This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
The simplest form of rendering uses escape times. Pixels are
coloured 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. Similar 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-colour fractals include
continuous potential, external decomposition and level-set methods
like Fractint's Bof60 and Bof61. Here the colour 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 colours. 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