How to get 2D graphics into 3D without background?

Question

Consider the following code:

g=Graphics[{EdgeForm[{Black, Thick}],
            {Green,Opacity[0.5],Rectangle[{0,0.5},{1,1}]},
            Red, Polygon[{{0, 0}, {1, 0}, {1/2, Sqrt[3]/2}}],
            White, Disk[{1/2, 1/(2Sqrt[3])}, 1/(2Sqrt[3])]},
           Background->None];
Graphics3D[{{Texture[g],
             Polygon[{{0,0,0},{0,1,0},{1,1,0},{1,0,0}},
                     VertexTextureCoordinates->{{0,0},{1,0},{1,1},{0,1}}]},
            {Texture[g],
             Polygon[{{0,0,1},{0,1,1},{1,1,1},{1,0,1}},
                     VertexTextureCoordinates->{{0,0},{1,0},{1,1},{0,1}}]}},
           Lighting->{{"Ambient", White}},
           ViewPoint->{1,4,7}]

This gives the following graphics:

As you can easily see, the part outside of the figures is opaque white, despite of the Background->None option. Also Background->RGBColor[1,1,1,0] didn't help, nor did passing the Graphics through Rasterize with the option Background->None

Here's roughly what I'd want to get (any inaccuracies are caused by my lacking GIMP-fu :-)):

Looking for a solution I found this code but couldn't get it work for my case. Also, if I understand it correctly, it doesn't derive the transparency from the graphics but just replaces a certain colour with fully transparent, which wouldn't work with the semi-transparent green rectangle.

Combining the 2D graphics with another one using Show does respect Background->None and Opacity on the green rectangle, so it's not a problem with the graphics itself.

Therefore my question: Is it possible to embed 2D graphics into 3D while keeping the transparency of the 2D image, and if so, how?

Answer 1

OK, I've now found the solution. The trick is to extract the data from the raster and pass it to the texture directly (for some reason I also have to reverse the list to get the same result):

gtexture=Texture[Rasterize[g, Background -> None][[1,1]]/255 //Reverse];
Graphics3D[{{gtexture,
             Polygon[{{0, 0, 0}, {0, 1, 0}, {1, 1, 0}, {1, 0, 0}},
                   VertexTextureCoordinates->{{0, 0}, {1, 0}, {1, 1}, {0, 1}}]},
            {gtexture,
             Polygon[{{0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {1, 0, 1}},
                   VertexTextureCoordinates->{{0, 0}, {1, 0}, {1, 1}, {0, 1}}]}},
           Lighting -> {{"Ambient", White}},
           ViewPoint -> {1, 4, 7}]

This gives the desired result:

Answer 2

Here is a solution partially based on this blog post. I needed to change some of your definitions, but I think the following should work. First, define a function that does some things for us:

to3d[plot_, height_, opacity_] := Module[{newplot}, newplot = First@Graphics[plot]; 
  newplot = N@newplot 
    /. {x_?AtomQ, y_?AtomQ} :> {x, y, height} 
    /. Disk[x_, y_] :> Cylinder[{x, x + {0, 0, 0.001}}, y];
  newplot /. GraphicsComplex[xx__] -> {Opacity[opacity], GraphicsComplex[xx]}]

Define your shapes separately now:

r = Graphics[{EdgeForm[{Black, Thick}], {Green, Opacity[0.5], 
     Polygon[{{0, 1}, {0, 0.5}, {1, 0.5}, {1, 1}}]}}, 
   Background -> None];
t = Graphics[{EdgeForm[{Black, Thick}], Red, 
    Polygon[{{0, 0}, {1, 0}, {1/2, Sqrt[3]/2}}], White, 
    Disk[{1/2, 1/(2 Sqrt[3])}, 1/(2 Sqrt[3])]}, Background -> None];

(Note the changed Rectangle -> Polygon definition. Polygon is valid for both 2D and 3D). Put everything together:

Graphics3D[{to3d[r, 1, 0.5], to3d[t, 1.01, 0.5], to3d[r, 0, 0.5], 
   to3d[t, .01, 0.5]}, Lighting -> "Neutral"]

Giving:

Answer 3

Here's... an approach:

g = Graphics[{EdgeForm[{Black, Thick}], {Green, Opacity[0.5],
     Rectangle[{0, 0.5}, {1, 1}]}, Red, 
    Polygon[{{0, 0}, {1, 0}, {1/2, Sqrt[3]/2}}], White, 
    Disk[{1/2, 1/(2 Sqrt[3])}, 1/(2 Sqrt[3])]}];

lifted = N[g] /. {
    {x_Real, y_Real} :> {z, y, x},
    Rectangle :> Cuboid,
    Disk :> Sphere};

zlevelI = 0;
zlevel[] := (zlevelI = zlevelI + .001);

Show[
 Graphics3D @@ (lifted /. z :> 1 + zlevel[]),
 Graphics3D @@ (lifted /. z :> 0 + zlevel[])]

You get the idea. It appears to be... moderately generalizable:

g = Graphics[{EdgeForm[{Black, Thick}], {Green, Opacity[0.5],
     Rectangle[{0, 0.5}, {1, 1}]}, Red, 
    Polygon[{{0, 0}, {1, 0}, {1/2, Sqrt[3]/2}}], White, 
    Disk[{1/2, 1/(2 Sqrt[3])}, 1/(2 Sqrt[3])]}];

g2 = Plot[{Sin[x*2], Sin[4 x], Sin[6 x]}, {x, 0, Pi/2}, 
   PlotStyle -> Thick];

lift = {
   {x_Real, y_Real} :> {z, y, x},
   Rectangle :> Cuboid,
   Disk :> Sphere};

lifted = N[g] /. lift;
lifted2 = N[g2] /. lift;

zlevelI = 0;
zlevel[] := (zlevelI = zlevelI + .001);

Show[
 Graphics3D @@ (lifted /. z :> 1 + zlevel[]),
 Graphics3D @@ (lifted2 /. z :> 0)]

But it's definitely not ideal.