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Fractal Software => 3D Fractal Generation => Topic started by: JosLeys on October 27, 2016, 10:43:02 PM




Title: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on October 27, 2016, 10:43:02 PM
Working on a special Mandelbox that generates 3D Kleinian limit sets, I got an idea for an escape-time version of 2D limit sets.
While these limit sets are usually drawn up by a tedious procedure of multiplication of two by two matrices representing Moebius transformations, this algorithm will draw up a finely detailed limit set in seconds.
It even allows deep zooming. The little film below goes up to a magnification of 10^10...

http://www.youtube.com/watch?v=epK2h-RG4SU

Different shapes can be obtained by circle inversions, like the image below...

I'm still fine-tuning the algorithm, so be patient...


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: KRAFTWERK on October 28, 2016, 03:18:07 PM
Super cool Jos!
Is it possible to get it into 3D space?  O0


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on October 28, 2016, 05:58:25 PM
Working on that also. See http://www.fractalforums.com/3d-fractal-generation/a-mandelbox-that-generates-kleinian-group-fractals/


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on October 29, 2016, 11:08:27 AM
Don't be cruel we are starving for your formula :'(
Post soon plzzz :D :D


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: hobold on October 29, 2016, 10:11:08 PM
If it were possible to compute good (i.e. fairly accurate) interior distance estimates per pixel, then one could speed this up even more. Basic idea is to fit a cone to a few sampled points to obtain an "interesting" hypothesis (i.e. a candidate with reasonably high probability of being accurate) for one disk. Fitting can be done efficiently with bivariate polynomial regression (using squared distances, so you'd actually fit a 2D quadric and "hope" for it to turn out as a circular paraboloid).


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on October 29, 2016, 11:23:41 PM
I think it would very hard to speed up the 2D code any further, nor would there be an interest :  the image below (with antialiasing) took less than 1 second to produce.
What I'm still struggling with is an efficient distance estimate for the 3D version


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on October 30, 2016, 10:23:48 AM
Actually this should not be a problem.
Mb3d uses one unique algo to estimate de for all ifs fractals and it is the normal mandelbox (Buddhi's? I suppose) analytic de. But I never cared as in mb I only write the actual formula and de is auto handled normally. Just difs are different as I have to manually evaluate de.
:)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on October 30, 2016, 12:45:19 PM
@Darkbeam.
Do you know how exactly M3D does the distance estimate for a Mandelbox type?
What I'm using now is what I believe M3D uses also, but in this case it is very inefficient, although it does the job in the end.
I'm getting small steps when 'far' from the object, and playing with the fudge factor does not help because then I get overstepping when close to the object.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on October 30, 2016, 01:59:10 PM
I am sorry  :sad1: but it is way too complicated for me to get into the source code... but you are much more expert in de methods so I am sure you will find a workaround.
An idea can be to write down a simplified fragmentarium script that people can study and improve? If it is not too complicated (in maths and number of temp variables) I'd love to implement it. Also you can use Mb3d's jit functionality to do preliminary tests. ;)
That's all I know :(


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: KRAFTWERK on October 30, 2016, 06:38:28 PM
Working on that also. See http://www.fractalforums.com/3d-fractal-generation/a-mandelbox-that-generates-kleinian-group-fractals/


Cool... but it looks like extruded 2d? But I might miss something? :hmh:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: 3dickulus on October 30, 2016, 09:32:46 PM
An idea can be to write down a simplified fragmentarium script that people can study and improve?

In the Fragmentarium/Examples/Knighty Collection there are 3 kleinian related frags and I've collected about 9 from the forum posts, there is a non-DE and a slow-DE version not included with the Fragmentarium distribution.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on October 30, 2016, 10:57:01 PM
@Kraftwerk, yes you are missing something.
Below is another view (with occasional overstepping due to inadequate DE)...and it's not extruded!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on October 30, 2016, 11:26:55 PM
@3dickulus
What I'm currently using is very close to one of the DE calculations in Fragmentarium that you mentioned.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: mclarekin on October 31, 2016, 12:08:07 AM


Here are  three analytic method DE calcs form Mandelbulber

pseudo klienian
double rxy = sqrt(z.x * z.x + z.y * z.y);
out->distance = max(rxy - 0.92784, fabs(rxy * z.z) / r) / (DE);

Mandelbox linear types
out->distance = r / fabs(DE);

Mandelbulb logrithmic types
out->distance = 0.5 * r * log(r) / DE


There is also the slower DeltaDE method for distance estimation, which  we use if the analytic methods are not suitable.

DE tweaks placed inside the formula can also help in some situations.




Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on October 31, 2016, 12:51:14 AM
Also read this Knighty script carefully;
http://www.fractalforums.com/other-types/escape-time-kleinian-(sort-of)/


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: pupukuusikko on October 31, 2016, 11:57:05 AM
..
Below is another view (with occasional overstepping due to inadequate DE)...and it's not extruded!

