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Author Topic: True 3D mandelbrot type fractal  (Read 279096 times)
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JosLeys
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« Reply #390 on: November 09, 2009, 10:45:52 AM »

I've read the article. Great overview!
As to the distance estimation formula that I'm using, it may still be buggy.
I was looking at the large size pictures of the degree 8 and comparing to what I'm getting in my Ultrafractal routine.
It seems that with my algorithm, the buds are on 'stalks', and I can see through them (see the images below).
I think this might be a rendering error, but I have not figured it out yet..
Any comments from the rendering experts?








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« Reply #391 on: November 09, 2009, 10:54:28 AM »

Article finished <wipes brow>:

Linky: The Unravelling of the Real 3D Mandelbulb

A deep zoom rendering of a certain spiral has been rendered too (featured on the 2nd page).

If anyone here would like to change a featured pic of theirs, let me know. Equally, any comments, suggestions, or even criticism is welcome!

Congratulations for this superb summary of the state-of-the-art.

I have 2 questions :

- In the 8-power monster, although the level of detail is infinite, I have the feeling that shapes do not vary as much as in the 2D M-set (2 or 8 power, it does not matter) when you zoom in. All is very self similar, and there is not such variery as you can find for example when you zoom near a minibrot in 2D. What do you think about that?

- Although I have not much time to play with the maths (I could, because I did a lot of maths when I was younger, but I prefer the beauty of images rather than the beauty of equations) I would love to explore this fractal. I understand you developped your own program. @David Makin (or anybody else) : do you use Ultrafractal for your beautiful images in this thread? If so, would you mind sharing the formula ?

Thanks!
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David Makin
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« Reply #392 on: November 09, 2009, 11:09:46 AM »

@David Makin (or anybody else) : do you use Ultrafractal for your beautiful images in this thread? If so, would you mind sharing the formula ?

Thanks!

Am still working on the formula, so far I've only given Trifox a copy and he'll probably attest to the fact that it's not really ready for public release yet - I've still to sort out the parameters correctly, add more options and get all the options I've added so far as parameters working properly - I'm hoping to have a "release version" by the end of this week.
Note that I didn't originally intend this formula for public release since I was just using it to fine-tune ideas for when I do a suite of class-based general ray-tracing formulas.
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David Makin
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« Reply #393 on: November 09, 2009, 11:11:53 AM »

I've read the article. Great overview!
As to the distance estimation formula that I'm using, it may still be buggy.
I was looking at the large size pictures of the degree 8 and comparing to what I'm getting in my Ultrafractal routine.
It seems that with my algorithm, the buds are on 'stalks', and I can see through them (see the images below).
I think this might be a rendering error, but I have not figured it out yet..
Any comments from the rendering experts?


Hi Jos, I'm confused by your renders, I seem to get the same results using your analytical method as I do with my delta method.
Are those renders of the degree 8 ?
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JosLeys
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« Reply #394 on: November 09, 2009, 11:23:26 AM »

Yes Dave, it is the degree 8.
So you do not see the 'holes' with both methods, is that what you are saying?
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David Makin
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« Reply #395 on: November 09, 2009, 11:37:22 AM »

Yes Dave, it is the degree 8.
So you do not see the 'holes' with both methods, is that what you are saying?

Am just checking what happens when I go closer to "inside" on the degree 8 than I've done previously.....
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David Makin
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« Reply #396 on: November 09, 2009, 11:51:37 AM »

Yes Dave, it is the degree 8.
So you do not see the 'holes' with both methods, is that what you are saying?

Am just checking what happens when I go closer to "inside" on the degree 8 than I've done previously.....

Hi Jos, no I definitely do not get the "holes" that you are getting, I suspect the problem is the one you mentioned to me previously where there are locations at which the analytical DE gets very large due to singularities in the derivative.
Here's a quick render of the degree 8, "straight on" view but zoomed to a bud on the edge - I get no holes using your analytical method even with the threshold set very small at 1e-6.



