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Author Topic: Quadray Sets  (Read 14885 times)
Description: Quadray representations of the mandelbrot set
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Jesse
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« Reply #15 on: April 28, 2011, 12:49:43 AM »

A foe??..  evil

i knew i better had fixed that typo  wink

Quote
Let me drop the Holy Fractal Grenade of Antioch  wink :
...
Complete with minibrots and spirals.. cheesy
...

Very nice, but where are the spirals?

Oh.. no foe??  embarrass
My english is not very good, thats why I misunderstood.. smiley

Wow, my english is not good either, i just noticed that you are the author of incendia and that 'foe' makes some sense... and now i understand also your following post  cheesy
But when i made the typo i did not knew, so my hand must be led from a kind of higher force  wink

Quote
Thanks for adding my formulas to M3D!!
As for the missing spirals, they are visible around the perimeter of the mandelbrot..
A good example is at: http://aexion.deviantart.com/art/MiniBrot-206346586
This is the largest minibrot located in the mandelbrot anntena..
I have zoomed on the elephant trunk valley and have seen them, but my main problem is the fact that they connections are very fine and they become visible thanks to the minibrot swarms.. Anyways, let me make a zoom into the easy ones.. smiley


Of course, the containing Mset must have some sort of spirals in zooms...
but for the pic competition here comes the battle ship  tease



in detail



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Aexion
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« Reply #16 on: April 28, 2011, 01:59:17 AM »

A foe??..  evil

i knew i better had fixed that typo  wink

Quote
Let me drop the Holy Fractal Grenade of Antioch  wink :
...
Complete with minibrots and spirals.. cheesy
...

Very nice, but where are the spirals?

Oh.. no foe??  embarrass
My english is not very good, thats why I misunderstood.. smiley

Wow, my english is not good either, i just noticed that you are the author of incendia and that 'foe' makes some sense... and now i understand also your following post  cheesy
But when i made the typo i did not knew, so my hand must be led from a kind of higher force  wink


Of course, the containing Mset must have some sort of spirals in zooms...
but for the pic competition here comes the battle ship  tease

in detail



Oh. I 'm happy to share some formulas with the Forums, since I consider that you people are doing a very good work on the escapetime arena, and systems like the Quadray can generate very good fractals!

As for your spaceship (this is out of the thread, but..smiley ), its very good..
But let me melt it with my fully armed Incendia generated Space Cannon.. tease
FIREEEE!!!!

« Last Edit: April 28, 2011, 02:01:12 AM by Aexion » Logged

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Jesse
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« Reply #17 on: April 28, 2011, 10:34:51 AM »

Oh. I 'm happy to share some formulas with the Forums, since I consider that you people are doing a very good work on the escapetime arena, and systems like the Quadray can generate very good fractals!

As for your spaceship (this is out of the thread, but..smiley ), its very good..
But let me melt it with my fully armed Incendia generated Space Cannon.. tease
FIREEEE!!!!
(cool image)

I SURRENDER!  (Even you shoot at your own ship, it was the 'Quadray with an offset' formula  wink )

Really cool to meet VIPs of fractal programming here in the forum.  Mostly all about that programming i got from this place, so i can be only thankful for all that sharing!
I not know much about the way Incendia is rendering, but i like the results very much.  Regards
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cKleinhuis
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« Reply #18 on: April 28, 2011, 10:58:53 AM »

indeed, i do not fully understand how incendia works, as far as i know it is using target regions, that somehow work like an IFS
but i never really understood what is the difference to standard IFS methods.....
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Aexion
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« Reply #19 on: April 28, 2011, 03:45:29 PM »

Oh. I 'm happy to share some formulas with the Forums, since I consider that you people are doing a very good work on the escapetime arena, and systems like the Quadray can generate very good fractals!

As for your spaceship (this is out of the thread, but..smiley ), its very good..
But let me melt it with my fully armed Incendia generated Space Cannon.. tease
FIREEEE!!!!
(cool image)

I SURRENDER!  (Even you shoot at your own ship, it was the 'Quadray with an offset' formula  wink )

Really cool to meet VIPs of fractal programming here in the forum.  Mostly all about that programming i got from this place, so i can be only thankful for all that sharing!
I not know much about the way Incendia is rendering, but i like the results very much.  Regards

Oh! that was the Quadray!! I didn't reconize it!
I will not shot another deadly ray.. cheesy
 
I think that I know why the M3D implementation of the QuadrayBrot doesn't look like the burger of the photo.. The init (x,y,z,w) values must be either (0,0,0,0) or (ctx,cty,ctz,ctw)!  not the original 3D ones, since its a full 4D space conversion.. Hope that Mr. DarkBean  update it.. 