This is absolutely wonderful, congrats on the great discovery!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: knighty on December 18, 2016, 03:48:20 PM
After a little (long) discussion with Jos Leys, some results:

(https://lh3.googleusercontent.com/5-1fTPCNJ8XsLTnWoOo7u5qE43FtXqwmwd6BWuyEKbixZJE2quoaDuOdQWG7SRYlnQp4_ZhFjWTd-HWETeaNXJTSkfKBiBMR=w1600-h900-no)
(https://lh3.googleusercontent.com/TYSyU8tgW5yZ5ZyOxcZZeHOTYLD9gmKqxqARs9ss9rzEadvATh2lFw-H45ykuZ0ZjvtWSTvTBztEILinXmO-HhQX_jDhI5kr=w1600-h900-no)
(https://lh3.googleusercontent.com/uq8oPKSYXdZck-JpCTuua3j1dJjuwwvOc7qY8x5akltJX3DiStBGZnXsgFeL6-0v7CpEdxO__GYPU1QccxPD0htSaANJLv-o=w1600-h900-no)

Still not perfect, but It is wonderful!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: quaz0r on December 18, 2016, 04:29:22 PM
DAMN those are great!   :o


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: hobold on December 18, 2016, 04:32:29 PM
Breathtaking!

I'm curious how this works. :)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: 3dickulus on December 18, 2016, 06:26:38 PM
Awesome, like watching a butterfly emerge from it's cocoon.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on December 18, 2016, 07:33:18 PM
Oh nooo too awesome... I am having a heart attack :'( gimmegimme  :embarrass: :-*


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: knighty on December 18, 2016, 07:58:30 PM
All the merit goes to Jos Leys. He did all the hard work.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: KRAFTWERK on December 19, 2016, 07:56:43 PM
HOLY.... Stuff!  O0 Fantastic work!  :beer:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: knighty on December 19, 2016, 08:41:22 PM
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.  :pray2:
The second one was "rewritten" in order to look better as glsl code. It may look obfuscated or better ;D.

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 (http://klein.math.okstate.edu/IndrasPearls/cusp.pdf) (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. :)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: M Benesi on December 19, 2016, 11:29:23 PM
Wow!@#!@#  :D


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on December 19, 2016, 11:52:20 PM
Well, the title of this series of posts has become a bit of a misnomer, as the actual algorithm that produces these 3D type images is not of the escape type.
Points do not escape outside some bailout, and in fact the distance estimate is a bit peculiar.

In parallel with the Fragmentarium code that Knighty and myself worked on, I also have an implementation in Ultrafractal, which of course is much slower. On the other hand it is stable, which cannot be said of the frag code. Overdoing the number of iterations there can easily run into trouble with the responsiveness of the graphics card. Having said that, further improvements are certainly possible.

The images below were done in Ultrafractal. The first shows the basic configuration. The second shows what can be achieved by a sphere inversion.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on December 20, 2016, 12:15:41 AM
Here is a small animation :  https://plus.google.com/+josleys/posts/QtfH3gjyXpg (https://plus.google.com/+josleys/posts/QtfH3gjyXpg)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: M Benesi on December 20, 2016, 03:20:06 AM
Elephants... 


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: KRAFTWERK on December 20, 2016, 08:33:16 PM
I
AM
SPEACHLESS...

LUCA!!!!!  

:horsie: :over: :music:

Great work Knighty & Jos!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: knighty on December 20, 2016, 08:37:20 PM
LUCAAAA!!!!!!!!  ;D


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on December 20, 2016, 11:04:29 PM
I was literally unable to turn on pc last two weeks. :'(
Wait patiently I also need some asm help :D I am preparing great stuff. I also pinned the post so it will stay very visible to posterity like bulatov.frag ;D
To Johan - do three renders with the noise before or I won't give the new toy tou! :sphappy: :over: :troll:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: mclarekin on December 21, 2016, 02:46:43 AM
cool formula :)

image rendered in Fragmentarium


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: KRAFTWERK on December 21, 2016, 06:43:39 PM
To Johan - do three renders with the noise before or I won't give the new toy tou! :sphappy: :over: :troll:

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


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on December 21, 2016, 08:33:15 PM
Lool just a joke  O0 one is 'nuff I am generous before xmas :tease: :-* :beer:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: trafassel on December 21, 2016, 08:38:17 PM
 :beer: :beer: :beer: :beer:

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.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on December 22, 2016, 09:51:45 AM
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.  :pray2:
The second one was "rewritten" in order to look better as glsl code. It may look obfuscated or better ;D.

