(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 43587, 1145]*) (*NotebookOutlinePosition[ 44285, 1169]*) (* CellTagsIndexPosition[ 44241, 1165]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Parametrizations of Curves", "Title", CellFrame->True, TextAlignment->Center, Background->RGBColor[0, 1, 0]], Cell["\<\ This notebook is by Steven Amgott. Please send any questions or comments to \ samgott1@swarthmore.edu. Feel free to use and distribute this notebook, but \ keep this author information in any copy you use or distribute.\ \>", "SmallText"], Cell[TextData[{ "In any input cell containing ", StyleBox["xxx", FontColor->RGBColor[1, 0, 1]], " , you must replace it with your input before evaluating the cell. In \ general, anything in ", StyleBox["magenta", FontColor->RGBColor[1, 0, 1]], " is something you can, and possibly should, change." }], "Text"], Cell[CellGroupData[{ Cell["\<\ Initialization. (Can be skipped, if you answer \"Yes\" to the initialization \ request.)\ \>", "Section"], Cell["\<\ The cells in this section (including the two subsections) are initialization \ cells, and will be automatically evaluated if you answer \"Yes\" to the \ initialization request. If you do not answer \"Yes\" to this request, you \ must evaluate them before creating the graphs in the sections below.\ \>", "Text"], Cell[BoxData[ \(Needs["\"]\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell["Eliminating some unnecessary warning messages.", "Subsection"], Cell[BoxData[{ \(\(Off[General::"\"];\)\), "\n", \(\(Off[General::"\"];\)\)}], "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell["Functions to create the graphs.", "Subsection"], Cell["\<\ The functions below create the t-y graphs, the x-t graphs, and the x-y graphs \ for all the 2-D figures below. \ \>", "Text"], Cell[BoxData[{\(Clear[t, s, tmin, tmax, ttPlot, tPlot, tyPlot, xtPlot, parPlot, parPlotComp]\), "\n", RowBox[{ RowBox[{\(ttPlot[t_]\), ":=", RowBox[{"ParametricPlot", "[", RowBox[{\({s, s}\), ",", \({s, tmin, tmax}\), ",", RowBox[{"AspectRatio", "\[Rule]", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], ",", \(DisplayFunction \[Rule] Identity\), ",", \(AxesLabel \[Rule] {"\", "\< t\>"}\), ",", RowBox[{"Epilog", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", StyleBox[".05", FontColor->RGBColor[1, 0, 1]], "]"}], ",", \(RGBColor[1, 0, 0]\), ",", \(Point[{t, t}]\)}], "}"}]}]}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{\(tPlot[t_]\), ":=", RowBox[{"ParametricPlot", "[", RowBox[{\({s, 0}\), ",", \({s, tmin, tmax}\), ",", RowBox[{"AspectRatio", "\[Rule]", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], ",", \(DisplayFunction \[Rule] Identity\), ",", \(Axes \[Rule] {True, False}\), ",", \(AxesLabel \[Rule] {"\", "\< \>"}\), ",", RowBox[{"Epilog", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", StyleBox[".05", FontColor->RGBColor[1, 0, 1]], "]"}], ",", \(RGBColor[1, 0, 0]\), ",", \(Point[{t, 0}]\)}], "}"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{\(tyPlot[t_]\), ":=", RowBox[{"ParametricPlot", "[", RowBox[{\({s, y[s]}\), ",", \({s, tmin, tmax}\), ",", RowBox[{"AspectRatio", "\[Rule]", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], ",", \(DisplayFunction \[Rule] Identity\), ",", \(AxesLabel \[Rule] {"\", "\"}\), ",", RowBox[{"Epilog", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", StyleBox[".05", FontColor->RGBColor[1, 0, 1]], "]"}], ",", \(RGBColor[1, 0, 0]\), ",", \(Point[{t, y[t]}]\)}], "}"}]}]}], "]"}]}], ";"}], "\n", RowBox[{\(xtPlot[t_]\), ":=", RowBox[{"ParametricPlot", "[", RowBox[{\({x[s], s}\), ",", \({s, tmin, tmax}\), ",", RowBox[{"AspectRatio", "\[Rule]", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], ",", \(DisplayFunction \[Rule] Identity\), ",", \(AxesLabel \[Rule] {"\", "\"}\), ",", RowBox[{"Epilog", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", StyleBox[".05", FontColor->RGBColor[1, 0, 1]], "]"}], ",", \(RGBColor[1, 0, 0]\), ",", \(Point[{x[t], t}]\)}], "}"}]}]}], "]"}]}], "\n", RowBox[{\(parPlot[t_]\), ":=", RowBox[{"ParametricPlot", "[", RowBox[{\({x[s], y[s]}\), ",", \({s, tmin - .001, t}\), ",", \(PlotRange \[Rule] {\(PlotRange[ xtPlot[t]]\)[\([1]\)], \(PlotRange[tyPlot[t]]\)[\([2]\)]}\), ",", RowBox[{"AspectRatio", "\[Rule]", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], ",", \(DisplayFunction \[Rule] Identity\), ",", \(AxesLabel \[Rule] {"\", "\"}\), ",", RowBox[{"Epilog", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", StyleBox[".05", FontColor->RGBColor[1, 0, 1]], "]"}], ",", \(RGBColor[1, 0, 0]\), ",", \(Point[{x[t], y[t]}]\)}], "}"}]}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{\(parPlotComp[t_]\), ":=", RowBox[{"ParametricPlot", "[", RowBox[{\({x[s], y[s]}\), ",", \({s, tmin, tmax}\), ",", \(PlotRange \[Rule] {\(PlotRange[ xtPlot[t]]\)[\([1]\)], \(PlotRange[tyPlot[t]]\)[\([2]\)]}\), ",", RowBox[{"AspectRatio", "\[Rule]", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], ",", \(DisplayFunction \[Rule] Identity\), ",", \(AxesLabel \[Rule] {"\", "\"}\), ",", RowBox[{"Epilog", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", StyleBox[".05", FontColor->RGBColor[1, 0, 1]], "]"}], ",", \(RGBColor[1, 0, 0]\), ",", \(Point[{x[t], y[t]}]\)}], "}"}]}]}], "]"}]}]}], "Input", InitializationCell->True] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["The unit circle.", "Section"], Cell[CellGroupData[{ Cell["A first parametrization.", "Subsection"], Cell["\<\ The input cell below creates a plot of the unit circle using the \ parametrization x = cos(t), y = sin(t), and animates the parametric plot. \ The parametric plot is created and drawn at the same time as the plots of \ x(t) and y(t) are traced, so you can see how the individual functions give \ rise to the final curve.\ \>", "Text"], Cell[TextData[{ "You can adjust the number of frames in the animation by changing the value \ of n. (You will need to reduce it if you are working on a slow computer.) \ ", "To see the animation, evaluate the cell, select the entire collection of \ output graphs, and choose ", StyleBox["Animate Selected Graphics", FontColor->RGBColor[0, 0, 1]], " from the ", StyleBox["Cell", FontColor->RGBColor[0, 0, 1]], " menu item. VCR like controls will appear at the bottom of the notebook \ window so that you can slow (or speed) the animation, pause it, and do other \ operations." }], "Text"], Cell[BoxData[{\(Clear[n, t, x, y, tmin, tmax]\), "\n", RowBox[{ RowBox[{"n", "=", StyleBox["50", FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[t]\), FontColor->RGBColor[1, 0, 1]]}]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{"tmax", "=", StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{"Animate", "[", RowBox[{\(Show[ Graphics[\ {Rectangle[{0, 0}, \ {1, 1}, ttPlot[t]], Rectangle[{0, 1}, \ {1, 2}, tyPlot[t]], Rectangle[{1, 1}, \ {2, 2}, parPlot[t]], \n\ \ \ \ \ \ \ \ Rectangle[{1, 0}, \ {2, 1}, xtPlot[t]]}]\ ]\), ",", \({t, tmin, tmax, \(tmax - tmin\)\/n}\), ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["500", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"], Cell["\<\ Here's a version which \"traces\" the entire curve rather than \"creating\" \ it (i.e. the entire curve is in each frame).