The Elven PalaceFireballs and IceMandy derivative point attractors
Fireballs and Ice
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Description: Standard z^p+c but rendered using its derivative so all the usual "inside" can easily be coloured in point attractor mode, in fact it turns the convergent areas into zero attractors.
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Keywords: mandelbrot attractor point zero attractor derivative calculus 
Posted by: FractalDave December 22, 2018, 04:06:10 AM

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gerson
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December 22, 2018, 09:04:24 PM
Very nice.
Sorry my ignorance but is "p" a complex variable? Value?
FractalDave
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December 22, 2018, 09:44:33 PM
Here it's just 2 as in z^2+c :) But it works for any positive integer >=2, haven't tested general reals or complex, but it won't work with negative real powers with repect to either the zero point attractor or possibly point attractors generally.
Also I haven't tested it for any standard polynomial P(z)+c given that the powers involved are positive integers but again I think zero should still be the major convergent point attractor - though you'd have to test directly for the zero attractor since in the general case there could well be other values occurring very close to convergence (i.e. magnitude z-zold may not work so well).
Last modified by: FractalDave December 22, 2018, 09:52:18 PM
gerson
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December 23, 2018, 02:11:21 AM
Ultrafractal?
Just to learn a little bit, in this image is inside the blue area? Is point attractors mode the method of coloring? And is convergent areas the areas where blue's "roots" shapes grow decreasing?
FractalDave
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December 24, 2018, 04:24:17 AM
Quote from: gerson
Ultrafractal?
Just to learn a little bit, in this image is inside the blue area? Is point attractors mode the method of coloring? And is convergent areas the areas where blue's "roots" shapes grow decreasing?


The blue is the standard divergent area (normally referred to as "outside") using Cilia style colouring and the fireballs are the convergent area (normally "inside" and not usually coloured).
The derivative turns the convergent areas into zero attractors rather than the normal periodic attractors which means that it becomes much easier to treat them as "outside" i.e. so they are areas that bailout like the divergent areas do - though here we test for approach to zero, in a similar way to how the Newton is normally tested for approach to a root i.e. testing the magnitude of either z-zold or z-attractor against a small bailout, if the magnitude is smaller then we colour as "outside", though of course it's usually easy to distinguish between divergent and convergent bailouts so allowing different colourings on each. Obviously here I'm just using one of the simplest that relies on varied iteration count for the convergent areas i.e. smooth iteration colouring. A key important fact is that it shows that the fraction used in the colouring works correctly so can be used to smooth many colourings that would normally have hard changes on the iteration band boundaries - basically the same way as this is done on the divergent areas or indeed on say Newton fractals.
I'm currently working on perfecting mapping images into the convergent areas in a way that produces controllable distortion, the hard part is getting it close to NO distortion ;)
Mapping them in in polar mode is straightforward-ish (i.e. so the image is rotated around the central point) but I want a rectangular mapping that is as close to no distortion as possible even when the image is tiled.
gerson
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December 24, 2018, 09:44:20 PM
Thanks, for your explanation.
If I understand, you are trying to improve incoloring method.
Are that white and black lines on convergent areas result of this method?
Is it to Ultrafractal?
FractalDave
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December 25, 2018, 03:26:51 AM
Quote from: gerson
Thanks, for your explanation.
If I understand, you are trying to improve incoloring method.
Are that white and black lines on convergent areas result of this method?
Is it to Ultrafractal?


Yes and the white is part of the convergent area colouring but the black is left as inside since neither divergent nor convergent bailout is detected there i.e. that area reaches max iterations without bailing out, this could be fixed by increasing maxiter but since this was mainly an example I didn't bother for render time reasons ;)

Edit: The inside is the black between the white/yellow/orange and the blue/white not the dark band between yellow/orange and white, that's actually dark red and just part of the convergent area.
Last modified by: FractalDave December 25, 2018, 01:13:36 PM

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