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    StyleBox[\(Pr\[AAcute]ctica\ 3 : \ Superficies\ I\),
      "Title",
      FontColor->RGBColor[1, 0, 0]]], "Input"],

Cell[TextData[{
  StyleBox["Objetivo:",
    FontFamily->"Book Antiqua",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox[" Aprender como dibujar con mathematica curvas planas definidas no \
param\[EAcute]tricamente: de manera implicita o en coordenadas polares. Tambi\
\[EAcute]n trataremos de reconstruir una curva de la que conocemos su \
curvatura y torsi\[OAcute]n. ",
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Cell[TextData[StyleBox["Los enlaces de internet a los que se hace referencia \
pueden encontrarse en:   www.ugr.es/~santimo/cursup.htm",
  FontFamily->"Book Antiqua"]], "Text",
  FontSize->14],

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Cell[TextData[StyleBox["Representaci\[OAcute]n gr\[AAcute]fica de \
superficies.",
  FontFamily->"Book Antiqua",
  FontColor->RGBColor[1, 0, 0]]], "Subsubtitle"],

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  StyleBox["Plot3D:",
    FontFamily->"Book Antiqua",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox[" Genera gr\[AAcute]ficas de funciones reales de dos variables. \
Busca en la ayuda de ",
    FontFamily->"Book Antiqua"],
  StyleBox["Mathematica",
    FontFamily->"Book Antiqua",
    FontSlant->"Italic"],
  StyleBox[" como se usa esta instrucci\[OAcute]n.",
    FontFamily->"Book Antiqua"]
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Cell[TextData[StyleBox["El paraboloide hiperb\[OAcute]lico es el grafo de \
f(x,y)=xy.",
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Cell[BoxData[
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conocidas como ViewPoint y PlotPoint. M\[AAcute]s opciones en la ayuda.",
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Cell[BoxData[
    \(\(Plot3D[\((x^2 + 3\ y^2)\)\ E^\((1 - x^2 - y^2)\), {x, \(-2\), 
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Cell[TextData[{
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    FontFamily->"Book Antiqua",
    FontWeight->"Bold"],
  StyleBox[" Dibuja la superficie silla de Mono: f(x,y)=x^3-3uv^2. Prueba \
algunas opciones de Plot3D",
    FontFamily->"Book Antiqua"]
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Cell[TextData[{
  StyleBox["ContourPlot:",
    FontFamily->"Book Antiqua",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox[" Puede generar graficos de contornos de una funci\[OAcute]n de \
dos variables. Busca m\[AAcute]s informaci\[OAcute]n en la ayuda.",
    FontFamily->"Book Antiqua"]
}], "Text"],

Cell[TextData[StyleBox["El paraboloide",
  FontFamily->"Book Antiqua"]], "Text"],

Cell[BoxData[
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Cell[TextData[StyleBox["Otra manera de ver el paraboloide:",
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Cell[TextData[{
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    FontFamily->"Book Antiqua",
    FontWeight->"Bold"],
  StyleBox[" Obt\[EAcute]n un gr\[AAcute]fico de contorno de la silla de \
mono.",
    FontFamily->"Book Antiqua"]
}], "Text"],

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  StyleBox["ParametricPlot3D:",
    FontFamily->"Book Antiqua",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox[" Dibuja un objeto en R3 dado en forma param\[EAcute]trica. Busca \
en la ayuda de ",
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  StyleBox["Mathematica",
    FontFamily->"Book Antiqua",
    FontSlant->"Italic"],
  StyleBox[" como se usa esta instrucci\[OAcute]n.",
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    FontFamily->"Book Antiqua",
    FontWeight->"Bold"],
  StyleBox[" Dibuja  un hiperboloide de una hoja: X(u,v)={cosh v  cos u, cosh \
v sen u, senh v}.",
    FontFamily->"Book Antiqua"]
}], "Text"],

Cell[BoxData[
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  StyleBox[" \[DownQuestion]Por qu\[EAcute] el anterior gr\[AAcute]fico no \
representa a una superficie?",
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  " (como grafo, como superficie de revoluci\[OAcute]n, a trav\[EAcute]s de \
la proyecci\[OAcute]n estereogr\[AAcute]fica)"
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Cell[TextData[StyleBox["Con coordenadas polares:",
  FontFamily->"Book Antiqua"]], "Text"],

