The problem is that calculation of the external rays - also known as field lines - breaks down at higher iterations.
You can use it in combination with either distance estimation or smooth iteration colouring to give x,y coords for mapping into the fractal - either calculated shapes/objects or indeed images used as source texture maps.
When done reasonably accurately this means you can get an "x" value or field-line "angle" from 0 to 360 around the "inside".
Examples:
http://makinmagic.deviantart.com/art/Potential-and-Field-Lines-103356230And see UF parameters below.
Using the values in polar form:
http://makinmagic.deviantart.com/art/Fractals-are-hard-to-swallow-102574524For more esoteric formulas than polynomials it's very difficult to get field lines anything like "correct", however the same technique can be achieved by using distance estimation values combined with an angle returned from slight additions to the DE calciulation - these angles do not match the field lines in the same way that DE doesn't match smooth iteration but the combination used for mapping can be effective:
http://makinmagic.deviantart.com/art/Magnetic-Blooms-104516876http://makinmagic.deviantart.com/art/Newton-Makin-104248676I've considered trying a similar technique to get colouring values for the surfaces of 3D+ fractals but always wimped out due to the extra complexity and the lack of definition of a 3D equivalent to trig - though I've been wondering if there's something similar using area ratios for irregular tetrahedrons...
UF example:
Fractal2 {
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