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Author Topic: True 3D mandelbrot type fractal  (Read 299152 times)
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twinbee
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« Reply #300 on: October 18, 2009, 03:53:39 AM »

Quote
I love it! Are these also 8th order Mandelbrots of your formula (White's formula)?

Yep, 8th order. I'm also working on some 3rd order pics to see if we can obtain similiar quality (since it's just above the 'noisy' 2nd power). So far, the best spots are revealing results that are weird to say the least, though perhaps not quite up to the standard of the higher powers.

By the way in the second pic, the 'stalk' is really like the equivalent of the 2D mandelbrot set, because it's smooth/coarse and bulbous at the tip. Adding more iterations lengthens the tip as one might expect. Here's that old pic from ages ago:

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David Makin
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« Reply #301 on: October 18, 2009, 03:58:31 AM »

Twinbee - truly awesome renders !
I really really need a faster machine so I can do global lighting ! (I haven't coded it up yet since I'm dreading how slow it will be on this old P4HT).

Garth - you are of course correct about acos !

I'm just rendering a 640*480 rotation around the main quadratic minibrot from my "new contender" formula, will upload to YouTube.
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Buddhi
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« Reply #302 on: October 18, 2009, 09:05:30 AM »

Great shape Garth - one can't get the resolution big enough for those methinks.

Here's another couple of renders (bigger resolutions are on the way). The 2nd one is from the inside!!

Twinbee, you made amazing renders! Shapes are very sharp and there is very good depth. Render from inside is also very interesting!
I think you have the best rendering method from our.
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David Makin
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« Reply #303 on: October 18, 2009, 03:48:39 PM »

Here's the "Mandelbrot Lightship" brought to you courtesy of the main minibrot for my new contender formula:

<a href="http://www.youtube.com/v/rIMqmDR30ks&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/rIMqmDR30ks&rel=1&fs=1&hd=1</a>

I reckon if someone created this so it could be converted to polygons and clipped a little more it would be a great ship or probe for Eve online smiley
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twinbee
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« Reply #304 on: October 19, 2009, 05:39:28 AM »

By accident, I rendered a close up of the old power 2 3D mandelbrot, and interestingly, after all, it would seem there's fractal detail that doesn't just resemble whipped cream. At least not completely smiley

If we zoom in on the noisy area at the top (perhaps illustrated most from Buddhi's render here), this is what we get:



and a further zoom in to the most promising area.......



Okay these don't compare to the higher powers, but I didn't think before there would be any fractal detail, and there may even be much juicier spots than the above (though I doubt it).

Quote
I think you have the best rendering method from our.

Thanks, I guess although my renderer is quite fast, it doesn't use global illumination or actually render light coming through underneath parts (in shapes such as bridges) yet, which I think may really help the speed for technical reasons. I think that's why I haven't seen occasions in my pictures such as that render from David where you could see 'through' areas (like that cool bridge).

Cheers everyone for the comments. I might see if I can get a bit more sleep now - got a driving exam in a couple of days!

***EDIT*** Haha, just realised I spelt "mandelbrot" as "mandlebrot" in the thread title - oops, took me a while wink
« Last Edit: October 19, 2009, 07:55:46 PM by twinbee » Logged
David Makin
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« Reply #305 on: October 20, 2009, 01:32:28 AM »

I decided to test the delta DE method a bit more:

Testing the Delta DE method on a quaternionic Julia under high magnification/small DE distance for solid.
Here you can see the original at *1.0 magnification and below a plain magnify of the part in the centre *10,378 with a delta DE solid value of 1e-7.
The render of the zoomed-in version took 4 mins, I intend to compare this to the analytical DE method for the same fractal zoom (which should be around 3* faster).


