Creating the colored mesh

### 4.2.10 Creating the colored mesh

If you think that these things we have been examining are advanced, then you haven't seen anything. Now comes real hard-core advanced POV-Ray code, so be prepared. This could be called The really advanced section.

We have now calculated the image into the array of colors. However, we still have to show these color "pixels" on screen, that is, we have to make POV-Ray to render our pixels so that it creates a real image.

There are several ways of doing this, all of them being more or less "kludges" (as there is currently no way of directly creating an image map from a group of colors). One could create colored boxes representing each pixel, or one could output to an ascii-formatted image file (mainly PPM) and then read it as an image map. The first one has the disavantage of requiring huge amounts of memory and missing bilinear interpolation of the image; the second one has the disadvantage of requiring a temporary file.

What we are going to do is to calculate a colored mesh2 which represents the "screen". As colors are interpolated between the vertices of a triangle, the bilinear interpolation comes for free (almost).

#### 4.2.10.1 The structure of the mesh

Although all the triangles are located in the x-y plane and they are all the same size, the structure of the mesh is quite complicated (so complicated it deserves its own section here).

The following image shows how the triangles are arranged for a 4x3 pixels image:

Triangle arrangement for a 4x3 image

The number pairs in parentheses represent image pixel coordinates (eg. `(0,0)` refers to the pixel at the lower left corner of the image and `(3,2)` to the pixel at the upper right corner). That is, the triangles will be colored as the image pixels at these points. The colors will then be interpolated between them along the surface of the triangles.

The filled and non-filled circles in the image represent the vertex points of the triangles and the lines connecting them show how the triangles are arranged. The smaller numbers near these circles indicate their index value (the one which will be created inside the `mesh2`).

We notice two things which may seem odd: Firstly there are extra vertex points outside the mesh, and secondly, there are extra vertex points in the middle of each square.

Let's start with the vertices in the middle of the squares: We could have just made each square with two triangles instead of four, as we have done here. However, the color interpolation is not nice this way, as there appears a clear diagonal line where the triangle edges go. If we make each square with four triangles instead, then the diagonal lines are less apparent, and the interpolation resembles a lot better a true bilinear interpolation. And what is the color of the middle points? Of course it's the average of the color of the four points in the corners.

Secondly: Yes, the extra vertex points outside the mesh are completely obsolete and take no part in the creation of the mesh. We could perfectly create the exact same mesh without them. However, getting rid of these extra vertex points makes our lives more difficult when creating the triangles, as it would make the indexing of the points more difficult. It may not be too much work to get rid of them, but they don't take any considerable amount of resources and they make our lives easier, so let's just let them be (if you want to remove them, go ahead).

#### 4.2.10.2 Creating the mesh

What this means is that for each pixel we create two vertex points, one at the pixel location and one shifted by "0.5" in the x and y directions. Then we specify the color for each vertex points: For the even vertex points it's directly the color of the correspondent pixel; for the odd vertex points it's the average of the four surrounding pixels.

Let's examine the creation of the mesh step by step:

##### 4.2.10.2.1 Creating the vertex points

    #default { finish { ambient 1 } }

    #debug "Creating colored mesh to show image...\n"
    mesh2
    { vertex_vectors
      { ImageWidth*ImageHeight*2,
        #declare IndY = 0;
        #while(IndY < ImageHeight)
          #declare IndX = 0;
          #while(IndX < ImageWidth)
            <(IndX/(ImageWidth-1)-.5)*ImageWidth/ImageHeight*2,
             IndY/(ImageHeight-1)*2-1, 0>,
            <((IndX+.5)/(ImageWidth-1)-.5)*ImageWidth/ImageHeight*2,
             (IndY+.5)/(ImageHeight-1)*2-1, 0>
            #declare IndX = IndX+1;
          #end
          #declare IndY = IndY+1;
        #end
      }

First of all we use a nice trick in POV-Ray: Since we are not using light sources and there's nothing illuminating our mesh, what we do is to set the ambient value of the mesh to 1. We do this by just making it the default with the `#default` command, so we don't have to bother later.

As we saw above, what we are going to do is to create two vertex points for each pixel. Thus we know without further thinking how many vertex vectors there will be: `ImageWidth*ImageHeight*2`

That was the easy part; now we have to figure out how to create the vertex points themselves. Each vertex location should correspond to the pixel location it's representing, thus we go through each pixel index (practically the number pairs in parentheses in the image above) and create vertex points using these index values. The location of these pixels and vertices are the same as we assumed when we calculated the image itself (in the previous part). Thus the y coordinate of each vertex point should go from -1 to 1 and similarly the x coordinate, but scaled with the aspect ratio.

If you look at the creation of the first vector in the code above, you'll see that it's almost identical to the direction vector we calculated when creating the image.

The second vector should be shifted by 0.5 in both directions, and that's exactly what is done there. The second vector definition is identical to the first one except that the index values are shifted by 0.5. This creates the points in the middle of the squares.

