How to connect a checkbox with part of the graphics?

Question

I'd like to have a checkbox in a Manipulate expression, governing whether part of the graphics is displayed or not. For instance, assume that I have a surface with two planes:

Show[Plot3D[x^2+y^2,{x,-1,1},{y,-1,1},PlotStyle->Opacity[0.5]],
  ParametricPlot3D[{0, u, v}, {u, -1, 1}, {v, -1, 1}, PlotStyle -> Opacity[0.5], Mesh -> None],
  ParametricPlot3D[{u, 0, v}, {u, -1, 1}, {v, -1, 1}, PlotStyle -> Opacity[0.5], Mesh -> None]]

Now I'd like to wrap it in a Manipulate with two checkboxes, enabling or disabling the planes. Currently, I'm doing something like this:

Manipulate[
 Apply[Show, 
  Join[{Plot3D[...]}, 
   If[xzero, {ParametricPlot3D[{0, u, v}, {u, -1, 1}, {v, -1, 1}, 
      PlotStyle -> Opacity[0.5], Mesh -> None]}, {}], 
   If[yzero, {ParametricPlot3D[{u, 0, v}, {u, -1, 1}, {v, -1, 1}, 
      PlotStyle -> Opacity[0.5], Mesh -> None]}, {}]]],
{{xzero, True, "x=0"}, {True, False}}, {{yzero, True, "y=0"}, {True, 
   False}}]

but somehow feel it's overly complicated. What is the right way to do this?

Answer 1

Discrete control via opacity:

Manipulate[
  Show[Plot3D[x^2 + y^2, {x, -1, 1}, {y, -1, 1},   PlotStyle -> Opacity[0.5]], 
  ParametricPlot3D[{0, u, v}, {u, -1, 1}, {v, -1, 1}, 
  PlotStyle -> Opacity[xzero], Mesh -> None], 
  ParametricPlot3D[{u, 0, v}, {u, -1, 1}, {v, -1, 1}, PlotStyle -> Opacity[yzero], 
  Mesh -> None], AxesLabel -> {X, Y, Z}],
 {{xzero, .5, "x=0"}, {0, .5}, Checkbox},
 {{yzero, .5, "y=0"}, {0, .5}, Checkbox},ControlPlacement -> Left]

But I usually prefer to control opacity continuously. Then you have more flexible control to "pronounce" one object and "denounce" another, still keeping it. If opacity is set to zero - plane disappear.

Manipulate[
 Show[Plot3D[x^2 + y^2, {x, -1, 1}, {y, -1, 1}, 
   PlotStyle -> Opacity[0.5], Mesh -> True, MeshFunctions -> {#3 &}, 
   ColorFunction -> "Rainbow"], 
  ParametricPlot3D[{0, u, v}, {u, -1, 1}, {v, -1, 1}, PlotStyle -> 
    Directive[col1, Specularity[White, 30], Opacity[xzero]], Mesh -> None], 
  ParametricPlot3D[{u, 0, v}, {u, -1, 1}, {v, -1, 1}, PlotStyle -> 
    Directive[col2, Specularity[White, 30], Opacity[xzero]], 
   Mesh -> None], AxesLabel -> {X, Y, Z}],
 {{xzero, .5, "x=0"}, 0, 1, Slider, ImageSize -> 75},
 {{yzero, .5, "y=0"}, 0, 1, Slider, ImageSize -> 75},
 {col1, Blue, ImageSize -> Tiny},
 {col2, Red, ImageSize -> Tiny},
 ControlPlacement -> Left]

Answer 2

p1 = ParametricPlot3D[{0, u, v}, {u, -1, 1}, {v, -1, 1}, 
                       PlotStyle -> Opacity[0.5], Mesh -> None];
p2 = ParametricPlot3D[{u, 0, v}, {u, -1, 1}, {v, -1, 1}, 
                       PlotStyle -> Opacity[0.5], Mesh -> None];

Manipulate[Show[
          Plot3D[x x + y y, {x, -1, 1}, {y, -1, 1}],
          If[xzero, p1, Sequence @@ {}], If[yzero, p2, Sequence @@ {}]],
         {{xzero, True, "x=0"}, {True, False}}, 
         {{yzero, True, "y=0"}, {True, False}}]