How to texturize a Disk/Circle/Rectangle?

Question

The documentation for Texture states that "other filled objects" can be texturized:

Texture[obj] is a graphics directive that specifies that obj should be
  used as a texture on faces of polygons and other filled graphics
  objects.

And also:

Texture can be used in FaceForm to texture front and back faces
  differently.

Though I fail to apply a simple texture to any of the following objects. It seems like that "other filled objects" only include Polygons and FilledPolygons, and FaceForm does not work with those.

img = Rasterize@
   DensityPlot[Sin@x Sin@y, {x, -4, 4}, {y, -3, 3}, 
    ColorFunction -> "BlueGreenYellow", Frame -> None, 
    ImageSize -> 100, PlotRangePadding -> 0];
{
 Graphics[{Texture@img, Disk[]}],
 Graphics[{FaceForm@Texture@img, Disk[]}],
 Graphics[{Texture@img, Rectangle[]}],
 Graphics[{FaceForm@Texture@img, Rectangle[]}],

(* Only this one works *)
 Graphics[{Texture@img, 
   Polygon[{{0, 0}, {1, 0}, {1, 1}, {0, 1}}, 
    VertexTextureCoordinates -> {{0, 0}, {1, 0}, {1, 1}, {0, 1}}]}],
 Graphics[{FaceForm@Texture@img, 
   Polygon[{{0, 0}, {1, 0}, {1, 1}, {0, 1}}, 
    VertexTextureCoordinates -> {{0, 0}, {1, 0}, {1, 1}, {0, 1}}]}]
 }

Edit:

It turns out that "Applying Texture to a disk directly isn't possible" (according to Heike, thanks s.s.o. for the link). This unfortunately means that:

the official documentation of Texture is wrong (or at least is misleading, as graphics objects usually include primitives);
either Texture is not fully integrated with the system, as it is not  applicable for such primitives as a Rectangle, which seems to be just a very specific Polygon; or Rectangle is something else and is defined some other way at the lowest level than a Polygon (maybe it is some OS-dependent object).

Frankly, it is quite hard to imagine what kept developers to include this functionality, but I must assume they had a good reason.

Silvia · Accepted Answer

I noticed an example in the document of Texture which used the alpha channel. So I think a disk-shape primitive may be simulated to a limited degree by mapping the image img, which has been set to 100% transparent outside of the circle, onto a rectangle-shape Polygon.

My code:

img = Rasterize[
                DensityPlot[Sin[x] Sin[y],
                            {x, -4, 4}, {y, -3, 3},
                            ColorFunction -> "BlueGreenYellow",
                            Frame -> None, ImageSize -> 100, PlotRangePadding -> 0
              ]];

imgdim = ImageDimensions[img]

alphamask = Array[
                  If[
                     Norm[{#1, #2} - imgdim/2] < imgdim[[1]]/2,
                     1,0]&,
                  imgdim];

alphaimg = MapThread[Append, {img // ImageData, alphamask}, 2];

Graphics[{
          Polygon[{{0, 0}, {1, 0}, {1, 1}, {0, 1}} + .3],
          Texture[alphaimg],
          Polygon[{{0, 0}, {1, 0}, {1, 1}, {0, 1}}, 
                  VertexTextureCoordinates -> {{0, 0}, {1, 0}, {1, 1}, {0, 1}}
                 ],
          Gray, Disk[{0, 0}, .5]
         }]

which gives result like this:

Mark McClure · Answer

Like RM, I've not been able to texture a Disk primitive.  We can create a textured disk using ParametricPlot, however.

ParametricPlot[{r*Cos[t], r*Sin[t]}, {r, 0, 1}, {t, 0, 2 Pi},
  Mesh -> False, BoundaryStyle -> None, Axes -> False,
  PlotStyle -> {Opacity[1], 
   Texture[ExampleData[{"ColorTexture", "LightCherry"}]]}]

s.s.o · Answer

If you click the disk and check Drawing tools from the menu there is only color fill options no texture options available. Also see Heike's related answers: in math group or mathematica group mainly she states "Applying Texture to a disk directly isn't possible, but you could for example use RegionPlot with the TextureCoordinateFunction option, e.g"

img = Rasterize@
   DensityPlot[Sin@x Sin@y, {x, -4, 4}, {y, -3, 3}, 
    ColorFunction -> "BlueGreenYellow", Frame -> None, 
    ImageSize -> 100, PlotRangePadding -> 0];

