3d Burning Ship & extension to 3d Mset for n=2,6,10..., odd n; & another formul

[Jesse]

I am way back in this thread:

  Do you think we should allow signs to vary (pixel signs) for the BS fractals so they are more like the traditional 2d BS fractals?  I like the harmony given by pixel absolute value assignments, but the "real" 2d BS fractal has that boring section on the bottom- which could be included if we allowed pixel variation (of course, this would also include z axis variations, which could be interesting).

I like the versions with absolute pixel addition more, the full body looks quite cool.

third formula:

  I like the way it looks- I'd check if it has an Mset cross section all the way through ; or is that what you mean by something different, the cross section at the axis has an Mset lower half and something else on top (forward sweeping Mset)?

Yes, the lower part contains (a half of) the Mset, but i noticed that i added also the absolute pixel values for y and z,
the version without absolute pixel and with condition "if sy>abs(sz)" instead of "if sy>sz" is attached.
But it looks still not right, must check it again.

  Also, I use the absolute value of the z component when checking whether to assign signs, rather than just the z component:

  Make sure your compiler doesn't give you the squared modulus of x when you put in |x| as well!  (I know CP does, which results in an entirely different sign assignment)...

Nope, abs() is making abs 
But i should make a similiar complex library than CP use to folllow your work faster.

  I think the fractal should have a complete 2d Mset cross section, and be fractally all over (no flat stretchy singularities). 

The Mset is now complete, but look yourself... (the tip is rotated towards the viewer, Z is from right to left):

[Jesse]

 @Jesse-  Want a trig re-write?  I don't know if trig is easier to implement or simply faster to calculate.  Trig versions are about 1/2 as fast in ChaosPro, so I tend to use complex numbers (and they are also... familiar to fracteologists) to save render time.  Anyways, if you want a re-write, we have top men working on it now.  Who?  Top men.  (I always thought they said "two top men" in the pinball game)....

Sounds good, i wish i could follow you though 

Complex math is nice for me, when i got it to work i break it even more down to more simpler code because you often dont need to calculate both parts, real + imag or you see that you dont have to squareroot before when using only the quadratic afterwards.

Does CP calculates the complex()^n  power function for arbitrary exponents?

[M Benesi]

 first:

Does CP calculates the complex()^n  power function for arbitrary exponents?

 Yeah.  Just checked for the first time.  lol... I always assumed I'd have to rewrite it trig style, but complex exponentiation isn't that complex (pardon the pun).  

 Speaking of rewriting trig style (victor, bravo, and cramden are now angle names)... after a brief syntax explanation (in case I haven't covered it previously):

  First of all, CP atan2 calculates the angle between the real and imaginary component of a complex number, if you aren't familiar with CP syntax.  Normally atan2 calculates the angle with y/x, but in CP it corresponds to the Arg (x + iy) function.  

http://en.wikipedia.org/wiki/Atan2
http://en.wikipedia.org/wiki/Arg_%28mathematics%29


victor=atan2 ;    

bravo=atan2 ;

cramden=atan2(sx,sy);

r=

nx=cos (victor * n) *r;
ny=-abs (sin (bravo * n)) *r;
nz=-abs (cos (bravo * n) * sin (cramden * n)) *r;

  //suppose I could use pipes for the abs | | but that does modulus^2 in some compilers


then add in your pixel components.    <--really had to resist the urge to write "yer" instead of your....

  I still am REALLY curious about whether the complex number method is faster compute wise in other compilers (besides CP's internal compiler).  Keep in mind that CP's internal compiler calculates z^4 as (z^2)^2 for added speed (which is part of the reason it's fast).  Lots of things can be done to streamline the process with complex numbers, but try out the trig... compare... and then let me know.  I am very curious.

[M Benesi]

I like the versions with absolute pixel addition more, the full body looks quite cool.

me too.  I suppose there should be an option to do non absolute pixels, but.... ya know.  The neater version is neater. 

Yes, the lower part contains (a half of) the Mset, but i noticed that i added also the absolute pixel values for y and z,
the version without absolute pixel and with condition "if sy>abs(sz)" instead of "if sy>sz" is attached.
But it looks still not right, must check it again.
...
The Mset is now complete, but look yourself... (the tip is rotated towards the viewer, Z is from right to left):

  Can't really tell what is going on.  It don't think it should have that big flat section (at least I don't recall seeing that anywhere...).  Maybe check that when you do your "if sy>abs(sz)" your first assignment uses positive r1, and if sy is not > abs(sz) then use negative r1.  You might have substituted sx or something in the check?  That flat section reminds me of the "if sy>sz ..." version (no abs(sz) version).  Don't know...