As for the Quadray Mandelbox, here is how it looks with a 4D delta (originally I posted a 1D delta value):

The init conditions also must be (0,0,0,0)..
smiley
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DarkBeam
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« Reply #20 on: April 28, 2011, 05:21:24 PM »

Jesse rewrote the formula but why? grin It gives the same exact output as mine, and both ones don't look like Incendia's images, dunno why! undecided shocked



* aaaaa iko.jpg (176.12 KB, 479x723 - viewed 489 times.)
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Jesse
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« Reply #21 on: April 28, 2011, 06:25:27 PM »

Jesse rewrote the formula but why? grin

I am not sure if you want an answer, nevertheless here are some...  Azn
 - you did not mentioned your formula in the final.zip update section, so i loaded it after my formula was already done
 - your formula seems to be fixed so that the vector starts at zero (or Ct.)
 - you have no offset parameter
 - mine is faster

 cheesy   enough?

Quote
It gives the same exact output as mine, and both ones don't look like Incendia's images, dunno why! undecided shocked


The output depends on many settings, and the general rendering technic.

Attached is a remake of the 6 bulbs with very small DEstop and an average iteration count of 32.5.
The thin surfaces are not shown up in Aexion renderings, this seems to be due to the different rendering methods.
 


* Aexion 6bulbs.jpg (84.32 KB, 533x533 - viewed 450 times.)
« Last Edit: April 28, 2011, 07:02:09 PM by Jesse, Reason: i am stupid » Logged
DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #22 on: April 28, 2011, 07:04:14 PM »

Small DEstop! embarrass The only thing that I didn't try

Effectively, your program is very sensible to thin surfaces; and it is not necessarily a defect, in some cases it reveals to be a great side-effect cheesy
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« Reply #23 on: April 28, 2011, 08:23:33 PM »

I've found a better DE-approximation:

Code:
float DE(vec3 p)
{
vec4 ct = abs(vec4(p.x+p.y+p.z,-p.x-p.y+p.z,-p.x+p.y-p.z,p.x-p.y-p.z))-Delta;
vec4 w = ct;
float dr = 1.0;
float r;
float  r2 = length(ct);
for (int iter=0;iter<Iterations;iter++) {
vec4 ww = w*w;
w = ww - ww.yxwz + 2.0*w.wwxx*w.zzyy + ct;
r = length(w);
dr = dr * (r/r2) *2.0 + 1.0;
r2 = r;
if (r > 8.0) break;
}
return r*log(r)/dr;
}

My images look more like Jesse than Aexion's, though. (The holes near the tip will disappear if the max ray number is increased).

I am curious: how do you raymarch your pictures - I guess you don't use an analytic DE? Do you use some sort of approximation based on the numerical gradient of the escapelength (like Makin/Buddhi's four-point method)? And what about coloring? A function of the escape-length/iteration count, or some kind of orbit traps?


* quadray2.jpg (43.46 KB, 605x567 - viewed 423 times.)
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Jesse
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« Reply #24 on: April 28, 2011, 09:19:36 PM »

I've found a better DE-approximation:
...

So you included the derivative of the vectorlength, looks interesting.
My attempts of analytic DE's based on only one parameter were not good enough for 'power' bulb formulas, but this should be tested.


Quote

My images look more like Jesse than Aexion's, though. (The holes near the tip will disappear if the max ray number is increased).

Max ray number?  Sounds like monte carlo or so  shocked

Quote
I am curious: how do you raymarch your pictures - I guess you don't use an analytic DE? Do you use some sort of approximation based on the numerical gradient of the escapelength (like Makin/Buddhi's four-point method)?

Looks like you asking me, you are fully right, i use mostly the 4 point method.  Based on the escaped length gradient at same iterationcounts.

Would be also nice to get a little insight of Aexions rendering method, especially for this bulbs.

Quote
And what about coloring? A function of the escape-length/iteration count, or some kind of orbit traps?

For this one i used the orbit trap to the origin, another implemented ones are smoothed iterations, last length increase and angles of escaped vector components, last ones are not very often used or useful.
There are far more possibilities, ask David for a wider summary, he has also much implemented in his UF formula file.
« Last Edit: April 28, 2011, 09:48:40 PM by Jesse, Reason: mixed up words » Logged
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« Reply #25 on: April 28, 2011, 10:10:34 PM »

So you included the derivative of the vectorlength, looks interesting.
My attempts of analytic DE's based on only one parameter were not good enough for 'power' bulb formulas, but this should be tested.

I'm pretty sure this is not ideal - a proper analytic DE would probably have a running 4-component derivative similar to the Quaternion Julia approach. And how do you multiply Quadray numbers anyway? Btw, the Wikipedia entries on Quadray and Synergetics are interesting reads - I hadn't heard about them, before Aexion introduced them.

Max ray number?  Sounds like monte carlo or so  shocked

I'm afraid it just means 'max number of ray steps'. But doing true distributed ray tracing, with proper DOF, soft shadows, and Image Based Lighting would be nice, though .-)


For this one i used the orbit trap to the origin, another implemented ones are smoothed iterations, last length increase and angles of escaped vector components, last ones are not very often used or useful.
There are far more possibilities, ask David for a wider summary, he has also much implemented in his UF formula file.