Note however that I am using the method of not stepping further at a given iteration depth on a given ray than was stepped previously at that depth on that ray *and using the array code that I sent you earlier* - I suspect your holes are due to more limited storage/checking of the step distances at iteration depth - see the code I sent you in the E-mail and see if that removes the holes.
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xenodreambuie
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« Reply #397 on: November 09, 2009, 12:01:00 PM »

Hi Jos,
I like the quality that your DE formula produces. The problem you have may be due to using z1=R.sin(ph), which only gives one hemisphere. Changing it to z1= -R.sin(ph) is necessary to flip the orbits between hemispheres.
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David Makin
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« Reply #398 on: November 09, 2009, 12:20:17 PM »

I don't get the holes Jos is getting even using positive sine - this is the closest I got to getting such holes (a much deeper zoom):



Here's virtually the same area using negative sine:

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JosLeys
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« Reply #399 on: November 09, 2009, 12:59:00 PM »

Well, I used the formula quoted in the article, which has z=R.cos(ph) !

And yes, Dave, I am limiting the max step to not be greater than the smallest observed for a particular iter count, but in a simpler way than what you sent me. I'll look at that again.
I'm sure I have a bug somewhere...thanks, I'll keep searching.
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David Makin
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« Reply #400 on: November 09, 2009, 01:14:44 PM »

Hi Daniel, it appears that you've quoted a different formula to the ones we've been using ?

For Paul's version of your formula I'm sure he quoted it as:

              r = magn^@mpwr
              dth = @mpwr*(th = atan2(zri))
              dph = @mpwr*(ph = asin(zj/magn))
              zri = r*(cos(dth)*cos(dph) + flip(cos(dph)*sin(dth))) + cri
              zj = -r*sin(dph) + cj

Then I also had two versions suggested by Garth:

              r = (magn=sqrt(magn))^@mpwr
              th = @mpwr*atan2(zri)
              ph = @mpwr*acos(zj/magn)
              zri = r*(cos(th)*sin(ph) + flip(sin(ph)*sin(th))) + cri
              zj = r*cos(ph) + cj

              r = magn^@mpwr
              dth = @mpwr*(th = atan2(zri))
              if (ph = acos(zj/magn))>0.5*#pi
                ph = #pi - ph
              endif
              dph = @mpwr*ph
              zri = r*(cos(dth)*sin(dph) + flip(sin(dph)*sin(dth))) + cri
              zj = r*cos(dph) + cj


I'm guessing but what Jos has rendered may well be correct for the formula you quoted in your article - I'm just about to try it....
« Last Edit: November 09, 2009, 01:16:29 PM by David Makin » Logged

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David Makin
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« Reply #401 on: November 09, 2009, 01:35:40 PM »

Hi Daniel, it appears that you've quoted a different formula to the ones we've been using ?

For Paul's version of your formula I'm sure he quoted it as:

              r = magn^@mpwr
              dth = @mpwr*(th = atan2(zri))
              dph = @mpwr*(ph = asin(zj/magn))
              zri = r*(cos(dth)*cos(dph) + flip(cos(dph)*sin(dth))) + cri
              zj = -r*sin(dph) + cj

Then I also had two versions suggested by Garth:

              r = (magn=sqrt(magn))^@mpwr
              th = @mpwr*atan2(zri)
              ph = @mpwr*acos(zj/magn)
              zri = r*(cos(th)*sin(ph) + flip(sin(ph)*sin(th))) + cri
              zj = r*cos(ph) + cj

              r = magn^@mpwr
              dth = @mpwr*(th = atan2(zri))
              if (ph = acos(zj/magn))>0.5*#pi
                ph = #pi - ph
              endif
              dph = @mpwr*ph
              zri = r*(cos(dth)*sin(dph) + flip(sin(dph)*sin(dth))) + cri
              zj = r*cos(dph) + cj


I'm guessing but what Jos has rendered may well be correct for the formula you quoted in your article - I'm just about to try it....


Apologies - I forgot, Paul essentially quoted ph = @mpwr*(atan(zj/cabs(zri)) - using the above was suggested by Garth as a faster alternative. Also don't worry about the dph/ph in the first version, I quoted from part of my code using Jos' analytical DE which requires the ph value multplied by (@mpwr-1).
« Last Edit: November 09, 2009, 01:38:39 PM by David Makin » Logged

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David Makin
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« Reply #402 on: November 09, 2009, 02:06:42 PM »

Sorry - I missed Paul's recent post where he quotes the triplex formula he's using - that's the formula I'm using too.

