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g1 = ParametricPlot3D[
{Cos[u]Cos[v],
Tan[v]Cos[u]Sin[v],
Sin[u]},
{u, -Pi/2, Pi/2},
{v, -Pi/2 + 0.01, Pi/2 - 0.01},
PlotRange -> {{-Pi/2, Pi/2}, {-Pi/2, Pi/2}, {-Pi/2, Pi/2}},
PlotPoints -> {50, 50},
DisplayFunction -> Identity];
g2 = ParametricPlot3D[
{Cos[u]Cos[v],
Tan[v]Cos[u]Sin[v],
Sin[u]},
{u, -Pi/2, Pi/2},
{v, Pi/2 + 0.01, 3 Pi/2 - 0.01},
PlotRange -> {{-Pi/2, Pi/2}, {-Pi/2, Pi/2}, {-Pi/2, Pi/2}},
PlotPoints -> {50, 50},
DisplayFunction -> Identity];
Show[{g1, g2},
Boxed -> False,
Axes -> None,
DisplayFunction -> $DisplayFunction]
ParametricPlot3D[
{Cos[2u]Cos[v]/3,
Sin[u]v/u + v/2,
Sin[u]Cos[2v]},
{u, -Pi, Pi},
{v, -Pi, Pi},
PlotPoints -> {50, 50},
Boxed -> False,
Axes -> None]
ParametricPlot3D[
{Cos[x + y^2] + Cos[x]Sin[y^2],
Sin[x] + Cos[x],
2Cos[y] + Sin[y^2]},
{x, 0, 2Pi},
{y, 0, 2Pi},
PlotPoints -> {50, 50},
Boxed -> False,
Axes -> None]
ParametricPlot3D[
{u Sin[t],
u Cos[t],
t/3},
{t, 0, 20},
{u, -1, 1},
PlotPoints -> {50, 25},
Boxed -> False,
Axes -> None]

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