\ \>", "Text"], Cell[BoxData[{\(Clear[n, t, x, y, tmin, tmax]\), "\n", RowBox[{ RowBox[{"n", "=", StyleBox["50", FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[t]\), FontColor->RGBColor[1, 0, 1]]}]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{"tmax", "=", StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{"Animate", "[", RowBox[{\(Show[ Graphics[\ {Rectangle[{0, 0}, \ {1, 1}, ttPlot[t]], Rectangle[{0, 1}, \ {1, 2}, tyPlot[t]], Rectangle[{1, 1}, \ {2, 2}, parPlotComp[t]], \n\ \ \ \ \ \ \ \ Rectangle[{1, 0}, \ {2, 1}, xtPlot[t]]}]\ ]\), ",", \({t, tmin, tmax, \(tmax - tmin\)\/n}\), ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["500", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"], Cell["\<\ If you only wish to see the final frame, you can use the following cell.\ \>", "Text"], Cell[BoxData[{\(Clear[t, x, y, tmin, tmax]\), "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"tmax", "=", StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Show", "[", RowBox[{\(Graphics[\ {Rectangle[{0, 0}, \ {1, 1}, ttPlot[tmax]], Rectangle[{0, 1}, \ {1, 2}, tyPlot[tmax]], Rectangle[{1, 1}, \ {2, 2}, parPlot[tmax]], \n\ \ \ \ \ \ \ \ Rectangle[{1, 0}, \ {2, 1}, xtPlot[tmax]]}]\), " ", ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["500", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Another parametrization.", "Subsection"], Cell[TextData[{ "The input cell below creates a different animation, drawing the unit \ circle using the parametrization x = cos(", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["t", FontSlant->"Plain"], "2"], TraditionalForm]]], "), y = sin(", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["t", FontSlant->"Plain"], "2"], TraditionalForm]]], "). The parametric plot is created and drawn at the same time as the plots \ of x(t) and y(t) are traced, so you can see how the individual functions give \ rise to the final curve. Notice that although the curve is the same circle \ as in the preceeding section, the way it is traced is quite different." }], "Text"], Cell[TextData[{ "You can adjust the number of frames in the animation by changing the value \ of n. (You will need to reduce it if you are working on a slow computer.) \ ", "To see the animation, evaluate the cell, select the entire collection of \ output graphs, and choose ", StyleBox["Animate Selected Graphics", FontColor->RGBColor[0, 0, 1]], " from the ", StyleBox["Cell", FontColor->RGBColor[0, 0, 1]], " menu item. VCR like controls will appear at the bottom of the notebook \ window so that you can slow (or speed) the animation, pause it, and do other \ operations." }], "Text"], Cell[BoxData[{\(Clear[n, t, x, y, tmin, tmax]\), "\n", RowBox[{ RowBox[{"n", "=", StyleBox["50", FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[t\^2]\), FontColor->RGBColor[1, 0, 1]]}]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[t\^2]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{"tmax", "=", StyleBox[ SqrtBox[ StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]], FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{"Animate", "[", RowBox[{\(Show[ Graphics[\ {Rectangle[{0, 0}, \ {1, 1}, ttPlot[t]], Rectangle[{0, 1}, \ {1, 2}, tyPlot[t]], Rectangle[{1, 1}, \ {2, 2}, parPlot[t]], \n\ \ \ \ \ \ \ \ Rectangle[{1, 0}, \ {2, 1}, xtPlot[t]]}]\ ]\), ",", \({t, tmin, tmax, \(tmax - tmin\)\/n}\), ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["500", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"], Cell["\<\ Here's a version which \"traces\" the entire curve rather than \"creating\" \ it (i.e. the entire curve is in each frame).\ \>", "Text"], Cell[BoxData[{\(Clear[n, t, x, y, tmin, tmax]\), "\n", RowBox[{ RowBox[{"n", "=", StyleBox["50", FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[t\^2]\), FontColor->RGBColor[1, 0, 1]]}]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[t\^2]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"tmax", "=", StyleBox[ SqrtBox[ StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]], FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"Animate", "[", RowBox[{\(Show[ Graphics[\ {Rectangle[{0, 0}, \ {1, 1}, ttPlot[t]], Rectangle[{0, 1}, \ {1, 2}, tyPlot[t]], Rectangle[{1, 1}, \ {2, 2}, parPlotComp[t]], \n\ \ \ \ \ \ \ \ Rectangle[{1, 0}, \ {2, 1}, xtPlot[t]]}]\ ]\), ",", \({t, tmin, tmax, \(tmax - tmin\)\/n}\), ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["500", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"], Cell["\<\ If you only wish to see the final frame, you can use the following cell.