The index values of these points will be arranged as shown in the image above.

##### 4.2.10.2.2 Creating the textures

      texture_list
      { ImageWidth*ImageHeight*2,
        #declare IndY = 0;
        #while(IndY < ImageHeight)
          #declare IndX = 0;
          #while(IndX < ImageWidth)
            texture { pigment { rgb Image[IndX][IndY] } }
            #if(IndX < ImageWidth-1 & IndY < ImageHeight-1)
              texture { pigment { rgb
                (Image[IndX][IndY]+Image[IndX+1][IndY]+
                 Image[IndX][IndY+1]+Image[IndX+1][IndY+1])/4 } }
            #else
              texture { pigment { rgb 0 } }
            #end
            #declare IndX = IndX+1;
          #end
          #declare IndY = IndY+1;
        #end
      }

Creating the textures is very similar to creating the vertex points (we could have done both inside the same loop, but due to the syntax of the `mesh2` we have to do it separately).

So what we do is to go through all the pixels in the image and create textures for each one. The first texture is just the pixel color itself. The second texture is the average of the four surrounding pixels.

Note: we can calculate it only for the vertex points in the middle of the squares; for the extra vertex points outside the image we just define a dummy black texture.

The textures have the same index values as the vertex points.

##### 4.2.10.2.3 Creating the triangles

This one is a bit trickier. Basicly we have to create four triangles for each "square" between pixels. How many triangles will there be?

Let's examine the creation loop first:

      face_indices
      { (ImageWidth-1)*(ImageHeight-1)*4,
        #declare IndY = 0;
        #while(IndY < ImageHeight-1)
          #declare IndX = 0;
          #while(IndX < ImageWidth-1)

            ...

            #declare IndX = IndX+1;
          #end
          #declare IndY = IndY+1;
        #end
      }

The number of "squares" is one less than the number of pixels in each direction. That is, the number of squares in the x direction will be one less than the number of pixels in the x direction. The same for the y direction. As we want four triangles for each square, the total number of triangles will then be `(ImageWidth-1)*(ImageHeight-1)*4`.

Then to create each square we loop the amount of pixels minus one for each direction.

Now in the inside of the loop we have to create the four triangles. Let's examine the first one:

            <IndX*2+  IndY    *(ImageWidth*2),
             IndX*2+2+IndY    *(ImageWidth*2),
             IndX*2+1+IndY    *(ImageWidth*2)>,
             IndX*2+  IndY    *(ImageWidth*2),
             IndX*2+2+IndY    *(ImageWidth*2),
             IndX*2+1+IndY    *(ImageWidth*2),

This creates a triangle with a texture in each vertex. The first three values (the indices to vertex points) are identical to the next three values (the indices to the textures) because the index values were exactly the same for both.

The `IndX` is always multiplied by 2 because we had two vertex points for each pixel and `IndX` is basically going through the pixels. Likewise `IndY` is always multiplied by `ImageWidth*2` because that's how long a row of index points is (ie. to get from one row to the next at the same x coordinate we have to advance `ImageWidth*2` in the index values).

These two things are identical in all the triangles. What decides which vertex point is chosen is the "+1" or "+2" (or "+0" when there's nothing). For `IndX` "+0" is the current pixel, "+1" chooses the point in the middle of the square and "+2" chooses the next pixel. For `IndY` "+1" chooses the next row of pixels.

Thus this triangle definition creates a triangle using the vertex point for the current pixel, the one for the next pixel and the vertex point in the middle of the square.

The next triangle definition is likewise:

            <IndX*2+  IndY    *(ImageWidth*2),
             IndX*2+  (IndY+1)*(ImageWidth*2),
             IndX*2+1+IndY    *(ImageWidth*2)>,
             IndX*2+  IndY    *(ImageWidth*2),
             IndX*2+  (IndY+1)*(ImageWidth*2),
             IndX*2+1+IndY    *(ImageWidth*2),

This one defines the triangle using the current point, the point in the next row and the point in the middle of the square.

The next two definitions define the other two triangles:

            <IndX*2+  (IndY+1)*(ImageWidth*2),
             IndX*2+2+(IndY+1)*(ImageWidth*2),
             IndX*2+1+IndY    *(ImageWidth*2)>,
             IndX*2+  (IndY+1)*(ImageWidth*2),
             IndX*2+2+(IndY+1)*(ImageWidth*2),
             IndX*2+1+IndY    *(ImageWidth*2),

            <IndX*2+2+IndY    *(ImageWidth*2),
             IndX*2+2+(IndY+1)*(ImageWidth*2),
             IndX*2+1+IndY    *(ImageWidth*2)>,
             IndX*2+2+IndY    *(ImageWidth*2),
             IndX*2+2+(IndY+1)*(ImageWidth*2),
             IndX*2+1+IndY    *(ImageWidth*2)