(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. 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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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[ 2949912, 79433]*) (*NotebookOutlinePosition[ 2958853, 79706]*) (* CellTagsIndexPosition[ 2957218, 79651]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Chapter", FontFamily->"Times New Roman", FontSize->18, FontSlant->"Italic", FontVariations->{"Underline"->True}], StyleBox[" 4: Visualization, Abstract Data Types; Electric Fields", FontFamily->"Times New Roman", FontSize->18, FontVariations->{"Underline"->True}] }], "Title", PageWidth->PaperWidth, TextAlignment->Center, TextJustification->0, FontFamily->"Times", FontSize->18, FontVariations->{"Underline"->True}, CellTags->"c:1"], Cell["4.1 Why Visualization?", "Section", PageWidth->PaperWidth, CellTags->"c:2"], Cell[TextData[{ "One of the most rewarding uses of computers is ", StyleBox["visualizing", FontSlant->"Italic"], " the results of calculations. This is done with 2-D and 3-D plots \ (especially with colored surfaces), with contour maps, and with animations. \ These types of visualization can be breathtakingly beautiful and often \ provide deep insight into a problem by letting you see and ``handle'' the \ functions with which you are working. Visualization also assists the program \ debugging process, the development of physical and mathematical intuition, \ and the all-around enjoyment of your work. Some of the reasons for this may \ arise from the fact that some large fraction (~50%) of our brain gets \ involved in visual processing, and if you can use this extra brain power in \ your scientific work, then you have extended what was otherwise possible with \ solely logical abilities.\nTraditionally, visualization of a scientific \ problem was the last step in problem solving. After studying table of \ numbers for hours and gaining confidence that they are right, a scientist \ might then go to the trouble of making a bunch of 2-D plots to examine \ various aspects of data. Well, in present times computational scientists \ have demonstrated how much there is to be gained by going beyond 2-D plots. \ Now it is regular practice to use surface plots, volume rendering (dicing and \ slicing), and animations (movies). In this chapter, we use some of these \ techniques within the context of visualizing the electric potential around \ charges. 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ool008l0oooo00<000@00?ooo`3oool0KP3oool00`001000oooo0?ooo`0o0?ooo`030004003oool0 oooo00d0oooo00<000@00?ooo`3oool0>`3oool008l0oooo00<000@00?ooo`3oool0KP3oool00`00 1000oooo0?ooo`100?ooo`030004003oool0oooo00/0oooo00<000@00?ooo`3oool0?03oool008l0 oooo00<000@00?ooo`3oool0KP3oool00`001000oooo0?ooo`110?ooo`030004003oool0oooo00T0 oooo00<000@00?ooo`3oool0?@3oool008l0oooo00<000@00?ooo`3oool0KP3oool00`001000oooo 0?ooo`120?ooo`030004003oool0oooo00L0oooo00<000@00?ooo`3oool0?P3oool008l0oooo00<0 00@00?ooo`3oool0KP3oool00`001000oooo0?ooo`130?ooo`030004003oool0oooo00D0oooo00<0 00@00?ooo`3oool0?`3oool008l0oooo00<000@00?ooo`3oool0KP3oool00`001000oooo0?ooo`14 0?ooo`030004003oool0oooo00@0oooo00<000@00?ooo`3oool0?`3oool008l0oooo00<000@00?oo o`3oool0KP3oool00`001000oooo0?ooo`140?ooo`030004003oool0oooo00<0oooo00<000@00?oo o`3oool0@03oool008l0oooo00<000@00?ooo`3oool0KP3oool00`001000oooo0?ooo`150?ooo`05 0004003oool0oooo0?ooo`001000@`3oool008l0oooo00<000@00?ooo`3oool0KP3oool00`001000 oooo0?ooo`160?ooo`030004003oool000@004@0oooo002?0?ooog8000@0B@3oool00`001000oooo 0?ooo`130?ooo`00o`3ooonA0?ooo`00o`3ooonA0?ooo`00o`3ooonA0?ooo`00\ \>"], ImageRangeCache->{{{99.25, 419}, {569.563, 512}} -> {-1.05278, 5.39701, \ 0.00985155, 0.0105693}, {{106.813, 202}, {568.188, 513.313}} -> {-3.4065, \ 18.4909, 0.022062, 0.0382685}, {{211.5, 306.688}, {568.188, 513.313}} -> \ {-4.28708, 13.8681, 0.0165464, 0.0287014}, {{316.188, 411.375}, {568.188, \ 513.313}} -> {-6.95048, 14.1181, 0.0191062, 0.0287014}}], Cell["\<\ Figure 4.1 Static configuration for two, three, and four charges \ (the charges are at the corners).\ \>", "Text", PageWidth->PaperWidth, TextAlignment->Center, TextJustification->0], Cell[TextData[{ "You are given some simple configurations of two, three, and four charge \ systems, as shown in the Fig. 4.1. The two charges are fixed on a line at \ coordinates (1,0), (-1,0); the three charges are fixed to the corners of an \ equilateral triangle at coordinates (0,1), (", Cell[BoxData[ \(TraditionalForm\`\@3\)]], "/2, -1/2), (-", Cell[BoxData[ \(TraditionalForm\`\@3\)]], "/2, -1/2); and the four charges are fixed to the corners of a square at \ coordinates (1,1), (1,-1), (-1,-1), (-1,1). The origin is at the center of \ each geometric figure. Your ", StyleBox["problem", FontWeight->"Bold"], " is to determine the electrical potential at the arbitrary point (", StyleBox["x,y", FontSlant->"Italic"], ") and see if there might be some points in space at which we can place a \ charge that is free to move and have it remain there even if perturbed. (For \ the equivalent gravitational problem these stable points are known as ", StyleBox["Lagrange", FontSlant->"Italic"], " points and are the location of asteroids for the earth-sun system.)" }], "Text", PageWidth->PaperWidth, TextAlignment->Left, TextJustification->0], Cell["4.3 Theory: Stability Criteria for Potential Energy", "Section", PageWidth->PaperWidth, CellTags->"c:4"], Cell[TextData[{ "Coulomb's law tells us that if we have a charge ", StyleBox["q", FontSlant->"Italic"], " at the origin, then the electric field ", StyleBox["E", FontWeight->"Bold"], " (the force per unit charge) at a distance ", StyleBox["r", FontSlant->"Italic"], " from that charge is" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["E", FontWeight->"Bold"], "(", "r", ")"}], " ", "=", " ", RowBox[{\(\(k\_e\ q\)\/r\^2\), StyleBox[ OverscriptBox[ StyleBox["r", FontWeight->"Bold"], "^"], FontWeight->"Bold"]}]}], TraditionalForm]], "DisplayFormula", PageWidth->PaperWidth, TextAlignment->Center, TextJustification->0], Cell[TextData[{ "where ", Cell[BoxData[ StyleBox[ OverscriptBox[ StyleBox["r", FontWeight->"Bold"], "^"], FontWeight->"Bold"]], PageWidth->WindowWidth, TextAlignment->Center, TextJustification->0], " is a unit vector in the radial ", StyleBox["(r) ", FontWeight->"Bold"], "direction. ", "Here ", Cell[BoxData[ \(TraditionalForm\`k\_e\)]], " = 8.9875 ", Cell[BoxData[ \(TraditionalForm\`10\^9\)]], "N ", Cell[BoxData[ \(TraditionalForm\`m\^2\)]], "/", Cell[BoxData[ \(TraditionalForm\`C\^2\)]], "is Coulomb's constant in SI units, and the electric force ", StyleBox["E", FontWeight->"Bold"], " is directed radially away from the charge. Because ", StyleBox["E", FontWeight->"Bold"], " is a vector, the electric force field about a charge is a vector field \ with both magnitude and direction at each point. However, no information is \ lost, and it's much simpler, if, instead of the electric field ", StyleBox["E", FontWeight->"Bold"], ",", " we consider the electric ", StyleBox["potential", FontSlant->"Italic"], " field" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(TraditionalForm\`V(r) = \ \(\(k\_e\ q\)\/r = \ q\/r\)\)], "DisplayFormula", PageWidth->PaperWidth, TextAlignment->Center, TextJustification->0], Cell[TextData[{ "In the second form of this equation we have left off the electric constant \ ", Cell[BoxData[ \(k\_e\)], PageWidth->WindowWidth, TextAlignment->Center, TextJustification->0], " for simplicity; since this affects just the magnitudes of the graphs and \ not their shapes, it will not change the conclusions we draw. We see that ", StyleBox["V(r) ", FontSlant->"Italic"], "falls off less rapidly than ", StyleBox["E", FontWeight->"Bold"], "(r) and is a scalar, that is, has no direction associated with it.\nOur \ problem requires us to determine the potentials for two- and three-charge \ systems, and then to look for stable points in these potentials. To determine \ the potential for two charges, we use Pythagorean's theorem to determine the \ distance to the charges," }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(TraditionalForm\`r\_1 = \ \@\(\((x - a)\)\^2\ + \ y\^2\), \ \ r\_2\ = \ \ \@\(\((x + a)\)\^2\ + \ y\^2\)\)], "DisplayFormula", PageWidth->PaperWidth, TextAlignment->Center, TextJustification->0], Cell["\<\ and then just add up the potentials from the individual \ charges:\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(TraditionalForm\`\(V\_2\)(x, y) = \(\(q1\)\(\ \)\)\/\@\(\((x - a)\)\^2 + y\^2\) + \ q2\/\@\(\((x + a)\)\^2 + y\^2\)\)], "DisplayFormula", PageWidth->PaperWidth, TextAlignment->Center, TextJustification->0], Cell["\<\ For three charges at the corners of the equilateral triangle, we \ know the coordinates are (0,a), (a cos \[Theta], -a sin \[Theta]), (-a cos \ \[Theta], -a sin \[Theta]), where \[Theta] = 30 degrees. Again we use \ Pythagorean's theorem and add the potentials from the individual charges to \ obtain\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(TraditionalForm\`\(V\_3\)(x, y)\ = q\_1\/\@\(x\^2\ + \((y - a)\)\^2\) + q\_2\/\@\(\((x - a\ cos\ \[Theta])\)\^2 + \((y\ + \ a\ sin\ \ \[Theta])\)\^2\) + q\_3\/\@\(\((x\ + \ a\ cos\ \[Theta])\)\^2 + \((y + a\ sin\ \ \[Theta])\)\^2\)\)], "DisplayFormula", PageWidth->PaperWidth, TextAlignment->Center, TextJustification->0], Cell[TextData[{ "These equations for the electric potentials are what we wish to visualize. \ To make them simpler to visualize, we set ", StyleBox["a ", FontSlant->"Italic"], "= 1 and substitute for \[Theta]:" }], "Text", PageWidth->PaperWidth], Cell[BoxData[{ \(TraditionalForm\`\(V\_1\)(x, y)\ = \ q\_1\/\@\(x\^2 + y\^2\), \ \ \ \ \ \ \(V\_2\)(x, y) = q\_1\/\@\(\((x - 1)\)\^2 + y\^2\)\ + \ q\_2\/\@\(\((x + 1)\)\^2 + y\^2\)\), "\[IndentingNewLine]", \(TraditionalForm\`\(V\_3\)(x, y)\ = q\_1\/\@\(x\^2 + \((y - 1)\)\^2\)\ + \ q\_2\/\@\(\((x - \@3\/2)\)\^2 + \((y + 1\/2)\)\^2\)\ + \(\(\ \ \)\(q\_3\)\)\/\@\(\((x + \@3\/2)\)\^2 + \((y + 1\/2)\)\^2\)\)}], \ "DisplayFormula", PageWidth->PaperWidth, CellMargins->{{87.5625, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->0], Cell["\<\ Owing to its two dimensional nature, a purely mathematical solution \ for the equilibrium points in these potential gets complicated. Instead, we \ will solve it graphically and rely on our intuitive understanding of how \ balls roll under the action of gravity. Specifically, we know that a ball \ released on a surface rolls downhill, and that if the ball is placed in a \ concave depression, it will remain there. Because the gravitational potential \ near the Earth's surface is proportional to height, our description of the \ ball on a surface is equivalent to a description of how a particle behaves in \ a potential energy field. It therefore follows that charges will ``roll \ down'' the electric potential surface, and will find a stable position at \ the concave minimum of the potential. So our problem translates into drawing \ pictures of the electric potential surfaces and looking for minima at the \ bottom of hills.\ \>", "Text", PageWidth->PaperWidth], Cell["4.4 Basic 2-D Plots: Plot", "Section", PageWidth->PaperWidth, CellTags->"c:5"], Cell[TextData[{ "Before we get to Maple's plotting commands, let us examine some general \ principles. First, keep in mind that the point of visualization is to make \ the science clearer and to better communicate your work to others. So it \ follows that when you produce a figure you should look at it and think if \ there are some better choices of units, ranges of axes colors, style, ", StyleBox["et cetera", FontSlant->"Italic"], " that might get the message across better and provide better insight. \ Taking into account that we are dealing with the complexity of human \ perception and cognition, there may not be one definite way to do things, and \ some trial and error is necessary to see what looks best.\nOur general \ recommendation for visualization is to make each figure as clear, informative \ and self-explanatory as possible. This means labels for various curves and \ data points, a title, and labels on the axes. We know, you are thinking that \ this is really a lot of work for a lousy assignment or report, and that you \ do not need all those time-consuming extras to comprehend what is going on. \ Yet the more often you do it, the quicker and better you get at it, and the \ more useful will your work be to others (and yourself in the future).\nThe \ convention when plotting is to have the independent variable, say ", StyleBox["x", FontSlant->"Italic"], ", along the abscissa (horizontal axis), and the dependent variable, say ", StyleBox["y=f(x)", FontSlant->"Italic"], ", along the ordinate. (Remember that your mouth spreads horizontally \ across when you say ``abscissa'', and that it puckers vertically up when you \ say ``ordinate''.) If you have trouble deciding which variable is \ independent, think of an experiment in which you measure the position or \ velocity of a ball as a function of time. Because you are free to pick the \ times at which you make the measurement, time is an independent variable. \ However, once you have chosen the time, nature picks what the position of the \ ball is at that particular time, so position and velocity are dependent \ variables." }], "Text", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " excels at easily producing graphs of all sorts, and indeed, visualization \ is one of the most valuable aspects of ", StyleBox["Mathematica", FontSlant->"Italic"], ". While we will discuss and give examples of a number of possible plots, \ ", StyleBox["Mathematica", FontSlant->"Italic"], " offers you more options than we discuss, and we recommend you browse the \ ", StyleBox["Help", FontFamily->"Courier"], " pages to create just the graph you want. Unlike Maple, where you need to \ load the graphics tools before you use them, the basic graphics tools in ", StyleBox["Mathematica", FontSlant->"Italic"], " are contained in the standard package and can be used without loading. \ However, we will need to load specific packages later as we go beyond the \ basics. We will first make a simple plot and then embellish it with things \ like labels and colors. 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]], Cell[TextData[{ "In summary, the first argument to the ", StyleBox["Plot", FontFamily->"Courier"], " command is the name of the function or expression to be plotted along the \ ordinate, that is, the dependent variable. The second argument is the range \ of values for abscissa, that is, the independent variable. The range is \ specified in a list containing the name of the independent variable, the \ minimum value, and the maximum value, ", StyleBox["e.g.", FontSlant->"Italic"], ", ", StyleBox["{x, -10, 10}", FontFamily->"Courier New"], ". In order to place a range on the ordinate, the ", StyleBox["PlotRange", FontFamily->"Courier"], " option must be used. Options are rules indicated by an arrow ->, with the \ arguments enclosed in curly braces. " }], "Text", PageWidth->PaperWidth, CellDingbat->None, CellMargins->{{12.75, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "After evaluating the command you may notice the ", StyleBox["-Graphics-", FontFamily->"Courier"], " description following the produced graph. This indicates that the plot \ is a ``graphics'' object, which can be manipulated. The description may be \ suppressed by placing a semicolon ; to the right of the ", StyleBox["Plot", FontFamily->"Courier"], " command. Try putting a semicolon after the last ", StyleBox["Plot", FontFamily->"Courier"], " command and re-evaluate the cell. We will do this for the rest of our \ examples." }], "Text", PageWidth->PaperWidth, CellDingbat->None, CellMargins->{{12.75, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Before we get on to embellishing the ", StyleBox["Plot", FontFamily->"Courier"], " command, let us have some fun with the pretty graph you just produced: \n\ \t- Click on the graph to select it.\n\t- Note how a box is formed around \ the selected graph and that there are dark little nodes at the corners and in \ the middle of the sides.\n\t- Use your mouse to resize the graph by grabbing \ one of the nodes, and then dragging it with the mouse button still depressed. \ Notice that \twhen a node is selected, a little two-sided arrow appears to \ show you the direction in which the frame can be resized. The resizing can \ be \tdone diagonally along the corners or horizontally and vertically along \ the edges. [If you find that you can only resize diagonally, with both \t\ sides having the same ratio, then the ", StyleBox["AspectRatioFixed", FontFamily->"Courier"], " option under ", StyleBox["Graphics Options", FontFamily->"Courier"], " needs to be changed to ", StyleBox["False", FontFamily->"Courier"], ". To \tdo this, make sure your graph is selected, then open the ", StyleBox["Format", FontFamily->"Courier"], " menu and choose ", StyleBox["Option Inspector....", FontFamily->"Courier"], " A separate window \twill appear where you can click on the arrow next to \ ", StyleBox["Graphics Options", FontFamily->"Courier"], ", then ", StyleBox["Image Bounding Box", FontFamily->"Courier"], ". You will then see the \tproperty that needs to be changed in order to \ resize your graph any way you wish.] " }], "Text", PageWidth->PaperWidth, CellDingbat->None, CellMargins->{{12.75, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "\t- Select the graph, copy it (it's placed on the ``clipboard''), and \ then paste it back to the notebook so that now you have two graphs.\n\t- \ Make one of your graphs into a long, narrow one, and the other into a high, \ tall one. Notice how the tall one emphasizes the variation in \tthe \ magnitude of ", StyleBox["V(r)", FontSlant->"Italic"], ", while the long one emphasizes the range of ", StyleBox["r", FontSlant->"Italic"], " values to which ", StyleBox["V(r)", FontSlant->"Italic"], " extends. Both are perfectly legitimate ways to \tview a function. One \ emphasizes the singular nature near the origin, the other the long range of \ the potential.