\ \>", "Text"], Cell[BoxData[{\(Clear[t, x, y, tmin, tmax]\), "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[t\^2]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[t\^2]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"tmax", "=", StyleBox[ SqrtBox[ StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]], FontColor->RGBColor[1, 0, 1]]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Show", "[", RowBox[{\(Graphics[\ {Rectangle[{0, 0}, \ {1, 1}, ttPlot[tmax]], Rectangle[{0, 1}, \ {1, 2}, tyPlot[tmax]], Rectangle[{1, 1}, \ {2, 2}, parPlot[tmax]], \n\ \ \ \ \ \ \ \ Rectangle[{1, 0}, \ {2, 1}, xtPlot[tmax]]}]\), " ", ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["500", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["A Lissajous figure.", "Section"], Cell["\<\ The input cell below creates a plot of a Lissajous figure using the \ parametrization x = cos(3t), y = sin(5t), and animates the parametric plot. \ The parametric plot is created and drawn at the same time as the plots of \ x(t) and y(t) are traced, so you can see how the individual functions give \ rise to the final curve.\ \>", "Text"], Cell[TextData[{ "You can adjust the number of frames in the animation by changing the value \ of n. (You will need to reduce it if you are working on a slow computer.) \ ", "To see the animation, evaluate the cell, select the entire collection of \ output graphs, and choose ", StyleBox["Animate Selected Graphics", FontColor->RGBColor[0, 0, 1]], " from the ", StyleBox["Cell", FontColor->RGBColor[0, 0, 1]], " menu item. VCR like controls will appear at the bottom of the notebook \ window so that you can slow (or speed) the animation, pause it, and do other \ operations." }], "Text"], Cell[BoxData[{\(Clear[n, t, x, y, tmin, tmax]\), "\n", RowBox[{ RowBox[{"n", "=", StyleBox["50", FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[3 t]\), FontColor->RGBColor[1, 0, 1]]}]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[5 t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{"tmax", "=", StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{"Animate", "[", RowBox[{\(Show[ Graphics[\ {Rectangle[{0, 0}, \ {1, 1}, ttPlot[t]], Rectangle[{0, 1}, \ {1, 2}, tyPlot[t]], Rectangle[{1, 1}, \ {2, 2}, parPlot[t]], \n\ \ \ \ \ \ \ \ Rectangle[{1, 0}, \ {2, 1}, xtPlot[t]]}]\ ]\), ",", \({t, tmin, tmax, \(tmax - tmin\)\/n}\), ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["500", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"], Cell["\<\ Here's a version which \"traces\" the entire curve rather than \"creating\" \ it (i.e. the entire curve is in each frame).\ \>", "Text"], Cell[BoxData[{\(Clear[n, t, x, y, tmin, tmax]\), "\n", RowBox[{ RowBox[{"n", "=", StyleBox["50", FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[3 t]\), FontColor->RGBColor[1, 0, 1]]}]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[5 t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"tmax", "=", StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"Animate", "[", RowBox[{\(Show[ Graphics[\ {Rectangle[{0, 0}, \ {1, 1}, ttPlot[t]], Rectangle[{0, 1}, \ {1, 2}, tyPlot[t]], Rectangle[{1, 1}, \ {2, 2}, parPlotComp[t]], \n\ \ \ \ \ \ \ \ Rectangle[{1, 0}, \ {2, 1}, xtPlot[t]]}]\ ]\), ",", \({t, tmin, tmax, \(tmax - tmin\)\/n}\), ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["500", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"], Cell["\<\ If you only wish to see the final frame, you can use the following cell.