(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' 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[ 13239, 429]*) (*NotebookOutlinePosition[ 48609, 1629]*) (* CellTagsIndexPosition[ 48565, 1625]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Based on Mathematica", FontSlant->"Italic"], " Notebooks for ATLAST\nby Richard Neidinger, Davidson College, 6/1/98." }], "Subsubtitle"], Cell[CellGroupData[{ Cell["\<\ Subset of ATLAST 5.2 Projects on Linear Transformations\ \>", "Subtitle"], Cell[CellGroupData[{ Cell["compositions & homogeneous coordinates", "Section"], Cell[CellGroupData[{ Cell["\<\ Initializations -- view, rotMat, transMat, scaleMat, and flag.\ \>", "Subsection"], Cell[BoxData[ \(Needs["\"]\)], "Input", InitializationCell->True], Cell["\<\ Clear[view] view[homoMat2D_, xyRange_:Automatic] := \tListPlot[ Transpose[ homoMat2D[[{1,2}]] ], \t\tPlotJoined->True, \t\tPlotRange->xyRange, \t\tAspectRatio->Automatic, \t\tPlotStyle->{Blue,Thickness[0.01]}, \t\tImageSize -> {600,400}]; view[homoMat2D_, xyRange_, ticks_] := \tListPlot[ Transpose[ homoMat2D[[{1,2}]] ], \t\tPlotJoined->True, \t\tPlotRange->xyRange, \t\tAspectRatio->Automatic, \t\tPlotStyle->{Blue,Thickness[0.01]}, \t\tTicks -> ticks, \t\tImageSize -> {600,400}]; view[homoMat2D_, radius_?NumberQ] := view[homoMat2D, {{-radius,radius},{-radius,radius}} ]\ \>", "Input", InitializationCell->True], Cell[BoxData[ \(rotMat[\[Theta]_]\ := \ N[{{Cos[\[Theta]], \(-Sin[\[Theta]]\), 0}, {Sin[\[Theta]], Cos[\[Theta]], 0}, {0, 0, 1}}]\)], "Input", InitializationCell->True], Cell[BoxData[ \(transMat[h_, k_]\ := \ N[{{1, 0, h}, {0, 1, k}, {0, 0, 1}}]\)], "Input",\ InitializationCell->True], Cell[BoxData[ \(scaleMat[s_, t_]\ := \ N[\ DiagonalMatrix[{s, t, 1}]\ ]\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(flag\ = \ Append[\n\t\tTranspose[{{0, 0}, {0, 2}, {2, 1.5}, {0, 1}}], \n\t\tTable[1, {4}]\n\t];\)\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell["Homogeneous 2D coordinates and the view function.", "Subsubsection"], Cell["\<\ For use in matrix transformations it is best to store the points in a 3\ \[Times]n matrix, where each point (x, y) is stored in a column with entries \ (x, y, 1).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(flag\ = \ Append[Transpose[pts], Table[1, {4}]];\)\), "\[IndentingNewLine]", \(MatrixForm[flag]\)}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "2", "0"}, {"0", "2", "1.5`", "1"}, {"1", "1", "1", "1"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell["\<\ For reference, I'll call any such matrix of points a \"homoMat2D.\" To plot \ a homoMat2D, we need to ListPlot the Transpose of the first two rows and use \ all of the options developed above. We can define such a function once and \ for all.\ \>", "Text"], Cell["\<\ Clear[view] view[homoMat2D_, xyRange_:Automatic] := \tListPlot[ Transpose[ homoMat2D[[{1,2}]] ], \t\tPlotJoined->True, \t\tPlotRange->xyRange, \t\tAspectRatio->Automatic, \t\tPlotStyle->Blue \t]; \t\t view[homoMat2D_, radius_?NumberQ] := view[homoMat2D, {{-radius,radius},{-radius,radius}} ]\ \>", "Input"], Cell["\<\ view has three options for the second argument as displayed below. Usually, \ you want to pick some xyRange, as in the first example, and stick with it \ throughout transformations so that you can correctly compare before and after \ transformation.