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David Makin
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« Reply #306 on: October 21, 2009, 12:37:23 PM »

Just gone back to the "roundy" true 3D method that was actually using 4D maths - here's a view of a Julia:



If no image above then go here:

http://makinmagic.deviantart.com/art/Under-Alien-Skies-140980746
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« Reply #307 on: October 26, 2009, 01:32:58 AM »

Your fractals in post #342 are amazing. The second looks like scrimshaw.
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David Makin
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« Reply #308 on: October 26, 2009, 05:44:53 AM »

A 3rd degree White/Nylander Julia:



If no image above then go here:

http://makinmagic.deviantart.com/art/Alien-Coral-141491974

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« Reply #309 on: October 26, 2009, 10:54:38 AM »

This has been a most rewarding experience. This is math as it should be, not hidebound by snobbish conventionality but collaborative and egalitarian and inclusive as well as playful. The use of the processor and algorithms for visualisation, shading , rendering etc show the mathematical toolset benefits by innovation and extension.

mathematic
c.1380 as singular, replaced by early 17c. by mathematics (1581), from L. mathematica (pl.), from Gk. mathematike tekhne "mathematical science," fem. sing. of mathematikos (adj.) "relating to mathematics, scientific," from mathema (gen. mathematos) "science, knowledge, mathematical knowledge," related to manthanein "to learn," from PIE base *mn-/*men-/*mon- "to think, have one's mind aroused" (cf. Gk. menthere "to care," Lith. mandras "wide-awake," O.C.S. madru "wise, sage," Goth. mundonsis "to look at," Ger. munter "awake, lively"). Mathematics (pl.) originally denoted the mathematical sciences collectively, including geometry, astronomy, optics. Math is the Amer.Eng. shortening, attested from 1890; the British preference, maths is attested from 1911.
 Online Etymology Dictionary free


So really! congratulations!! As you may know i think the foundations of maths can now be revised, but not in terms of unifying, an old value and goal.RATHER IN TERMS OF THE ITERATIVE NATURE of all things i perceive. I find you practitioners more stimulating than dry elegant text. wink  \pi
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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« Reply #310 on: October 30, 2009, 12:44:03 PM »

I've been re-reading everything that has been posted on the spherical coordinates type 3D fractals, and have been looking for my own way of rendering them.
Dave Makin's "delta DE" works fine, but I thought that if we can get the derivative of the transformation, things would be faster.

So I've been reading up on Jacobians and all that stuff. Getting the Jacobian matrix for the transformation is no problem, but then I did not know what to do with it :-))..

In general one can get a distance estimate from DE= |z|.log(|z|)/|dz|, z and dz being the values after bailout and dz can be calculated iteratively (for  z=z^p+c ) as dz=p.z^(p-1).dz+1  (starting with dz=1).

So I tried (maybe a bit naively, to do something similar with the 'spherical transformation'.
Let's do a Mandelbrot of degree p.

We have point on the ray x1,y1,z1 or x1=R.cos(ph).cos(th), y1=R.cos(ph).sin(th), z1=R.sin(ph), R=sqrt(x1^2+y1^2+z1^2),ph=atan(y1/x1), th=asin(z1/R)
Put dzx=1,dzy=0,dzz=0 . Express this in spherical coordinates also: ph_dz=0, th_dz=0, R_dz=1
We iterate this point:
First calculate dz:
dzx=p.R^(p-1).R_dz.cos((p-1).ph+ph_dz).cos((p-1).th+th_dz)+1
dzy=p.R^(p-1).R_dz.cos((p-1).ph+ph_dz).sin((p-1).th+th_dz)
dzz=p.R^(p-1).R_dz.sin((p-1).ph+ph_dz)
Calculate new R_dz and ph_dz and th_dz.
Now the new x,y,z:
x=R^p.cos(p.ph).cos(p.th)+x1
y=R^p.cos(p.ph).sin(p.th)+y1
z=R^p.sin(p.ph)+z1
Calculate new R and ph and th.

If R>bailout or maxiters reached, calculate DE=R.log(R)/R_dz

Move the point along the ray a distance f.DE (f<1) and start over until DE<epsilon.

Believe it or not, but this works! Here is the order 12 Mandelbrot :


..and here is a Julia :



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David Makin
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« Reply #311 on: October 30, 2009, 10:31:49 PM »

I've been re-reading everything that has been posted on the spherical coordinates type 3D fractals, and have been looking for my own way of rendering them.
Dave Makin's "delta DE" works fine, but I thought that if we can get the derivative of the transformation, things would be faster.

