Background image in a polygon / CountryData

Question

Is it possible to use a background image in a polygon without rasterizing the polygon?

I tried with textures, but they seem to be viable only for simple polygons.

I want to put a flag as the background of the polygon that represents a country shape. I can do this for one country by rasterizing the polygon:

ImageAdd[
    Graphics[CountryData["Monaco", "Polygon"]], 
    CountryData["Monaco", "Flag"]
]

But by rasterizing the polygon, I make it difficult to apply this procedure to several neigboring countries and assemble them together.

I would like to put each country flag in the background of, for example:

Graphics[
    {EdgeForm[White], 
     CountryData[#, "Polygon"] & /@ {"Venezuela", "Colombia"}}]  

---

Solution

After seeing cormullion answer, I was able to do what I wanted:

Graphics[
 {EdgeForm[Black],
  {{Texture[ImageReflect[CountryData[#, "Flag"], Top -> Right]], 
    CountryData[#, "Polygon"] /. 
      {Polygon[a_] :> Polygon[First[a], VertexTextureCoordinates -> 
       Transpose[Rescale /@ Transpose[First[CountryData[#, "Coordinates"]]]]]}} & /@ 
CountryData["SouthAmerica"]}}]

The key part is the use of Rescale[].

Answer 1

This example is from the documentation for Texture in Mathematica version 8:

With[{vc = 
 Transpose[
  Rescale /@ Transpose[First[CountryData[#, "Coordinates"]]]]}, 
Show[CountryData[#, "Shape"], 
 ImageSize -> {{100}, {100}}] /. {RGBColor[__] :> 
  Texture[ImageReflect[Image[CountryData[#, "Flag"]], 
    Top -> Right]], 
 Polygon[a_] :> 
  Polygon[First[a], VertexTextureCoordinates -> vc]}] & /@ 
CountryData["SouthAmerica"]

but I don't know whether it does what you need. Still, it looks pretty.

Answer 2

Try:

Graphics[{EdgeForm[White], {{Texture[Image[CountryData[#, "Flag"]]], 
  Polygon[First@First[CountryData[#, "Polygon"]], 
   VertexTextureCoordinates ->Transpose[
     Rescale /@ Transpose[First@First[CountryData[#, 
         "Polygon"]]]]]}} & /@ {"Venezuela", "Colombia"}}]

which gives

Answer 3

For coastal countries with a lot of islands, CountryData[tag,"Polygon"] gives something of the form Polygon[{island1, island2, island3,...}], where island1 and the like is another list of latitudes and longitudes.

The case is simpler for inland countries. CountryData[tag,"Polygon"] gives Polygon[{island1}], where island1 is simply the mainland, and is itself a list of latitudes and longitudes.

An example can be given using UK and Venezuela:

Table[Length /@ First[CountryData[country, "Polygon"]], {country, {"UnitedKingdom", "Venezuela"}}]

The result is:

{{20, 25, 31, 54, 52, 63, 141, 195, 1927}, {491}}

which means that UK is made up of 9 polygons/islands, and Venezuela only 1.

To fill a country's land with its own flag, we must be careful to apply the flag texture to all the islands. The important thing is to treat each island of a country one by one.

To apply a texture to any polygon, we must first come up with a function that can yield the correct VertexTextureCoordinates. For example:

countryRescale1[polygon_] := (Transpose[Rescale /@ Transpose[First[polygon]]])

The function accepts as argument a polygon of an island of a country, which is of the form Polygon[{{x1,y1},{x2,y2},...}}]. We obtain the list of coordinates with First[].

We apply transpose to the list of coordinates {{x1,y1},{x2,y2},...}}, so that we group all the x's together, and y's together: {{x1, x2, x3, ...}, {y1, y2, y3, ...}}. We apply to it Rescale[], so that x's and y's are both rescaled from 0 to 1 separately. Then apply transpose to group them back into {{x1,y1},{x2,y2},...}}, where x and y is now in [0,1].

Then we can overlay the flag onto a country. Let's use Hong Kong (though not a country, but it has many islands) as an example. The important thing is to treat each island of a country one by one.

Graphics[{EdgeForm[Black], Texture[CountryData["HongKong", "Flag"]], 
  Polygon[#, VertexTextureCoordinates -> countryRescale1[Polygon[#]]] & /@
  First[CountryData["HongKong", "Polygon"]]}]

We map the pure function Polygon[#, VertexTextureCoordinates -> countryRescale1[Polygon[#]]]& to the expression First[CountryData["HongKong", "Polygon"]], which in itself is a list of islands { {{x1, y1}, {x2, y2}, ..., {xn,yn}}, {{x1, y1}, {x2, y2}, ...}, ... }.

So it's just like:

Graphics[{EdgeForm[Black], Texture[CountryData["HongKong", "Flag"]],
  {Polygon[], Polygon[], Polygon[] ... }]

The result looks like this:

For Argentina, the island province Tierra del Fuego (the bottom right triangular piece of land) is also treated with the flag:

The final thing left to do is to apply this to all countries to make a world flag map:

Graphics[{EdgeForm[Black], Table[{Texture[CountryData[c, "Flag"]], 
  Polygon[#, VertexTextureCoordinates -> countryRescale1[Polygon[#]]] & /@  
  First[CountryData[c, "FullPolygon"]]}, {c, CountryData[]}]}, ImageSize -> 700]

The result looks like this:

For better flags, one can refer to this post: Why do some CountryData flags render at low resolution?