\ \>", "Text"], Cell[BoxData[ \(\(view[ flag, {{\(-2\), 2}, {0, 2}}, {Range[\(-2\), 2], Range[\(-2\), 2]}];\)\)], "Input"], Cell[BoxData[ \(\(view[flag];\)\)], "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Transformations", "Subsection"], Cell["\<\ To perform any linear transformation T on all the points in a homoMat2D, use \ a 3\[Times]3 matrix that includes the standard 2\[Times]2 matrix of the \ transformation in the upper left and a one in the lower right (zeros \ elsewhere). Here is a function that returns the rotation matrix. The N \ makes computation with such a matrix more efficient.\ \>", "Text"], Cell[BoxData[{ \(rotMat[\[Theta]_]\ := \ N[{{Cos[\[Theta]], \(-Sin[\[Theta]]\), 0}, {Sin[\[Theta]], Cos[\[Theta]], 0}, {0, 0, 1}}]\), "\[IndentingNewLine]", \(MatrixForm[rotMat[theta]]\)}], "Input"], Cell[BoxData[{ \(\(Ar\ = \ rotMat[Pi/4];\)\), "\[IndentingNewLine]", \(MatrixForm[Ar]\)}], "Input"], Cell[BoxData[{ \(\(pts\ = \ {{0, 0, 1}, {0, 2, 1}, {2, 1.5, 1}, {0, 1, 1}};\)\), "\n", \(MatrixForm[pts]\)}], "Input"], Cell[BoxData[{ \(\(flag\ = Transpose[pts];\)\), "\[IndentingNewLine]", \(MatrixForm[flag]\)}], "Input"], Cell[BoxData[ \(\(view[flag, {{\(-3\), 3}, {0, 3}}];\)\)], "Input"], Cell[BoxData[ \(MatrixForm[Ar . flag]\)], "Input"], Cell["\<\ Notice how Ar multiples the x,y coordinates by the upper left 2\[Times]2 and \ leaves the \"3rd coordinate\" as one. This 3rd coordinate has no purpose yet.\ \>", "Text"], Cell[BoxData[ \(\(view[ Ar . flag, {{\(-3\), 3}, {0, 3}}, {Range[\(-3\), 3], Range[1, 3]}];\)\)], "Input"], Cell["\<\ The extra 1 enables translations (not a linear transformation in 2D) to be \ performed by matrix multiplication (a 3 coordinate linear transformation!) as shown in Lay, p. 160. \ \>", "Text"], Cell[BoxData[ \(transMat[h_, k_]\ := \ N[{{1, 0, h}, {0, 1, k}, {0, 0, 1}}]\)], "Input"], Cell[BoxData[{ \(\(At\ = \ transMat[1, .5];\)\), "\[IndentingNewLine]", \(MatrixForm[At]\)}], "Input"], Cell[BoxData[ \(MatrixForm[flag]\)], "Input"], Cell[BoxData[ \(MatrixForm[At . flag]\)], "Input"], Cell[BoxData[ \(\(view[At . flag, {{\(-3\), 3}, {0, 3}}];\)\)], "Input"], Cell["Let's save this shifted flag.", "Text"], Cell[BoxData[{ \(\(flag1\ = \ At . flag;\)\), "\[IndentingNewLine]", \(MatrixForm[flag1]\)}], "Input"], Cell["\<\ We also need a scaling transformation that will multiply x coordinates by one \ factor s and y coordinates by another factor t. If 0 < s = t < 1, then the \ transformation just shrinks the image toward the origin. With negatives, you \ can perform reflections. Let's try s=-0.5 and t=0.5 on the shifted flag1.\ \>", "Text"], Cell[BoxData[ \(scaleMat[s_, t_]\ := \ N[\ DiagonalMatrix[{s, t, 1}]\ ]\)], "Input"], Cell[BoxData[{ \(\(As = \ scaleMat[\(- .5\), .5];\)\), "\[IndentingNewLine]", \(MatrixForm[As]\)}], "Input"], Cell[BoxData[ \(MatrixForm[As . flag1]\)], "Input"], Cell[BoxData[ \(\(view[ As . flag1, {{\(-3\), 3}, {0, 3}}, {Range[\(-3\), 3], Range[1, 3]}];\)\)], "Input"], Cell["\<\ Here is how you can superimpose any two (or more) plots. We use it to see \ before and after together. You can ignore the \"friendly\" possible spelling \ errors, they indicate that Before already has a definition.