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 (http://klein.math.okstate.edu/IndrasPearls/cusp.pdf) (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?... :o


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on December 22, 2016, 10:34:47 AM
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.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on December 22, 2016, 11:40:35 AM
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 :D . But idk how to find those points just now. :D
Thanks so much for the share!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: knighty on December 22, 2016, 01:39:15 PM
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.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: mclarekin on December 23, 2016, 09:22:18 AM
WOW, this may be the best Christmas ever!!   Thanks guys ;D   


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: lycium on December 23, 2016, 02:42:49 PM
Thanks very much for these Jos + Knighty! Best 3D fractal ever, by far :)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: lycium on December 24, 2016, 03:36:42 AM
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):

(http://img12.deviantart.net/2706/i/2016/359/c/2/kleinian_i_by_lyc-dasrsfs.png) (http://orig04.deviantart.net/923b/f/2016/359/f/7/kleinian_i_by_lyc-dasrsfs.png)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Crist-JRoger on December 24, 2016, 06:24:54 AM
Beautiful fractal! Jos and Knighty thank you guys!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Sabine on December 24, 2016, 11:26:10 AM
I am quite speechless...

Thank you all!  :beer: :beer: :beer:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Crist-JRoger on December 24, 2016, 12:35:56 PM
DE-Kn render with reflections, volumetric light and bloom

(http://pre00.deviantart.net/a52d/th/pre/f/2016/359/c/8/begin_by_c_jr-dast8xj.jpg) (http://orig14.deviantart.net/3ef1/f/2016/359/c/8/begin_by_c_jr-dast8xj.jpg)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on December 24, 2016, 02:00:19 PM
I made a version of the frag file that allows to rotate objects obtained by sphere inversions. (and eventually turn it into an animation)
The following two animations were made with this:

https://dl.dropboxusercontent.com/u/107725368/teapot01.mov (https://dl.dropboxusercontent.com/u/107725368/teapot01.mov)
https://dl.dropboxusercontent.com/u/107725368/curls01.mov (https://dl.dropboxusercontent.com/u/107725368/curls01.mov)

Merry Christmas!

Jos


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Crist-JRoger on December 24, 2016, 02:35:14 PM
That's great! Thank you very much  :dink:
Another one, quick render (can't stop  ;D)

(http://img00.deviantart.net/82f9/i/2016/359/5/9/2016_12_24_163028_by_c_jr-dastihf.jpg) (http://orig01.deviantart.net/9f32/f/2016/359/d/f/2016_12_24_163028_by_c_jr-dastihf.jpg)

Merry Christmas!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Crist-JRoger on December 24, 2016, 03:14:44 PM
Final_Iterations helps like alternative of FudgeFactor
FourGen kills shadows  :- strange



Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Aexion on December 24, 2016, 03:15:17 PM
This Fractal is Incredible!!  :)
Many Thanks to Jos and and Knighty for this creation!!
I toyed with it and made a little variation with the sphere inversion, rotating p by an angle in one of the axes.

Merry Christmas!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Sabine on December 25, 2016, 12:20:41 AM
(http://orig08.deviantart.net/30d5/f/2016/359/e/6/tentacle_by_sabine62-dasuxqx.png)

Gotta love this formula! Thanks again! :)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: knighty on December 26, 2016, 01:08:09 PM
Wow! So beautiful renders! Klein and Poincaré Would have loved to see such pictures by their own eyes.

@Crist-JRoger :
The shadows look like that because the spheres are very thin. Cool effect nevertheless. I guess, you are using Eiffie's shadow.

@lycium:
This is probably the most accurate view of a 3D kleinian group limit set. How much time did it take to render?

Happy holidays and new year!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on December 26, 2016, 03:45:27 PM
Knighty Lyc is experimenting with his own soft but idk well what it does and how it works :)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: taurus on December 26, 2016, 04:58:25 PM
Wowzzze, congrats guys! this is incredible! I was abscent some time here and now this...
Seldomly saw an innovation with that aesthetic potential.  :beer:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Crist-JRoger on December 26, 2016, 08:30:01 PM
Hybrid of course  :embarrass: Quick render
Replaced sphere to Knighty's voxel. This fractal more natural and organic, great for hybrids )

(http://orig15.deviantart.net/ce3b/f/2016/361/e/e/2016_12_26_222446_by_c_jr-dat2iua.jpg)

and again  :)

(http://img14.deviantart.net/8769/i/2016/361/3/c/new_dimention_by_c_jr-dat2nsa.png) (http://orig14.deviantart.net/7777/f/2016/361/5/e/new_dimention_by_c_jr-dat2nsa.png)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: M Benesi on December 26, 2016, 11:01:46 PM
very nice Crist!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: trafassel on December 27, 2016, 10:00:33 AM
Knighty, thank you for the spirals.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Tim Emit on December 27, 2016, 12:16:51 PM
This is fantastic, really beautiful., here was I mucking about with the Mandelcup and this was going on.. Thanks everyone, this will be fun :D Seasons best y'all O0