\n\t- Another way to view a function, especially one that has \ orders of magnitude variation in value, is with a semilog plot or a log-log \ plot. \t(Note, however, that you cannot take the log of 0). In the ", StyleBox["Input", FontFamily->"Lucida sans typewriter"], " cell below, use the ", StyleBox["Log[10, z]", FontFamily->"Courier"], " function to see how a semilog plot \tchanges the appearance of the same \ ", StyleBox["f(x)", FontSlant->"Italic"], " we have been viewing. Follow the instructions in the comment fields \ below:" }], "Text", PageWidth->PaperWidth, CellDingbat->None, CellMargins->{{12.75, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ RowBox[{" ", StyleBox[\( (*\ \ Repeat\ plot\ with\ Log[10, V \((r)\)]\ \ *) \), FontSize->10, FontWeight->"Plain"]}]], "Input", CellLabel->"In[12]:=", PageWidth->PaperWidth], Cell[TextData[{ "Next try the explicit semilog plot function, ", StyleBox["LogPlot", FontFamily->"Courier"], ". " }], "Text", PageWidth->PaperWidth], Cell["4.4.1 Loading Graphics Package", "Subsection", PageWidth->PaperWidth, CellTags->"c:6"], Cell[TextData[{ "Before attempting to use that function, we need to load the ", StyleBox["Graphics", FontFamily->"Courier"], " package, which contains it. For many calculations the standard ", StyleBox["Mathematica", FontSlant->"Italic"], " package is sufficient, but often you may need more specialized functions \ and procedures that are not loaded with ", StyleBox["Mathematica", FontSlant->"Italic"], ", but are found in various ", StyleBox["Mathematica", FontSlant->"Italic"], " packages. They are loaded with the << command:" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(<< Graphics`Graphics`\)], "Input", CellLabel->"In[1]:=", PageWidth->PaperWidth], Cell[TextData[{ "where ", StyleBox["Graphics", FontFamily->"Courier New"], " is the name of the package (folder) in ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ", StyleBox["StandardPackages,", FontFamily->"Courier New"], " and ", StyleBox["Graphics.m ", FontFamily->"Courier New"], "is the name of the file being loaded. You may also load this file from a \ specific location (if you know the absolute path of the package), using the \ ", StyleBox["Get", FontFamily->"Courier"], " command: " }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \( (*\ Get["\", Path \[Rule] \ {"\"\ }]\ \ *) \ \)], "Input", CellLabel->"In[2]:=", PageWidth->PaperWidth], Cell[TextData[{ "Notice how both arguments are enclosed in quotes, and the ", StyleBox["Path", FontFamily->"Courier"], " option in curly braces.\nOnce a package is loaded you can get a list of \ the functions it contains by using the ", StyleBox["Names", FontFamily->"Courier"], " command:" }], "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[BoxData[ \(Names["\"]\)], "Input", CellLabel->"In[2]:=", PageWidth->PaperWidth], Cell[BoxData[ \({"BarChart", "BarEdges", "BarEdgeStyle", "BarGroupSpacing", "BarLabels", "BarOrientation", "BarSpacing", "BarStyle", "BarValues", "DisplayTogether", "DisplayTogetherArray", "ErrorListPlot", "GeneralizedBarChart", "Histogram", "LabeledListPlot", "LinearLogListPlot", "LinearLogPlot", "LinearScale", "ListAndCurvePlot", "LogGridMajor", "LogGridMinor", "LogLinearListPlot", "LogLinearPlot", "LogListPlot", "LogLogListPlot", "LogLogPlot", "LogPlot", "LogScale", "PercentileBarChart", "PieChart", "PieExploded", "PieLabels", "PieLineStyle", "PieStyle", "PiScale", "PolarListPlot", "PolarPlot", "ScaledListPlot", "ScaledPlot", "SkewGraphics", "StackedBarChart", "TextListPlot", "TransformGraphics", "UnitScale"}\)], "Output", CellLabel->"Out[2]="] }, Open ]], Cell[TextData[{ "An easy mistake is trying to use a command before its package has been \ loaded, then loading the package, and then trying to use the command again. \ This leads to an error message due to the way the command was initially \ entered into the symbol table. You need issue the ", StyleBox["Remove[", FontFamily->"Courier"], StyleBox["command name", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier"], "command before loading the package. For example, enter:" }], "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[Graphics3D[Cylinder[0.5, 0.5]]];\)\)], "Input", CellLabel->"In[3]:=", PageWidth->PaperWidth], Cell[BoxData[ \(Graphics3D::"gprim" \(\(:\)\(\ \)\) "\!\(Cylinder[\(\(0.5`, 0.5`\)\)]\) was encountered where a \ \!\(Graphics3D\) primitive or directive was expected."\)], "Message", CellLabel->"From In[3]:="] }, Open ]], Cell[TextData[{ "The blue error message is a reminder that ", StyleBox["Shapes", FontFamily->"Lucida sans typewriter"], " package containing the ", StyleBox["Cylinder", FontFamily->"Courier"], " graphics primitive was not loaded, so we do it now:" }], "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[BoxData[{ \(<< Graphics`Shapes`\), "\[IndentingNewLine]", \(Show[Graphics3D[Cylinder[0.5, 0.5]]]\)}], "Input", CellLabel->"In[4]:=", PageWidth->PaperWidth], Cell[BoxData[ \(Graphics3D::"gprim" \(\(:\)\(\ \)\) "\!\(Cylinder[\(\(0.5`, 0.5`\)\)]\) was encountered where a \ \!\(Graphics3D\) primitive or directive was expected."\)], "Message", CellLabel->"From In[4]:="], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics3D \[SkeletonIndicator]\), False, Editable->False]], "Output", CellLabel->"Out[5]="] }, Open ]], Cell[TextData[{ "Typing the command again does not give us the expected graph because we \ now have two symbols ", StyleBox["Cylinder", FontFamily->"Courier"], ", and the one introduced before the loading of the package takes \ precedence. 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A string is just a literal recording of whatever characters are within \ the quotes; the computer does not try to interpret a string as either \ variables or numbers. You can put anything you want within the string and it \ will get pasted into your plot. Modify the previous command so that the \ words ``abscissa'' and ``ordinate'' appear in the appropriate places and so \ that the actual expression being plotted appears in the title:" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ RowBox[{" ", StyleBox[\( (*\ \ Plot\ with\ your\ modified\ labels\ and\ title\ \ *) \ \), FontSize->10, FontWeight->"Plain"]}]], "Input", CellLabel->"In[13]:=", PageWidth->PaperWidth], Cell[TextData[{ "In a later section we will discuss how to place a ", StyleBox["legend", FontSlant->"Italic"], " on your graphs to identify an individual curve on a multicurve graph. You \ may have already noticed that we have placed ", StyleBox["r", FontSlant->"Italic"], " along the \"x'' axis and ", StyleBox["V(r)", FontSlant->"Italic"], " along the \"y'' axis. Since this is a free world, we may also call ", StyleBox["y", FontSlant->"Italic"], " the independent variable at times. So you see already why the words \ ordinate and abscissa were invented (to avoid confusion)." }], "Text", PageWidth->PaperWidth], Cell[TextData[{ "4.5 Compound (Abstract) Data Types, ", StyleBox["[..]", FontSlant->"Italic"], " and ", StyleBox["{..}", FontSlant->"Italic"] }], "Section", PageWidth->PaperWidth, CellTags->"c:8"], Cell[TextData[{ "As we proceed with our exercises in visualization, you will see how to \ enter arguments to the ", StyleBox["Plot", FontFamily->"Courier New"], " commands using different types of parentheses. We recognize that some \ users may prefer just following the rules without questioning them. \ Nevertheless, the commands will make more sense, and will be easier to \ generalize, if you have some understanding of the method behind the madness. \ And so we now take a little excursion in which we define some terms that are \ frequently used in mathematics and computer science and employed by ", StyleBox["Mathematica", FontSlant->"Italic"], " commands.\nWe have already seen a number of ways in which ", StyleBox["Mathematica", FontSlant->"Italic"], " displays data. Sometimes there's just a single symbol, sometime's there's \ a bunch of symbols separated by commas, sometimes there's a bunch of things \ in curly braces, and sometimes not. 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Note that the arguments given to this ", StyleBox["Solve", FontFamily->"Courier"], " command, and indeed to most ", StyleBox["Mathematica", FontSlant->"Italic"], " commands, are variables or rules separated by commas, and accordingly \ form sequences. The square brackets indicate an argument to the function (as \ discussed in Chapter 2), so in these cases the function argument is a \ sequence.\n", StyleBox["List:", FontWeight->"Bold"], " In ", StyleBox["Mathematica", FontSlant->"Italic"], " the way to make collections of objects is with a ", StyleBox["list", FontWeight->"Bold", FontSlant->"Italic"], ". Lists are important and general structures (objects themselves) that \ contain comma-separated objects within curly braces {...}. Sometimes the \ order of objects within a list matters, and sometimes it doesn't. 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The ", StyleBox["PlotStyle", FontFamily->"Courier"], " option is also used to specify the style of line used for various curves. \ We can add a list of styles containing the type of lines for each graph. An \ especially effective way to distinguish different curves on the same plot \ without the use of color, is to draw them with different thickness, which you \ can try below:" }], "Text", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["Dashing[{", FontWeight->"Bold"], StyleBox["r1, r2, ...", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["}]", FontWeight->"Bold"], " shows the lines as a sequence of dashed segments and spaces of lengths, \ ", StyleBox["r1, r2,", FontSlant->"Italic"], " ... repeated cyclically. Each ", StyleBox["r", FontSlant->"Italic"], " value is given as a fraction of the total width of the graph. If you \ wish to distinguish curves with dashing, they each need their own ", StyleBox["Dashing", FontFamily->"Courier"], " function. 