\ \>", "Text"], Cell[BoxData[ \(\(before\ = \ view[flag1, {{\(-3\), 3}, {0, 3}}];\)\)], "Input"], Cell[BoxData[ \(\(after\ = \ view[As . flag1, {{\(-3\), 3}, {0, 3}}];\)\)], "Input"], Cell[BoxData[ \(\(Show[before, after];\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Flag waving", "Subsection"], Cell[BoxData[{ \(\(initA\ = \ rotMat[\(-Pi\)/5];\)\), "\[IndentingNewLine]", \(MatrixForm[initA]\)}], "Input"], Cell[BoxData[{ \(\(rotStep\ = \ rotMat[Pi/20];\)\), "\[IndentingNewLine]", \(MatrixForm[rotStep]\)}], "Input"], Cell[BoxData[{ \(\(view[ initA . flag, {{\(-3\), 3}, {0, 3}}, {Range[\(-3\), 3], Range[1, 3]}];\)\), "\n", \(\(view[ rotStep . initA . flag, {{\(-3\), 3}, {0, 3}}, {Range[\(-3\), 3], Range[1, 3]}];\)\), "\n", \(\(view[ rotStep . rotStep . initA . flag, {{\(-3\), 3}, {0, 3}}, {Range[\(-3\), 3], Range[1, 3]}];\)\)}], "Input"], Cell["\<\ Now, let's use Do to wave the flag by viewing each step and producing the \ matrix for the next step in the sequence. We use the matrix name A to denote \ the constantly changing matrix that will transform the homoMat2D. It is just \ a variable, the matrices we designed are initA, the initial value of A, and \ stepA (called rotStep here) that is the change on each step.\ \>", "Text"], Cell[BoxData[{ \(\(A\ = \ initA;\)\), "\n", \(\(Do[ view[A . flag, {{\(-3\), 3}, {0, 3}}, {Range[\(-3\), 3], Range[1, 3]}]; \n\t A\ = \ rotStep . A, \[IndentingNewLine]{i, 1, 10}];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Rising spinning pinwheel.", "Subsection"], Cell["p = N[1/Sqrt[2]];", "Input"], Cell[BoxData[{ \(\(pinwheelpts = {{0, 0, 1}, {0, 1, 1}, {p, p, 1}, {0, 0, 1}, {1, 0, 1}, {p, \(-p\), 1}, {0, 0, 1}, {0, \(-1\), 1}, {\(-p\), \(-p\), 1}, {0, 0, 1}, {\(-1\), 0, 1}, {\(-p\), p, 1}, {0, 0, 1}};\)\), "\n", \(MatrixForm[pinwheelpts]\)}], "Input"], Cell[BoxData[{ \(\(pinwheel = Transpose[pinwheelpts];\)\), "\[IndentingNewLine]", \(MatrixForm[pinwheel]\)}], "Input"], Cell["view[pinwheel,1.2];", "Input"], Cell["\<\ Let's rotate the pinwheel. Of course, you'll want to animate (Ctrl-Y) the \ result of all of these movie frames.\ \>", "Text"], Cell["\<\ rotStep = rotMat[Pi/10]; MatrixForm[rotStep]\ \>", "Input"], Cell[BoxData[{ \(\(A\ = \ IdentityMatrix[3];\)\), "\n", \(\(Do[view[A . pinwheel, 1.2]; \n\t A\ = \ rotStep . A, \[IndentingNewLine]{i, 1, 10}];\)\)}], "Input"], Cell["Let's make it rise straight up.", "Text"], Cell["\<\ upStep = {{1,0,0},{0,1,.2},{0,0,1}}; MatrixForm[upStep]\ \>", "Input"], Cell[BoxData[{ \(\(A\ = \ IdentityMatrix[3];\)\), "\n", \(\(Do[ view[A . pinwheel, {{\(-1.2\), 1.2}, {\(-1\), 3}}, {{\(-1\), 1}, Range[1, 3]}]; \n\t A\ = \ upStep . A, \n\ \ \ \ \ \ \ \ {i, 1, 10}];\)\)}], "Input"], Cell["\<\ The real objective of this problem is to make the pinwheel do both \ simultaneously. Your first idea might be to just multiply by both matrices, \ rot and up, on each step. But you may be surprised what this does.\ \>", "Text"], Cell[BoxData[{ \(\(A\ = \ IdentityMatrix[3];\)\), "\n", \(\(Do[ view[A . pinwheel, {{\(-2.5\), 2.5}, {\(-1\), 2}}, {Range[\(-2\), 2], Range[\(-1\), 2]}]; \n\t A\ = \ rotStep . upStep . A, {i, 1, 10}];\)\)}], "Input"], Cell["Why doesn't the pinwheel move straight up?", "Text"], Cell["\<\ This really is a problem because matrix multiplication is not commutative. \ Why is the action of matrix upStep.rotStep different than the action of \ matrix rotStep.upStep?\ \>", "Text"], Cell["\<\ If we want to rise straight up, we really want to do all of the rotations \ first and then all of the ups. In particular, after two steps the image \ should be accomplished by upStep.upStep.rotStep.rotStep and not by \ upStep.rotStep.upStep.rotStep. The idea here is to accumulate the net result \ of n rotations in Ar and the net result of n shifts up in Au. These are both \ variable matrices that have initial values and a change on each step.