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Cell[TextData[StyleBox["Como superficie de revoluci\[OAcute]n:",
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Cell[BoxData[
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          2  Pi}, \ ViewPoint \[Rule] {1, 2, 1}];\)\)], "Input"],

Cell[TextData[StyleBox["Parametrizada por la proyecci\[OAcute]n estereogr\
\[AAcute]fica:",
  FontFamily->"Book Antiqua"]], "Text"],

Cell[BoxData[{
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Cell[TextData[{
  StyleBox["Ejercicio:",
    FontFamily->"Book Antiqua",
    FontWeight->"Bold"],
  StyleBox[" Consulta el enlace 5 a internet que muestra una galer\[IAcute]a \
de distintas superficies. \[DownQuestion]Son todas ellas superficies seg\
\[UAcute]n la definici\[OAcute]n dada en clase?",
    FontFamily->"Book Antiqua"]
}], "Text"],

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  FontColor->RGBColor[1, 0, 0]]], "Text",
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  FontSize->16,
  FontWeight->"Plain",
  FontVariations->{"CompatibilityType"->0}],

Cell[BoxData[{
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Cell[BoxData[{
    \(\(t1 = 0;\)\), "\[IndentingNewLine]", 
    \(\(t2 = 8\ Pi;\)\)}], "Input"],

Cell[BoxData[
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          0, 2  Pi}, {t, t1, t2}, AspectRatio \[Rule] Automatic, 
        ViewPoint \[Rule] {1, 1, 1}, PlotRange \[Rule] All, 
        PlotPoints \[Rule] 50];\)\)], "Input"],

Cell[TextData[{
  StyleBox["Ejercicio:",
    FontFamily->"Book Antiqua",
    FontWeight->"Bold"],
  StyleBox[" Dibujar por el m\[EAcute]todo anterior las siguientes \
superficies de revoluci\[OAcute]n:",
    FontFamily->"Book Antiqua"]
}], "Text"],

Cell["\<\

paraboloide:    f[t]=t, g[t]=t^2
hiperboloide de una hoja: f[t]=Cosh[t], g[t]=Sinh[t]
hiperboloide de dos hojas: f[t]= Sinh[t], g[t]=Cohs[t]
cilindro: f[t]=1,g[t]=t
cono= f[t]=t,g[t]=-t+1
elipse: f[t]= 2 Cos[t], g[t]= Sin[t]
toro: f[t]=2+ Cos[t],g[t]=Sin[t]
\
\>", "Text",
  FontFamily->"Book Antiqua"],

Cell[TextData[{
  StyleBox["Ejercicio:",
    FontFamily->"Book Antiqua",
    FontWeight->"Bold"],
  StyleBox[" Consulta el enlace 4 a internet sobre algunos ejemplos de \
superficies. \[DownQuestion]Qu\[EAcute] tiene de peculiar la superficie \
llamada pseudoesfera?",
    FontFamily->"Book Antiqua"]
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Isometr\[IAcute]a entre el catenoide y el helicoide.",
  
  FontFamily->"Book Antiqua",
  FontColor->RGBColor[1, 0, 0]]], "Subsubtitle"],

Cell[TextData[{
  StyleBox["Ejercicio:",
    FontFamily->"Book Antiqua",
    FontWeight->"Bold"],
  StyleBox[" Dibujar el siguiente helicoide: X(u,v)=(v cos u, v sen u, u).",
    FontFamily->"Book Antiqua"]
}], "Text"],

Cell[TextData[{
  StyleBox["Ejercicio:",
    FontFamily->"Book Antiqua",
    FontWeight->"Bold"],
  StyleBox[" Dibujar el siguiente catenoide: X(u,v)=(cos u cosh v,sen u cosh \
v, v).",
    FontFamily->"Book Antiqua"]
}], "Text"],

Cell[TextData[StyleBox["Considerar la siguiente familia de superficies \
dependiente del parametro \[Theta]. Esta familia forma una \
deformaci\[OAcute]n isom\[EAcute]trica del catenoide en el helicoide para \
\[Theta] entre 0 y \[Pi]/2.",
  FontFamily->"Book Antiqua"]], "Text"],