\ \>", "Text"], Cell[BoxData[{\(Clear[t, x, y, tmin, tmax]\), "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[3 t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[5 t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"tmax", "=", StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Show", "[", RowBox[{\(Graphics[\ {Rectangle[{0, 0}, \ {1, 1}, ttPlot[tmax]], Rectangle[{0, 1}, \ {1, 2}, tyPlot[tmax]], Rectangle[{1, 1}, \ {2, 2}, parPlot[tmax]], \n\ \ \ \ \ \ \ \ Rectangle[{1, 0}, \ {2, 1}, xtPlot[tmax]]}]\), " ", ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["500", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["A helix (3D).", "Section"], Cell["\<\ This gives a parametrization for a helix, and animates the creation of the \ parametric plot. Since it would be difficult to display and follow all three \ individual functions together with the parametric plot, only the latter is \ drawn and a point \"creating\" the curve is animated.\ \>", "Text"], Cell[TextData[{ "You can adjust the number of frames in the animation by changing the value \ of n. (You will need to reduce it if you are working on a slow computer.) \ You can also adjust the viewpoint using the ", StyleBox["3D Viewpoint Selector", FontColor->RGBColor[0, 0, 1]], " item in the ", StyleBox["Input", FontColor->RGBColor[0, 0, 1]], " menu. ", "To see the animation, evaluate the cell, select the entire collection of \ output graphs, and choose ", StyleBox["Animate Selected Graphics", FontColor->RGBColor[0, 0, 1]], " from the ", StyleBox["Cell", FontColor->RGBColor[0, 0, 1]], " menu item. VCR like controls will appear at the bottom of the notebook \ window so that you can slow (or speed) the animation, pause it, and do other \ operations." }], "Text"], Cell[BoxData[{\(Clear[n, t, s, x, y, tmin, tmax, plotRange, parPlot3D]\), "\n", RowBox[{ RowBox[{"n", "=", StyleBox["50", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{\(z[t_]\), ":=", StyleBox["t", FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"tmax", "=", StyleBox[\(8\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", \(plotRange = {\(PlotRange[ ParametricPlot[{t, x[t]}, {t, tmin, tmax}, DisplayFunction \[Rule] Identity]]\)[\([2]\)], \(PlotRange[ ParametricPlot[{t, y[t]}, {t, tmin, tmax}, DisplayFunction \[Rule] Identity]]\)[\([2]\)], \(PlotRange[ ParametricPlot[{t, z[t]}, {t, tmin, tmax}, DisplayFunction \[Rule] Identity]]\)[\([2]\)]};\), "\[IndentingNewLine]", RowBox[{ RowBox[{\(parPlot3D[t_]\), ":=", RowBox[{"ParametricPlot3D", "[", RowBox[{\({x[s], y[s], z[s]}\), ",", RowBox[{"{", RowBox[{"s", ",", \(tmin - .001\), StyleBox[",", FontColor->GrayLevel[0]], "t"}], "}"}], ",", \(DisplayFunction \[Rule] Identity\), ",", RowBox[{"AspectRatio", "\[Rule]", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], ",", \(PlotRange \[Rule] plotRange\), ",", StyleBox[\(ViewPoint -> {0.617, \ \(-0.861\), \ 3.214}\), FontColor->RGBColor[1, 0, 1]]}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{"Animate", "[", RowBox[{ RowBox[{"Show", "[", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{\(Rectangle[{0, 0}, {2, .3}, tPlot[t]]\), ",", RowBox[{"Rectangle", "[", RowBox[{\({0, .3}\), ",", \({2, 1}\), ",", RowBox[{"Show", "[", RowBox[{\(parPlot3D[t]\), ",", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", StyleBox[".03", FontColor->RGBColor[1, 0, 1]], "]"}], ",", \(RGBColor[1, 0, 0]\), ",", \(Point[{x[t], y[t], z[t]}]\)}], "}"}], "]"}]}], "]"}]}], "]"}]}], "}"}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "tmin", StyleBox[",", FontColor->GrayLevel[0]], "tmax", StyleBox[",", FontColor->GrayLevel[0]], FractionBox[\(tmax - tmin\), StyleBox["n", FontColor->GrayLevel[0]]]}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["600", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"], Cell["\<\ Here's a version which \"traces\" the entire curve rather than \"creating\" \ it (i.e. the entire curve is in each frame).