\ \>", "Text"], Cell[BoxData[{ \(\(Ar\ = \ IdentityMatrix[3];\)\), "\n", \(\(Au\ = \ IdentityMatrix[3];\)\), "\n", \(\(Do[ view[Au . Ar . pinwheel, {{\(-2\), 2}, {\(-1\), 3}}, {Range[\(-2\), 2], Range[\(-1\), 3]}]; \n\tAr\ = \ rotStep . 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10}, {2, 3}}], Cell[StyleData["Input", "Printout"], CellMargins->{{39, 0}, {4, 6}}, LinebreakAdjustments->{0.85, 2, 10, 1, 1}] }, Closed]], Cell[StyleData["InputOnly"], Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", StyleMenuListing->None, FontSize->18, FontWeight->"Bold"], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{47, 10}, {7, 5}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->InputForm, CounterIncrements->"Output", FontSize->18], Cell[StyleData["Output", "Presentation"], CellMargins->{{72, Inherited}, {10, 8}}, LineSpacing->{1, 0}], Cell[StyleData["Output", "Condensed"], CellMargins->{{41, Inherited}, {3, 2}}], Cell[StyleData["Output", "Printout"], CellMargins->{{39, 0}, {6, 4}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontSize->18, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Message", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Message", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}], Cell[StyleData["Message", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None, FontSize->18], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Print", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{4, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Graphics", ImageMargins->{{43, Inherited}, {Inherited, 0}}, StyleMenuListing->None, FontFamily->"Courier", FontSize->18], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Condensed"], ImageMargins->{{38, Inherited}, {Inherited, 0}}, Magnification->0.6], Cell[StyleData["Graphics", "Printout"], ImageMargins->{{30, Inherited}, {Inherited, 0}}, Magnification->0.8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->18, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"]], Cell[StyleData["CellLabel", "Condensed"]], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Inline Formatting", "Section"], Cell["\<\ These styles are for modifying individual words or letters in a cell \ exclusive of the cell tag.\ \>", "Text"], Cell[StyleData["RM"], StyleMenuListing->None, FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["BF"], StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["IT"], StyleMenuListing->None, FontSlant->"Italic"], Cell[StyleData["TR"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["TI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["TB"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["TBI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["MR"], HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["MO"], HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["MB"], HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["MBO"], HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["SR"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["SO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SB"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["SBO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[StyleData["SO10"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "Printout"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "EnhancedPrintout"], StyleMenuListing->None, FontFamily->"Futura", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["InlineFormula", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["DisplayFormula", "Condensed"], LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["DisplayFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellFrame->{{0, 