Fractal Related Discussion > Fractal Mathematics And New Theories

Plotting field lines during iteration...

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Chris_M_Thomasson:
When I first started plotting field lines on fractal points, I was literally plotting a charge per point, then running them through one of my vector field plotters in order to gain the fractals overall field, and associated perpendicular equipotential field. This required two steps. Also, I was using IFS because they more naturally support the dynamic generation of a fractal point, instead of checking to see if a given point is “valid” or not. However, during a late night of experimentation, I discovered a very simple, perhaps naïve method for generating a field and equipotential plot without using an IFS.

So far, it seems to be compatible with basically any escape time fractal. Here is some quick and dirty pseudo-code of the basic algorithm, where Z is the current point; ZP is the previous point; C is the constant point; I is the iteration count; N is the threshold of I, and G is the “granularity level” of the resulting “field lines”:
_________________________________________________
Z = ZP = C;
G = 100.0;

for (I = 0; I < N; ++I)
{
// Mutate Z, e.g. (Z = Z^2 + C)

if (Z.real() / ZP.real() > G && Z.imag() / ZP.imag() > G)
{
// The point C escapes.

return;
}

ZP = Z;
}

// We assume that C does not escape.
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Divide-by-zero conditions aside for a moment, this experimental escape condition sure seems to end up producing plots that contain some fairly detailed field lines. I attached some examples. If I remember correctly, G has to be adjusted to build the field on deeper zooms or else some interesting artifacts appear. I came up with this technique a while back, 2015 iirc.

unassigned:
Wow! Some of those images definitely look pretty interesting. I think your method is quite similar to that of 'binary decomposition' or 'stripe average coloring'. These methods look at the angle (or angles) that escaped points follow as they escape the set.

This method has a stopping condition which is based on the combined individual size of the real and imaginary parts and therefore will preferentially stop when the angles are away from the real/imaginary axis (e.g. 45 degrees) which causes the striped bands of external rays to appear.

C0ryMcG:
I like this coloring method!
I got it to work for me, here it is with a Frame Robertson julia set, looking extra fancy.