Am 23.07.2017 um 08:21 schrieb omniverse:

> Thanks for the explanation, guess I thought there must still be some "blending"
> going on in there somewhere, from one color entry to next, especially if the
> actual blending is limited only by usable memory.

The /actual blending/ is not even limited by usable memory -- it is
computed on the fly while rendering.

> Ohhh! Maybe I get it now. Reduce the number of entries/indices and the between
> zones should exist more visibly...?
> Um, nope. Not it. At least I'm unable to reduce it down to fewer colors and see
> the purple lessen in the middle of blue and red.
> So I'm still not understanding why I can't get blend_mode 2 the same as when the
> color_map includes only a 0 and 1 entry by using only a few instead. Main
> difference being the #for loop, only thing I can imagine causing this.

Suppose you have the followng object:

    box { <-1,-1,-1>, <1,1,1>
      texture {
        pigment {
          gradient x
          colour_map {
            blend_mode BLENDMODE
            [0.0 colour rgb <0,0,1>]
            [1.0 colour rgb <1,0,0>]
          }
        }
      }
    }

Now assume that for one particular pixel the camera ray hits the box
surface at <0.4,-1,0>; that's halfway along the gradient, giving a
pattern value of 0.4.

POV-Ray then looks up the pattern value in the colour map: It finds 0.0
and 1.0 as the two closest index values, with colours rgb <0,0,1> and
rgb <1,0,0> respectively. It computes the colour at pattern value 0.4 as
a weighted average of the two colours:

    P1 = 0.0; C1 = <0,0,1>
    P2 = 1.0; C2 = <1,0,0>
    P  = 0.4

    W1 = (P2-P)/(P2-P1)
    W2 = 1.0-W1
    Result = Average(W1,C1, W2,C2)

For `blend_mode 0`, we have:

    Average(W1,C1, W2,C2) = W1*C1 + W2*C2

And thus:

    W1 = (1.0-0.4)/(1.0-0.0) = 0.6/1.0 = 0.6
    W2 = 1.0-0.6 = 0.4
    Result = 0.6*<0,0,1>+0.4*<1,0,0> = <0,0,0.6>+<0.4,0,0>
           = <0.4,0,0.6>

However, for the other blend modes the formula is more complicated. The
only property that's guaranteed for all blend modes is that

    Average(1.0,C1, 0.0,C2) = C1
    Average(0.0,C1, 1.0,C2) = C2

but that's not the case here, so for a pattern value of 0.4 the result
depends on the blend mode.


Now suppose you specify interim entries in the colour map:

    box { <-1,-1,-1>, <1,1,1>
      texture {
        pigment {
          gradient x
          colour_map {
            blend_mode BLENDMODE
            #for(I,0,1,0.1)
              [I colour rgb (1-I)*<0,0,1> + I*<1,0,0>]
            #end
          }
        }
      }
    }

It does not matter that this uses a `#for` loop; the following is
entirely equivalent:

    box { <-1,-1,-1>, <1,1,1>
      texture {
        pigment {
          gradient x
          colour_map {
            blend_mode BLENDMODE
            [0.0 colour <0.0,0.0,1.0>]
            [0.1 colour <0.1,0.0,0.9>]
            [0.2 colour <0.2,0.0,0.8>]
            [0.3 colour <0.3,0.0,0.7>]
            [0.4 colour <0.4,0.0,0.6>]
            ... // I'm too lazy here
            [1.0 colour <1.0,0.0,0.0>]
          }
        }
      }
    }

In this case, when POV-Ray looks up the pattern value 0.4 in the colour
map, it finds 0.3 and 0.4 as the two closest index values, with colours
rgb <0.3,0.0,0.7> and rgb <0.7,0.0,0.3> respectively. Again, it computes
the colour at pattern value 0.4 as a weighted average of the two colours:

    P1 = 0.3; C1 = <0.3,0.0,0.7>
    P2 = 0.4; C2 = <0.4,0.0,0.6>
    P  = 0.4

    W1 = (P2-P)/(P2-P1)
    W2 = 1.0-W1
    Result = Average(W1,C1,W2,C2)

We get:

    W1 = (0.4-0.4)/(0.4-0.3) = 0.0/0.1 = 0.0
    W2 = 1.0-0.0 = 0.0
    Result = Average(0.0,<0.3,0.0,0.7>, 1.0,<0.4,0.0,0.6>)

Now remember that for all blend modes we have:

    Average(0.0,C1, 1.0,C2) = C2

And thus:

    Result = <0.4,0.0,0.6>

Note that this result is /independent/ of the blend mode.

For pattern values in between the interim entries (e.g. 0.44) you'd
still get results that depend on the blend mode, but those differences
would be far less pronounced as the interim entry colours are already
quite close to each other.


To achieve the same effect with interim entries as you'd get with just
two colour map entries, you'd have to modify the formula for the interim
entry colours to match the averaging formula used for the particular
blend mode.

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