Jos I'd be interested to know exactly what your formula was that was producing the "holes" since it's quite different from what I get from any versions of the formula - I also tried the formula that Daniel has quoted in his article (were ph and th are reversed etc.) but still didn't get a result with holes like yours.
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JosLeys
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« Reply #403 on: November 09, 2009, 03:10:44 PM »

Quote
Jos I'd be interested to know exactly what your formula was that was producing the "holes" since it's quite different from what I get from any versions of the formula

OK, here it is : (the first part handles what is needed for the derivative)
   dzx= @pow*(R^(@pow-1))*Rdz*sin(phdz+(@pow-1)*ph)*cos(thdz+(@pow-1)*th)+c
   dzy= @pow*(R^(@pow-1))*Rdz*sin(phdz+(@pow-1)*ph)*sin(thdz+(@pow-1)*th)
   dzz= @pow*(R^(@pow-1))*Rdz*cos(phdz+(@pow-1)*ph)
     Rdz=sqrt(dzx*dzx+dzy*dzy+dzz*dzz)
     thdz=atan2(dzx+i*dzy)
     phdz=acos(dzz/Rdz)

   zx= (R^@pow)*sin(@pow*ph)*cos(@pow*th)+cx
   zy= (R^@pow)*sin(@pow*ph)*sin(@pow*th)+cy
   zz=(R^@pow)*cos(@pow*ph)+cz
     R=sqrt(zx*zx+zy*zy+zz*zz)
     th=atan2(zx+i*zy)
     ph=atan2(sqrt(zx*zx+zy*zy)+i*zz)

..but remember, as I said, it could still well be a weird rendering error
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David Makin
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« Reply #404 on: November 09, 2009, 04:57:18 PM »

Quote
Jos I'd be interested to know exactly what your formula was that was producing the "holes" since it's quite different from what I get from any versions of the formula

OK, here it is : (the first part handles what is needed for the derivative)
   dzx= @pow*(R^(@pow-1))*Rdz*sin(phdz+(@pow-1)*ph)*cos(thdz+(@pow-1)*th)+c
   dzy= @pow*(R^(@pow-1))*Rdz*sin(phdz+(@pow-1)*ph)*sin(thdz+(@pow-1)*th)
   dzz= @pow*(R^(@pow-1))*Rdz*cos(phdz+(@pow-1)*ph)
     Rdz=sqrt(dzx*dzx+dzy*dzy+dzz*dzz)
     thdz=atan2(dzx+i*dzy)
     phdz=acos(dzz/Rdz)

   zx= (R^@pow)*sin(@pow*ph)*cos(@pow*th)+cx
   zy= (R^@pow)*sin(@pow*ph)*sin(@pow*th)+cy
   zz=(R^@pow)*cos(@pow*ph)+cz
     R=sqrt(zx*zx+zy*zy+zz*zz)
     th=atan2(zx+i*zy)
     ph=atan2(sqrt(zx*zx+zy*zy)+i*zz)

..but remember, as I said, it could still well be a weird rendering error

Hi, in Daniel's article he actually put:

------------------------
r = sqrt(x^2 + y^2 + z^2 )
theta = atan2(sqrt(x^2 + y^2) , z)
phi = atan2(y,x) ;

newx = r^n * sin(theta*n) * cos(phi*n)
newy = r^n * sin(theta*n) * sin(phi*n)
newz = r^n * cos(theta*n)
------------------------

I presume for phi he meant atan(y/x) or atan2(x+flip(y)).

Here's what I used based on Daniel's above version:

              dr = @mpwr*(r=sqrt(|dzri|+sqr(dzj)))*(magn=sqrt(magn))^(@mpwr-1.0)
              dph = (@mpwr-1.0)*ph + atan2(dzri)
              dth = (@mpwr-1.0)*th + asin(dzj/r)
              dzri = dr*(cos(dph)*sin(dth) + flip(sin(dth)*sin(dph)))
              dzj = dr*cos(dth)
              if @fractaltype==1 ; 1==Mandy
                dzri = dzri + 1.0
              endif
              r = magn^@mpwr
              dph = @mpwr*(ph = atan2(zri))
              dth = @mpwr*(th = asin(zj/magn))
              zri = r*(sin(dth)*cos(dph) + flip(sin(dth)*sin(dph))) + cri
              zj = r*cos(dth) + cj
              magn = |zri| + sqr(zj)

You should note that for my version magn, ph and th are precalculated before entry into the iteration loop.
It turns out that z^2+c for the above gives a 180 degree rotation of Garth Thornton's suggested formula (the "correct" one) but is quite different from Garth's at higher powers.
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