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When presenting \ several curves in one graph, it is important that the viewer not only be able \ to tell them apart by the different color or line style used for each, but \ also be given information as to what the different curves represent. It is \ good practice to explain in the caption below a graph what each curve means, \ as well as in the text (or in your talk) when the graph gets referenced. \ However, it is also good practice to have a ", StyleBox["legend", FontSlant->"Italic"], " in the plot itself explaining what each curve means. Captions and text \ may get removed, but it is a lot harder to remove a good legend.\nThe legend \ option for the ", StyleBox["Plot", FontFamily->"Courier"], " command is ", StyleBox["PlotLegend[{", FontFamily->"Courier"], StyleBox["text1, text2, ...", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["}]", FontFamily->"Courier"], " which attaches strings ", StyleBox["text1, text2, ...", FontFamily->"Courier", FontSlant->"Italic"], " to each description in ", StyleBox["PlotStyle", FontFamily->"Courier"], ", in the order of the curves. If there are more ", StyleBox["PlotStyle", FontFamily->"Courier"], " descriptions than text descriptions, the text will be repeated \ cyclically. 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The default \ position for the legend is in the lower left corner of the graph and will \ most likely cover important portions of the plots. Hence, you will want to \ tweak the position coordinates in the command to place the legend where you \ wish. Other display options for the legend can be found in the ", StyleBox["Graphics`Legend`", FontFamily->"Courier"], " package." }], "Text", PageWidth->PaperWidth], Cell["4.5.4 Other Options", "Subsection", PageWidth->PaperWidth, CellTags->"c:12"], Cell[TextData[{ "There are many ways to customize your graph, and the ", StyleBox["Help", FontFamily->"Courier"], " browser is a good place to find out about them. Here we look at a few. \ When ", StyleBox["Mathematica", FontSlant->"Italic"], " plots, it tries to scale the ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " axes to show only the most interesting features of the plot. 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Is it better or worse? \ Remember that the larger the value of ", StyleBox["PlotPoints", FontFamily->"Courier"], ", the longer it will take ", StyleBox["Mathematica", FontSlant->"Italic"], " to plot the function, but it can be most helpful at times. " }], "Text", PageWidth->PaperWidth], Cell["4.6 3D (Surface) Plots of Analytic Functions", "Section", PageWidth->PaperWidth, CellTags->"c:13"], Cell[TextData[{ "\nWe have examined the potential field ", StyleBox["V(r) = 1/r", FontSlant->"Italic"], " surrounding a single charge as a function of ", StyleBox["r", FontSlant->"Italic"], ". A 2-D plot is fine for this since there is only one independent \ variable ", StyleBox["r", FontSlant->"Italic"], ". However, when the same potential is expresses as a function of the \ Cartesian coordinate ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y,", FontSlant->"Italic"] }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(TraditionalForm\`\(\(V(x, y)\ = \ 1\/\@\(x\^2\ + y\^2\)\)\(,\)\)\)], "DisplayFormula", PageWidth->PaperWidth, TextAlignment->Center, TextJustification->0], Cell[TextData[{ "we have two independent variables, ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], ", and so need a 3-D visualization. We get that by creating a world in \ which the ", StyleBox["z", FontSlant->"Italic"], " dimension (mountain height) is the value of the potential, and ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " lie on a flat plane below the mountain. Because the surface we are \ creating is a 3-D object it is not possible to draw it on a flat screen, \ and so different techniques are used to give the impression of three \ dimensions to our brains. We do that by rotating the object, shading it, \ employing parallax, and so forth. 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r .93362 .44612 .93954 .45445 .95542 .44931 .94956 .44089 Metetra .673 .757 .914 r .94956 .44089 .95542 .44931 .97138 .44414 .96559 .43564 Metetra .649 .749 .923 r .38898 .59928 .39692 .60624 .41013 .60267 .40222 .59572 Metetra .649 .75 .923 r .40222 .59572 .41013 .60267 .4234 .5991 .41554 .59215 Metetra .648 .751 .924 r .41554 .59215 .4234 .5991 .43674 .59552 .42892 .58857 Metetra .648 .752 .925 r .42892 .58857 .43674 .59552 .45015 .59193 .44237 .58499 Metetra .647 .753 .926 r .44237 .58499 .45015 .59193 .46363 .58833 .4559 .5814 Metetra .647 .754 .927 r .4559 .5814 .46363 .58833 .47718 .58472 .46949 .57779 Metetra .646 .755 .929 r .46949 .57779 .47718 .58472 .4908 .58108 .48316 .57418 Metetra .646 .757 .93 r .48316 .57418 .4908 .58108 .5045 .57743 .4969 .57054 Metetra .646 .758 .931 r .4969 .57054 .5045 .57743 .51826 .57375 .51071 .56688 Metetra .646 .76 .932 r .51071 .56688 .51826 .57375 .5321 .57003 .5246 .56319 Metetra .646 .762 .933 r .5246 .56319 .5321 .57003 .54602 .56629 .53856 .55946 Metetra .647 .763 .934 r .53856 .55946 .54602 .56629 .56 .56249 .5526 .5557 Metetra .647 .765 .935 r .5526 .5557 .56 .56249 .57406 .55865 .56671 .55187 Metetra .648 .767 .936 r .56671 .55187 .57406 .55865 .5882 .55476 .58089 .54799 Metetra .65 .769 .936 r .58089 .54799 .5882 .55476 .60241 .5508 .59515 .54405 Metetra .651 .771 .936 r .59515 .54405 .60241 .5508 .61669 .54677 .60949 .54002 Metetra .653 .773 .937 r .60949 .54002 .61669 .54677 .63104 .54267 .6239 .53591 Metetra .655 .774 .937 r .6239 .53591 .63104 .54267 .64547 .53848 .63839 .53171 Metetra .657 .775 .936 r .63839 .53171 .64547 .53848 .65997 .53421 .65295 .52741 Metetra .659 .776 .936 r .65295 .52741 .65997 .53421 .67455 .52986 .66758 .52302 Metetra .661 .777 .935 r .66758 .52302 .67455 .52986 .6892 .52542 .68228 .51852 Metetra .664 .777 .934 r .68228 .51852 .6892 .52542 .70392 .52089 .69706 .51393 Metetra .666 .777 .933 r .69706 .51393 .70392 .52089 .71871 .51628 .71191 .50925 Metetra .668 .777 .931 r .71191 .50925 .71871 .51628 .73357 .5116 .72683 .50448 Metetra .669 .776 .93 r .72683 .50448 .73357 .5116 .74851 .50684 .74182 .49963 Metetra .671 .776 .928 r .74182 .49963 .74851 .50684 .76352 .50201 .75689 .4947 Metetra .672 .774 .927 r .75689 .4947 .76352 .50201 .7786 .49712 .77203 .48972 Metetra .673 .773 .926 r .77203 .48972 .7786 .49712 .79375 .49218 .78724 .48467 Metetra .674 .772 .924 r .78724 .48467 .79375 .49218 .80899 .48719 .80253 .47957 Metetra .674 .77 .923 r .80253 .47957 .80899 .48719 .82429 .48216 .8179 .47444 Metetra .675 .769 .921 r .8179 .47444 .82429 .48216 .83967 .47709 .83334 .46926 Metetra .675 .767 .92 r .83334 .46926 .83967 .47709 .85513 .47199 .84885 .46406 Metetra .675 .766 .919 r .84885 .46406 .85513 .47199 .87067 .46686 .86445 .45883 Metetra .675 .764 .918 r .86445 .45883 .87067 .46686 .88629 .4617 .88013 .45358 Metetra .675 .763 .917 r .88013 .45358 .88629 .4617 .90198 .45653 .89588 .4483 Metetra .674 .761 .916 r .89588 .4483 .90198 .45653 .91776 .45133 .91172 .44302 Metetra .674 .76 .916 r .91172 .44302 .91776 .45133 .93362 .44612 .92764 .43771 Metetra .674 .759 .915 r .92764 .43771 .93362 .44612 .94956 .44089 .94365 .4324 Metetra .673 .758 .914 r .94365 .4324 .94956 .44089 .96559 .43564 .95974 .42707 Metetra .648 .749 .923 r .38096 .59227 .38898 .59928 .40222 .59572 .39425 .5887 Metetra .647 .75 .924 r .39425 .5887 .40222 .59572 .41554 .59215 .4076 .58514 Metetra .647 .751 .925 r .4076 .58514 .41554 .59215 .42892 .58857 .42102 .58157 Metetra .646 .752 .926 r .42102 .58157 .42892 .58857 .44237 .58499 .43452 .578 Metetra .645 .753 .927 r .43452 .578 .44237 .58499 .4559 .5814 .44808 .57442 Metetra .645 .754 .929 r .44808 .57442 .4559 .5814 .46949 .57779 .46172 .57084 Metetra .644 .756 .93 r .46172 .57084 .46949 .57779 .48316 .57418 .47543 .56724 Metetra .644 .757 .931 r .47543 .56724 .48316 .57418 .4969 .57054 .48922 .56363 Metetra .644 .759 .933 r .48922 .56363 .4969 .57054 .51071 .56688 .50308 .56 Metetra .644 .761 .934 r .50308 .56 .51071 .56688 .5246 .56319 .51702 .55635 Metetra .644 .763 .935 r .51702 .55635 .5246 .56319 .53856 .55946 .53103 .55265 Metetra .644 .765 .936 r .53103 .55265 .53856 .55946 .5526 .5557 .54511 .54892 Metetra .645 .767 .937 r .54511 .54892 .5526 .5557 .56671 .55187 .55928 .54513 Metetra .646 .769 .938 r .55928 .54513 .56671 .55187 .58089 .54799 .57352 .54128 Metetra .647 .772 .939 r .57352 .54128 .58089 .54799 .59515 .54405 .58783 .53735 Metetra .649 .774 .94 r .58783 .53735 .59515 .54405 .60949 .54002 .60222 .53333 Metetra .651 .776 .94 r .60222 .53333 .60949 .54002 .6239 .53591 .61669 .52922 Metetra .653 .778 .94 r .61669 .52922 .6239 .53591 .63839 .53171 .63123 .52501 Metetra .656 .779 .939 r .63123 .52501 .63839 .53171 .65295 .52741 .64585 .52069 Metetra .659 .781 .939 r .64585 .52069 .65295 .52741 .66758 .52302 .66054 .51625 Metetra .661 .781 .938 r .66054 .51625 .66758 .52302 .68228 .51852 .6753 .51171 Metetra .664 .782 .937 r .6753 .51171 .68228 .51852 .69706 .51393 .69014 .50705 Metetra .666 .782 .935 r .69014 .50705 .69706 .51393 .71191 .50925 .70505 .50228 Metetra .669 .781 .934 r .70505 .50228 .71191 .50925 .72683 .50448 .72003 .49741 Metetra .671 .78 .932 r .72003 .49741 .72683 .50448 .74182 .49963 .73508 .49246 Metetra .672 .779 .93 r .73508 .49246 .74182 .49963 .75689 .4947 .7502 .48743 Metetra .674 .778 .928 r .7502 .48743 .75689 .4947 .77203 .48972 .7654 .48232 Metetra .675 .776 .927 r .7654 .48232 .77203 .48972 .78724 .48467 .78067 .47716 Metetra .675 .775 .925 r .78067 .47716 .78724 .48467 .80253 .47957 .79602 .47195 Metetra .676 .773 .923 r .79602 .47195 .80253 .47957 .8179 .47444 .81144 .46669 Metetra .676 .771 .922 r .81144 .46669 .8179 .47444 .83334 .46926 .82694 .4614 Metetra .676 .769 .921 r .82694 .4614 .83334 .46926 .84885 .46406 .84251 .45609 Metetra .676 .767 .919 r .84251 .45609 .84885 .46406 .86445 .45883 .85817 .45075 Metetra .676 .766 .918 r .85817 .45075 .86445 .45883 .88013 .45358 .8739 .44539 Metetra .676 .764 .917 r .8739 .44539 .88013 .45358 .89588 .4483 .88972 .44002 Metetra .676 .763 .916 r .88972 .44002 .89588 .4483 .91172 .44302 .90562 .43463 Metetra .675 .761 .916 r .90562 .43463 .91172 .44302 .92764 .43771 .9216 .42923 Metetra .675 .76 .915 r .9216 .42923 .92764 .43771 .94365 .4324 .93767 .42383 Metetra .674 .759 .914 r .93767 .42383 .94365 .4324 .95974 .42707 .95382 .41841 Metetra .647 .749 .924 r .37287 .58519 .38096 .59227 .39425 .5887 .38619 .58163 Metetra .646 .75 .925 r .38619 .58163 .39425 .5887 .4076 .58514 .39959 .57807 Metetra .645 .751 .926 r .39959 .57807 .4076 .58514 .42102 .58157 .41305 .57452 Metetra .644 .752 .927 r .41305 .57452 .42102 .58157 .43452 .578 .42659 .57096 Metetra .644 .753 .929 r .42659 .57096 .43452 .578 .44808 .57442 .4402 .56741 Metetra .643 .754 .93 r .4402 .56741 .44808 .57442 .46172 .57084 .45388 .56385 Metetra .642 .756 .931 r .45388 .56385 .46172 .57084 .47543 .56724 .46764 .56028 Metetra .641 .758 .933 r .46764 .56028 .47543 .56724 .48922 .56363 .48147 .55671 Metetra .641 .76 .934 r .48147 .55671 .48922 .56363 .50308 .56 .49537 .55312 Metetra .641 .762 .936 r .49537 .55312 .50308 .56 .51702 .55635 .50936 .54951 Metetra .641 .764 .938 r .50936 .54951 .51702 .55635 .53103 .55265 .52342 .54586 Metetra .641 .766 .939 r .52342 .54586 .53103 .55265 .54511 .54892 .53756 .54217 Metetra .642 .769 .94 r .53756 .54217 .54511 .54892 .55928 .54513 .55177 .53843 Metetra .643 .772 .942 r .55177 .53843 .55928 .54513 .57352 .54128 .56607 .53462 Metetra .644 .775 .943 r .56607 .53462 .57352 .54128 .58783 .53735 .58044 .53073 Metetra .646 .777 .943 r .58044 .53073 .58783 .53735 .60222 .53333 .59489 .52674 Metetra .648 .78 .944 r .59489 .52674 .60222 .53333 .61669 .52922 .60941 .52264 Metetra .651 .782 .944 r .60941 .52264 .61669 .52922 .63123 .52501 .62401 .51842 Metetra .654 .784 .943 r .62401 .51842 .63123 .52501 .64585 .52069 .63869 .51408 Metetra .658 .786 .942 r .63869 .51408 .64585 .52069 .66054 .51625 .65344 .5096 Metetra .661 .787 .941 r .65344 .5096 .66054 .51625 .6753 .51171 .66826 .505 Metetra .664 .787 .94 r .66826 .505 .6753 .51171 .69014 .50705 .68315 .50026 Metetra .667 .787 .938 r .68315 .50026 .69014 .50705 .70505 .50228 .69812 .4954 Metetra .67 .786 .936 r .69812 .4954 .70505 .50228 .72003 .49741 .71316 .49042 Metetra .672 .785 .934 r .71316 .49042 .72003 .49741 .73508 .49246 .72827 .48535 Metetra .674 .784 .932 r .72827 .48535 .73508 .49246 .7502 .48743 .74345 .48019 Metetra .676 .782 .93 r .74345 .48019 .7502 .48743 .7654 .48232 .75871 .47495 Metetra .677 .78 .928 r .75871 .47495 .7654 .48232 .78067 .47716 .77404 .46966 Metetra .677 .778 .926 r .77404 .46966 .78067 .47716 .79602 .47195 .78944 .46431 Metetra .678 .775 .924 r .78944 .46431 .79602 .47195 .81144 .46669 .80492 .45893 Metetra .678 .773 .922 r .80492 .45893 .81144 .46669 .82694 .4614 .82048 .45351 Metetra .678 .771 .921 r .82048 .45351 .82694 .4614 .84251 .45609 .83611 .44807 Metetra .678 .769 .92 r .83611 .44807 .84251 .45609 .85817 .45075 .85182 .44262 Metetra .678 .767 .918 r .85182 .44262 .85817 .45075 .8739 .44539 .86762 .43715 Metetra .677 .765 .917 r .86762 .43715 .8739 .44539 .88972 .44002 .8835 .43166 Metetra .677 .764 .916 r .8835 .43166 .88972 .44002 .90562 .43463 .89946 .42618 Metetra .676 .762 .915 r .89946 .42618 .90562 .43463 .9216 .42923 .9155 .42068 Metetra .676 .76 .915 r .9155 .42068 .9216 .42923 .93767 .42383 .93163 .41518 Metetra .675 .759 .914 r .93163 .41518 .93767 .42383 .95382 .41841 .94785 .40967 Metetra .646 .749 .924 r .3647 .57806 .37287 .58519 .38619 .58163 .37806 .5745 Metetra .645 .75 .926 r .37806 .5745 .38619 .58163 .39959 .57807 .3915 .57095 Metetra .644 .751 .927 r .3915 .57095 .39959 .57807 .41305 .57452 .405 .56741 Metetra .642 .752 .928 r .405 .56741 .41305 .57452 .42659 .57096 .41858 .56388 Metetra .641 .753 .93 r .41858 .56388 .42659 .57096 .4402 .56741 .43223 .56035 Metetra .64 .754 .931 r .43223 .56035 .4402 .56741 .45388 .56385 .44596 .55682 Metetra .639 .756 .933 r .44596 .55682 .45388 .56385 .46764 .56028 .45976 .5533 Metetra .639 .758 .935 r .45976 .5533 .46764 .56028 .48147 .55671 .47364 .54977 Metetra .638 .76 .937 r .47364 .54977 .48147 .55671 .49537 .55312 .48759 .54623 Metetra .637 .762 .938 r .48759 .54623 .49537 .55312 .50936 .54951 .50162 .54268 Metetra .637 .765 .94 r .50162 .54268 .50936 .54951 .52342 .54586 .51573 .5391 Metetra .637 .768 .942 r .51573 .5391 .52342 .54586 .53756 .54217 .52992 .53547 Metetra .637 .771 .944 r .52992 .53547 .53756 .54217 .55177 .53843 .54419 .53179 Metetra .638 .774 .946 r .54419 .53179 .55177 .53843 .56607 .53462 .55854 .52805 Metetra .64 .778 .947 r .55854 .52805 .56607 .53462 .58044 .53073 .57297 .52421 Metetra .642 .781 .948 r .57297 .52421 .58044 .53073 .59489 .52674 .58747 .52026 Metetra .645 .784 .948 r .58747 .52026 .59489 .52674 .60941 .52264 .60206 .51619 Metetra .648 .787 .948 r .60206 .51619 .60941 .52264 .62401 .51842 .61672 .51198 Metetra .652 .79 .948 r .61672 .51198 .62401 .51842 .63869 .51408 .63145 .50762 Metetra .656 .792 .947 r .63145 .50762 .63869 .51408 .65344 .5096 .64626 .5031 Metetra .661 .793 .945 r .64626 .5031 .65344 .5096 .66826 .505 .66115 .49843 Metetra .665 .794 .944 r .66115 .49843 .66826 .505 .68315 .50026 .6761 .4936 Metetra .668 .793 .941 r .6761 .4936 .68315 .50026 .69812 .4954 .69113 .48863 Metetra .672 .792 .939 r .69113 .48863 .69812 .4954 .71316 .49042 .70623 .48353 Metetra .674 .791 .936 r .70623 .48353 .71316 .49042 .72827 .48535 .7214 .47832 Metetra .677 .789 .934 r .7214 .47832 .72827 .48535 .74345 .48019 .73664 .47301 Metetra .678 .787 .931 r .73664 .47301 .74345 .48019 .75871 .47495 .75195 .46762 Metetra .679 .784 .929 r .75195 .46762 .75871 .47495 .77404 .46966 .76734 .46217 Metetra .68 .781 .927 r .76734 .46217 .77404 .46966 .78944 .46431 .7828 .45668 Metetra .68 .779 .925 r .7828 .45668 .78944 .46431 .80492 .45893 .79834 .45115 Metetra .68 .776 .923 r .79834 .45115 .80492 .45893 .82048 .45351 .81395 .44559 Metetra .68 .773 .921 r .81395 .44559 .82048 .45351 .83611 .44807 .82964 .44002 Metetra .68 .771 .92 r .82964 .44002 .83611 .44807 .85182 .44262 .84542 .43443 Metetra .679 .769 .918 r .84542 .43443 .85182 .44262 .86762 .43715 .86127 .42884 Metetra .679 .767 .917 r .86127 .42884 .86762 .43715 .8835 .43166 .87721 .42324 Metetra .678 .765 .916 r .87721 .42324 .8835 .43166 .89946 .42618 .89323 .41765 Metetra .678 .763 .915 r .89323 .41765 .89946 .42618 .9155 .42068 .90934 .41205 Metetra .677 .761 .914 r .90934 .41205 .9155 .42068 .93163 .41518 .92553 .40644 Metetra .676 .76 .914 r .92553 .40644 .93163 .41518 .94785 .40967 .94181 .40084 Metetra .644 .748 .925 r .35645 .57086 .3647 .57806 .37806 .5745 .36985 .56731 Metetra .643 .749 .926 r .36985 .56731 .37806 .5745 .3915 .57095 .38333 .56377 Metetra .642 .75 .928 r .38333 .56377 .3915 .57095 .405 .56741 .39687 .56025 Metetra .64 .751 .929 r .39687 .56025 .405 .56741 .41858 .56388 .41049 .55674 Metetra .639 .753 .931 r .41049 .55674 .41858 .56388 .43223 .56035 .42419 .55324 Metetra .638 .754 .933 r .42419 .55324 .43223 .56035 .44596 .55682 .43796 .54976 Metetra .636 .756 .935 r .43796 .54976 .44596 .55682 .45976 .5533 .4518 .54628 Metetra .635 .758 .937 r .4518 .54628 .45976 .5533 .47364 .54977 .46573 .54282 Metetra .634 .761 .939 r .46573 .54282 .47364 .54977 .48759 .54623 .47973 .53935 Metetra .633 .763 .941 r .47973 .53935 .48759 .54623 .50162 .54268 .49381 .53587 Metetra .632 .766 .944 r .49381 .53587 .50162 .54268 .51573 .5391 .50797 .53237 Metetra .632 .77 .946 r .50797 .53237 .51573 .5391 .52992 .53547 .52221 .52883 Metetra .632 .773 .948 r .52221 .52883 .52992 .53547 .54419 .53179 .53653 .52524 Metetra .633 .777 .95 r .53653 .52524 .54419 .53179 .55854 .52805 .55093 .52158 Metetra .635 .782 .952 r .55093 .52158 .55854 .52805 .57297 .52421 .56542 .51782 Metetra .637 .786 .953 r .56542 .51782 .57297 .52421 .58747 .52026 .57998 .51394 Metetra .641 .79 .954 r .57998 .51394 .58747 .52026 .60206 .51619 .59463 .50992 Metetra .645 .794 .954 r .59463 .50992 .60206 .51619 .61672 .51198 .60935 .50573 Metetra .649 .797 .953 r .60935 .50573 .61672 .51198 .63145 .50762 .62415 .50136 Metetra .655 .799 .952 r .62415 .50136 .63145 .50762 .64626 .5031 .63902 .4968 Metetra .66 .801 .95 r .63902 .4968 .64626 .5031 .66115 .49843 .65397 .49205 Metetra .665 .802 .948 r .65397 .49205 .66115 .49843 .6761 .4936 .66899 .48712 Metetra .67 .801 .945 