\ \>", "Text"], Cell[BoxData[{\(Clear[n, t, s, x, y, tmin, tmax, parPlot3D]\), "\n", RowBox[{ RowBox[{"n", "=", StyleBox["50", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{\(z[t_]\), ":=", StyleBox["t", FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"tmax", "=", StyleBox[\(8\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"parPlot3D", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{\({x[t], y[t], z[t]}\), ",", RowBox[{"{", RowBox[{"t", ",", "tmin", StyleBox[",", FontColor->GrayLevel[0]], "tmax"}], "}"}], ",", \(DisplayFunction \[Rule] Identity\), ",", RowBox[{"AspectRatio", "\[Rule]", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], ",", StyleBox[\(ViewPoint -> {0.617, \ \(-0.861\), \ 3.214}\), FontColor->RGBColor[1, 0, 1]]}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{"Animate", "[", RowBox[{ RowBox[{"Show", "[", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{\(Rectangle[{0, 0}, {2, .3}, tPlot[t]]\), ",", RowBox[{"Rectangle", "[", RowBox[{\({0, .3}\), ",", \({2, 1}\), ",", RowBox[{"Show", "[", RowBox[{"parPlot3D", ",", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", StyleBox[".03", FontColor->RGBColor[1, 0, 1]], "]"}], ",", \(RGBColor[1, 0, 0]\), ",", \(Point[{x[t], y[t], z[t]}]\)}], "}"}], "]"}]}], "]"}]}], "]"}]}], "}"}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "tmin", StyleBox[",", FontColor->GrayLevel[0]], "tmax", StyleBox[",", FontColor->GrayLevel[0]], FractionBox[\(tmax - tmin\), StyleBox["n", FontColor->GrayLevel[0]]]}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["600", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Other curves.", "Section"], Cell[CellGroupData[{ Cell["2D plots.", "Subsection"], Cell[TextData[{ "You can easily create graphs of curves by entering the parametrization in \ the cell below, and evaluating them. You may wish to change items in ", StyleBox["magenta", FontColor->RGBColor[1, 0, 1]], ". (If you don't change the parametrization, why is the result what it \ is?)" }], "Text"], Cell["\<\ The parametric plot is created and drawn at the same time as the plots of \ x(t) and y(t) are traced, so you can see how the individual functions give \ rise to the final curve.\ \>", "Text"], Cell[TextData[{ "You can adjust the number of frames in the animation by changing the value \ of n. (You will need to reduce it if you are working on a slow computer.) \ ", "To see the animation, evaluate the cell, select the entire collection of \ output graphs, and choose ", StyleBox["Animate Selected Graphics", FontColor->RGBColor[0, 0, 1]], " from the ", StyleBox["Cell", FontColor->RGBColor[0, 0, 1]], " menu item. VCR like controls will appear at the bottom of the notebook \ window so that you can slow (or speed) the animation, pause it, and do other \ operations." }], "Text"], Cell[BoxData[{\(Clear[n, t, x, y, tmin, tmax]\), "\n", RowBox[{ RowBox[{"n", "=", StyleBox["50", FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[t]\^2\), FontColor->RGBColor[1, 0, 1]]}]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[t]\^2\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{"tmax", "=", StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{"Animate", "[", RowBox[{\(Show[ Graphics[\ {Rectangle[{0, 0}, \ {1, 1}, ttPlot[t]], Rectangle[{0, 1}, \ {1, 2}, tyPlot[t]], Rectangle[{1, 1}, \ {2, 2}, parPlot[t]], \n\ \ \ \ \ \ \ \ Rectangle[{1, 0}, \ {2, 1}, xtPlot[t]]}]\ ]\), ",", \({t, tmin, tmax, \(tmax - tmin\)\/n}\), ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["500", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"], Cell["\<\ Here's a version which \"traces\" the entire curve rather than \"creating\" \ it (i.e. the entire curve is in each frame).