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Caleidoscope on December 27, 2016, 03:52:46 PM
It is definitly if somebody opened a hidden treasurybox.  It stunningly beautiful and I enjoy each and every one of them!  Have fun ;)  :beer:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on December 28, 2016, 11:58:53 PM
Here are three Shadertoy implementations:
https://www.shadertoy.com/view/XlVXzh (https://www.shadertoy.com/view/XlVXzh)
https://www.shadertoy.com/view/MtKXRh (https://www.shadertoy.com/view/MtKXRh)
https://www.shadertoy.com/view/llKXRh (https://www.shadertoy.com/view/llKXRh)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Imagyx on December 29, 2016, 12:14:27 PM
Thank you for making a shadertoy implementation  :)
This makes it a lot easier for me to look for errors in my own implementation.
There must be something I'm missing.
I'm using the code from shadertoy as well as the fragmentarium script from the Kleinian thread
instead of the mandelbox distance estimation, which works fine.
And all I get is this image with my code looking like that:

Code:
kl = new V2D(1.95, 0.03);
b_a = kl.y / kl.x;
box = new V2D(1.0);
extraIter = 5;
balls = true;
fourGen = false;
clamp = new V2D(0.3, 2.0);
doInv = false;
invCenter = new V3D(1, 0.96, 0);
invR = 1;
recenter = new V3D(0);

void DE() {
V3D p = previous.ergebnis;
V3D q = p.copy();
V3D lz = q.getAdd(V3D.V1);
V3D llz = q.getSub(V3D.V1);
DF = 1.0;
double DE = 1e10;
double f = Math.signum(kl.y);
for(int i = 0 ; i < maxIterM ; i++){
q.x = q.x + b_a * q.y;
if(fourGen){
q.x = wrap(q.x, 2.0 * box.x, -box.x);
q.x = wrap(q.y, kl.x, 0);
q.x = wrap(q.z, 2.0 * box.y, -box.y);
}else{
q.x = wrap(q.x, 2.0 * box.x, -box.x);
q.x = wrap(q.z, 2.0 * box.y, -box.y);
}
q.x = q.x - b_a * q.y;
if(q.y >= kl.x * (0.5 + f * (2.0 * kl.x - 1.95) * 0.25 * Math.signum(q.x + kl.y * 0.5) * (1.0 - Math
.exp(-(7.2 - (1.95 - kl.x) * 15.0) * Math.abs(q.x + kl.y * 0.5))))){
q.x = -kl.y - q.x;
q.y = kl.x - q.y;
q.z = -q.z;
}
q = TransA(q);
V3D tmp = q.getSub(llz);
if(tmp.dot(tmp) < 1e-5){
break;
}
llz = lz;
lz = q;
}
double y = balls ? Math.min(q.y, kl.x - q.y) : q.y;
DE = Math.min(DE, Math.min(y, clamp.x) / Math.max(DF, clamp.y));
dist = DE;
hitMat = mat;
}

private V3D TransA(V3D q) {
double r = 1.0 / q.dot(q);
q.mult(-r);
q.x = -kl.y - q.x;
q.y = kl.x + q.y;
DF *= r;
return q;
}

private double wrap(double x, double a, double s) {
x -= s;
return x - a * Math.floor(x / a) + s;
}

I have a near (10) and far (50) clipping plane which results in the same behaviour as the slice_end and start variables in shadertoy.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on December 29, 2016, 01:39:52 PM
At first sight the 'wrapping' you do seems strange.
Compare with the Shadertoy or frag code....


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Aexion on December 29, 2016, 02:53:26 PM
Hello,

I made an small change to the fragmentarium shader that Knighty provided in order to add an animated KleinJ parameter (KleinR and KleinI are also animated).
You can try it on all examples..its fun. :)
I can only preview in a very small window due to GPU limitation (old gpu on a laptop), but I suppose that it will look better on standard machines.

Aexion  

Edit: I forgot to mention that the animations are only visible if you press the Animation(time) button.  :)



Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: knighty on January 03, 2017, 11:47:22 AM
Knighty, thank you for the spirals.

Thank Jos Leys. :)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: knighty on January 03, 2017, 11:48:36 AM
Hello,

I made an small change to the fragmentarium shader that Knighty provided in order to add an animated KleinJ parameter (KleinR and KleinI are also animated).
You can try it on all examples..its fun. :)
I can only preview in a very small window due to GPU limitation (old gpu on a laptop), but I suppose that it will look better on standard machines.

Aexion  

Edit: I forgot to mention that the animations are only visible if you press the Animation(time) button.  :)

Awesome! Other possibilities. Thank you.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on January 03, 2017, 12:46:16 PM
Cool coloring can be seen using a simpler option

Code:
	saveOTColor(lzy/(i+1)); // good enuff :)

At cycle end, to see polygons :angel1:

Added to Mb3D!!!!!!!!! :cantor_dance: :spiral:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Tim Emit on January 03, 2017, 01:05:14 PM
@ Axion .. doubtless this is just something I am doing wrong but when I try to run your frag I get an  error 'Could not create fragment shader: 0(1215) : error C1038: declaration of "time" conflicts with previous declaration at 0(46)'
any hints anyone?  ... ok sorry scrap that ... i just deleted line39 uniform float-time and it runs fine !


