SuperFractalThing: Arbitrary precision mandelbrot set rendering in Java.

[Kalles Fraktaler]

(In this example one can skip safely about 400 iterations. The video shows skipping from 390 to 2750 iterations. The lesson to be learned: Skipping degradation happens in jumps and not any distorted picture is obviously distorted at first sight.)

I should have recognized it. I did a similar image with series approximation all the way
http://www.fractalforums.com/images-showcase-(rate-my-fractal)/series-approximation/
I had a hope that it would be possible to have arrays of A,B and C for every iteration and then find the bailout iteration by binary search on these array. But the result is a bunch of large circles, not looking like a Mandelbrot fractal at all.

[hapf]

I should have recognized it. I did a similar image with series approximation all the way
http://www.fractalforums.com/images-showcase-(rate-my-fractal)/series-approximation/
I had a hope that it would be possible to have arrays of A,B and C for every iteration and then find the bailout iteration by binary search on these array. But the result is a bunch of large circles, not looking like a Mandelbrot fractal at all.

That does not work because the error is not continuously going up or down, it jumps around.

[Mircode]

Hey, guys!

First: Amazing work! I stumbled upon it while looking for some arbitrary precision Mandelbrot implementation in CUDA. It felt like finding happiness while searching for ways to make more money

Some remarks on this whole issue:

It takes some time to read the whole thread, it would be cool if someone with some insight could summarize all the findings so far or maybe even publish them on Wikipedia.

I am also interested in the progress of Pauldelbrot's program, are there any updates on that?

And I also have some simple feature requests for Kalles Fraktaler 2:

- Maybe I am just too stupid but it feels like it is not possible to change the aspect ratio of the image. That would be very useful.
- Also, rotation of the rendering and better zoomlevel control would be nice. How about: aspect ratio of the image is determined by the aspect ratio of the window or can be predefined, the mouse-down-point determines the center of the next image, the mouse-up-point determines the the upper right corner of the image (which also defines the rotation angle)

I hate people who only demand stuff, so I will try to make a contribution myself by creating a WebGL based player that imports zoom-out pictures from Kalles Fraktaler 2. One will be able to choose between scrolling and auto-zoom.

@Kalles Fractaler:
It would also be very cool if one was able to export not just rendered color images, but also "raw data" like iteration, estimated distances, ... per pixel. That way the coloring can be done dynamically using WebGL and GLSL shaders which will allow super sexy effects, for instance an emboss effect with the mouse as lightsource.

Looking forward to this!

[Pauldelbrot]

I am also interested in the progress of Pauldelbrot's program, are there any updates on that?

Sorry; it's been on the back burner lately. I'll report any future progress...

[simon.snake]

Excuse my possibly stupid question but is this method of rendering able to work with other fractal types, burning ship, etc.?

Would be pretty impressive for those other types of fractal.

I guess there would be limits, but having had the reasoning why this works fly over my head with a large whoosh, I wouldn't know where to begin.

[hapf]

How many percentages of iterations can you skip here? 
-1.9998932137214844260699049350864386104439321454783886763742616969527203348379237560E+00
0.0
horizontal size: 4.086827785E-73

[Kalles Fraktaler]

How many percentages of iterations can you skip here? 
-1.9998932137214844260699049350864386104439321454783886763742616969527203348379237560E+00
0.0
horizontal size: 4.086827785E-73

87.5 

[hapf]

87.5 

How do you decide?

[Kalles Fraktaler]

How do you decide?

By comparing the series approximation function with the perturbation function for a couple of pixels, I use the corners. If the difference is mor than some 0.01% then it is time to stop.

[hapf]

By comparing the series approximation function with the perturbation function for a couple of pixels, I use the corners. If the difference is mor than some 0.01% then it is time to stop.

I think you are way too restrictive with 0.01%. An error in the iterations of 0.1 % or even 1 % is usually invisible. This region has a very small range for iterations. Using the error relative to the range (which maximises error) and keeping the max error < 1 % one can skip 99.95% of iterations here. 

[Kalles Fraktaler]

Unfortunately other areas with very low iteration range are very sensitive to error from series approximation.
I am thinking of the "surfing along curved edges"
Re: -1.99909583197699652777778166894532
Im: 0.00001369052463107638888599999998
Magnification 1.04857E6 or size=0.0000038147
The big empty area with horizontal stripes gets incorrectly one-colored.
I don't know if there is a way to identify the difference.

Also your location, if I calculated the magnification correctly, is a shape like a "wasp waist". I you zoom on the top or bottom edge of the waist, the color outside the waist will be incorrect.

[hapf]

Unfortunately other areas with very low iteration range are very sensitive to error from series approximation.
I am thinking of the "surfing along curved edges"
Re: -1.99909583197699652777778166894532
Im: 0.00001369052463107638888599999998
Magnification 1.04857E6 or size=0.0000038147
The big empty area with horizontal stripes gets incorrectly one-colored.
I don't know if there is a way to identify the difference.

(This is an area that needs no perturbation.) I see no horizontal stripes. I see a vertical mandelbrot set filament and the colours curving around the tip. I'm not using integer iterations, I always use smooth iterations. I can skip 14 iterations if I force
the use of perturbation. All looks fine.

Also your location, if I calculated the magnification correctly, is a shape like a "wasp waist". I you zoom on the top or bottom edge of the waist, the color outside the waist will be incorrect.

It's part of the real axis and shows a cross. Whether I skip 59614 iterations or 0, it looks the same with smooth iterations. With integer iterations it might not work exactly the same. It might also depend on how you colour given the iteration type. As long as the order of iterations is not changed my colouring stays the same. Yours might change with the same order.

[Kalles Fraktaler]

Are you using a bailout value higher than 2?

[hapf]

Are you using a bailout value higher than 2?

Yes. Otherwise the distance measure/smooth iterations show artifacts when zooming in deeper.

[Kalles Fraktaler]

Yes. Otherwise the distance measure/smooth iterations show artifacts when zooming in deeper.

Thank you so much hapf!
I have not consider this or tried it.
If I allow 1% mismatch I still get a correct image on the edge of the "waist" of your image - which is not a waist but a cross with a high bailout value.
And then the approximation allow me to skip 59624 iterations - the same as the lowest iteration number in the image.
That allows me to skip 100%

I need to make a new version of my program soon, and I would like to re-render flora-fantasy some day, which required a lot of manual work because of these approximation glitches.