\ \>", "Text"], Cell[BoxData[{\(Clear[n, t, x, y, tmin, tmax]\), "\n", RowBox[{ RowBox[{"n", "=", StyleBox["50", FontColor->RGBColor[1, 0, 1]]}], ";", "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[t]\^2\), FontColor->RGBColor[1, 0, 1]]}]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[t]\^2\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"tmax", "=", StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"Animate", "[", RowBox[{\(Show[ Graphics[\ {Rectangle[{0, 0}, \ {1, 1}, ttPlot[t]], Rectangle[{0, 1}, \ {1, 2}, tyPlot[t]], Rectangle[{1, 1}, \ {2, 2}, parPlotComp[t]], \n\ \ \ \ \ \ \ \ Rectangle[{1, 0}, \ {2, 1}, xtPlot[t]]}]\ ]\), ",", \({t, tmin, tmax, \(tmax - tmin\)\/n}\), ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["500", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"], Cell["\<\ If you only wish to see the final frame, you can use the following cell.\ \>", "Text"], Cell[BoxData[{\(Clear[t, x, y, tmin, tmax]\), "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[t]\^2\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[t]\^2\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"tmax", "=", StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Show", "[", RowBox[{\(Graphics[\ {Rectangle[{0, 0}, \ {1, 1}, ttPlot[tmax]], Rectangle[{0, 1}, \ {1, 2}, tyPlot[tmax]], Rectangle[{1, 1}, \ {2, 2}, parPlot[tmax]], \n\ \ \ \ \ \ \ \ Rectangle[{1, 0}, \ {2, 1}, xtPlot[tmax]]}]\), " ", ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["500", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["3D plots.", "Subsection"], Cell[TextData[{ "You can easily create graphs of curves by entering the parametrization in \ the cell below, and evaluating it. You may wish to change items in ", StyleBox["magenta", FontColor->RGBColor[1, 0, 1]], ". Since it would be difficult to display and follow all three individual \ functions together with the parametric plot, only the latter is drawn and a \ point \"creating\" the curve is animated." }], "Text"], Cell[TextData[{ "You can adjust the number of frames in the animation by changing the value \ of n. (You will need to reduce it if you are working on a slow computer.) \ You can also adjust the viewpoint using the ", StyleBox["3D Viewpoint Selector", FontColor->RGBColor[0, 0, 1]], " item in the ", StyleBox["Input", FontColor->RGBColor[0, 0, 1]], " menu. ", "To see the animation, evaluate the cell, select the entire collection of \ output graphs, and choose ", StyleBox["Animate Selected Graphics", FontColor->RGBColor[0, 0, 1]], " from the ", StyleBox["Cell", FontColor->RGBColor[0, 0, 1]], " menu item. VCR like controls will appear at the bottom of the notebook \ window so that you can slow (or speed) the animation, pause it, and do other \ operations." }], "Text"], Cell[BoxData[{\(Clear[n, t, x, y, tmin, tmax, plotRange, parPlot3D]\), "\n", RowBox[{ RowBox[{"n", "=", StyleBox["50", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[5 t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[3 t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{\(z[t_]\), ":=", StyleBox[\(Sin[t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"tmax", "=", StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", \(plotRange = {\(PlotRange[ ParametricPlot[{t, x[t]}, {t, tmin, tmax}, DisplayFunction \[Rule] Identity]]\)[\([2]\)], \(PlotRange[ ParametricPlot[{t, y[t]}, {t, tmin, tmax}, DisplayFunction \[Rule] Identity]]\)[\([2]\)], \(PlotRange[ ParametricPlot[{t, z[t]}, {t, tmin, tmax}, DisplayFunction \[Rule] Identity]]\)[\([2]\)]};\), "\[IndentingNewLine]", RowBox[{ RowBox[{\(parPlot3D[t_]\), ":=", RowBox[{"ParametricPlot3D", "[", RowBox[{\({x[s], y[s], z[s]}\), ",", RowBox[{"{", RowBox[{"s", ",", \(tmin - .001\), StyleBox[",", FontColor->GrayLevel[0]], "t"}], "}"}], ",", \(DisplayFunction \[Rule] Identity\), ",", RowBox[{"AspectRatio", "\[Rule]", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], ",", \(PlotRange \[Rule] plotRange\), ",", StyleBox[\(ViewPoint -> {1.915, \ \(-1.827\), \ 2.108}\), FontColor->RGBColor[1, 0, 1]]}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{"Animate", "[", RowBox[{ RowBox[{"Show", "[", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{\(Rectangle[{0, 0}, {2, .3}, tPlot[t]]\), ",", RowBox[{"Rectangle", "[", RowBox[{\({0, .3}\), ",", \({2, 1}\), ",", RowBox[{"Show", "[", RowBox[{\(parPlot3D[t]\), ",", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", StyleBox[".03", FontColor->RGBColor[1, 0, 1]], "]"}], ",", \(RGBColor[1, 0, 0]\), ",", \(Point[{x[t], y[t], z[t]}]\)}], "}"}], "]"}]}], "]"}]}], "]"}]}], "}"}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "tmin", StyleBox[",", FontColor->GrayLevel[0]], "tmax", StyleBox[",", FontColor->GrayLevel[0]], FractionBox[\(tmax - tmin\), StyleBox["n", FontColor->GrayLevel[0]]]}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["600", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"], Cell["\<\ Here's a version which \"traces\" the entire curve rather than \"creating\" \ it (i.e. the entire curve is in each frame).\ \>", "Text"], Cell[BoxData[{\(Clear[n, t, x, y, tmin, tmax, parPlot3D]\), "\n", RowBox[{ RowBox[{"n", "=", StyleBox["50", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{\(x[t_]\), ":=", StyleBox[\(Cos[5 t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{\(y[t_]\), ":=", StyleBox[\(Sin[3 t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{\(z[t_]\), ":=", StyleBox[\(Sin[t]\), FontColor->RGBColor[1, 0, 1]]}], "\n", RowBox[{ RowBox[{"tmin", "=", StyleBox["0", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"tmax", "=", StyleBox[\(2\ \[Pi]\), FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"parPlot3D", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{\({x[t], y[t], z[t]}\), ",", RowBox[{"{", RowBox[{"t", ",", "tmin", StyleBox[",", FontColor->GrayLevel[0]], "tmax"}], "}"}], ",", \(DisplayFunction \[Rule] Identity\), ",", RowBox[{"AspectRatio", "\[Rule]", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], ",", StyleBox[\(ViewPoint -> {1.915, \ \(-1.827\), \ 2.108}\), FontColor->RGBColor[1, 0, 1]]}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{"Animate", "[", RowBox[{ RowBox[{"Show", "[", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{\(Rectangle[{0, 0}, {2, .3}, tPlot[t]]\), ",", RowBox[{"Rectangle", "[", RowBox[{\({0, .3}\), ",", \({2, 1}\), ",", RowBox[{"Show", "[", RowBox[{"parPlot3D", ",", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", StyleBox[".03", FontColor->RGBColor[1, 0, 1]], "]"}], ",", \(RGBColor[1, 0, 0]\), ",", \(Point[{x[t], y[t], z[t]}]\)}], "}"}], "]"}]}], "]"}]}], "]"}]}], "}"}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "tmin", StyleBox[",", FontColor->GrayLevel[0]], "tmax", StyleBox[",", FontColor->GrayLevel[0]], FractionBox[\(tmax - tmin\), StyleBox["n", FontColor->GrayLevel[0]]]}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", StyleBox["600", FontColor->RGBColor[1, 0, 1]]}]}], "]"}], ";"}]}], "Input"] }, Closed]] }, Closed]] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 971}}, AutoGeneratedPackage->None, WindowToolbars->"EditBar", WindowSize->{689, 631}, WindowMargins->{{2, Automatic}, {Automatic, 0}} ] (******************************************************************* Cached data follows. 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