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: knighty on January 03, 2017, 04:39:46 PM
DarkBeam:
Where is the MB3D formula?  :spgloomy:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: KRAFTWERK on January 03, 2017, 08:43:02 PM
Cool coloring can be seen using a simpler option

Code:
	saveOTColor(lzy/(i+1)); // good enuff :)

At cycle end, to see polygons :angel1:

Added to Mb3D!!!!!!!!! :cantor_dance: :spiral:

:wtf: :stfu: :whatever: :agree: :boooo: :search: :smooth: :stop: :w00t: :waassup: :what: :help: :read: :howdy: :hrmm: :surrender: :nono: :ok: :jam:

And yes... WHERE IS THE FORMULA?  :hmh: :hmh: :hmh:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: bib on January 03, 2017, 09:40:40 PM
KRAFTWERK you should know where the formula is! And you're going to like it! (see image)

This is crazy! Gives me motivation to start playing again with M3D :D

Thanks Jos, Knighty and Luca. Awesome collaboration!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on January 03, 2017, 10:01:44 PM
Johan are you okay!!! I cannot recognize you  :o 88) :beer:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: KRAFTWERK on January 03, 2017, 10:21:50 PM
Johan are you okay!!! I cannot recognize you  :o 88) :beer:

Sorry... found it... thanks Bib!  O0
And THANK YOU LUCA!!!
 :beer: :beer: :beer: :beer: :beer: :beer: :beer:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: KRAFTWERK on January 04, 2017, 08:23:29 PM
Ohhh Yeah, there's art hidden in this formula too... Tried it out first time this evening.
Have a couple of more traditional Kleinian renders too... later, but this is "Sun Hole":


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: KRAFTWERK on January 04, 2017, 08:42:45 PM
...and this is to Jos Leys, Knighty and Luca. I have been waiting for this ever since I read Indras Pearls...  :beer:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Crist-JRoger on January 05, 2017, 08:52:52 AM
Tried to add "Thingy" from PseudoKleinian03.frag, to get this "things"
(http://orig07.deviantart.net/50f2/f/2017/004/4/2/2017_01_05_105015_by_c_jr-daua7t4.jpg)

but got another  ;D

(http://pre12.deviantart.net/7f29/th/pre/f/2017/004/2/2/the_way_to_go_by_c_jr-daua6f8.jpg) (http://orig04.deviantart.net/0e2f/f/2017/004/2/2/the_way_to_go_by_c_jr-daua6f8.jpg)



Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on January 05, 2017, 10:29:32 AM
I have accidentally found Twitter's page of mister Yoshiaki Araki, one of the inventors of the pionieristic "Quasifuchsian fractal 3D" years ago and he retweeted this very topic :D


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: bib on January 05, 2017, 02:48:32 PM
... mister Yoshiaki Araki...

http://www.fractalforums.com/fractal-news-across-the-world/3d-fractal-meeting-in-tokyo/
:D


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on January 05, 2017, 03:08:33 PM
But he is still not registered in here :'( please Dr. Araki join us :beer:
I am almost sure he rendered some images directly using the .frag script ;)
Too bad the language barrier is not surpassable for me!

https://mobile.twitter.com/alytile/status/813727903854653440


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: pupukuusikko on January 05, 2017, 03:34:26 PM
EDIT: This post is not relevant, Aexion's script on the previous page does what I propose and more.

It would be nice if the spirals spiralled in all three dimensions for cork-screw type of things.
I tried to implement this and it kind of works by adding a variable to transA() to modify z-axis:

(http://i.imgur.com/jummJ8G.jpg)





Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: eiffie on January 06, 2017, 02:10:25 AM
I just wanted to add these are absolutely wonderful, but you all knew that already.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: KRAFTWERK on January 08, 2017, 10:50:00 AM
3D print ready to order :) Thank you guys!!!
(http://nocache-nocookies.digitalgott.com/gallery/19/1002_07_01_17_5_24_46.jpeg) (https://www.shapeways.com/product/RCYCM9BNU/kleinian-pearls-fractal-pendant)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Tim Emit on January 09, 2017, 07:45:47 PM
Must say everything is looking sweet folks.. Aexions latest is great fun, will look at yours pupukuusikko : )


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on January 10, 2017, 02:08:09 PM
For those who want more info on the rapid calculation of Kleinian group limit sets in 2D, here is an article for you to read: http://www.josleys.com/articles/Kleinian%20escape-time_2.pdf (http://www.josleys.com/articles/Kleinian%20escape-time_2.pdf)

(it was actually working on this 2D algorithm that gave me the idea for the 3D one)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: trafassel on January 11, 2017, 11:40:50 PM
JosLeys Thank you for sharing you ideas. I like you easy to understand style of describing you geometrical and mathematical research. It would be cool to see some of this in a scientific paper.



Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: pupukuusikko on January 12, 2017, 09:03:56 AM
 :joy: breaking news! :joy:

It looks like KleinR, I (and J) can be interpreted as Julia coordinates, and there exists a kleinbrot.
All kinds of problems, but I managed to take this terrible shot:

(http://i.imgur.com/dU8hI7h.jpg)



Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: bib on January 12, 2017, 10:24:42 AM
Not surprisingly it looks very much like the scale 1 amazing box M-set
http://bib993.deviantart.com/art/Underwater-amazing-object-208457489


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on January 12, 2017, 01:01:36 PM
It is a troble because kleinr must be > kleini and kleinj. Else the skewness becomes too high.
The algo should be redesigned avoiding to use a skew factor into wrap() somehow to make it more flexible... don't know how btw.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: pupukuusikko on January 12, 2017, 09:19:24 PM
I think it's going to be fine  ^-^

In the previous pic I handled the separation line part wrong, so it became more box-like than it should have.
I still have no idea of the correct way, but the real kleinbrot will probably look more like this:

(http://i.imgur.com/SsBpNf0.jpg)

EDIT: added frag, including brot and Julia mode, as well as 3d boxfold.
There might still be gross misunderstandings as well as minor sign issues,
but it's now quite clean and explorable, although slowish.
 


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: trafassel on January 13, 2017, 12:04:10 AM
I was not able to implement the box foldings an line of separation definition, given in Reply #81. So my research is still based on the Fragment code Kleinian_test15-bis.frag, presented in Reply #23.

I did some simplifications and scale the line of separation, but the fractal set seems compatible to the picture in Reply #55 .

The following images should demonstrate the difference in using the line of separation and using classic box folding. The only image without cut is the classical mandelbox julia without using the line of separation.

Default Kleinian Formula:
Code:
public override bool GetBool(double x, double y, double z)
{
  double KleinR=1.94;
  double KleinI=0.033;
  for (int i = 0;i < 44;  i++)
  {
    x = x-Math.Floor(x+0.5);
    z = z-Math.Floor(z+0.5);
    if ( y >= KleinR*(0.5+0.1*Math.Sin(-2.0*Math.PI*x)) )
    {
      x = KleinI-x;
      y = KleinR-y;
      z = -z;
    }
    double r=x*x+y*y+z*z;
    if(r>0)
    {
      x /= r;
      y /= r;
      z /= r;
      if(r<0.2) return false;
    }
    x = KleinI + x;
    y = KleinR - y;
    z = -z;
  }
  return true;
}

Same code without line of separation:
Code:
public override bool GetBool(double x, double y, double z)
{
  double KleinR=1.94;
  double KleinI=0.033;
  for (int i = 0;i < 44;  i++)
  {
    x = x-Math.Floor(x+0.5);
    z = z-Math.Floor(z+0.5);
    double r=x*x+y*y+z*z;
    if(r>0)
    {
      x /= r;
      y /= r;
      z /= r;
      if(r<0.2) return false;
    }
    x = KleinI + x;
    y = KleinR - y;
    z = -z;
  }
  return true;
}

With boxfold and line of Separation:
Code:
public override bool GetBool(double x, double y, double z)
{
  double KleinR=1.94;
  double KleinI=0.033;
  for (int i = 0;i < 44;  i++)
  {
    x=x%2;
    z=z%2;
    while(Math.Abs(x)>0.5)
    {
      if(x>0.5) x = 1-x;
      if(x<-0.5) x = -1-x;
    }
    while(Math.Abs(z)>0.5)
    {
      if(z>0.5) z = 1-z;
      if(z<-0.5) z = -1-z;
    }
    if ( y >= KleinR*(0.5+0.1*Math.Sin(-2.0*Math.PI*x)) )
    {
      x = KleinI-x;
      y = KleinR-y;
      z = -z;
    }
    double r=x*x+y*y+z*z;
    if(r>0)
    {
      x /= r;
      y /= r;
      z /= r;
      if(r<0.2) return false;
    }
    x = KleinI + x;
    y = KleinR - y;
    z = -z;
  }
  return true;
}

With boxfold (Mandelbox Julia with scale=1):
Code:
public override bool GetBool(double x, double y, double z)
{
  double KleinR=1.94;
  double KleinI=0.033;
  for (int i = 0;i < 44;  i++)
  {
    x=x%2;
    z=z%2;
    while(Math.Abs(x)>0.5)
    {
      if(x>0.5) x = 1-x;
      if(x<-0.5) x = -1-x;
    }
    while(Math.Abs(z)>0.5)
    {
      if(z>0.5) z = 1-z;
      if(z<-0.5) z = -1-z;
    }
    double r=x*x+y*y+z*z;
    if(r>0)
    {
      x /= r;
      y /= r;
      z /= r;
      if(r<0.2) return false;
    }
    x = KleinI + x;
    y = KleinR - y;
    z = -z;
  }
  return true;
}


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: pupukuusikko on January 13, 2017, 08:15:28 PM
The following images should demonstrate the difference in using the line of separation and using classic box folding. The only image without cut is the classical mandelbox julia without using the line of separation.