RegionPlot[x^2 + y^2 < 1, {x, -1, 1}, {y, -1, 1}, 
 BoundaryStyle -> None, 
 Axes -> False, Frame -> False, 
 PlotStyle -> Directive[Opacity[1], Texture[img]], 
 TextureCoordinateFunction -> ({#1, #2} &)]

István Zachar · Answer

My fallback method for the moment is the following: approximate a circle with a polygon, fill the latter with the texture and finally conceal the angular edge with an overlaid Circle. If the whole image is small, the number of nodes of the polygon can be further reduced. One annoying sideeffect is though that the Circle is not antialiased...

img = Rasterize@
   DensityPlot[Sin@x Sin@y, {x, -4, 4}, {y, -3, 3}, 
    ColorFunction -> "BlueGreenYellow", Frame -> None, 
    ImageSize -> 200, PlotRangePadding -> 0];
coord = Block[{n = 100}, 
   Table[{Cos[2 [Pi] k/n], Sin[2 [Pi] k/n]}, {k, 0, n - 1}]];
Graphics[{Texture@img, EdgeForm@None, 
  Polygon[coord, VertexTextureCoordinates -> (coord/2 + .5)], Black, 
  Thick, Circle[]}, ImageSize -> 200, Background -> GrayLevel@.9]

Simon Woods · Answer

Here's an extension of Silvia's method of setting an alpha channel in the texture. The alpha mask is obtained directly from the shape using Rasterize, allowing the code to work with ellipses, rectangles, etc.

img = Rasterize@
   DensityPlot[Sin@x Sin@y, {x, -4, 4}, {y, -3, 3}, 
    ColorFunction -> "BlueGreenYellow", Frame -> None, 
    ImageSize -> 300, PlotRangePadding -> 0];

texturedShape[img_, shape_] := Module[{g, p, ar, i},
  g = Graphics[shape, PlotRangePadding -> 0];
  p = Polygon[
    AbsoluteOptions[g, PlotRange][[1, 
       2]] /. {{l_, r_}, {b_, t_}} :> {{l, b}, {l, t}, {r, t}, {r, 
        b}}, VertexTextureCoordinates -> {{0, 0}, {0, 1}, {1, 1}, {1, 
       0}}];
  ar = AbsoluteOptions[g, AspectRatio][[1, 2]];
  i = SetAlphaChannel[img, 
    ColorNegate@Rasterize[g, ImageSize -> ImageDimensions@img]];
  i = ImageCrop[i, 
    Round[If[ar > 1, {1/ar, 1}, {1, ar}] ImageDimensions@img]];
  {Texture[ImageData@i], p}]

The Rasterize gets its ImageDimensions from the original texture image, so I've increased the size of that to 300 to get cleaner edges.

Examples:

Graphics[{texturedShape[img,Disk[{0,0},2]],Red,Disk[{0,1},1]}]

Graphics[{texturedShape[img,Disk[{0,0},{2,1}]],Red,Disk[{0,1},1]}]

Graphics[{Red,Disk[{0,0},0.5],texturedShape[img,Rectangle[]]}]

Mr.Wizard · Answer

Mathematica 10 introduced Regions which make this kind of operation much easier.

img = Rasterize@
   DensityPlot[Sin@x Sin@y, {x, -4, 4}, {y, -3, 3}, 
    ColorFunction -> "BlueGreenYellow", Frame -> None, ImageSize -> 100, 
    PlotRangePadding -> 0];

RegionPlot[
  RegionUnion[Disk[{0, 0}, {2, 2}], Disk[{0, 0}, {3.5, 0.7}]]
  , PlotStyle -> Texture[img]
][[{1}]]

kglr · Answer

As of version 12.1, we can use (yet undocumented) SurfaceAppearance["TextureShading", Texture[img]] to texturize 2D primitives such as Disk, Rectangle as well as 3D primitives like Sphere, Tube etc.
lena = ExampleData[{"TestImage", "Lena"}];
mandrill = ExampleData[{"TestImage", "Mandrill"}];

Graphics[{SurfaceAppearance["TextureShading", Texture[ImageMultiply[lena, Green]]], 
  Disk[], 
  SurfaceAppearance["TextureShading", Texture[mandrill]], 
  Rectangle[{2, -3/2}, {4, 1/2}], 
  BoundaryDiscretizeRegion @ RegionUnion[Disk[{1, 1}], Rectangle[{3/2, 1}]]}]

How to texturize a Disk/Circle/Rectangle?

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