0}, {0.5, 0.5}}, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, Hyphenation->False, LanguageCategory->"Formula", ScriptLevel->1, FontFamily->"Courier"], Cell[StyleData["Program", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Program", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["Program", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->9] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext ButtonBoxes. The \ \"Hyperlink\" style is for links within the same Notebook, or between \ Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], FontSize->16], Cell[StyleData["Hyperlink", "Condensed"], FontSize->11], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line help \ system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"], FontSize->16], Cell[StyleData["MainBookLink", "Condensed"], FontSize->11], Cell[StyleData["MainBookLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"], FontSize->16], Cell[StyleData["AddOnsLink", "Condensed"], FontSize->11], Cell[StyleData["AddOnsLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLink", "Condensed"], FontSize->11], Cell[StyleData["RefGuideLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"], FontSize->16], Cell[StyleData["GettingStartedLink", "Condensed"], FontSize->11], Cell[StyleData["GettingStartedLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"], FontSize->16], Cell[StyleData["OtherInformationLink", "Condensed"], FontSize->11], Cell[StyleData["OtherInformationLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->9, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard ButtonFunctions, for use \ in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ 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FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder objects in \ palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Placeholder->True, StyleMenuListing->None, FontSlant->"Italic", FontColor->RGBColor[0.890623, 0.864698, 0.384756], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PrimaryPlaceholder"], StyleMenuListing->None, DrawHighlighted->True, FontSlant->"Italic", Background->RGBColor[0.912505, 0.891798, 0.507774], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["PrimaryPlaceholder", "Presentation"]], Cell[StyleData["PrimaryPlaceholder", "Condensed"]], Cell[StyleData["PrimaryPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles of most \ cells. If a cell's FormatType matches the name of one of the styles defined \ below, then that style is applied between the cell's style and its own \ options. This is particularly true of Input and Output.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, Hyphenation->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", FontSize->12, Background->GrayLevel[1]], Cell[StyleData["InputForm"], InputAutoReplacements->{}, AllowInlineCells->False, Hyphenation->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{0.6, 1}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an inline cell \ within another.\ \>", "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]], Cell[CellGroupData[{ Cell["Automatic Styles", "Section"], Cell["\<\ The cells below define styles that are used to affect the display of certain \ types of objects in typeset expressions. For example, \"UnmatchedBracket\" \ style defines how unmatched bracket, curly bracket, and parenthesis \ characters are displayed (typically by coloring them to make them stand out).