Colourings and Mappings

[David Makin]

Hi all,

There have been one or two questions recently with respect to colouring methods and potential mapping methods (specifically texture but could just as easily be bump) and someone was also discussing "fractal" controlled animation, well these two things reminded me of something...

I first saw this suggested as a method of getting colouring values for IFS fractals (I think suggested by Barnsley but I'm not sure).

Anyhow the method I'm referring to is extensible to any number of variables to be used for colouring or mapping onto the fractal surface (or for a given pixel in 2D).

One simply adds an extra dimension to the formulas for each extra variable required, whether they be convergent IFS or standard escape-time.

For instance if using an index into a palette for colouring the surface of a 3D fractal then find the surface point/normals etc. in the usual way but when computing the orbit for the surface point specifically to get a colouring then use 4 dimensions instead of 3 such that the value from the 4th dimension does not feed back into *any* of the other 3 but is affected by one or more of them based on the iteration formula (under any rules you like).
One could either assume that the initial value in this extra dimension is zero for all points or use a computed value as the initial value for a given surface point based on any algorithm you like.
In the case of IFS one would modify all the transforms to be used to get the colouring/mapping in a similar manner i.e. such that the value/s in any extra dimensions change based on the others (and themselves) but do not affect the lower dimensions in which the attractor exists.

[Softology]

Interesting idea.

This is sort of how the Scott Draves coloring works for the 2D fractal flames.  Each transformation is given a value that maps to the gradient.  As the points are iterated each pixel is given a new color palette index based on the last value averaged with which transformation it uses.  The color index values are kept separate from the parameters that control the fractal shape.

Using the same theory could lead to new methods to color the bulbs and other 3D fractals.

Jason.