### (Other) Into The Beauty - Mandelbrot Set Zoom vs Julia Set

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#### mfcc64

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#### Into The Beauty - Mandelbrot Set Zoom vs Julia Set

« on: September 27, 2020, 01:19:49 AM »

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#### Re: Into The Beauty - Mandelbrot Set Zoom vs Julia Set

« Reply #1 on: September 27, 2020, 04:08:26 PM »
looks nice. thx

I like that you show both ( dynamic and parameter ) planes
Parameter is changing along circle. Visual effect is cool

What about going along external/ internal  rays and equipotential lines ?

#### mfcc64

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#### Re: Into The Beauty - Mandelbrot Set Zoom vs Julia Set

« Reply #2 on: September 29, 2020, 05:05:50 AM »
Interesting idea. But I really have no experience to calculate internal/external ray.

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#### Re: Into The Beauty - Mandelbrot Set Zoom vs Julia Set

« Reply #3 on: September 29, 2020, 04:34:21 PM »
Interesting idea. But I really have no experience to calculate internal/external ray.

ray ( or equipotential curve) is a curve.

Here is c code. It computes series of c points on the external ray.

One can start with drawing ( near Mandelbrot set) ray 1/3 and 2/3 ( symmetrical ) with it's landing point c = -3/4

#### mfcc64

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#### Re: Into The Beauty - Mandelbrot Set Zoom vs Julia Set

« Reply #4 on: October 07, 2020, 04:15:19 PM »
How to make ray landing on a specific location (e.g. a minibrot)?

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#### Re: Into The Beauty - Mandelbrot Set Zoom vs Julia Set

« Reply #5 on: October 07, 2020, 05:53:54 PM »

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#### Re: Into The Beauty - Mandelbrot Set Zoom vs Julia Set

« Reply #6 on: October 08, 2020, 10:33:33 PM »
You need to know an external angle (conveniently represented as a binary expansion measured in turns) of a ray that lands on it.  The two periodic (under angle doubling, modulo 1 full turn) rays of lowest period land together at its cusp, and they have period the same as the period (the number of steps before iteration returns to 0+0i, starting from 0+0i) of the minibrot's cardioid's nucleus (the special C value at its center which returns exactly to 0+0i).  The period P can be found by iterating the corners of a box and seeing when it surrounds the origin, or alternatively with ball arithmetic (KF can use both methods for period finding).

Finding the external angles of a minibrot given the C value can be complicated.  I do it by finding the size of the minibrot's restricted atom domain (the region where the P'th iteration reaches a new minimum), then tracing rays outwards from probe points on a circle about that size.  Collect bits when crossing dwell bands according to whether the inner point was in the left or right half of the outer point's binary decomposition, and when reaching the limit ("near infinity", 0 iterations left with this escape radius) you can reverse the bitstring and round it to the desired periodic expansion.

Then you can trace the ray back inwards using a slightly simpler algorithm than the outwards tracing algorithm.  But do note that ray tracing is O(n^2), not even counting the higher precision you need for higher iteration counts (as the dwell bands get closer together the close you get to the boundary of the m-set), so it's not really feasible beyond a few 1000 steps, and you need an infinite number of steps to actually reach the edge of the mini.

#### mfcc64

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#### Re: Into The Beauty - Mandelbrot Set Zoom vs Julia Set

« Reply #7 on: October 15, 2020, 04:36:43 PM »
I don't know whether I can do that. Anyway, thank's for the explanation.

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#### Re: Into The Beauty - Mandelbrot Set Zoom vs Julia Set

« Reply #8 on: October 15, 2020, 08:48:36 PM »
The algorithm is complex but there are easy to use programs

https://en.wikibooks.org/wiki/Fractals/mandelbrot-perturbator

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