\ \ \>", "Text"], Cell[StyleData["UnmatchedBracket"], StyleMenuListing->None, FontColor->RGBColor[0.760006, 0.330007, 0.8]] }, Closed]] }, Open ]] }] ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1754, 51, 171, 4, 71, "Subsubtitle"], Cell[CellGroupData[{ Cell[1950, 59, 85, 3, 93, "Subtitle"], Cell[CellGroupData[{ Cell[2060, 66, 57, 0, 69, "Section"], Cell[CellGroupData[{ Cell[2142, 70, 92, 2, 50, "Subsection"], Cell[2237, 74, 90, 2, 58, "Input", InitializationCell->True], Cell[2330, 78, 632, 20, 449, "Input", InitializationCell->True], Cell[2965, 100, 195, 4, 58, "Input", InitializationCell->True], Cell[3163, 106, 123, 3, 58, "Input", InitializationCell->True], Cell[3289, 111, 117, 2, 35, "Input", InitializationCell->True], Cell[3409, 115, 187, 4, 113, "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[3621, 123, 74, 0, 48, "Subsubsection"], Cell[3698, 125, 186, 4, 62, "Text"], Cell[CellGroupData[{ Cell[3909, 133, 147, 3, 61, "Input"], Cell[4059, 138, 246, 7, 73, "Output"] }, Open ]], Cell[4320, 148, 268, 5, 62, "Text"], Cell[4591, 155, 320, 12, 265, "Input"], Cell[4914, 169, 274, 5, 86, "Text"], Cell[5191, 176, 129, 3, 35, "Input"], Cell[5323, 181, 48, 1, 35, "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[5420, 188, 37, 0, 34, "Subsection"], Cell[5460, 190, 375, 6, 86, "Text"], Cell[5838, 198, 228, 4, 84, "Input"], Cell[6069, 204, 109, 2, 84, "Input"], Cell[6181, 208, 128, 2, 84, "Input"], Cell[6312, 212, 113, 2, 84, "Input"], Cell[6428, 216, 71, 1, 58, "Input"], Cell[6502, 219, 54, 1, 35, "Input"], Cell[6559, 222, 182, 4, 62, "Text"], Cell[6744, 228, 129, 3, 58, "Input"], Cell[6876, 233, 203, 4, 62, "Text"], Cell[7082, 239, 93, 1, 58, "Input"], Cell[7178, 242, 113, 2, 84, "Input"], Cell[7294, 246, 49, 1, 58, "Input"], Cell[7346, 249, 54, 1, 58, "Input"], Cell[7403, 252, 76, 1, 58, "Input"], Cell[7482, 255, 45, 0, 38, "Text"], Cell[7530, 257, 112, 2, 84, "Input"], Cell[7645, 261, 336, 5, 86, "Text"], Cell[7984, 268, 89, 1, 58, "Input"], Cell[8076, 271, 118, 2, 84, "Input"], Cell[8197, 275, 55, 1, 58, "Input"], Cell[8255, 278, 130, 3, 58, "Input"], Cell[8388, 283, 240, 4, 62, "Text"], Cell[8631, 289, 85, 1, 58, "Input"], Cell[8719, 292, 89, 1, 58, "Input"], Cell[8811, 295, 57, 1, 58, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[8905, 301, 33, 0, 34, "Subsection"], Cell[8941, 303, 120, 2, 84, "Input"], Cell[9064, 307, 120, 2, 84, "Input"], Cell[9187, 311, 407, 10, 110, "Input"], Cell[9597, 323, 398, 6, 86, "Text"], Cell[9998, 331, 236, 5, 136, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[10271, 341, 47, 0, 34, "Subsection"], Cell[10321, 343, 34, 0, 58, "Input"], Cell[10358, 345, 306, 5, 136, "Input"], Cell[10667, 352, 127, 2, 84, "Input"], Cell[10797, 356, 36, 0, 58, "Input"], Cell[10836, 358, 137, 3, 38, "Text"], Cell[10976, 363, 69, 3, 81, "Input"], Cell[11048, 368, 179, 3, 136, "Input"], Cell[11230, 373, 47, 0, 38, "Text"], Cell[11280, 375, 80, 3, 81, "Input"], Cell[11363, 380, 254, 5, 136, "Input"], Cell[11620, 387, 239, 4, 62, "Text"], Cell[11862, 393, 256, 5, 110, "Input"], Cell[12121, 400, 58, 0, 38, "Text"], Cell[12182, 402, 198, 4, 62, "Text"], Cell[12383, 408, 473, 7, 110, "Text"], Cell[12859, 417, 340, 7, 162, "Input"] }, Closed]] }, Open ]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)