Thanks. I use a combination of different orbit traps (including origin) as well, but I find it difficult to get as good results as M3D. Guess that palette editor helps a lot :-)
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Aexion
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« Reply #26 on: April 29, 2011, 12:49:58 AM »

Its very good to see that the formula is finally implemented on M3D.
Well, hope to see some interesting renders. smiley

About the questions:
Quote
I am curious: how do you raymarch your pictures - I guess you don't use an analytic DE? Do you use some sort of approximation based on the numerical gradient of the escapelength (like Makin/Buddhi's four-point method)?
Looks like you asking me, you are fully right, i use mostly the 4 point method.  Based on the escaped length gradient at same iterationcounts.

Would be also nice to get a little insight of Aexions rendering method, especially for this bulbs.


I just writed a very basic raymarcher, nothing special (thats why I didn't saw the very fine tendrils). It outputs a buffer for Incendia and I made the final render from it.
(really a very crude process.. I have a lot to learn!)

I'm pretty sure this is not ideal - a proper analytic DE would probably have a running 4-component derivative similar to the Quaternion Julia approach. And how do you multiply Quadray numbers anyway? Btw, the Wikipedia entries on Quadray and Synergetics are interesting reads - I hadn't heard about them, before Aexion introduced them.


Quadray coordinates are often overlooked but they produce very interesting results, in special for those higher dimensional objects. I have used them for years for plotting many interesting fractals (ifs attractors and sphere inversion fractals, for example:http://aexion.deviantart.com/art/Apollonian-Gasket-III-30135208). 

These links has some interesting quadray information:
http://minortriad.com/quadray.html
http://www.grunch.net/synergetics/quadintro.html

I will try to port some of the other 3D escape time fractals to the quadray, to see how they look..
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Kali
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« Reply #27 on: April 29, 2011, 02:38:05 PM »

@Aexion: Very nice formulas! thanks for sharing and welcome to FF!

I've found an interesting thing: My "Mandelbrot on real numbers" patterns are on the thin surface of the 6-bulbs version!!  huh?

This is the M3D implementation:



And this is from my method for generating real number mandelbrot patterns:



I couldn't get the same coloring, but I added some skew to match the other image and comparing the patterns similarty.

Any idea why is this happening??

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David Makin
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« Reply #28 on: April 29, 2011, 05:25:38 PM »


Quote
And what about coloring? A function of the escape-length/iteration count, or some kind of orbit traps?

For this one i used the orbit trap to the origin, another implemented ones are smoothed iterations, last length increase and angles of escaped vector components, last ones are not very often used or useful.
There are far more possibilities, ask David for a wider summary, he has also much implemented in his UF formula file.

Effectively *any* colouring that works for 2D fractals can be extended to colour the 3D surface - with the exception of DE if your surface threshold is based on DE or smooth iteration if your surface threshold is based on smooth iteration count and you can even use multiple "colourings" to say add bump-mapping or such.
Ideally said colourings would be proper 3D+ counterparts of the 2D ones e.g. orbit trapping to (a,b,c....) instead of just (a,b) but in order to allow more variety quickly my formula for UF allows you to specify a conversion of 3D values to 2D which can then be plugged into any of UF's (class based) complex colouring formulas.
Of course "any" is usually restricted to ones that work "Outside" (or "Inside") but not ones that work "Inside" only because generally the rendered surface is still mathematically "Outside" - that is unless of course you're rendering with "Inside" hollow and "Outside" solid in which case the reverse is true wink
For more IFS-style fractals you can also use the "genetics" for the colouring, in the case of KIFS or Mandelbox style fractals that means if you have say 3 conditionals per iteration each with 2 choices then you have 8 values per genetic "level" (iteration depth) and on doing the iteration for the surface pixel you can convert said genetic code to a colour in a manner of your own choosing. For instance with 8 values per iteration then at the end of each iteration loop I would use:

s = s/8 (divide size factor by 8, start value was 1)
code = code + newcode*s

Where newcode is 0 to 7 based on the path through the 3 conditionals.
Then use code for the colouring say as a simple index into palette/gradient (raw value of code here would be min 0 max<1).
Of course if there were say 12 possible routes through the iteration loop then newcode would be 0 to 11 and you'd divide s by 12 instead of 8.
Also you can make this a lot more complex giving the user finer control over the way the final code is created e.g. if there are 8 values from the loop but you only divide s by say 6 then the distribution of the colouring will be such that the variation between the primary fractal areas will be more graduated - though of course you lose the "ideal" range from 0 to <1 so you'd have to adjust accordingly.
« Last Edit: April 29, 2011, 05:46:42 PM by David Makin » Logged

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« Reply #29 on: April 29, 2011, 05:44:02 PM »

Of course you can make a more sophisticated genetics colouring by borrowing tricks from orbit trapping, e.g. skip 1st n iterations as far as the genetic code is concerned, or only use values from n iterations, or somehow specify a varying scale value or just plain different scale values for the genetic code from each iteration.

If you want a "final value angle" style colouring then a trick I though of just now but haven't tried is take the final (3D or 4D) value of "z" at bailout and get an angle-based value (magnitude) from it using either standard quaternionic or bi-complex reverse trig functions - or maybe do this for trapped values when orbit trapping.
« Last Edit: April 29, 2011, 06:47:30 PM by David Makin » Logged

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