Coloring Methods
Q: What are the rendering methods commonly used for 256-color fractals?
A: The simplest form of rendering uses escape times. Pixels are colored according to the number of iterations it takes for a pixel to "blow-up" or escape the loop. Different criteria may be chosen to speed a pixel to its blow-up point and therefore change the rendering of a fractal. These include the biomorph method and epsilon-cross method, both developed by Clifford Pickover. Akin to the escape-time methods are Fractint's "real", "imag" and "summ" options. These add the real and/or imaginary values of a points Z-potential (at the blow-up time) to the escape time. Normally, escape-time fractals exhibit a flat 2-D appearance with "banding" quite apparent at the lowest escape times. The addition of z-potential to the escape times tends to reduce banding and simulate 3-D effects in the outer bands.
Other traditional rendering methods for 256-color fractals include continuous potential, external decomposition and level-set methods like Fractint's Bof60 and Bof61. Here the color of a point is based on its Z-potential and/or exit angle. The potential may be obtained for when it is at its lowest or at its last value, or some other criteria. The potential is scaled then applied to the palette used. Scaling may be linear or logarithmic, as for example palettes are defined in Fractint. Orbit-trap fractals make extensive use of level curves, which are based on z-potentials scaled linearly. Decomposition uses exit angles to define colors. Exit angles are derived from the polar notation of a point's complex value. Akin to decomposition is Paul Carlson's atan method (which uses an average of the last two angles) and the "atan" (single angle) method in Fractint. All of these methods can be used to simulated 3-D effects because of the continuous shadings possible.
Q: How does rendering differ for true-color fractals?
A: The problem with true-color rendering is that computers use a 3D approach to simulating 16 million colors. The basic components for addressing true color are red, green and blue (256 shades each.) There is no logical way to determine a one-dimensional index which can be used to address all the RGB colors available in true color. Palettes can be simulated in true color but are limited to about 65000 colors(256X256.) Even so, this is enough to eliminate most banding found in 256-color fractals due to limited color spread.
Because of the flexibility in choosing colors from an expanded "palette", the best rendering methods will use a combination of level curves and exit angles. While escape times can be fractionalized using interpolated iteration, the result is still very flat. One promising addition to true-color rendering is achieved by accumulating data about a point as it is iterated. The data is then used as an offset to the color normally calculated by other methods. Depending on the algorithm used, the "filter" can intensify, fragment or add interesting details to a picture.