So I've been reading up on Jacobians and all that stuff. Getting the Jacobian matrix for the transformation is no problem, but then I did not know what to do with it :-))..

In general one can get a distance estimate from DE= |z|.log(|z|)/|dz|, z and dz being the values after bailout and dz can be calculated iteratively (for  z=z^p+c ) as dz=p.z^(p-1).dz+1  (starting with dz=1).

So I tried (maybe a bit naively, to do something similar with the 'spherical transformation'.
Let's do a Mandelbrot of degree p.

We have point on the ray x1,y1,z1 or x1=R.cos(ph).cos(th), y1=R.cos(ph).sin(th), z1=R.sin(ph), R=sqrt(x1^2+y1^2+z1^2),ph=atan(y1/x1), th=asin(z1/R)
Put dzx=1,dzy=0,dzz=0 . Express this in spherical coordinates also: ph_dz=0, th_dz=0, R_dz=1
We iterate this point:
First calculate dz:
dzx=p.R^(p-1).R_dz.cos((p-1).ph+ph_dz).cos((p-1).th+th_dz)+1
dzy=p.R^(p-1).R_dz.cos((p-1).ph+ph_dz).sin((p-1).th+th_dz)
dzz=p.R^(p-1).R_dz.sin((p-1).ph+ph_dz)
Calculate new R_dz and ph_dz and th_dz.
Now the new x,y,z:
x=R^p.cos(p.ph).cos(p.th)+x1
y=R^p.cos(p.ph).sin(p.th)+y1
z=R^p.sin(p.ph)+z1
Calculate new R and ph and th.

If R>bailout or maxiters reached, calculate DE=R.log(R)/R_dz

Move the point along the ray a distance f.DE (f<1) and start over until DE<epsilon.



Thanks for that Jos, I think I know where I was going wrong when I tried using the Jacobian to get the derivative for DE when I tried it on the 4D "true 3D" formula that I suggested a while ago - I'm going to try again now......
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David Makin
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« Reply #312 on: October 31, 2009, 03:53:59 AM »



A degree 5 White/Nylander Julia.

This shows that the difference in renders using the delta DE and analytical DE is sometimes not that great especially with a formula such as the White/Nylander where calculating the derivative is as complicated as calculating the normal iterate
The time differnce is not so great - note that how often the viewing rays approach the surface without striking it also has a bearing on the timing, this generally causes more slow down in the analytical method than the delta method.

You should note that the final analytical DE calculation was modified to correct for scale differences for different powers/degrees, the UF calculation used (after trial and error based on getting the normals correct - the normals being calculated from adjacent DE values with no extra ray-tracing) was:

 dist = 0.5*sqrt(@mpwr-1.0)*log(magn)*sqrt(magn/(|dzri|+sqr(dzj)))

i.e. the usual calculation but scaled by the square root of the power/degree minus one.
« Last Edit: October 31, 2009, 04:30:00 AM by David Makin » Logged

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« Reply #313 on: October 31, 2009, 11:48:27 AM »

Hi

I wrote completely new program for rendering 3D fractals. I resigned with rendering slices and fixed grid. Now fractal is calculated directly in 3D space using scanline algorithm. DE is not implemented yet but for accurate searching of fractal boundary I use binary search algorithm. I included all shading algorithms which I used in old program: global illumination, hard shadows and normal vector calculation (angle of incidence of light). Without volumetric fog (opacity proportional to number of iterations) rendering is much more faster than in old program and quality is higher. Details are sharper because I didn't have to use any interpolation algorithms. This image was rendered 25 minutes in 2560x2560 resolution and max. 20 iterations (rendered on Intel Core 2 Duo Quad 8200)


http://www.fractalforums.com/gallery/?sa=view;id=1029



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twinbee
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« Reply #314 on: October 31, 2009, 03:38:10 PM »

Congrats on the renderer speedup - that looks really ace!

(nice golden brot before too).

Here is an animation showing cross sections of the old Mandelbulb gateau I did:

<a href="http://www.youtube.com/v/mTdQ8jTHQ7I&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/mTdQ8jTHQ7I&rel=1&fs=1&hd=1</a>
« Last Edit: October 31, 2009, 03:47:25 PM by twinbee » Logged
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