An escape time algorithm for Kleinian group limit sets

[DarkBeam]

I was literally unable to turn on pc last two weeks.
Wait patiently I also need some asm help I am preparing great stuff. I also pinned the post so it will stay very visible to posterity like bulatov.frag
To Johan - do three renders with the noise before or I won't give the new toy tou!

[mclarekin]

cool formula

image rendered in Fragmentarium

[KRAFTWERK]

To Johan - do three renders with the noise before or I won't give the new toy tou!

Three!?!???!!!! OK, I will work on that this weekend!

[DarkBeam]

Lool just a joke  one is 'nuff I am generous before xmas

[trafassel]

 

I remember that (2d) Kleinian are described in the famous book of Benoît Mandelbrot, and 3d Kleinian was once called the "closest thing to the 3d Mandelbrot" from Daniel White. Very cool images.

[DarkBeam]

Here are the fragmentarium shaders:
The first is from Jos Leys which don't work on my PC for unknown reason. I put it here especially for darkbeam in order to make a formula for MB3D and  mclarekin to include it in mandelbulber. 
The second one was "rewritten" in order to look better as glsl code. It may look obfuscated or better .

It is still work in progress. There are some ideas to try in order to make it faster / have better DE.

The algorithm is very simple. The main idea is to use a "line of separation" in order to choose the transformation to apply. In those shaders, the line is approximated by a simple sinusoid. IMHO, it is related to "Schottky blobs" as described in this article (see for example figure 2). The drawback of this method is that in most cases the fractal will have cuts... but the cases where it works well is already huge.

Further question.
Just for speedup, may I replace sin() with a polynome accurate to 1e-5?
And what may be the "exact" expression of the line? Tangent circle?...

[JosLeys]

The sinus as a separation line is just an approximation that works ok in most cases.
You can replace it with anything else that works.

In fact, getting the real line is difficult. We can collect points on the separation line by iteratively transforming, in the nomenclature of the frag file, the point (box_size_x-b/2, a/2) until we get to an x value below -b/2, en then taking the symmetric line, as in the picture below.

However this is probably too expensive to do in Fragmentarium.

[DarkBeam]

At this point we can try any curve with S shape like atan(x) (also slow) and fractional polynomes (faster). I will do some tests definitely.
Also it is not necessary imho to find all points; there are methods that find an interpolating polynome given n points - polynomes work great for most anything and are CPU friendly . But idk how to find those points just now.
Thanks so much for the share!

[knighty]

Update: new frag attached.
Fixed: discontinuity.
New option: fourGenerators check box. Default to 3 generators.

Now the "line of separation" uses an exponential function as suggested by Jos Leys... Still WIP.

[mclarekin]

WOW, this may be the best Christmas ever!!   Thanks guys    

[lycium]

Thanks very much for these Jos + Knighty! Best 3D fractal ever, by far

[lycium]

Such an awesome DE! I had to set the fudge factor down a bit (e.g. 0.125) to stop it overstepping, and it looks pretty good with this setting

Also, in the code I saw:

Code:

	float r=length(p);
	float r2=r*r;

which can be improved to:

Code:

	float r2=dot(p, p);
	float r=sqrt(r2);

Finally, my first test render (click for 5120x2880 resolution):

[Crist-JRoger]

Beautiful fractal! Jos and Knighty thank you guys!

[Sabine]

I am quite speechless...

Thank you all! 

[Crist-JRoger]

DE-Kn render with reflections, volumetric light and bloom