Mdot .44911 .38546 Mdot .5 Mabswid .2305 .51101 m .21019 .51592 L s .01 w .4682 .38537 Mdot .47339 .38562 Mdot .45877 .38476 Mdot .26629 .46477 Mdot .65681 .83412 Mdot .44115 .38495 Mdot .40925 .38827 Mdot .54507 .39827 Mdot .19241 .60565 Mdot .29413 .43891 Mdot .6077 .42906 Mdot .44816 .38367 Mdot .30053 .4337 Mdot .44559 .38323 Mdot .45647 .38278 Mdot .55153 .39937 Mdot .44461 .38289 Mdot .34336 .4077 Mdot .46134 .38215 Mdot .19641 .58049 Mdot .45642 .38184 Mdot .46022 .38166 Mdot .52818 .91582 Mdot .47796 .38207 Mdot .5 Mabswid .24163 .83851 m .24072 .81706 L s .01 w .43038 .38258 Mdot .48501 .92491 Mdot .59237 .89136 Mdot .33797 .40693 Mdot .22123 .51332 Mdot .46052 .37727 Mdot .29384 .43084 Mdot .42253 .37885 Mdot .55595 .91146 Mdot .45372 .37645 Mdot .4508 .37624 Mdot .51296 .38099 Mdot .54585 .91753 Mdot .5 Mabswid .47313 .37396 m .49162 .36197 L s .55944 .92887 m .56219 .91242 L s .01 w .60044 .41195 Mdot .25236 .4621 Mdot .44124 .37252 Mdot .49291 .37392 Mdot .3489 .39361 Mdot .20559 .52818 Mdot .22529 .49411 Mdot .52908 .38022 Mdot .5 Mabswid .44996 .37099 m .44057 .3554 L s .01 w .48983 .37269 Mdot .5 Mabswid .75257 .6282 m .73756 .61593 L s .01 w .17558 .66791 Mdot .5 Mabswid .21019 .51592 m .19141 .52178 L s .01 w .39475 .3762 Mdot .27522 .86953 Mdot .29198 .42253 Mdot .21463 .80021 Mdot .43946 .36813 Mdot .5 Mabswid .24406 .8588 m .24163 .83851 L s .01 w .23921 .83651 Mdot .18245 .73116 Mdot .36502 .38116 Mdot .70289 .50357 Mdot .43122 .36662 Mdot .21244 .50317 Mdot .46402 .36493 Mdot .2214 .48857 Mdot .41424 .36773 Mdot .47472 .36486 Mdot .18225 .73816 Mdot .54419 .37674 Mdot .5795 .91446 Mdot .28196 .42253 Mdot .61743 .89366 Mdot .51969 .36934 Mdot .54769 .37686 Mdot .74639 .6221 Mdot .74797 .63719 Mdot .40259 .36525 Mdot .17131 .70936 Mdot .5 Mabswid .49162 .36197 m .51002 .35186 L s .01 w .59145 .39237 Mdot .55142 .37482 Mdot .24193 .45389 Mdot .70764 .49861 Mdot .47413 .35919 Mdot .23232 .46435 Mdot .63001 .41341 Mdot .50213 .36101 Mdot .27585 .88332 Mdot .18455 .76321 Mdot .5 Mabswid .55595 .94351 m .55944 .92887 L s .01 w .39284 .36367 Mdot .5 Mabswid .76566 .6407 m .75257 .6282 L s .01 w .73761 .55549 Mdot .49941 .35845 Mdot .5 Mabswid .19141 .52178 m .1743 .52857 L s .01 w .62315 .89815 Mdot .5 Mabswid .44057 .3554 m .4312 .34168 L s .01 w .46664 .35451 Mdot .65683 .43077 Mdot .5 Mabswid .248 .87778 m .24406 .8588 L s .01 w .21702 .47635 Mdot .59047 .91831 Mdot .33958 .92515 Mdot .17064 .57205 Mdot .41877 .94703 Mdot .43332 .35447 Mdot .64452 .41916 Mdot .39437 .36009 Mdot .17827 .54593 Mdot .7563 .63575 Mdot .16049 .6856 Mdot .67447 .44506 Mdot .39889 .35785 Mdot .5376 .36284 Mdot .47752 .95107 Mdot .15744 .64302 Mdot .33616 .9262 Mdot .60677 .91256 Mdot .58771 .92292 Mdot .70553 .48056 Mdot .6201 .90538 Mdot .25004 .43286 Mdot .67436 .86179 Mdot .39222 .35609 Mdot .73835 .76459 Mdot .15503 .64591 Mdot .48907 .34985 Mdot .42564 .95286 Mdot .53774 .35824 Mdot .5 Mabswid .51002 .35186 m .52823 .34371 L s .01 w .6718 .43521 Mdot .75092 .57026 Mdot .75054 .56841 Mdot .43339 .34782 Mdot .36287 .36191 Mdot .18733 .79138 Mdot .29305 .39498 Mdot .57203 .36834 Mdot .19429 .80546 Mdot .76279 .62852 Mdot .68698 .44805 Mdot .69856 .84162 Mdot .76299 .68626 Mdot .59949 .37794 Mdot .6452 .40666 Mdot .36479 .35725 Mdot .38155 .35247 Mdot .5 Mabswid .77671 .65336 m .76566 .6407 L s .01 w .20034 .48129 Mdot .75258 .56016 Mdot .5 Mabswid .1743 .52857 m .15902 .53624 L s .55173 .95621 m .55595 .94351 L s .25345 .8953 m .248 .87778 L s .01 w .58986 .37095 Mdot .28418 .90735 Mdot .58101 .36678 Mdot .76662 .62961 Mdot .17078 .53391 Mdot .14811 .66831 Mdot .5 Mabswid .4312 .34168 m .42191 .32995 L s .01 w .73452 .51039 Mdot .57256 .36235 Mdot .52527 .95413 Mdot .2889 .38941 Mdot .7628 .58712 Mdot .17141 .77431 Mdot .21905 .85213 Mdot .34853 .35795 Mdot .23471 .86954 Mdot .14579 .62888 Mdot .41322 .34067 Mdot .75448 .55025 Mdot .68923 .43941 Mdot .74263 .78138 Mdot .44968 .33645 Mdot .62808 .38653 Mdot .5 Mabswid .52823 .34371 m .54612 .33761 L s .01 w .65309 .89725 Mdot .56195 .94756 Mdot .1444 .68356 Mdot .71504 .83193 Mdot .19031 .8172 Mdot .55812 .35145 Mdot .17998 .50029 Mdot .72115 .82425 Mdot .16409 .53364 Mdot .52069 .34071 Mdot .14731 .71014 Mdot .14479 .60519 Mdot .67771 .87742 Mdot .62369 .38069 Mdot .40649 .33726 Mdot .17595 .79646 Mdot .45018 .33272 Mdot .2029 .83988 Mdot .14323 .60076 Mdot .69568 .43794 Mdot .76472 .73599 Mdot .5 Mabswid .78563 .66611 m .77671 .65336 L s .26039 .91124 m .25345 .8953 L s .15902 .53624 m .14567 .54476 L s .01 w .27148 .38997 Mdot .41603 .96715 Mdot .65948 .40135 Mdot .5801 .94581 Mdot .28626 .37929 Mdot .42822 .96892 Mdot .5 Mabswid .5468 .96687 m .55173 .95621 L s .01 w .68161 .42051 Mdot .77696 .67821 Mdot .71152 .84598 Mdot .75884 .76068 Mdot .60265 .93623 Mdot .23439 .88065 Mdot .77881 .65125 Mdot .64859 .39141 Mdot .5 Mabswid .42191 .32995 m .41275 .3203 L s .01 w .27614 .91556 Mdot .50048 .33072 Mdot .58293 .35326 Mdot .45434 .32742 Mdot .33161 .35237 Mdot .58134 .35135 Mdot .47482 .32711 Mdot .16643 .50744 Mdot .5 Mabswid .54612 .33761 m .56356 .33362 L s .01 w .37557 .96335 Mdot .48939 .97258 Mdot .27367 .91729 Mdot .58883 .35197 Mdot .52536 .33123 Mdot .73512 .81909 Mdot .34261 .34464 Mdot .69791 .86969 Mdot .51144 .3275 Mdot .34021 .34468 Mdot .34034 .34382 Mdot .68027 .40952 Mdot .2115 .86481 Mdot .52551 .32888 Mdot .14419 .74535 Mdot .5 Mabswid .2688 .92545 m .26039 .91124 L s .01 w .46729 .32117 Mdot .61443 .93727 Mdot .1386 .57146 Mdot .52101 .32695 Mdot .56533 .33894 Mdot .33653 .34383 Mdot .27223 .37801 Mdot .5 Mabswid .79232 .67884 m .78563 .66611 L s .01 w .51641 .97228 Mdot .5 Mabswid .14567 .54476 m .1344 .55409 L s .01 w .1724 .48496 Mdot .18659 .46188 Mdot .37297 .96751 Mdot .71816 .85084 Mdot .5 Mabswid .54116 .97537 m .5468 .96687 L s .01 w .46628 .97921 Mdot .23518 .40371 Mdot .42456 .31933 Mdot .15108 .77441 Mdot .5 Mabswid .41275 .3203 m .40381 .31286 L s .01 w .12797 .61299 Mdot .49424 .31919 Mdot .42482 .3187 Mdot .7728 .75146 Mdot .63555 .92856 Mdot .50247 .97746 Mdot .66868 .90476 Mdot .43026 .31774 Mdot .5 Mabswid .56356 .33362 m .58043 .33181 L s .01 w .17097 .8169 Mdot .27201 .37272 Mdot .57456 .33724 Mdot .12554 .67719 Mdot .66292 .9109 Mdot .60915 .3509 Mdot .21322 .87597 Mdot .36352 .32731 Mdot .42085 .31575 Mdot .14146 .53853 Mdot .23906 .39476 Mdot .69681 .41424 Mdot .13869 .75051 Mdot .75152 .80884 Mdot .61952 .3548 Mdot .5188 .31859 Mdot .13589 .7428 Mdot .12523 .60263 Mdot .5 Mabswid .27865 .93782 m .2688 .92545 L s .01 w .72341 .4435 Mdot .14745 .51861 Mdot .55355 .32603 Mdot .6422 .36729 Mdot .5 Mabswid .79668 .69148 m .79232 .67884 L s .01 w .2276 .40254 Mdot .32193 .95592 Mdot .5 Mabswid .1344 .55409 m .12531 .56418 L s .01 w .12398 .60324 Mdot .30724 .94913 Mdot .70819 .42212 Mdot .74088 .46482 Mdot .44025 .31057 Mdot .29461 .94369 Mdot .15992 .80786 Mdot .5 Mabswid .53486 .98161 m .54116 .97537 L s .01 w .75321 .48318 Mdot .7919 .59323 Mdot .35128 .32583 Mdot .5 Mabswid .58043 .33181 m .5966 .33223 L s .01 w .76489 .50472 Mdot .5 Mabswid .40381 .31286 m .39514 .3077 L s .01 w .50256 .31137 Mdot .33546 .96404 Mdot .76691 .50816 Mdot .13071 .7392 Mdot .5452 .31862 Mdot .12006 .60348 Mdot .30157 .34457 Mdot .14088 .52058 Mdot .55491 .32031 Mdot .5 Mabswid .28989 .94822 m .27865 .93782 L s .01 w .54647 .97665 Mdot .36412 .97552 Mdot .72783 .43853 Mdot .27395 .35915 Mdot .75067 .82247 Mdot .44271 .9886 Mdot .23974 .91065 Mdot .77816 .52829 Mdot .36206 .31852 Mdot .5 Mabswid .79867 .70393 m .79668 .69148 L s .01 w .44668 .30499 Mdot .38731 .98195 Mdot .5 Mabswid .12531 .56418 m .11849 .57497 L s .01 w .35013 .97214 Mdot .61009 .34006 Mdot .11547 .62526 Mdot .69772 .40234 Mdot .23642 .38481 Mdot .79652 .69749 Mdot .11901 .70199 Mdot .48714 .98863 Mdot .28406 .35045 Mdot .77861 .76835 Mdot .11451 .66906 Mdot .5 Mabswid .5966 .33223 m .61195 .3349 L s .52792 .9855 m .53486 .98161 L s .01 w .63626 .35263 Mdot .79758 .69863 Mdot .26255 .93018 Mdot .31441 .33342 Mdot .59492 .33087 Mdot .25264 .36975 Mdot .18686 .43359 Mdot .53268 .98245 Mdot .5 Mabswid .39514 .3077 m .38681 .30492 L s .01 w .23184 .38616 Mdot .73408 .85179 Mdot .48476 .98994 Mdot .46927 .30171 Mdot .14703 .79677 Mdot .27413 .35389 Mdot .80225 .65954 Mdot .70847 .88302 Mdot .37361 .98114 Mdot .46392 .30083 Mdot .73012 .85818 Mdot .5 Mabswid .30246 .95655 m .28989 .94822 L s .01 w .56283 .31715 Mdot .1119 .64989 Mdot .59841 .96173 Mdot .48347 .99083 Mdot .77642 .51249 Mdot .5307 .30801 Mdot .54495 .31138 Mdot .46693 .99178 Mdot .69656 .89602 Mdot .1621 .4656 Mdot .42233 .99024 Mdot .5 Mabswid .7982 .71611 m .79867 .70393 L s .01 w .59866 .32982 Mdot .5 Mabswid .11849 .57497 m .11405 .58639 L s .01 w .27946 .34879 Mdot .13102 .52861 Mdot .80274 .67788 Mdot .27784 .94238 Mdot .72286 .86896 Mdot .56689 .31638 Mdot .5 Mabswid .61195 .3349 m .62635 .33985 L s .01 w .80291 .61346 Mdot .60621 .33202 Mdot .41302 .98974 Mdot .49602 .99044 Mdot .28337 .94586 Mdot .15337 .81307 Mdot .57398 .31834 Mdot .5 Mabswid .52039 .98698 m .52792 .9855 L s .01 w .11069 .6226 Mdot .57161 .3171 Mdot .34693 .31581 Mdot .63424 .94449 Mdot .5 Mabswid .38681 .30492 m .37889 .30456 L s .01 w .47841 .29811 Mdot .71156 .88251 Mdot .77229 .49761 Mdot .65117 .35632 Mdot .80513 .63804 Mdot .50936 .98905 Mdot .5 Mabswid .31629 .96272 m .30246 .95655 L s .01 w .36927 .98164 Mdot .80276 .68737 Mdot .31102 .32927 Mdot .14552 .49049 Mdot .69612 .89821 Mdot .3321 .32027 Mdot .76752 .80294 Mdot .30174 .3336 Mdot .22415 .38655 Mdot .10972 .66304 Mdot .63103 .94662 Mdot .43941 .29721 Mdot .11715 .71889 Mdot .5 Mabswid .79526 .72791 m .7982 .71611 L s .11405 .58639 m .11205 .59838 L s .01 w .23307 .91118 Mdot .13654 .50906 Mdot .13504 .77674 Mdot .16417 .45657 Mdot .37078 .30724 Mdot .40425 .30024 Mdot .6055 .32922 Mdot .7946 .73205 Mdot .62149 .33721 Mdot .5 Mabswid .62635 .33985 m .63968 .34707 L s .01 w .30056 .33334 Mdot .22998 .38033 Mdot .60417 .96043 Mdot .2497 .92421 Mdot .515 .30065 Mdot .4529 .99278 Mdot .76221 .47542 Mdot .67651 .37329 Mdot .5 Mabswid .51229 .98599 m .52039 .98698 L s .01 w .80419 .6104 Mdot .68271 .37835 Mdot .17927 .4341 Mdot .5 Mabswid .37889 .30456 m .37144 .30669 L s .33131 .96665 m .31629 .96272 L s .01 w .387 .98537 Mdot .20874 .88824 Mdot .10888 .64632 Mdot .60004 .32614 Mdot .50203 .9896 Mdot .80483 .61823 Mdot .79877 .57445 Mdot .62015 .95205 Mdot .35462 .97687 Mdot .12369 .54356 Mdot .40123 .98775 Mdot .79536 .55962 Mdot .12208 .54915 Mdot .80137 .58918 Mdot .3556 .97696 Mdot .10984 .6197 Mdot .72607 .86524 Mdot .5 Mabswid .78981 .73925 m .79526 .72791 L s .01 w .32076 .96407 Mdot .5 Mabswid .11205 .59838 m .11255 .61085 L s .01 w .43514 .99112 Mdot .23417 .37635 Mdot .3789 .98285 Mdot .23073 .37927 Mdot .59235 .96451 Mdot .79863 .71228 Mdot .31895 .96298 Mdot .48435 .29652 Mdot .11312 .69514 Mdot .5 Mabswid .63968 .34707 m .65182 .35656 L s .01 w .30588 .95665 Mdot .63751 .34582 Mdot .61451 .95363 Mdot .25578 .92676 Mdot .27802 .34533 Mdot .40073 .30019 Mdot .12816 .75521 Mdot .55425 .97673 Mdot .63019 .94443 Mdot .5 Mabswid .5037 .98248 m .51229 .98599 L s .34741 .96825 m .33131 .96665 L s .01 w .74669 .83498 Mdot .73329 .43242 Mdot .54282 .97943 Mdot .22816 .90369 Mdot .11023 .64055 Mdot .35449 .9744 Mdot .78327 .52389 Mdot .5 Mabswid .37144 .30669 m .36453 .31134 L s .01 w .43931 .29638 Mdot .17994 .85135 Mdot .72412 .86457 Mdot .24025 .91349 Mdot .27703 .93924 Mdot .11049 .64797 Mdot .78638 .75231 Mdot .24939 .92028 Mdot .3889 .98263 Mdot .32191 .96185 Mdot .61169 .95294 Mdot .66946 .91669 Mdot .47049 .98898 Mdot .51536 .30111 Mdot .62183 .33777 Mdot .20223 .87719 Mdot .80283 .6673 Mdot .79761 .7071 Mdot .5 Mabswid .78186 .75003 m .78981 .73925 L s .11255 .61085 m .11559 .62372 L s .01 w .47664 .29698 Mdot .80071 .68541 Mdot .32129 .96073 Mdot .74649 .45264 Mdot .37417 .30675 Mdot .54647 .9766 Mdot .5 Mabswid .65182 .35656 m .66267 .36828 L s .01 w .45986 .98825 Mdot .62918 .34253 Mdot .57651 .31802 Mdot .34434 .969 Mdot .17626 .84343 Mdot .27806 .93776 Mdot .61587 .94917 Mdot .78778 .74237 Mdot .68185 .90456 Mdot .1992 .41312 Mdot .61489 .94904 Mdot .80156 .66313 Mdot .5644 .96988 Mdot .77327 .50425 Mdot .48301 .98593 Mdot .14887 .79586 Mdot .74507 .45295 Mdot .3814 .30616 Mdot .5 Mabswid .36451 .9675 m .34741 .96825 L s .01 w .20232 .41082 Mdot .80105 .63678 Mdot .42072 .9838 Mdot .5 Mabswid .49465 .97644 m .5037 .98248 L s .01 w .12591 .73319 Mdot .44315 .98525 Mdot .22221 .89212 Mdot .5 Mabswid .36453 .31134 m .35821 .31851 L s .01 w .15185 .79817 Mdot .15407 .8024 Mdot .76827 .49758 Mdot .12025 .70431 Mdot .12241 .71466 Mdot .48201 .29986 Mdot .58333 .32305 Mdot .54633 .97228 Mdot .5 Mabswid .77141 .76016 m .78186 .75003 L s .01 w .43906 .98326 Mdot .68143 .89988 Mdot .5 Mabswid .11559 .62372 m .1212 .63689 L s .01 w .19466 .4223 Mdot .5 Mabswid .66267 .36828 m .67212 .38217 L s .01 w .344 .31897 Mdot .57356 .96273 Mdot .12225 .70901 Mdot .57307 .9628 Mdot .38475 .97526 Mdot .75435 .47343 Mdot .52722 .30743 Mdot .7048 .40682 Mdot .55561 .96762 Mdot .13422 .74998 Mdot .1521 .48949 Mdot .57025 .96252 Mdot .17376 .82989 Mdot .38786 .97455 Mdot .37471 .97152 Mdot .7821 .74255 Mdot .1252 .71513 Mdot .48801 .97957 Mdot .34217 .96078 Mdot .19688 .42301 Mdot .54056 .97015 Mdot .17012 .45974 Mdot .37696 .31112 Mdot .30614 .94501 Mdot .33592 .958 Mdot .5 Mabswid .38247 .96435 m .36451 .9675 L s .01 w .41121 .97717 Mdot .3798 .97105 Mdot .17173 .45806 Mdot .76842 .77525 Mdot .40053 .97468 Mdot .18055 .44574 Mdot .67224 .37948 Mdot .55246 .96533 Mdot .38508 .97097 Mdot .5 Mabswid .48522 .96787 m .49465 .97644 L s .01 w .59151 .9504 Mdot .16888 .46555 Mdot .1258 .57813 Mdot .5 Mabswid .35821 .31851 m .35256 .3282 L s .01 w .27368 .3584 Mdot .2396 .38378 Mdot .12332 .68202 Mdot .5 Mabswid .75852 .76955 m .77141 .76016 L s .01 w .25063 .90576 Mdot .5 Mabswid .67212 .38217 m .68007 .39817 L s .1212 .63689 m .12936 .65028 L s .01 w .79301 .63045 Mdot .26531 .91603 Mdot .13198 .72425 Mdot .68385 .88723 Mdot .1245 .67941 Mdot .15875 .79229 Mdot .75599 .48915 Mdot .46496 .97473 Mdot .17092 .81314 Mdot .39426 .96863 Mdot .55131 .96077 Mdot .51073 .96967 Mdot .79004 .61658 Mdot .74946 .80075 Mdot .41037 .97013 Mdot .6188 .93146 Mdot .5 Mabswid .40118 .95879 m .38247 .96435 L s .01 w .51017 .96837 Mdot .35088 .32501 Mdot .66377 .90023 Mdot .593 .94261 Mdot .60321 .93754 Mdot .15436 .77499 Mdot .6743 .38937 Mdot .78796 .62072 Mdot .44099 .97044 Mdot .43992 .30914 Mdot .78729 .65895 Mdot .73442 .81991 Mdot .37242 .3201 Mdot .3519 .32629 Mdot .4595 .97001 Mdot .35739 .3248 Mdot .7674 .52725 Mdot .27073 .91178 Mdot .60141 .93616 Mdot .18157 .45851 Mdot .14005 .72959 Mdot .5 Mabswid .47546 .95679 m .48522 .96787 L s .01 w .17191 .8042 Mdot .68628 .40328 Mdot .31735 .93741 Mdot .13097 .68273 Mdot .5 Mabswid .68007 .39817 m .68643 .41617 L s .35256 .3282 m .34763 .34037 L s .74323 .77812 m .75852 .76955 L s .01 w .4512 .31129 Mdot .4225 .96561 Mdot .34935 .94916 Mdot .5 Mabswid .12936 .65028 m .14007 .66377 L s .01 w .15909 .50285 Mdot .23222 .40101 Mdot .41314 .31468 Mdot .47234 .96655 Mdot .52082 .9604 Mdot .41025 .31543 Mdot .32394 .93764 Mdot .54292 .95437 Mdot .69769 .85969 Mdot .33597 .94221 Mdot .14427 .54922 Mdot .18643 .8198 Mdot .50413 .96035 Mdot .5 Mabswid .4205 .95083 m .40118 .95879 L s .01 w .60375 .34986 Mdot .45319 .96302 Mdot .5468 .94985 Mdot .47463 .9614 Mdot .29308 .91705 Mdot .49227 .95978 Mdot .42208 .95974 Mdot .62915 .91188 Mdot .68168 .87038 Mdot .69823 .85261 Mdot .52231 .95412 Mdot .22366 .41743 Mdot .41125 .95675 Mdot .26186 .38346 Mdot .15765 .52237 Mdot .60157 .92541 Mdot .75903 .52959 Mdot .40777 .95535 Mdot .77076 .57296 Mdot .73249 .47476 Mdot .14154 .68986 Mdot .5 Mabswid .68643 .41617 m .69114 .43606 L s .01 w .51238 .95304 Mdot .76189 .73355 Mdot .53446 .94821 Mdot .5 Mabswid .46546 .94323 m .47546 .95679 L s .01 w .34828 .33873 Mdot .4589 .9573 Mdot .6784 .86768 Mdot .77542 .62626 Mdot .69202 .85347 Mdot .5 Mabswid .72564 .78577 m .74323 .77812 L s .01 w .38065 .94734 Mdot .47812 .95572 Mdot .5 Mabswid .14007 .66377 m .15328 .67727 L s .01 w .51809 .95036 Mdot .5 Mabswid .34763 .34037 m .34346 .35498 L s .01 w .27313 .89633 Mdot .5544 .93984 Mdot .46292 .955 Mdot .14168 .66698 Mdot .45444 .95475 Mdot .71225 .45134 Mdot .69242 .84929 Mdot .4874 .95253 Mdot .6091 .91434 Mdot .76703 .6936 Mdot .5 Mabswid .44027 .94049 m .4205 .95083 L s .01 w .54102 .33494 Mdot .77029 .66367 Mdot .40606 .94774 Mdot .67925 .41676 Mdot .24616 .40637 Mdot .71592 .46244 Mdot .36013 .93481 Mdot .54161 .93852 Mdot .4205 .94802 Mdot .20919 .44808 Mdot .15121 .69943 Mdot .33511 .92491 Mdot .23122 .85215 Mdot .30531 .90897 Mdot .55183 .93288 Mdot .51939 .94092 Mdot .41063 .94367 Mdot .74267 .75595 Mdot .75028 .73542 Mdot .4826 .94559 Mdot .5 Mabswid .69114 .43606 m .69413 .45772 L s .01 w .48238 .94529 Mdot .35205 .92728 Mdot .3947 .33594 Mdot .76443 .64942 Mdot .47092 .33047 Mdot .4752 .94398 Mdot .5475 .9308 Mdot .5 Mabswid .70585 .79244 m .72564 .78577 L s .01 w .3522 .92574 Mdot .42328 .94227 Mdot .46544 .94389 Mdot .5754 .92022 Mdot .5 Mabswid .15328 .67727 m .1689 .69065 L s .01 w .45861 .33112 Mdot .58921 .91265 Mdot .46277 .94263 Mdot .45663 .94214 Mdot .39769 .9362 Mdot .59032 .91127 Mdot .29111 .38185 Mdot .52904 .93303 Mdot .5 Mabswid .34346 .35498 m .34011 .37194 L s .01 w .36672 .9277 Mdot .74055 .52709 Mdot .15314 .64054 Mdot .4312 .93999 Mdot .43633 .93877 Mdot .48191 .9384 Mdot .42669 .93748 Mdot .45252 .93846 Mdot .41773 .93544 Mdot .47238 .93761 Mdot .53969 .9264 Mdot .40716 .93376 Mdot .46464 .93756 Mdot .35743 .91961 Mdot .4421 .93515 Mdot .41928 .93301 Mdot .72793 .76154 Mdot .39454 .92811 Mdot .4494 .93429 Mdot .38371 .92481 Mdot .5 Mabswid .69413 .45772 m .69536 .48098 L s .01 w .4539 .933 Mdot .69554 .81213 Mdot .69278 .81576 Mdot .44813 .93231 Mdot .45144 .93237 Mdot .62199 .88183 Mdot .74063 .71776 Mdot .45367 .93146 Mdot .4566 .93141 Mdot .71662 .77613 Mdot .26538 .86042 Mdot .69794 .46728 Mdot .31448 .37907 Mdot .52593 .92248 Mdot .39228 .92325 Mdot .73621 .54481 Mdot .59028 .8984 Mdot .5 Mabswid .68399 .79806 m .70585 .79244 L s .45699 .9301 m .4748 .91742 L .47102 .91693 L closepath p .668 .759 .919 r F P 0 g s .45706 .93006 m .46546 .94323 L s .45699 .9301 m .47102 .91693 L .46722 .91647 L closepath p .666 .756 .917 r F P 0 g s .45699 .9301 m .46722 .91647 L .4634 .91603 L closepath p .664 .752 .916 r F P 0 g s .44027 .94049 m .45709 .92989 L s .45699 .9301 m .4634 .91603 L .45957 .91561 L closepath p .661 .749 .915 r F P 0 g s .45699 .9301 m .45957 .91561 L .45571 .91521 L closepath p .658 .746 .915 r F P 0 g s .4588 .92881 m .45884 .92879 L s .45893 .92873 m .45907 .92864 L s .4593 .9285 m .46036 .92784 L s .45907 .92864 m .4593 .9285 L s .45884 .92879 m .45893 .92873 L s 2 Mabswid .45702 .91535 m .45739 .64534 L s .5 Mabswid .45699 .9301 m .45571 .91521 L .45185 .91483 L closepath p .655 .744 .915 r F P 0 g s .45699 .9301 m .45185 .91483 L .44797 .91447 L closepath p .651 .741 .916 r F P 0 g s .45529 .92728 m .45533 .92735 L s .45533 .92735 m .45564 .92783 L s .45594 .9283 m .45601 .92842 L s .45606 .92848 m .45608 .92852 L s .45601 .92842 m .45606 .92848 L s .45582 .92812 m .45594 .9283 L s .45564 .92783 m .45582 .92812 L s .01 w .4571 .92999 Mdot .46525 .92951 Mdot .5 Mabswid .1689 .69065 m .18686 .70382 L s .01 w .61171 .88536 Mdot .46731 .9293 Mdot .31433 .38097 Mdot .45311 .92884 Mdot .45128 .92872 Mdot .61151 .88454 Mdot .6543 .85192 Mdot .6298 .40101 Mdot .74698 .60297 Mdot .5 Mabswid .46036 .92784 m .46299 .92589 L s .46299 .92589 m .46409 .92509 L s .46409 .92509 m .46507 .92436 L s .46507 .92436 m .46566 .92392 L s .01 w .2562 .42559 Mdot .5 Mabswid .4529 .92303 m .45529 .92728 L s .4526 .92249 m .4529 .92303 L s .01 w .46704 .92701 Mdot .47263 .92662 Mdot .33029 .89768 Mdot .53613 .91496 Mdot .4051 .92128 Mdot .4575 .92589 Mdot .74541 .65838 Mdot .36439 .91022 Mdot .58462 .89549 Mdot .5 Mabswid .34011 .37194 m .33762 .39115 L s .01 w .43573 .92434 Mdot .36722 .36173 Mdot .46631 .92439 Mdot .5137 .91866 Mdot .31324 .38723 Mdot .43871 .34949 Mdot .20481 .77294 Mdot .49469 .3522 Mdot .47251 .35047 Mdot .20253 .50567 Mdot .67307 .82311 Mdot .45302 .92148 Mdot .47652 .35163 Mdot .46077 .92044 Mdot .5 Mabswid .69536 .48098 m .69478 .50568 L s .01 w .43019 .91862 Mdot .55884 .90084 Mdot .45006 .9193 Mdot .4377 .91871 Mdot .5 Mabswid .47102 .91693 m .4748 .91742 L .49262 .90304 L p .686 .745 .896 r F P 0 g s .01 w .32825 .88664 Mdot .67849 .80997 Mdot .5 Mabswid .49262 .90304 m .48508 .90224 L .47102 .91693 L p .686 .745 .896 r F P 0 g s .46722 .91647 m .47102 .91693 L .48508 .90224 L p .683 .743 .896 r F P 0 g s .01 w .3769 .90536 Mdot .5 Mabswid .48508 .90224 m .47748 .90142 L .46722 .91647 L p .683 .743 .896 r F P 0 g s .47794 .91488 m .47924 .91392 L s .47924 .91392 m .48058 .91293 L s .4634 .91603 m .46722 .91647 L .47748 .90142 L p .679 .74 .897 r F P 0 g s .01 w .37618 .90425 Mdot .5 Mabswid .47748 .90142 m .46983 .90059 L .4634 .91603 L p .679 .74 .897 r F P 0 g s .45957 .91561 m .4634 .91603 L .46983 .90059 L p .675 .737 .897 r F P 0 g s .46983 .90059 m .46215 .89974 L .45957 .91561 L p .675 .737 .897 r F P 0 g s .45571 .91521 m .45957 .91561 L .46215 .89974 L p .672 .734 .897 r F P 0 g s .01 w .55393 .89824 Mdot .51066 .36124 Mdot .5 Mabswid .46215 .89974 m .45443 .89888 L .45571 .91521 L p .672 .734 .897 r F P 0 g s .45185 .91483 m .45571 .91521 L .45443 .89888 L p .669 .731 .896 r F P 0 g s .01 w .71423 .52636 Mdot .5 Mabswid .45443 .89888 m .44668 .898 L .45185 .91483 L p .669 .731 .896 r F P 0 g s .44797 .91447 m .45185 .91483 L .44668 .898 L p .666 .727 .895 r F P 0 g s .66022 .80256 m .68399 .79806 L s .44668 .898 m .43891 .89711 L .44797 .91447 L p .666 .727 .895 r F P 0 g s .44633 .91134 m .44637 .91141 L s .44503 .90903 m .44633 .91134 L s .18686 .70382 m .20704 .71665 L s .48058 .91293 m .48215 .9116 L s .48215 .9116 m .48459 .90955 L s .48459 .90955 m .48554 .90874 L s .01 w .18317 .68525 Mdot .21078 .50764 Mdot .73409 .63472 Mdot .43457 .91054 Mdot .36269 .89478 Mdot .43279 .90929 Mdot .46238 .91028 Mdot .5 Mabswid .44251 .90401 m .44503 .90903 L s .01 w .22542 .48902 Mdot .47564 .90845 Mdot .38912 .90045 Mdot .67047 .80599 Mdot .71783 .71656 Mdot .43414 .90749 Mdot .69637 .76611 Mdot .5314 .89765 Mdot .39028 .8992 Mdot .6798 .79119 Mdot .5 Mabswid .33762 .39115 m .33601 .41248 L s .01 w .5593 .88601 Mdot .5 Mabswid .69478 .50568 m .69239 .53161 L s .01 w .20951 .74513 Mdot .72748 .62606 Mdot .20539 .53635 Mdot .5 Mabswid .48508 .90224 m .49262 .90304 L .51034 .88704 L p .706 .741 .879 r F P 0 g s .01 w .55042 .88845 Mdot .38554 .89536 Mdot .52584 .8953 Mdot .37418 .89101 Mdot .50048 .90034 Mdot .5 Mabswid .51034 .88704 m .49908 .88608 L .48508 .90224 L p .706 .741 .879 r F P 0 g s .47748 .90142 m .48508 .90224 L .49908 .88608 L p .699 .736 .879 r F P 0 g s .49908 .88608 m .48771 .88501 L .47748 .90142 L p .699 .736 .879 r F P 0 g s .49725 .89886 m .50078 .89588 L s .49713 .89896 m .49725 .89886 L s .46983 .90059 m .47748 .90142 L .48771 .88501 L p .693 .731 .879 r F P 0 g s .01 w .21227 .52911 Mdot .67688 .78548 Mdot .5 Mabswid .48771 .88501 m .47626 .88384 L .46983 .90059 L p .693 .731 .879 r F P 0 g s .46215 .89974 m .46983 .90059 L .47626 .88384 L p .687 .725 .879 r F P 0 g s .01 w .51392 .89386 Mdot .5 Mabswid .47626 .88384 m .46473 .88256 L .46215 .89974 L p .687 .725 .879 r F P 0 g s .45443 .89888 m .46215 .89974 L .46473 .88256 L p .681 .719 .879 r F P 0 g s .01 w .46553 .89959 Mdot .5 Mabswid .50078 .89588 m .50492 .89193 L s .46473 .88256 m .45315 .88118 L .45443 .89888 L p .681 .719 .879 r F P 0 g s .44668 .898 m .45443 .89888 L .45315 .88118 L p .675 .714 .878 r F P 0 g s .63472 .80589 m .66022 .80256 L s .01 w .7199 .6206 Mdot .5 Mabswid .45315 .88118 m .44154 .8797 L .44668 .898 L p .675 .714 .878 r F P 0 g s .43891 .89711 m .44668 .898 L .44154 .8797 L p .669 .708 .877 r F P 0 g s .01 w .355 .87765 Mdot .5 Mabswid .20704 .71665 m .22928 .72904 L s .44154 .8797 m .4299 .87812 L .43891 .89711 L p .669 .708 .877 r F P 0 g s .4358 .89065 m .43662 .89228 L s .43476 .88858 m .4358 .89065 L s .01 w .54511 .38803 Mdot .70587 .71647 Mdot .20203 .69346 Mdot .67542 .77639 Mdot .44513 .89456 Mdot .71034 .68896 Mdot .42427 .89127 Mdot .4713 .89279 Mdot .5 Mabswid .49908 .88608 m .51034 .88704 L .52785 .86952 L p .722 .736 .862 r F P 0 g s .69239 .53161 m .68819 .55858 L s .01 w .54995 .87366 Mdot .29208 .83052 Mdot .21603 .54912 Mdot .5 Mabswid .52785 .86952 m .51295 .86855 L .49908 .88608 L p .722 .736 .862 r F P 0 g s .51628 .8811 m .52079 .8768 L s .48771 .88501 m .49908 .88608 L .51295 .86855 L p .713 .729 .863 r F P 0 g s .43266 .88393 m .43476 .88858 L s .01 w .35999 .87003 Mdot .49112 .88687 Mdot .31143 .84202 Mdot .5 Mabswid .33601 .41248 m .33532 .43579 L s .01 w .49753 .38502 Mdot .5 Mabswid .51295 .86855 m .49786 .86732 L .48771 .88501 L p .713 .729 .863 r F P 0 g s .47626 .88384 m .48771 .88501 L .49786 .86732 L p .705 .721 .863 r F P 0 g s .01 w .49657 .88344 Mdot .55351 .86665 Mdot .61168 .83265 Mdot .5 Mabswid .52079 .8768 m .52393 .87345 L s .49786 .86732 m .48263 .86585 L .47626 .88384 L p .705 .721 .863 r F P 0 g s .46473 .88256 m .47626 .88384 L .48263 .86585 L p .696 .714 .862 r F P 0 g s .01 w .59686 .42684 Mdot .21358 .57672 Mdot .58336 .84849 Mdot .67121 .50813 Mdot .5 Mabswid .48263 .86585 m .4673 .86414 L .46473 .88256 L p .696 .714 .862 r F P 0 g s .45315 .88118 m .46473 .88256 L .4673 .86414 L p .688 .706 .861 r F P 0 g s .01 w .40781 .87756 Mdot .38562 .8719 Mdot .50119 .87827 Mdot .64518 .4729 Mdot .48035 .88027 Mdot .5 Mabswid .60768 .80802 m .63472 .80589 L s .4673 .86414 m .45188 .86219 L .45315 .88118 L p .688 .706 .861 r F P 0 g s .44154 .8797 m .45315 .88118 L .45188 .86219 L p .679 .698 .86 r F P 0 g s .01 w .36904 .86498 Mdot .5 Mabswid .22928 .72904 m .25342 .74087 L s .45188 .86219 m .43643 .86001 L .44154 .8797 L p .679 .698 .86 r F P 0 g s .4299 .87812 m .44154 .8797 L .43643 .86001 L p .671 .69 .859 r F P 0 g s .51295 .86855 m .52785 .86952 L .54503 .85059 L p .736 .731 .846 r F P 0 g s .01 w .62205 .81615 Mdot .5 Mabswid .43643 .86001 m .42098 .8576 L .4299 .87812 L p .671 .69 .859 r F P 0 g s .42458 .86609 m .42688 .87116 L s .01 w .64066 .79414 Mdot .21486 .65937 Mdot .5 Mabswid .68819 .55858 m .6822 .58636 L s .01 w .41977 .39405 Mdot .63156 .80241 Mdot .35593 .41396 Mdot .5 Mabswid .54503 .85059 m .5266 .84973 L .51295 .86855 L p .736 .731 .846 r F P 0 g s .53536 .86124 m .54043 .85584 L s .49786 .86732 m .51295 .86855 L .5266 .84973 L p .725 .722 .847 r F P 0 g s .01 w .62298 .80884 Mdot .3644 .85594 Mdot .69671 .61641 Mdot .55297 .85375 Mdot .65021 .4937 Mdot .67825 .72284 Mdot .6947 .64606 Mdot .68512 .69949 Mdot .59241 .82874 Mdot .5 Mabswid .54043 .85584 m .54251 .85337 L s .5266 .84973 m .50788 .84843 L .49786 .86732 L p .725 .722 .847 r F P 0 g s .48263 .86585 m .49786 .86732 L .50788 .84843 L p .715 .712 .847 r F P 0 g s .01 w .30788 .81755 Mdot .67934 .71328 Mdot .5 Mabswid .42298 .8622 m .42458 .86609 L s .5266 .84973 m .54503 .85059 L .56176 .83034 L p .749 .726 .831 r F P 0 g s 2 Mabswid .57865 .80991 m .59483 .83186 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