Nice, the boxfold version could well become quite popular 'general' fractal, if Luca were to implement a hybridisable version in mb3d. I would definitely explore it :) 

(also added the brot frag to previous post)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on January 13, 2017, 11:30:19 PM
If you can use bitwise operators on the integer that derives from floor(), like I do in assembly, do a bw AND 1.
If the result is 1 change the sign of the corresponding variable after wrap() to obtain infinite fold. Else do a Tglad fold after wrap() using an appropriate wrap constant...
Sorry but I feel bad now, must sleep badly


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on January 17, 2017, 05:20:53 PM
The 'separation line' that is now in the algorithm works fine for most cases, but there are other!
For certain values of the Klein parameter the limit set becomes 'cusped'. The complement, that is , the part that normally remains empty, now consist of circles also.
In such a case, the separation line has to be very curvy and thus makes it very difficult to determine whether a point is 'under' or 'above' that line.
The image below is for the so-called 33/185 cusp. I got help from Geoff Smith, a Kleinian 'connaisseur' is determining the line.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on January 17, 2017, 06:04:11 PM
A fractal separation line!!! Does not look easy to implement it at all :hurt: ... ouch


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Aexion on January 17, 2017, 06:44:13 PM
A fractal separation line!!! Does not look easy to implement it at all :hurt: ... ouch
Hmmm, you can draw the line on a texture (or any bitmap buffer), use a paint algorithm, and then read the transformation from the buffer... It may works for 2d or simple 3d cases.. (it's just an idea).. :)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on January 17, 2017, 08:08:26 PM
Ramiro it is really not helpful :) ... I don't have access to images normally... :'( And even if, it would mean a dramatic slowdown to read from it!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on January 17, 2017, 08:26:02 PM
#Darkbeam: if you think reading from an image or texture slows things down, you don't want to know how the points on this goofy separation line are calculated :)
That will really slow things down...


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on January 17, 2017, 10:56:34 PM
I do imagine that ;)
The formula must stay as is - "brute" and working for most cases - people simply ignores slow formulas. Sad truth but.  :sad1:
Little corrections can be done but not really much else. In my opinion. :beer:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on January 17, 2017, 11:14:48 PM
@Darkbeam : I fully agree


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on January 19, 2017, 01:21:02 PM
Here is the result when using the exact separation line. In this example the line consists of about 150 points.
As the image shows, the horizontal 'wrapping' cannot be just by a straight line anymore either.

Not good for a 3D version I'm afraid: taking into account all those points at every iteration and step will be very expensive indeed.

Also: for this level of detail, it required 800 iterations!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on January 20, 2017, 05:21:31 PM
A side note: the separation line looks self similar! :D Can we compute a single segment of it, then "tile" it somewhat?


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on January 20, 2017, 05:40:31 PM
@Darkbeam : I don't think so, I'm afraid....


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on January 23, 2017, 12:16:26 PM
I have a new version ready of my article on the subject: http://www.josleys.com/articles/Kleinian%20escape-time_3.pdf (http://www.josleys.com/articles/Kleinian%20escape-time_3.pdf)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Imagyx on January 24, 2017, 04:58:28 PM
I have to ask again, because it doesn't seem to be working in my program yet.
The Distance Estimation
Code:
float  JosKleinian(vec3 z)
{...}
from one of the posted .frag files
can be inserted into any raymarcher that works well with for example:
Code:
dist = z.getLength() - r;
for tracing a sphere where z is the point the distance estimation is calculated for and r the radius of the sphere.
Right ?


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on January 24, 2017, 07:18:01 PM
The de is returned into a single float value. Read again the frags for details.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Crist-JRoger on January 29, 2017, 12:11:14 AM
Kleinian_test23_zrot.frag from Pupukuusikko, De-Kn based renderer, wanted to see what happens when FudgeFactor over than 1  ;D

(http://pre09.deviantart.net/9f4d/th/pre/f/2017/028/1/3/coral_reefs_by_c_jr-dax290n.jpg) (http://orig06.deviantart.net/fa03/f/2017/028/1/3/coral_reefs_by_c_jr-dax290n.jpg)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Crist-JRoger on January 30, 2017, 08:14:47 AM
And GI from Eiffie  :)

(http://pre05.deviantart.net/bace/th/pre/f/2017/029/a/d/sea_treasures_by_c_jr-dax8ld0.png) (http://orig13.deviantart.net/9b18/f/2017/029/a/d/sea_treasures_by_c_jr-dax8ld0.png)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Sabine on January 30, 2017, 12:07:03 PM
@C-JR (http://st.deviantart.net/emoticons/f/faint.gif)(http://st.deviantart.net/emoticons/f/faint.gif)(http://st.deviantart.net/emoticons/f/faint.gif)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: utak3r on February 10, 2017, 01:07:39 AM
Ok, my turn on Jos' "escape time", this time with some traps.
And ShaderToy's version with some zooming:
https://www.shadertoy.com/view/MdfcRr




Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Imagyx on February 10, 2017, 07:05:54 PM
I had a look at the very first .frag
(http://www.fractalforums.com/index.php?action=dlattach;topic=24728.0;attach=13852)
and the Java code I wrote from that works well.
Therefore I'm not totally stupid  :embarrass:
But I must have misinterpreted something in the later used glsl style code
(http://www.fractalforums.com/index.php?action=dlattach;topic=24728.0;attach=13853)
But I don't know yet what exactly.... :hmh:

https://imagyx.files.wordpress.com/2017/02/p3d_2017-02-10_18-55-29.png


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: 3dickulus on February 11, 2017, 02:53:22 AM
@utak3r I converted your ShaderToy script to Fragmentarium :)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: utak3r on February 11, 2017, 12:56:06 PM
Great  :D  I haven't looked at Fragmentarium yet, unfortunately. Probably will have to do it finally  ^-^

Anyway, here's a 3D version, with some fake ambient occlusion and very fake environment mapping.
Fullsize can be seen here (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=20034) or here (http://utak3r.deviantart.com/art/Kleinian-Jewelry-662752787).
BTW, I have really big problems with a gallery here in our forum... many images just don't show (404). Strange...  :hurt: Also, most of the times I'm unable to attach images.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Crist-JRoger on March 03, 2017, 08:58:05 PM
Tried add this (http://www.fractalforums.com/fragmentarium-b210/coloring-add-on-t14522/) coloring  :embarrass:
2 archives: DE-Kn2 and mine DE-Kn10 (slower)

(http://pre03.deviantart.net/10e7/th/pre/f/2017/062/8/4/park_by_c_jr-db1208s.jpg) (http://orig15.deviantart.net/8bfe/f/2017/062/8/4/park_by_c_jr-db1208s.jpg)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: mclarekin on March 04, 2017, 07:36:50 AM
@ crist-JRoger.


This image looks GREAT O0 O0 O0.

I love the circles amongst the standard  klienian formations


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Efoia on March 04, 2017, 05:40:14 PM
Looks very ornate, thanks for the DE!!  (Also managed to figure out sphere inversion thanks to that frag)

(http://www.davidbyrd.io/wp-content/uploads/2016/07/etk-reflective.png)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Sabine on March 05, 2017, 12:58:32 PM
@C-JR Fantastic frag :yes:, I will take a closer look! ;)

@Efoia Very nicely done render! Which program did you use?


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Efoia on March 05, 2017, 01:10:28 PM
@Efoia Very nicely done render! Which program did you use?

Thanks!  I used my own software I'm developing.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Sabine on March 05, 2017, 01:19:19 PM
@Efoia Good luck then, looks very promising!


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: DarkBeam on March 05, 2017, 10:39:05 PM
Efoia I remember your program pls finish it  :beer:


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: Crist-JRoger on March 09, 2017, 12:08:00 AM
Played some with Kleinian 2D here above, added some colors, and then desided to convert Jos Leys shader (https://www.shadertoy.com/view/llKXRh) by myself ) So add more colors too. Thanks to Jos Leys and Kali for great shaders and fractals!

(http://img03.deviantart.net/7828/i/2017/067/e/6/08_03_2017_2_by_c_jr-db1n4cv.png) (http://orig06.deviantart.net/bfb9/f/2017/067/3/8/08_03_2017_2_by_c_jr-db1n4cv.png)


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: ericr on March 23, 2017, 11:16:23 AM
if A=mat(t,-i,-i,0) and B=mat(1,2,o,1)
then tr(A^n*B^p)=2  give us a polynome p(t)
if I find t a have a cuspid

I use  wxMaxima a free software like "maple" formal cacul
It is a god think ??


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on March 23, 2017, 11:32:18 AM
I can send you a list of 19.000 values for t that generate cusps if you are interested.
No need to calculate them yourself.
Mind you, the current state of the algorithm as you find it in Fragmentarium or M3d will not allow to draw 'cusped' situations correctly as the 'separation line' gets very complicated.


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: ericr on March 23, 2017, 12:29:51 PM
Thé liste or à part it Interest me.
I will is what happend in. M3d.  For. Séparation Line
That you in. Advence
















Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: SamTiba on March 23, 2017, 12:37:22 PM
give this man someone to speak french with :D


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: bib on March 23, 2017, 12:44:55 PM
give this man someone to speak french with :D
Or perhaps a proper keyboard? :D


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: JosLeys on March 23, 2017, 01:30:31 PM
@ericr
Tu parles français?


Title: Re: An escape time algorithm for Kleinian group limit sets
Post by: ericr on March 23, 2017, 01:38:46 PM
Je suis. Francais. J écris sur une tablette. D ou les problèmes en anglais. Pardon!!!