Cell[BoxData[
    \(x[r_, 
        t_, \[Theta]_] := {\(-\(\(Cos[t - \[Theta]] + 
                r\ \((\(-2\)\ Cos[\[Theta]] + 
                      r\ Cos[t + \[Theta]])\)\)\/\(2\ r\)\)\), \(-\(\(Sin[
                  t - \[Theta]] + r\^2\ Sin[t + \[Theta]]\)\/\(2\ r\)\)\), 
        Cos[\[Theta]]\ Log[r] - t\ Sin[\[Theta]]}\)], "Input",
  Background->None],

Cell[BoxData[
    \(gr[\[Theta]_] := 
      ParametricPlot3D[
        x[r^2, t, \[Theta]], {r, 1/Sqrt[6], Sqrt[6]}, {t, 0, 2\ Pi}, 
        PlotPoints \[Rule] 35, Axes \[Rule] False, 
        ViewPoint \[Rule] {\(-1.481\), 2.293, 2.000}]\)], "Input",
  Background->None],

Cell[BoxData[
    \(gr[0]; gr[Pi/2];\)], "Input"],

Cell[BoxData[
    \(Table[gr[\[Theta]], {\[Theta], 0, Pi/2, \((Pi/2)\)/6}]\)], "Input",
  Background->None],

Cell[TextData[StyleBox["Selecciona la celda del resultado anterior, pulsa \
\"cell\" en el men\[UAcute] y despu\[EAcute]s \"Animate selected...\" para \
visualizar una animaci\[OAcute]n de la deformaci\[OAcute]n.",
  FontFamily->"Book Antiqua"]], "Text"],

Cell[TextData[{
  StyleBox["Ejercicio:",
    FontFamily->"Book Antiqua",
    FontWeight->"Bold"],
  StyleBox[" Modifica las instrucciones anteriores para crear una animaci\
\[OAcute]n con m\[AAcute]s fotogramas, y as\[IAcute] mostrar una animaci\
\[OAcute]n m\[AAcute]s suave.",
    FontFamily->"Book Antiqua"]
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Superficies no orientables:",
  FontFamily->"Book Antiqua",
  FontColor->RGBColor[1, 0, 0]]], "Subsubtitle"],

Cell[TextData[StyleBox["Cinta de Moebius",
  FontColor->RGBColor[1, 0, 0]]], "Text",
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  FontSize->16,
  FontWeight->"Plain",
  FontVariations->{"CompatibilityType"->0}],

Cell[TextData[{
  StyleBox["Ejercicio:",
    FontFamily->"Book Antiqua",
    FontWeight->"Bold"],
  StyleBox[" Dibujar la siguiente cinta de Moebius: X(u,v)=(Cos u+v Cos(u/2) \
Cos u, Sen u +v Cos(u/2) Sen u, v Sen(u/2)) con u entre 0 y 2Pi. Consultar \
sus propiedades y utilidades pr\[AAcute]cticas en los enlaces 1, 2 y 3.",
    FontFamily->"Book Antiqua"]
}], "Text"],

Cell[TextData[StyleBox["Botella de Klein",
  FontColor->RGBColor[1, 0, 0]]], "Text",
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    FontWeight->"Bold"],
  StyleBox["  Consulta en los enlaces 6, 7 y 8 como es esta superficie.",
    FontFamily->"Book Antiqua"]
}], "Text"]
}, Open  ]],

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Cell[TextData[StyleBox["Superficies minimales y superficies con curvatura \
media constante:",
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Cell[TextData[StyleBox["Superficie de Scherck",
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  FontWeight->"Plain",
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Cell["\<\
Las superficies de Scherck son grafos de la funci\[OAcute]n z=Log[Cos[c \
y+d]/Cos[c x+e]]\
\>", "Subtitle",
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Cell["Haciendo c=1, d=0 y variando e", "Text",
  FontFamily->"Book Antiqua"],

Cell[BoxData[
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      Table[ParametricPlot3D[
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            Pi/2 - e}, {y, \(-Pi\)/2 - 1, Pi/2 - 1}], {e, 0, 2}]\)], "Input"],

Cell[BoxData[
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Cell[TextData[StyleBox["Consulta los enlaces 9 y 10.",
  FontFamily->"Book Antiqua"]], "Text"]
}, Open  ]]
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