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Notebook[{

Cell[CellGroupData[{
Cell["Parametric Plots in 3D, Part 2", "Title"],

Cell[CellGroupData[{

Cell["Initializations", "Section"],

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["\<x\>", FontSize \[Rule] 18], {1.85,  .15, 
                    0}], \ Text[
                  StyleForm["\<y\>", FontSize \[Rule] 18], { .25, 1.85, 
                    0}], \ Text[
                  StyleForm["\<z\>", 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],

Cell[BoxData[{
    \(Clear[drawCylindrical]\), "\[IndentingNewLine]", 
    \(\(drawCylindrical[] := 
        Module[{curve1, line1, 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]line1 = 
            Graphics3D[
              Line[{{0, 0, 0}, {1.5, 1.5, 0}}]]; \[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["\<r\>", FontSize \[Rule] 18], { .7,  .9, 
                  0}]]; \[IndentingNewLine]text2 = 
            Graphics3D[
              Text[StyleForm["\<\[Theta]\>", FontSize \[Rule] 18], {1.1,  .5, 
                  0}]]; \[IndentingNewLine]text3 = 
            Graphics3D[
              Text[StyleForm["\<z\>", FontSize \[Rule] 18], {1.6, 1.6, 
                  1}]]; \[IndentingNewLine]axeslabels = 
            Graphics3D[{Text[
                  StyleForm["\<x\>", FontSize \[Rule] 18], {1.85,  .15, 
                    0}], \ Text[
                  StyleForm["\<y\>", FontSize \[Rule] 18], { .25, 1.85, 
                    0}], \ Text[
                  StyleForm["\<z\>", FontSize \[Rule] 18], { .2, 0, 
                    1.85}]}]; \[IndentingNewLine]Show[curve1, line1, line3, 
            point, text1, text2, text3, axeslabels, 
            DisplayFunction \[Rule] $DisplayFunction];];\)\)}], "Input",
  InitializationCell->True]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
Parametric surfaces - variations on cylindrical coordinates \
(continued)\
\>", "Section"],

Cell[TextData[{
  "Recall that in our previous class period, we modified the basic \
cylindrical coordinate equations below to get some interesting parametric \
surfaces.  Most of these shapes are based on making ",
  StyleBox["r",
    FontSlant->"Italic"],
  " a function of ",
  StyleBox["z",
    FontSlant->"Italic"],
  " or \[Theta]."
}], "Text"],

Cell[TextData[{
  StyleBox["x",
    FontSlant->"Italic"],
  " = ",
  StyleBox["r",
    FontSlant->"Italic"],
  " cos(\[Theta])\n",
  StyleBox["y",
    FontSlant->"Italic"],
  " = ",
  StyleBox["r",
    FontSlant->"Italic"],
  " sin(\[Theta])\n",
  StyleBox["z",
    FontSlant->"Italic"],
  " = ",
  StyleBox["z",
    FontSlant->"Italic"]
}], "Text"],

Cell["The most basic form was a cylinder.", "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, 10}, ViewPoint \[Rule] {3, 2, 2}, 
        AxesLabel \[Rule] {"\<x\>", "\<y\>", "\<z\>"}, 
        PlotPoints \[Rule] {60, 30}];\)\)}], "Input"],

Cell[TextData[{
  "We worked our way up to a surface where ",
  StyleBox["r",
    FontSlant->"Italic"],
  " was a function of ",
  StyleBox["z",
    FontSlant->"Italic"],
  "."
}], "Text"],

Cell[BoxData[{
    \(Clear[r, x, y, z, u, v]\), "\[IndentingNewLine]", 
    \(r[u_, v_] := 2 + Sin[6  v]\), "\[IndentingNewLine]", 
    \(x[u_, v_] := r[u, v]\ Cos[u]\), "\[IndentingNewLine]", 
    \(y[u_, 
        v_] := \(\(r[u, v]\)\(\ \)\(Sin[
          u]\)\(\ \)\)\), "\[IndentingNewLine]", 
    \(z[u_, v_] := v\), "\[IndentingNewLine]", 
    \(\(ParametricPlot3D[{x[u, v], y[u, v], z[u, v]}, {u, 0, 2  \[Pi]}, {v, 
          0, 8}, ViewPoint -> {7, \ \(-5\), \ 4}, 
        AxesLabel \[Rule] {"\<x\>", "\<y\>", "\<z\>"}, 
        PlotPoints \[Rule] {20, 120}];\)\)}], "Input"],

Cell["\<\
Now, let's create a fish.  Try to resist the temptation to look \
ahead in this notebook.  Follow along with the presentation to the rest of \
the class.  We will derive some of the mathematical formulas before you look \
at the answers below.\
\>", "Text"],

Cell[CellGroupData[{

Cell["A fish", "Subsection",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[TextData[{
  "We will create a reef fish by rotating a piece of a parabola about a the \
",
  StyleBox["x",
    FontSlant->"Italic"],
  " axis in an elliptical path.  Our fish will have the same basic body shape \
as these Queen Angels from MarineBio, at \
http://marinebio.org/species.asp?id=285."
}], "Text"],

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