END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

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:

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.

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

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).


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

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

A 3rd degree White/Nylander Julia:



If no image above then go here:

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

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 :

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.

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>