r .66899 .48712 .6761 .4936 .69113 .48863 .68407 .48202 Metetra .674 .8 .942 r .68407 .48202 .69113 .48863 .70623 .48353 .69923 .47677 Metetra .677 .798 .939 r .69923 .47677 .70623 .48353 .7214 .47832 .71446 .47139 Metetra .68 .795 .936 r .71446 .47139 .7214 .47832 .73664 .47301 .72976 .46591 Metetra .681 .792 .933 r .72976 .46591 .73664 .47301 .75195 .46762 .74513 .46034 Metetra .682 .789 .93 r .74513 .46034 .75195 .46762 .76734 .46217 .76058 .45472 Metetra .683 .785 .928 r .76058 .45472 .76734 .46217 .7828 .45668 .77609 .44905 Metetra .683 .782 .925 r .77609 .44905 .7828 .45668 .79834 .45115 .79169 .44336 Metetra .683 .779 .923 r .79169 .44336 .79834 .45115 .81395 .44559 .80736 .43764 Metetra .683 .776 .921 r .80736 .43764 .81395 .44559 .82964 .44002 .82311 .43192 Metetra .682 .773 .92 r .82311 .43192 .82964 .44002 .84542 .43443 .83895 .4262 Metetra .681 .77 .918 r .83895 .4262 .84542 .43443 .86127 .42884 .85486 .42047 Metetra .681 .768 .917 r .85486 .42047 .86127 .42884 .87721 .42324 .87086 .41475 Metetra .68 .766 .916 r .87086 .41475 .87721 .42324 .89323 .41765 .88694 .40904 Metetra .679 .764 .915 r .88694 .40904 .89323 .41765 .90934 .41205 .90311 .40333 Metetra .678 .762 .914 r .90311 .40333 .90934 .41205 .92553 .40644 .91937 .39762 Metetra .677 .76 .914 r .91937 .39762 .92553 .40644 .94181 .40084 .93571 .39192 Metetra .643 .748 .925 r .34812 .56358 .35645 .57086 .36985 .56731 .36157 .56004 Metetra .641 .749 .927 r .36157 .56004 .36985 .56731 .38333 .56377 .37508 .55652 Metetra .64 .75 .928 r .37508 .55652 .38333 .56377 .39687 .56025 .38867 .55302 Metetra .638 .751 .93 r .38867 .55302 .39687 .56025 .41049 .55674 .40233 .54954 Metetra .637 .752 .932 r .40233 .54954 .41049 .55674 .42419 .55324 .41607 .54609 Metetra .635 .754 .934 r .41607 .54609 .42419 .55324 .43796 .54976 .42988 .54265 Metetra .633 .756 .936 r .42988 .54265 .43796 .54976 .4518 .54628 .44377 .53924 Metetra .631 .758 .939 r .44377 .53924 .4518 .54628 .46573 .54282 .45773 .53584 Metetra .63 .761 .941 r .45773 .53584 .46573 .54282 .47973 .53935 .47178 .53246 Metetra .628 .764 .944 r .47178 .53246 .47973 .53935 .49381 .53587 .48591 .52907 Metetra .627 .767 .947 r .48591 .52907 .49381 .53587 .50797 .53237 .50012 .52568 Metetra .626 .771 .95 r .50012 .52568 .50797 .53237 .52221 .52883 .51441 .52226 Metetra .625 .775 .953 r .51441 .52226 .52221 .52883 .53653 .52524 .52879 .5188 Metetra .626 .78 .955 r .52879 .5188 .53653 .52524 .55093 .52158 .54325 .51526 Metetra .627 .786 .958 r .54325 .51526 .55093 .52158 .56542 .51782 .55779 .51161 Metetra .63 .791 .959 r .55779 .51161 .56542 .51782 .57998 .51394 .57242 .50783 Metetra .634 .796 .96 r .57242 .50783 .57998 .51394 .59463 .50992 .58712 .50389 Metetra .639 .801 .961 r .58712 .50389 .59463 .50992 .60935 .50573 .60191 .49974 Metetra .645 .806 .96 r .60191 .49974 .60935 .50573 .62415 .50136 .61677 .49537 Metetra .652 .809 .959 r .61677 .49537 .62415 .50136 .63902 .4968 .63171 .49077 Metetra .659 .811 .956 r .63171 .49077 .63902 .4968 .65397 .49205 .64672 .48593 Metetra .666 .812 .953 r .64672 .48593 .65397 .49205 .66899 .48712 .6618 .48088 Metetra .672 .811 .95 r .6618 .48088 .66899 .48712 .68407 .48202 .67695 .47561 Metetra .677 .809 .946 r .67695 .47561 .68407 .48202 .69923 .47677 .69217 .47018 Metetra .681 .806 .942 r .69217 .47018 .69923 .47677 .71446 .47139 .70746 .4646 Metetra .683 .802 .938 r .70746 .4646 .71446 .47139 .72976 .46591 .72282 .4589 Metetra .685 .798 .935 r .72282 .4589 .72976 .46591 .74513 .46034 .73825 .45313 Metetra .686 .794 .931 r .73825 .45313 .74513 .46034 .76058 .45472 .75375 .4473 Metetra .687 .79 .928 r .75375 .4473 .76058 .45472 .77609 .44905 .76932 .44143 Metetra .687 .786 .926 r .76932 .44143 .77609 .44905 .79169 .44336 .78498 .43555 Metetra .686 .782 .923 r .78498 .43555 .79169 .44336 .80736 .43764 .80071 .42966 Metetra .685 .778 .921 r .80071 .42966 .80736 .43764 .82311 .43192 .81652 .42378 Metetra .685 .775 .92 r .81652 .42378 .82311 .43192 .83895 .4262 .83241 .4179 Metetra .684 .772 .918 r .83241 .4179 .83895 .4262 .85486 .42047 .84838 .41204 Metetra .683 .769 .917 r .84838 .41204 .85486 .42047 .87086 .41475 .86444 .40618 Metetra .681 .767 .916 r .86444 .40618 .87086 .41475 .88694 .40904 .88059 .40035 Metetra .68 .764 .915 r .88059 .40035 .88694 .40904 .90311 .40333 .89682 .39452 Metetra .679 .763 .914 r .89682 .39452 .90311 .40333 .91937 .39762 .91314 .38871 Metetra .679 .761 .913 r .91314 .38871 .91937 .39762 .93571 .39192 .92954 .38291 Metetra .641 .748 .926 r .33972 .55624 .34812 .56358 .36157 .56004 .3532 .55271 Metetra .64 .748 .928 r .3532 .55271 .36157 .56004 .37508 .55652 .36675 .54921 Metetra .638 .749 .929 r .36675 .54921 .37508 .55652 .38867 .55302 .38038 .54573 Metetra .636 .751 .931 r .38038 .54573 .38867 .55302 .40233 .54954 .39408 .54229 Metetra .634 .752 .933 r .39408 .54229 .40233 .54954 .41607 .54609 .40786 .53888 Metetra .632 .754 .936 r .40786 .53888 .41607 .54609 .42988 .54265 .42172 .5355 Metetra .629 .756 .938 r .42172 .5355 .42988 .54265 .44377 .53924 .43565 .53216 Metetra .627 .758 .941 r .43565 .53216 .44377 .53924 .45773 .53584 .44966 .52885 Metetra .624 .761 .944 r .44966 .52885 .45773 .53584 .47178 .53246 .46375 .52557 Metetra .622 .764 .947 r .46375 .52557 .47178 .53246 .48591 .52907 .47793 .52231 Metetra .62 .768 .951 r .47793 .52231 .48591 .52907 .50012 .52568 .49219 .51905 Metetra .618 .772 .954 r .49219 .51905 .50012 .52568 .51441 .52226 .50653 .51579 Metetra .617 .778 .958 r .50653 .51579 .51441 .52226 .52879 .5188 .52096 .51248 Metetra .616 .783 .961 r .52096 .51248 .52879 .5188 .54325 .51526 .53548 .50911 Metetra .618 .79 .964 r .53548 .50911 .54325 .51526 .55779 .51161 .55008 .50563 Metetra .62 .797 .967 r .55008 .50563 .55779 .51161 .57242 .50783 .56477 .50199 Metetra .625 .804 .968 r .56477 .50199 .57242 .50783 .58712 .50389 .57954 .49816 Metetra .631 .811 .969 r .57954 .49816 .58712 .50389 .60191 .49974 .5944 .49409 Metetra .639 .816 .968 r .5944 .49409 .60191 .49974 .61677 .49537 .60933 .48974 Metetra .648 .821 .966 r .60933 .48974 .61677 .49537 .63171 .49077 .62433 .48509 Metetra .658 .823 .963 r .62433 .48509 .63171 .49077 .64672 .48593 .63941 .48016 Metetra .666 .824 .959 r .63941 .48016 .64672 .48593 .6618 .48088 .65455 .47494 Metetra .674 .823 .955 r .65455 .47494 .6618 .48088 .67695 .47561 .66977 .46948 Metetra .681 .82 .95 r .66977 .46948 .67695 .47561 .69217 .47018 .68505 .46381 Metetra .685 .816 .945 r .68505 .46381 .69217 .47018 .70746 .4646 .70039 .45798 Metetra .688 .811 .941 r .70039 .45798 .70746 .4646 .72282 .4589 .71581 .45203 Metetra .69 .805 .936 r .71581 .45203 .72282 .4589 .73825 .45313 .73129 .446 Metetra .691 .8 .932 r .73129 .446 .73825 .45313 .75375 .4473 .74685 .43993 Metetra .691 .795 .929 r .74685 .43993 .75375 .4473 .76932 .44143 .76249 .43383 Metetra .691 .789 .926 r .76249 .43383 .76932 .44143 .78498 .43555 .7782 .42774 Metetra .69 .785 .923 r .7782 .42774 .78498 .43555 .80071 .42966 .79398 .42165 Metetra .689 .781 .921 r .79398 .42165 .80071 .42966 .81652 .42378 .80985 .41558 Metetra .687 .777 .919 r .80985 .41558 .81652 .42378 .83241 .4179 .8258 .40954 Metetra .686 .773 .918 r .8258 .40954 .83241 .4179 .84838 .41204 .84184 .40353 Metetra .685 .77 .916 r .84184 .40353 .84838 .41204 .86444 .40618 .85796 .39753 Metetra .683 .768 .915 r .85796 .39753 .86444 .40618 .88059 .40035 .87416 .39157 Metetra .682 .765 .914 r .87416 .39157 .88059 .40035 .89682 .39452 .89046 .38562 Metetra .681 .763 .913 r .89046 .38562 .89682 .39452 .91314 .38871 .90684 .3797 Metetra .68 .761 .913 r .90684 .3797 .91314 .38871 .92954 .38291 .92331 .37379 Metetra .64 .747 .926 r .33123 .54881 .33972 .55624 .3532 .55271 .34475 .5453 Metetra .638 .748 .928 r .34475 .5453 .3532 .55271 .36675 .54921 .35834 .54181 Metetra .636 .749 .93 r .35834 .54181 .36675 .54921 .38038 .54573 .37201 .53837 Metetra .633 .75 .932 r .37201 .53837 .38038 .54573 .39408 .54229 .38575 .53496 Metetra .631 .751 .934 r .38575 .53496 .39408 .54229 .40786 .53888 .39957 .53161 Metetra .628 .753 .937 r .39957 .53161 .40786 .53888 .42172 .5355 .41347 .5283 Metetra .625 .755 .94 r .41347 .5283 .42172 .5355 .43565 .53216 .42745 .52504 Metetra .622 .757 .943 r .42745 .52504 .43565 .53216 .44966 .52885 .4415 .52183 Metetra .618 .76 .947 r .4415 .52183 .44966 .52885 .46375 .52557 .45564 .51868 Metetra .615 .764 .951 r .45564 .51868 .46375 .52557 .47793 .52231 .46986 .51557 Metetra .611 .768 .955 r .46986 .51557 .47793 .52231 .49219 .51905 .48417 .51248 Metetra .608 .773 .959 r .48417 .51248 .49219 .51905 .50653 .51579 .49857 .50941 Metetra .605 .779 .964 r .49857 .50941 .50653 .51579 .52096 .51248 .51305 .50633 Metetra .603 .786 .968 r .51305 .50633 .52096 .51248 .53548 .50911 .52763 .50318 Metetra .603 .794 .972 r .52763 .50318 .53548 .50911 .55008 .50563 .54229 .49993 Metetra .606 .803 .975 r .54229 .49993 .55008 .50563 .56477 .50199 .55704 .49651 Metetra .611 .812 .978 r .55704 .49651 .56477 .50199 .57954 .49816 .57188 .49285 Metetra .619 .821 .978 r .57188 .49285 .57954 .49816 .5944 .49409 .5868 .48889 Metetra .629 .83 .977 r .5868 .48889 .5944 .49409 .60933 .48974 .6018 .48459 Metetra .642 .836 .975 r .6018 .48459 .60933 .48974 .62433 .48509 .61688 .4799 Metetra .655 .84 .971 r .61688 .4799 .62433 .48509 .63941 .48016 .63202 .47484 Metetra .667 .84 .966 r .63202 .47484 .63941 .48016 .65455 .47494 .64723 .46942 Metetra .678 .838 .96 r .64723 .46942 .65455 .47494 .66977 .46948 .66251 .46369 Metetra .686 .834 .954 r .66251 .46369 .66977 .46948 .68505 .46381 .67785 .45772 Metetra .692 .828 .948 r .67785 .45772 .68505 .46381 .70039 .45798 .69326 .45157 Metetra .695 .821 .943 r .69326 .45157 .70039 .45798 .71581 .45203 .70873 .4453 Metetra .697 .814 .938 r .70873 .4453 .71581 .45203 .73129 .446 .72428 .43897 Metetra .697 .807 .933 r .72428 .43897 .73129 .446 .74685 .43993 .73989 .43261 Metetra .697 .8 .929 r .73989 .43261 .74685 .43993 .76249 .43383 .75558 .42624 Metetra .695 .794 .926 r .75558 .42624 .76249 .43383 .7782 .42774 .77135 .41991 Metetra .694 .788 .923 r .77135 .41991 .7782 .42774 .79398 .42165 .78719 .4136 Metetra .692 .783 .921 r .78719 .4136 .79398 .42165 .80985 .41558 .80312 .40733 Metetra .69 .779 .919 r .80312 .40733 .80985 .41558 .8258 .40954 .81913 .40111 Metetra .689 .775 .917 r .81913 .40111 .8258 .40954 .84184 .40353 .83522 .39493 Metetra .687 .771 .916 r .83522 .39493 .84184 .40353 .85796 .39753 .8514 .38879 Metetra .685 .768 .915 r .8514 .38879 .85796 .39753 .87416 .39157 .86767 .38269 Metetra .684 .766 .914 r .86767 .38269 .87416 .39157 .89046 .38562 .88403 .37662 Metetra .682 .764 .913 r .88403 .37662 .89046 .38562 .90684 .3797 .90048 .37059 Metetra .681 .761 .912 r .90048 .37059 .90684 .3797 .92331 .37379 .91701 .36457 Metetra .638 .746 .927 r .32266 .54131 .33123 .54881 .34475 .5453 .33622 .5378 Metetra .636 .747 .929 r .33622 .5378 .34475 .5453 .35834 .54181 .34985 .53434 Metetra .633 .748 .931 r .34985 .53434 .35834 .54181 .37201 .53837 .36356 .53092 Metetra .631 .749 .933 r .36356 .53092 .37201 .53837 .38575 .53496 .37734 .52756 Metetra .628 .75 .935 r .37734 .52756 .38575 .53496 .39957 .53161 .3912 .52426 Metetra .624 .752 .938 r .3912 .52426 .39957 .53161 .41347 .5283 .40514 .52103 Metetra .62 .754 .942 r .40514 .52103 .41347 .5283 .42745 .52504 .41916 .51787 Metetra .616 .756 .945 r .41916 .51787 .42745 .52504 .4415 .52183 .43326 .51478 Metetra .611 .759 .95 r .43326 .51478 .4415 .52183 .45564 .51868 .44744 .51178 Metetra .606 .763 .954 r .44744 .51178 .45564 .51868 .46986 .51557 .46171 .50884 Metetra .6 .767 .959 r .46171 .50884 .46986 .51557 .48417 .51248 .47607 .50598 Metetra .595 .773 .965 r .47607 .50598 .48417 .51248 .49857 .50941 .49051 .50316 Metetra .59 .78 .971 r .49051 .50316 .49857 .50941 .51305 .50633 .50505 .50036 Metetra .585 .788 .976 r .50505 .50036 .51305 .50633 .52763 .50318 .51968 .49752 Metetra .583 .798 .981 r .51968 .49752 .52763 .50318 .54229 .49993 .53441 .49459 Metetra .583 .809 .985 r .53441 .49459 .54229 .49993 .55704 .49651 .54923 .49148 Metetra .588 .821 .988 r .54923 .49148 .55704 .49651 .57188 .49285 .56413 .48809 Metetra .598 .834 .989 r .56413 .48809 .57188 .49285 .5868 .48889 .57913 .48432 Metetra .612 .846 .988 r .57913 .48432 .5868 .48889 .6018 .48459 .5942 .4801 Metetra .63 .855 .985 r .5942 .4801 .6018 .48459 .61688 .4799 .60935 .47537 Metetra .65 .86 .98 r .60935 .47537 .61688 .4799 .63202 .47484 .62457 .47014 Metetra .668 .861 .973 r .62457 .47014 .63202 .47484 .64723 .46942 .63985 .46445 Metetra .682 .858 .966 r .63985 .46445 .64723 .46942 .66251 .46369 .65519 .45837 Metetra .693 .852 .959 r .65519 .45837 .66251 .46369 .67785 .45772 .6706 .452 Metetra .7 .843 .951 r .6706 .452 .67785 .45772 .69326 .45157 .68606 .44544 Metetra .704 .833 .944 r .68606 .44544 .69326 .45157 .70873 .4453 .70159 .43876 Metetra .705 .823 .938 r .70159 .43876 .70873 .4453 .72428 .43897 .71719 .43205 Metetra .704 .814 .933 r .71719 .43205 .72428 .43897 .73989 .43261 .73286 .42534 Metetra .703 .806 .929 r .73286 .42534 .73989 .43261 .75558 .42624 .7486 .41867 Metetra .701 .798 .925 r .7486 .41867 .75558 .42624 .77135 .41991 .76442 .41205 Metetra .698 .791 .922 r .76442 .41205 .77135 .41991 .78719 .4136 .78033 .4055 Metetra .696 .785 .92 r .78033 .4055 .78719 .4136 .80312 .40733 .79631 .39902 Metetra .694 .78 .918 r .79631 .39902 .80312 .40733 .81913 .40111 .81238 .3926 Metetra .691 .776 .916 r .81238 .3926 .81913 .40111 .83522 .39493 .82853 .38625 Metetra .689 .772 .915 r .82853 .38625 .83522 .39493 .8514 .38879 .84478 .37996 Metetra .687 .769 .914 r .84478 .37996 .8514 .38879 .86767 .38269 .86111 .37371 Metetra .685 .766 .913 r .86111 .37371 .86767 .38269 .88403 .37662 .87753 .36752 Metetra .684 .764 .912 r .87753 .36752 .88403 .37662 .90048 .37059 .89404 .36136 Metetra .682 .762 .912 r .89404 .36136 .90048 .37059 .91701 .36457 .91065 .35524 Metetra .636 .745 .927 r .31401 .53371 .32266 .54131 .33622 .5378 .32761 .53021 Metetra .634 .746 .929 r .32761 .53021 .33622 .5378 .34985 .53434 .34128 .52677 Metetra .631 .747 .931 r .34128 .52677 .34985 .53434 .36356 .53092 .35503 .52339 Metetra .628 .748 .933 r .35503 .52339 .36356 .53092 .37734 .52756 .36885 .52008 Metetra .624 .749 .936 r .36885 .52008 .37734 .52756 .3912 .52426 .38275 .51684 Metetra .62 .75 .939 r .38275 .51684 .3912 .52426 .40514 .52103 .39673 .51369 Metetra .615 .752 .943 r .39673 .51369 .40514 .52103 .41916 .51787 .41079 .51063 Metetra .609 .754 .947 r .41079 .51063 .41916 .51787 .43326 .51478 .42493 .50769 Metetra .603 .757 .952 r .42493 .50769 .43326 .51478 .44744 .51178 .43915 .50485 Metetra .596 .761 .958 r .43915 .50485 .44744 .51178 .46171 .50884 .45347 .50214 Metetra .587 .765 .964 r .45347 .50214 .46171 .50884 .47607 .50598 .46787 .49954 Metetra .578 .771 .971 r .46787 .49954 .47607 .50598 .49051 .50316 .48237 .49704 Metetra .569 .778 .978 r .48237 .49704 .49051 .50316 .50505 .50036 .49696 .49461 Metetra .559 .788 .984 r .49696 .49461 .50505 .50036 .51968 .49752 .51164 .49219 Metetra .551 .799 .99 r .51164 .49219 .51968 .49752 .53441 .49459 .52643 .48972 Metetra .547 .812 .995 r .52643 .48972 .53441 .49459 .54923 .49148 .54132 .48706 Metetra .549 .828 .997 r .54132 .48706 .54923 .49148 .56413 .48809 .5563 .48408 Metetra .56 .845 .997 r .5563 .48408 .56413 .48809 .57913 .48432 .57137 .48062 Metetra .58 .863 .995 r .57137 .48062 .57913 .48432 .5942 .4801 .58653 .47654 Metetra .608 .877 .991 r .58653 .47654 .5942 .4801 .60935 .47537 .60176 .47178 Metetra .638 .887 .986 r .60176 .47178 .60935 .47537 .62457 .47014 .61705 .46631 Metetra .666 .889 .979 r .61705 .46631 .62457 .47014 .63985 .46445 .6324 .46022 Metetra .688 .884 .97 r .6324 .46022 .63985 .46445 .65519 .45837 .64781 .45365 Metetra .703 .874 .962 r .64781 .45365 .65519 .45837 .6706 .452 .66327 .44673 Metetra .711 .861 .953 r .66327 .44673 .6706 .452 .68606 .44544 .67879 .43963 Metetra .715 .848 .945 r .67879 .43963 .68606 .44544 .70159 .43876 .69438 .43244 Metetra .715 .834 .938 r .69438 .43244 .70159 .43876 .71719 .43205 .71003 .42525 Metetra .713 .822 .933 r .71003 .42525 .71719 .43205 .73286 .42534 .72575 .41813 Metetra .71 .811 .928 r .72575 .41813 .73286 .42534 .7486 .41867 .74155 .41109 Metetra .707 .802 .924 r .74155 .41109 .7486 .41867 .76442 .41205 .75743 .40416 Metetra .704 .794 .921 r .75743 .40416 .76442 .41205 .78033 .4055 .77339 .39734 Metetra .7 .787 .918 r .77339 .39734 .78033 .4055 .79631 .39902 .78943 .39062 Metetra .697 .782 .917 r .78943 .39062 .79631 .39902 .81238 .3926 .80556 .384 Metetra .694 .777 .915 r .80556 .384 .81238 .3926 .82853 .38625 .82177 .37746 Metetra .692 .773 .914 r .82177 .37746 .82853 .38625 .84478 .37996 .83808 .37101 Metetra .689 .769 .913 r .83808 .37101 .84478 .37996 .86111 .37371 .85447 .36462 Metetra .687 .766 .912 r .85447 .36462 .86111 .37371 .87753 .36752 .87096 .3583 Metetra .685 .764 .911 r .87096 .3583 .87753 .36752 .89404 .36136 .88753 .35202 Metetra .683 .762 .911 r .88753 .35202 .89404 .36136 .91065 .35524 .90421 .34579 Metetra .635 .744 .927 r .30527 .52601 .31401 .53371 .32761 .53021 .31891 .52253 Metetra .632 .745 .929 r .31891 .52253 .32761 .53021 .34128 .52677 .33262 .51911 Metetra .629 .746 .931 r .33262 .51911 .34128 .52677 .35503 .52339 .34641 .51576 Metetra .625 .746 .934 r .34641 .51576 .35503 .52339 .36885 .52008 .36027 .51249 Metetra .621 .747 .937 r .36027 .51249 .36885 .52008 .38275 .51684 .37421 .50932 Metetra .616 .749 .94 r .37421 .50932 .38275 .51684 .39673 .51369 .38823 .50626 Metetra .61 .75 .945 r .38823 .50626 .39673 .51369 .41079 .51063 .40233 .50332 Metetra .602 .752 .949 r .40233 .50332 .41079 .51063 .42493 .50769 .41651 .50052 Metetra .594 .755 .955 r .41651 .50052 .42493 .50769 .43915 .50485 .43078 .49789 Metetra .584 .758 .961 r .43078 .49789 .43915 .50485 .45347 .50214 .44513 .49542 Metetra .572 .762 .968 r .44513 .49542 .45347 .50214 .46787 .49954 .45958 .49314 Metetra .557 .767 .976 r .45958 .49314 .46787 .49954 .48237 .49704 .47412 .49104 Metetra .54 .774 .984 r .47412 .49104 .48237 .49704 .49696 .49461 .48877 .48909 Metetra .521 .783 .992 r .48877 .48909 .49696 .49461 .51164 .49219 .50351 .48725 Metetra .502 .793 .998 r .50351 .48725 .51164 .49219 .52643 .48972 .51836 .48542 Metetra .486 .807 1 r .51836 .48542 .52643 .48972 .54132 .48706 .53331 .48344 Metetra .478 .824 .998 r .53331 .48344 .54132 .48706 .5563 .48408 .54837 .48109 Metetra .486 .846 .992 r .54837 .48109 .5563 .48408 .57137 .48062 .56353 .47813 Metetra .513 .871 .986 r .56353 .47813 .57137 .48062 .58653 .47654 .57877 .47432 Metetra .559 .896 .982 r .57877 .47432 .58653 .47654 .60176 .47178 .59409 .46952 Metetra .612 .915 .979 r .59409 .46952 .60176 .47178 .61705 .46631 .60946 .46371 Metetra .66 .922 .975 r .60946 .46371 .61705 .46631 .6324 .46022 .62489 .45703 Metetra .696 .916 .969 r .62489 .45703 .6324 .46022 .64781 .45365 .64036 .44973 Metetra .717 .902 .961 r .64036 .44973 .64781 .45365 .66327 .44673 .65588 .44205 Metetra .727 .883 .952 r .65588 .44205 .66327 .44673 .67879 .43963 .67146 .4342 Metetra .73 .864 .944 r .67146 .4342 .67879 .43963 .69438 .43244 .6871 .42634 Metetra .728 .846 .937 r .6871 .42634 .69438 .43244 .71003 .42525 .7028 .41857 Metetra .724 .83 .931 r .7028 .41857 .71003 .42525 .72575 .41813 .71857 .41095 Metetra .719 .817 .926 r .71857 .41095 .72575 .41813 .74155 .41109 .73443 .4035 Metetra .714 .806 .922 r .73443 .4035 .74155 .41109 .75743 .40416 .75036 .39621 Metetra .709 .797 .919 r .75036 .39621 .75743 .40416 .77339 .39734 .76637 .38909 Metetra .705 .789 .917 r .76637 .38909 .77339 .39734 .78943 .39062 .78247 .38212 Metetra .701 .783 .915 r .78247 .38212 .78943 .39062 .80556 .384 .79866 .37528 Metetra .697 .778 .914 r .79866 .37528 .80556 .384 .82177 .37746 .81494 .36856 Metetra .694 .773 .912 r .81494 .36856 .82177 .37746 .83808 .37101 .8313 .36194 Metetra .691 .77 .912 r .8313 .36194 .83808 .37101 .85447 .36462 .84776 .35541 Metetra .689 .767 .911 r .84776 .35541 .85447 .36462 .87096 .3583 .86431 .34895 Metetra .686 .764 .91 r .86431 .34895 .87096 .3583 .88753 .35202 .88096 .34255 Metetra .685 .762 .91 r .88096 .34255 .88753 .35202 .90421 .34579 .89769 .33621 Metetra .634 .743 .927 r .29645 .51821 .30527 .52601 .31891 .52253 .31013 .51474 Metetra .63 .744 .929 r .31013 .51474 .31891 .52253 .33262 .51911 .32388 .51134 Metetra .627 .744 .932 r .32388 .51134 .33262 .51911 .34641 .51576 .33771 .50802 Metetra .622 .745 .934 r .33771 .50802 .34641 .51576 .36027 .51249 .35161 .50479 Metetra .617 .745 .937 r .35161 .50479 .36027 .51249 .37421 .50932 .36559 .50169 Metetra .611 .746 .941 r .36559 .50169 .37421 .50932 .38823 .50626 .37964 .49871 Metetra .604 .747 .946 r .37964 .49871 .38823 .50626 .40233 .50332 .39378 .4959 Metetra .595 .749 .951 r .39378 .4959 .40233 .50332 .41651 .50052 .408 .49327 Metetra .584 .751 .957 r .408 .49327 .41651 .50052 .43078 .49789 .42231 .49085 Metetra .57 .753 .964 r .42231 .49085 .43078 .49789 .44513 .49542 .43671 .48867 Metetra .553 .756 .972 r .43671 .48867 .44513 .49542 .45958 .49314 .4512 .48676 Metetra .53 .76 .981 r .4512 .48676 .45958 .49314 .47412 .49104 .46578 .48514 Metetra .502 .764 .99 r .46578 .48514 .47412 .49104 .48877 .48909 .48047 .48381 Metetra .466 .769 .996 r .48047 .48381 .48877 .48909 .50351 .48725 .49527 .48274 Metetra .422 .773 .996 r .49527 .48274 .50351 .48725 .51836 .48542 .51018 .48182 Metetra .376 .776 .985 r .51018 .48182 .51836 .48542 .53331 .48344 .5252 .48086 Metetra .337 .781 .96 r .5252 .48086 .53331 .48344 .54837 .48109 .54034 .47954 Metetra .326 .795 .928 r .54034 .47954 .54837 .48109 .56353 .47813 .55559 .47744 Metetra .36 .826 .906 r .55559 .47744 .56353 .47813 .57877 .47432 .57093 .47411 Metetra .439 .873 .905 r .57093 .47411 .57877 .47432 .59409 .46952 .58635 .46926 Metetra .543 .922 .922 r .58635 .46926 .59409 .46952 .60946 .46371 .60181 .4629 Metetra .638 .951 .942 r .60181 .4629 .60946 .46371 .62489 .45703 .61731 .45529 Metetra .703 .953 .953 r .61731 .45529 .62489 .45703 .64036 .44973 .63285 .44687 Metetra .736 .935 .953 r .63285 .44687 .64036 .44973 .65588 .44205 .64843 .43807 Metetra .749 .908 .947 r .64843 .43807 .65588 .44205 .67146 .4342 .66405 .42921 Metetra .749 .881 .94 r .66405 .42921 .67146 .4342 .6871 .42634 .67974 .42048 Metetra .744 .858 .933 r .67974 .42048 .6871 .42634 .7028 .41857 .69549 .41199 Metetra .736 .838 .927 r .69549 .41199 .7028 .41857 .71857 .41095 .71132 .40378 Metetra .729 .822 .923 r .71132 .40378 .71857 .41095 .73443 .4035 .72722 .39585 Metetra .722 .809 .919 r .72722 .39585 .73443 .4035 .75036 .39621 .74321 .38818 Metetra .715 .799 .916 r .74321 .38818 .75036 .39621 .76637 .38909 .75928 .38073 Metetra .71 .79 .914 r .75928 .38073 .76637 .38909 .78247 .38212 .77543 .3735 Metetra .705 .783 .913 r .77543 .3735 .78247 .38212 .79866 .37528 .79168 .36644 Metetra .7 .778 .912 r .79168 .36644 .79866 .37528 .81494 .36856 .80802 .35952 Metetra .696 .773 .911 r .80802 .35952 .81494 .36856 .8313 .36194 .82445 .35274 Metetra .693 .769 .91 r .82445 .35274 .8313 .36194 .84776 .35541 .84097 .34606 Metetra .69 .766 .91 r .84097 .34606 .84776 .35541 .86431 .34895 .85759 .33947 Metetra .688 .764 .909 r .85759 .33947 .86431 .34895 .88096 .34255 .8743 .33295 Metetra .686 .761 .909 r .8743 .33295 .88096 .34255 .89769 .33621 .89111 .3265 Metetra .632 .742 .927 r .28755 .51031 .29645 .51821 .31013 .51474 .30127 .50684 Metetra .629 .742 .929 r .30127 .50684 .31013 .51474 .32388 .51134 .31506 .50345 Metetra .625 .742 .932 r .31506 .50345 .32388 .51134 .33771 .50802 .32892 .50015 Metetra .62 .743 .934 r .32892 .50015 .33771 .50802 .35161 .50479 .34286 .49697 Metetra .614 .743 .938 r .34286 .49697 .35161 .50479 .36559 .50169 .35688 .49393 Metetra .607 .744 .941 r .35688 .49393 .36559 .50169 .37964 .49871 .37097 .49104 Metetra .599 .744 .946 r .37097 .49104 .37964 .49871 .39378 .4959 .38515 .48835 Metetra .588 .745 .952 r .38515 .48835 .39378 .4959 .408 .49327 .39941 .48588 Metetra .574 .746 .958 r .39941 .48588 .408 .49327 .42231 .49085 .41376 .48369 Metetra .556 .747 .966 r .41376 .48369 .42231 .49085 .43671 .48867 .42819 .48183 Metetra .531 .748 .975 r .42819 .48183 .43671 .48867 .4512 .48676 .44272 .48034 Metetra .498 .748 .984 r .44272 .48034 .4512 .48676 .46578 .48514 .45734 .47929 Metetra .452 .746 .992 r .45734 .47929 .46578 .48514 .48047 .48381 .47207 .47873 Metetra .386 .739 .991 r .47207 .47873 .48047 .48381 .49527 .48274 .48692 .47868 Metetra .294 .72 .971 r .48692 .47868 .49527 .48274 .51018 .48182 .50188 .47906 Metetra .179 .681 .91 r .50188 .47906 .51018 .48182 .5252 .48086 .51697 .47967 Metetra .068 .626 .804 r .51697 .47967 .5252 .48086 .54034 .47954 .5322 .48004 Metetra .5322 .48004 .54034 .47954 .55559 .47744 .54755 .47946 Metetra .54755 .47946 .55559 .47744 .57093 .47411 .56301 .47705 Metetra .56301 .47705 .57093 .47411 .58635 .46926 .57853 .47217 Metetra .37 .823 .713 r .57853 .47217 .58635 .46926 .60181 .4629 .5941 .46481 Metetra .568 .937 .821 r .5941 .46481 .60181 .4629 .61731 .45529 .60968 .4556 Metetra .702 .981 .896 r .60968 .4556 .61731 .45529 .63285 .44687 .62527 .44539 Metetra .761 .969 .927 r .62527 .44539 .63285 .44687 .64843 .43807 .6409 .43493 Metetra .777 .934 .934 r .6409 .43493 .64843 .43807 .66405 .42921 .65657 .42466 Metetra .773 .899 .931 r .65657 .42466 .66405 .42921 .67974 .42048 .6723 .41481 Metetra .762 .868 .926 r .6723 .41481 .67974 .42048 .69549 .41199 .6881 .40545 Metetra .75 .844 .921 r .6881 .40545 .69549 .41199 .71132 .40378 .70397 .39656 Metetra .739 .825 .918 r .70397 .39656 .71132 .40378 .72722 .39585 .71993 .3881 Metetra .729 .811 .915 r .71993 .3881 .72722 .39585 .74321 .38818 .73597 .38001 Metetra .721 .799 .913 r .73597 .38001 .74321 .38818 .75928 .38073 .7521 .37223 Metetra .714 .79 .911 r .7521 .37223 .75928 .38073 .77543 .3735 .76831 .36472 Metetra .708 .783 .91 r .76831 .36472 .77543 .3735 .79168 .36644 .78462 .35744 Metetra .703 .777 .91 r .78462 .35744 .79168 .36644 .80802 .35952 .80102 .35033 Metetra .699 .773 .909 r .80102 .35033 .80802 .35952 .82445 .35274 .81752 .34338 Metetra .695 .769 .909 r .81752 .34338 .82445 .35274 .84097 .34606 .8341 .33656 Metetra .692 .766 .908 r .8341 .33656 .84097 .34606 .85759 .33947 .85079 .32984 Metetra .689 .763 .908 r .85079 .32984 .85759 .33947 .8743 .33295 .86757 .32321 Metetra .687 .761 .908 r .86757 .32321 .8743 .33295 .89111 .3265 .88445 .31665 Metetra .631 .741 .927 r .27856 .50228 .28755 .51031 .30127 .50684 .29232 .49881 Metetra .627 .741 .929 r .29232 .49881 .30127 .50684 .31506 .50345 .30614 .49543 Metetra .623 .741 .931 r .30614 .49543 .31506 .50345 .32892 .50015 .32005 .49216 Metetra .618 .741 .934 r .32005 .49216 .32892 .50015 .34286 .49697 .33403 .48901 Metetra .611 .741 .937 r .33403 .48901 .34286 .49697 .35688 .49393 .34808 .48602 Metetra .603 .741 .941 r .34808 .48602 .35688 .49393 .37097 .49104 .36222 .48321 Metetra .593 .741 .946 r .36222 .48321 .37097 .49104 .38515 .48835 .37643 .48063 Metetra .581 .74 .952 r .37643 .48063 .38515 .48835 .39941 .48588 .39073 .47833 Metetra .564 .74 .959 r .39073 .47833 .39941 .48588 .41376 .48369 .40511 .47637 Metetra .541 .739 .967 r .40511 .47637 .41376 .48369 .42819 .48183 .41958 .47483 Metetra .509 .737 .976 r .41958 .47483 .42819 .48183 .44272 .48034 .43414 .4738 Metetra .462 .731 .985 r .43414 .4738 .44272 .48034 .45734 .47929 .4488 .4734 Metetra .39 .718 .988 r .4488 .4734 .45734 .47929 .47207 .47873 .46357 .47376 Metetra .276 .685 .97 r .46357 .47376 .47207 .47873 .48692 .47868 .47845 .475 Metetra .101 .609 .894 r .47845 .475 .48692 .47868 .50188 .47906 .49346 .47719 Metetra 0 .465 .712 r .49346 .47719 .50188 .47906 .51697 .47967 .50861 .48021 Metetra .306 0 0 r .50861 .48021 .51697 .47967 .5322 .48004 .52392 .4835 Metetra .374 0 0 r .52392 .4835 .5322 .48004 .54755 .47946 .53939 .48583 Metetra .324 0 0 r .53939 .48583 .54755 .47946 .56301 .47705 .55499 .48533 Metetra .168 0 0 r .55499 .48533 .56301 .47705 .57853 .47217 .57066 .48042 Metetra .57066 .48042 .57853 .47217 .5941 .46481 .58633 .47109 Metetra .58633 .47109 .5941 .46481 .60968 .4556