[Jesse]

 I still am REALLY curious about whether the complex number method is faster compute wise in other compilers (besides CP's internal compiler).  Keep in mind that CP's internal compiler calculates z^4 as (z^2)^2 for added speed (which is part of the reason it's fast).  Lots of things can be done to streamline the process with complex numbers, but try out the trig... compare... and then let me know.  I am very curious.

The trig version must be slower  , especially if the complex version makes optimizations for the integer powers like you mentioned.

I think the arbitrary exponent functions also has to compute the angle with arctan2 and uses sin and cos functions?

So it would be more fair to compare the arbitrary exponent function with the trig version on exponents like 2.5.


Btw, the trig version that you wrote is not the 3rd formula with conditional sign changes?

Because my results on this are more compareable with the 2nd formula, i guess.. or i still messed up things.

Cheers!

[M Benesi]

  Jesse- you're correct.  It's the second formula.  Thought that is what you wanted... too blitzed now for a coherent rewrite (problems typing even)...

  So tomorrow.  Anyways, the second is fricken awesome man.  Seriously, it uses rotation to create squares and stuff:


  Ok, 3rd formula rewrite.
  Use this formula for z^2,6,10,14.....   Not any other n. 


if ((sy)>abs(sz))
{
   victor=atan2 ;
} else {
   victor=atan2 ;
}
cramden=atan2 ;
bravo=atan2(sx,sy);
r=(sx^2+sy^2+sz^2)^{n/2};

nx=cos(victor*n)*r;
ny=sin(victor*n)*cos(bravo*n)*r;
nz=-sin(cramden*n)*sin(bravo*n)*r;

  pixel values, or julia, etc...

For odd n, z^3,5,7,9.....  do:

victor=atan2 ;
bravo=atan2(sx,sy);
r=(sx^2+sy^2+sz^2)^{n/2};

nx=cos(victor*n)*r;
ny=sin(victor*n)*cos(bravo*n)*r;
nz=sin(victor*n)*sin(bravo*n)*r;

[M Benesi]

  For the second formula (the one with the cool shapes), if you use the absolute value of the new x component:

nx= -abs (part_r(victor));

  You get a different fractal, although I don't find it as new (different from other fractals, whats the correct word?) as the version without using the absolute value of the x component.  Of course this implies that removing the absolute value of the other components (and giving to the nx) could generate interesting variations as well... although my favorite is the one I originally posted (and being my "favorite" formula is a very ephemeral thing... not exactly something that's going to last forever). 

[Jesse]

Thank you Matthew, i was so busy with different stuff that i have not seen your update.
Will test it in some free minutes!

[M Benesi]

  Awesome Jesse.  I hope you include the formula with the z^6 brambles and z^2 towers... I like those things...

[cKleinhuis]

btw, have you seen my tryouts with subblues skript:
http://www.fractalforums.com/mandelbulb-renderings/burning-bulb/

[M Benesi]

Nice images Trifox,

   It looks like you used one of the other formula types for your base fractal?  

  Interesting how it has similar swirls to the second (non-BS) formula, although the formula types are totally different.

  Something I noticed (which you might want to apply to your variant) is that NOT taking the absolute value of the x component produces a MUCH more interesting fractal, at least for the BS variant I implemented (first formula).  


  The second formula makes awesome julias, especially if you select seeds near where awesomeness happens in the main formula.

  Near the great towers (z^2 pointy end, click to enlarge):


  z^6 shrine of the grail end (click to enlarge):

[Tabasco Raremaster]

Those are very wonderful creations Matthew.

[Jesse]

The third formula seems not to be discovered by me 
... so i made the power 6 version of the second formula, looks like one of your julias, so could be alright:

[M Benesi]

Those are very wonderful creations Matthew.

  lol.. I just stared at your whatchacallit (icon under your name.. avatar?  can a fractal be an avatar?) through a whole phone call...  anyways, likewise (about your avatar), and thanks.

 @Jesse:
  I'm likin' the second formula the most anyways. 
Looks pretty good, rotated differently than I've looked at it though...  Check out the - x axis end... that's where the neat tubes are.  Also, setting the x axis julia component to close to the negative x axis end gives super cool fractals (for even n, odd n... are different).
Top *I think* of the 6th julia at x=-1.1 y=0.03 z= 0.0: