(************** 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[ 1100461, 21034]*) (*NotebookOutlinePosition[ 1101184, 21059]*) (* CellTagsIndexPosition[ 1101140, 21055]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Section 10.5 Parameterized Surfaces", "Title"], Cell["H. Edward Donley", "Subtitle"], Cell[CellGroupData[{ Cell["Initializations", "Subsection"], Cell[BoxData[{ \(Clear[drawSpherical]\), "\[IndentingNewLine]", \(\(drawSpherical[] := Module[{curve1, curve2, line1, line2, line3, point, text1, text2, text3, axeslabels}, curve1 = ParametricPlot3D[{Cos[t], Sin[t], 0}, {t, 0, \[Pi]/4}, PlotRange \[Rule] {{0, 2}, {0, 2}, {0, 2}}, ViewPoint \[Rule] {3, 1, 1}, Axes \[Rule] False, DisplayFunction \[Rule] Identity]; \[IndentingNewLine]curve2 = ParametricPlot3D[{Sin[t], Sin[t], Cos[t]}, {t, 0, \[Pi]/4}, PlotRange \[Rule] {{0, 2}, {0, 2}, {0, 2}}, ViewPoint \[Rule] {3, 1, 1}, Axes \[Rule] False, DisplayFunction \[Rule] Identity]; \[IndentingNewLine]line1 = Graphics3D[ Line[{{0, 0, 0}, {1.5, 1.5, 0}}]]; \[IndentingNewLine]line2 = Graphics3D[{RGBColor[1, 0, 0], Line[{{0, 0, 0}, {1.5, 1.5, 1.5}}]}]; \[IndentingNewLine]line3 = Graphics3D[ Line[{{1.5, 1.5, 0}, {1.5, 1.5, 1.5}}]]; \[IndentingNewLine]point = Graphics3D[{PointSize[0.03], RGBColor[1, 0, 0], Point[{1.5, 1.5, 1.5}]}]; \[IndentingNewLine]text1 = Graphics3D[ Text[StyleForm["\<\[Rho]\>", FontSize \[Rule] 18], {1.2, 1.2, 1.3}]]; \[IndentingNewLine]text2 = Graphics3D[ Text[StyleForm["\<\[Theta]\>", FontSize \[Rule] 18], {1.1, .5, 0}]]; \[IndentingNewLine]text3 = Graphics3D[ Text[StyleForm["\<\[Phi]\>", FontSize \[Rule] 18], { .6, .6, 1}]]; \[IndentingNewLine]axeslabels = Graphics3D[{Text[ StyleForm["\", FontSize \[Rule] 18], {1.85, .15, 0}], \ Text[ StyleForm["\", FontSize \[Rule] 18], { .25, 1.85, 0}], \ Text[ StyleForm["\", FontSize \[Rule] 18], { .2, 0, 1.85}]}]; \[IndentingNewLine]Show[curve1, curve2, line1, line2, line3, point, text1, text2, text3, axeslabels, DisplayFunction \[Rule] $DisplayFunction];];\)\)}], "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "Surfaces are two-dimensional objects embedded in three-dimensional space.\n\ x = ", Cell[BoxData[ \(TraditionalForm\`f\_1\)]], "(u, v)\n y = ", Cell[BoxData[ \(TraditionalForm\`f\_2\)]], "(u, v)\n z = ", Cell[BoxData[ \(TraditionalForm\`f\_3\)]], "(u, v)" }], "Text"], Cell["\<\ For example, you can describe a cylinder by letting u represent the angle \ around the cylinder and v represent the distance along the cylinder's axis.\ \>", "Text"], Cell[BoxData[{ \(Clear[x, y, z, u, v]\), "\[IndentingNewLine]", \(x[u_, v_] := 3 Cos[u]\), "\[IndentingNewLine]", \(y[u_, v_] := 3 Sin[u]\), "\[IndentingNewLine]", \(z[u_, v_] := v\), "\[IndentingNewLine]", \(\(ParametricPlot3D[{x[u, v], y[u, v], z[u, v]}, {u, 0, 2 \[Pi]}, {v, 0, 5}];\)\)}], "Input"], Cell["\<\ How would you draw a cylinder of radius 2 oriented along the y axis?\ \>", "Text"], Cell[TextData[{ "What happens when you make the radius a function of v", "? For example, try a radius of ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\( .1 v\)\)]], ". 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Define a quadratic function with ", StyleBox["x ", FontSlant->"Italic"], "intercepts at 0 and 1. Adjust the height of the parabola so that it \ roughly matches the outline of the doctor fish." }], "Text", TextAlignment->Left], Cell[BoxData[{ \(Clear[f]\), "\[IndentingNewLine]", \(\(f[x_] := \[IndentingNewLine]Plot[f[x], {x, \(- .1\), 1}, AspectRatio \[Rule] Automatic];\)\)}], "Input"], Cell[TextData[{ "Now rotate this section of the parabola about the ", StyleBox["x", FontSlant->"Italic"], " axis in an elliptical path." }], "Text"], Cell[BoxData[{ \(Clear[x, y, z, u, v]\), "\[IndentingNewLine]", \(\(x[u_, v_] := \[IndentingNewLine]\(y[u_, v_] := \[IndentingNewLine]\(z[u_, v_] := \[IndentingNewLine]ParametricPlot3D[{x[u, v], y[u, v], z[u, v]}, {u, \(- .1\), 1}, {v, 0, 2 \[Pi]}]\)\);\)\)}], "Input"], Cell["\<\ In our next few examples, we will use spherical coordinates, where \[Rho] \ represents the distance from the origin, \[Phi] is the angle that the point's \ position vector makes with the z axis (latitude), and \[Theta] is the angle \ that it makes with the xz plane (longitude).\ \>", "Text"], Cell[BoxData[ \(drawSpherical[]\)], "Input"], Cell["\<\ The following equations convert from spherical coordinates to rectangular \ coordinates. x = \[Rho] sin(\[Phi]) cos(\[Theta]) y = \[Rho] sin(\[Phi]) sin(\[Theta]) z = \[Rho] cos(\[Phi])\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Snail Shell", "Section"], Cell[CellGroupData[{ Cell["Creating the Snail Shell", "Subsection"], Cell["\<\ We will create a snail shell similar to the following Ross Land Snail from \ New Zealand. 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The snail shell spiral increases exponentially because \ exponential functions have constant percent growth rates. Here is a graph of \ \[Rho] = ", Cell[BoxData[ \(\[ExponentialE]\^\( .1 \[Theta]\)\)]], "." }], "Text"], Cell[BoxData[{ \(Clear[x, y, z, \[Theta], \[Phi]]\), "\[IndentingNewLine]", \(\(x[\[Theta]_, \[Phi]_] = \[ExponentialE]\^\( .1 \[Theta]\)\ \ \ Sin[\ \[Phi]]\ Cos[\[Theta]];\)\), "\[IndentingNewLine]", \(\(y[\[Theta]_, \[Phi]_] = \[ExponentialE]\^\( .1 \[Theta]\)\ \ \ Sin[\ \[Phi]]\ Sin[\[Theta]];\)\), "\[IndentingNewLine]", \(\(z[\[Theta]_, \[Phi]_] = \[ExponentialE]\^\( .1 \[Theta]\)\ \ \ Cos[\ \[Phi]];\)\), "\[IndentingNewLine]", \(\(snail = ParametricPlot3D[{x[\[Theta], \[Phi]], y[\[Theta], \[Phi]], z[\[Theta], \[Phi]]}, {\[Phi], 0, \[Pi]}, {\[Theta], 0, 3 \[Pi]}, PlotPoints \[Rule] {15, 60}, Boxed \[Rule] False, Axes \[Rule] False, ViewPoint \[Rule] {\(-0.091\), \ \(-1.372\), \ 3.092}];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Pierced Snail Shell", "Subsection"], Cell["\<\ We can view the interior of the snail by shrinking the sizes of the polygonal \ faces.\ \>", "Text"], Cell[BoxData[ \(InputForm[snail]\)], "Input"], Cell[BoxData[ \(average[list_List] := Apply[Plus, list]/Length[list]\)], "Input"], Cell[BoxData[ \(shrink[vertices_List, shrinkfactor_] := Module[{centroid}, centroid = Map[average, Transpose[vertices]]; Map[\((\((# - centroid)\)*shrinkfactor\ + \ centroid)\) &, \ vertices]]\)], "Input"], Cell[BoxData[{ \(\(piercedSnail = snail /. Polygon[vertices_] \[Rule] Polygon[shrink[vertices, .75]];\)\), "\[IndentingNewLine]", \(\(Show[piercedSnail];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Virtual Reality Modeling Language", "Subsection"], Cell["\<\ Virtual Reality Modeling Language (VRML) is a 3D environment for the Web. \ With VRML, we can navigate around and through the graphic.\ \>", "Text"], Cell[BoxData[ \(Needs["\"]\)], "Input"], Cell[BoxData[ \(VRMLConvert[ piercedSnail, \ "\"]\)], "Input"], Cell[TextData[{ "See ", ButtonBox["snail.wrl", ButtonData:>{ URL[ "http://www.math.iup.edu/~hedonley/snail.wrl"], None}, ButtonStyle->"Hyperlink"], "." }], "Text"], Cell[TextData[{ "Another technique for viewing the interior of the shell is to make the \ surface semi-transparent. ", StyleBox["Mathematica", FontSlant->"Italic"], " does not support transparency, but Virtual Reality Modeling Language \ (VRML) does." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(VRMLConvert[ snail, \ "\", VRMLtransparency \[Rule] .8]\)], "Input"], Cell[BoxData[ \("C:\\Windows\\Desktop\\SigmaXi\\snailtransparent.wrl"\)], "Output"] }, Open ]], Cell[TextData[{ "See ", ButtonBox["snailtransparent.wrl", ButtonData:>{ URL[ "http://www.math.iup.edu/~hedonley/snailtransparent.wrl"], None}, ButtonStyle->"Hyperlink"], "." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Another snail shell", "Subsection"], Cell["\<\ Here is a snail shell created by Stewart Dickens, California Institute of the \ Arts, http://emsh.calarts.edu/~mathart/sw/Color_3D_Prints.html.\ \>", "Text"], Cell[BoxData[{ \(Clear[rad, pitch, x, y, z]\), "\[IndentingNewLine]", \(\(rad[\[Theta]_] := \[ExponentialE]\^\(\[Theta]/\((6 \[Pi])\)\) - 1\ \ ;\)\), "\[IndentingNewLine]", \(\(pitch = 2/3;\)\), "\[IndentingNewLine]", \(\(x[\[Theta]_, \[Phi]_] := rad[\[Theta]] \((1 + Cos[\[Phi]])\)\ Cos[\[Theta]];\)\), "\[IndentingNewLine]", \(\(y[\[Theta]_, \[Phi]_] := rad[\[Theta]]\ \((1 + Cos[\[Phi]])\)\ Sin[\[Theta]];\)\), "\[IndentingNewLine]", \(\(z[\[Theta]_, \[Phi]_] := \(-pitch\)\ \ \((\[ExponentialE]\^\(\[Theta]/\((3 \[Pi])\)\) - 1)\) + rad[\[Theta]]\ Sin[\[Phi]];\)\), "\[IndentingNewLine]", \(\(ParametricPlot3D[{x[\[Theta], \[Phi]], y[\[Theta], \[Phi]], z[\[Theta], \[Phi]]}, {\[Theta], 0, 6\ \[Pi]}, {\[Phi], 0, 2\ \[Pi]}, \ PlotPoints \[Rule] {70, \ 20}, PlotRange \[Rule] {{\(-4\), 4}, {\(-4\), 4}, {\(-5\), 0}}, Boxed \[Rule] False, Axes \[Rule] False];\)\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Periwinkle Seashell", "Section"], Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgHVFnJVWBIVGNHUg^>SGR@ Sg]YJeIMGdYnP6YkM6AKG5AHHEaYKVMiM62=PVJ>PfMnN5ieLEQgLEV7PFV9 PfV0NEb:Q6:9PEb9PEZ/Y82i/XRCT6EgM5F;RGZ;RXEhNVm>DclT9R4O81/b 0T8fEXj=gPXHW<3L?51XE6A`92`X>3P`N7A/] ;BD`C1;BSE]IDjGRWVBV7ifN61YIdieKeN:Q6bSWHFQWhJ: RW1nOFQmO6YPHE=0 O65cKU5ECcDc?RhS;1/b>bY[LV2BUh>0PfiQHdeJG4AeM628QG:>QgBCR7B: P6M[I4QVHD=gM5EoNUacKeAhMf:7RWNAU8>LWH^8Qg>9Qg:8R6iZK5M:C450 @C]PH5R;R7FHTgFCRfJQT6BQT6:ZV6JgYG>bYG6]Xg:TX76TfZC U6bARFBcLg@3/l@c]8Bd1@DD5MFTUf Ke:DTG:QXX>TZHVHWh2@UgQgNf1ZKE:7PF6ITg>KUWV2OF1lMeVSX7g1_iJi ^8bf^96d]Hg?bjGhl/kflcK7=cZjRXh63P5mkM5N@QV^LVX>YZiFb]j>e_:Zg`:f] ]J29SWMaMEZ2Q5nEUFnLVW>HU6j9QV=gMe]iOVR1R7J?TW>PX8BKVH:QWhZM V8BFT7ZR`h>2`g>2Xo?2eQe>ESEbDU7RGVH6EV8NGVHj9Qg]eL5el LeJRUG>XW6RXWV^VXW:DTfE/KDE@D2a@CBiHDcI5?C1@AcQME4=[I564PVf; SW]UKUdkAcTc>CLX;BLN81<271d=0C8d9`000?lj>R15AB]GF3UdLeB?Sf^G V72HUf^LVfnEUg:EUWJQXHNRXi6RXiEhNVeQHeIaLfJ@RGVDUHelQH0k@3TO 6PHe:PiWG420NV1gLeYOFD=XHTb0NVB9PVR>ShV9SGmCFd@/;R/X:2Hf=C=/JfUZJVQ2AdB]9BBmWH4BAQFf> TgedN65ZJ5=jLUfFRgFXWXFGT7AfKe=_L625QgYbLf]DEe1SJ6:@Uhn5ShAg PWAiPH>DWYfZ];>:Ti1GFEIKEeAfJfUeIFEbKU>8QVn:Rg]nP7=mOg6@TWfH Ug^0OEikO5aWIU8k=S0hRVn:QFnTWXRh/9W1]Z2/XXV7PFMlNen0PfAjNV1XJUA]LUikOg1k Og6?Sh>=RGfESh6HUWeiN6AIFDm8Ad9MFU>3OV^=QVV@QV:QS5N`VfB_V6:e X6Vn[gRcZGJNW6b?TF:MVV>ZYg:UXG6^ZGnl^IJUZ8]nQVmSKe]HGEIJHEYj Ph2GXYb0S7icOFA_Me1nQ5AoPTf6Q5>5O563MU=ICc@hE4QoNEnEU7ROXXFS[8nIY8JBVgijP6I^LUVA SfJETfbETVn9Q6J8PfF/Z8G=b:;?bZ:h]iRf/i:g/Hg7_i[3^iJi/hnIUWEm O5ebKe20O5VDSfNNUf^FSf=oNeEYJ4ULGdAgL62FTGjf]9_2`jBMW7f5PFB2 Nf6BQVjMW8Zc]ZG0akO2cKfc_ZfHXHaZLeQYKe>6R66LWG>VYGNUXWFGUFb8 RFZBUh>RZ9j`]9VTYXjFV8>RXHnIUXFKUX>@RW@005aiLe]=BT4_;bLU9AdQ 8ATL6A0O5`XX6`/h:AI=BQeKEc5gLDnAS6JHU6FGT6J5Nf1JCT0c@D4c?C`m @S`^:b8m=b]`JEUjLf5hLEn6R6>a/Hfm^YNYYXF0Of5WJE=JGTeSJeimN5^5 OV5lLe:4O5R7Pf2CTW>BUgV;U7N?UG^BVXFNYYVSZZ>^]:Rb]YnQXgjGUVJ? 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