Title: Two further spherical coordinate systems Post by: Mircode on March 23, 2010, 10:04:08 PM Hi there!
I want to suggest two (I hope) new spherical coordinate systems for dealing with the rotation. The colors are X, Y, Z, phi1, phi2 and the yellow point is the point of interest P. The 3rd component R is still the distance between P and the origin. I hope it is understandable. The second method even preserves the original Mandelbrot fractal in the x-y-plane. As soon as I find some time I can implement this, I just wanted to create some eagerness with this post :D Greetings, Mirko Title: Re: Two further spherical coordinate systems Post by: cKleinhuis on March 23, 2010, 10:58:11 PM this looks scary :D
we are excited ;) :alien: Title: Re: Two further spherical coordinate systems Post by: hobold on March 23, 2010, 10:58:26 PM I think both these constructions lead to variants of the Riemann sphere. Probably with different parameterization than what I used for the Riemandelettuce, but that would not be a bad thing. I tried to preserve a 2D Mandelbrot at least in a specific cutting plane, but that might not actually be a good choice for a 3D fractal.
Title: Re: Two further spherical coordinate systems Post by: reesej2 on March 23, 2010, 11:44:49 PM Looks like fun! Also looks challenging to code. Good luck!
Title: Re: Two further spherical coordinate systems Post by: Timeroot on March 24, 2010, 02:39:58 AM Trippy! I suppose there should be some cool results coming from these... can't wait to see!
Hobold: I wonder if anyone has tried to create a 3D Mandelbrot, deliberately destroying any connection with it? So far, all of the most popular contenders are, in my opinion, just spherical-ized versions of the 2D Mandelbrot, with nothing really new to offer. I would like to see some that are intentionally trying to achieve the diversity that we originally wanted with this, not some copycat of the Mset... :angry: Title: Re: Two further spherical coordinate systems Post by: Mircode on March 24, 2010, 10:31:18 AM Quote just spherical-ized versions of the 2D Mandelbrot, with nothing really new to offer I thought the goal was not just to find a diverse 3D fractal, but a 3D version of the Mandelbrot... Isnt it a good idea then to try to transform the 2D Mandelbrot principles into 3D Space? And even to preserve the original Mandelbrot set in one cutting plane is totally retro and back to the roots from my point of view :D I would like to know if twinbee tried all the coordinate systems he posted here http://www.physicsforums.com/showthread.php?t=331883, at least the working ones. Are the results somewhere? Title: Re: Two further spherical coordinate systems Post by: hobold on March 24, 2010, 10:58:19 AM Different people are looking for different things. Some of us have a specific idea how a voluminous Mandelbrot set was supposed to look - msltoe's newest juliabulb comes very very close.
Some of us are looking more abstractly for a 3D fractal based on a "simple" formula, that exhibits a large variety of structures in all three dimensions. Something that is analogue to the Mandelbrot set in spirit. The original power 8 Mandelbulb is the prime example, with three dimensional seahorse valleys, spirals, and all kinds of beautiful structures. The simplicity of the formula is an aesthetic requirement that would justify such a fractal as being fundamental, or otherwise of singular significance in a sea of more arbitrary fractals. And some of us don't even care much about the formula, or similarities to the Mandelbrot set, but just want to be blown away by sights never before seen. I guess Tglad's Mandelbox is an example of such "constructed" fractals, and later got topped by the merged formulas that manage to blend completely different kinds of fractals into one single shape. (Mind blowing really, but words failed me, so I never commented in any of those threads.) And then there are people like me, who explore different approaches mostly for the sake of exploration; question the unquestioned stuff, such as "Okay, zero poles isn't possible - but do we really need two?". I didn't have any further goal, just wanted to know. Title: Re: Two further spherical coordinate systems Post by: Mircode on March 24, 2010, 12:19:36 PM What a very nice response. I really love the spirit here.
Could you support your references with links? "msltoe's newest juliabulb" "Tglad's Mandelbox is an example of such "constructed" fractals" "merged formulas that manage to blend completely different kinds of fractals into one single shape" I guess it is easy to find with the forum search but it would be nice if I just had to click :) Quote "Okay, zero poles isn't possible - but do we really need two?". I didn't have any further goal, just wanted to know. Yeah I know that drive :D And I dont know if this is just an example or if you are still looking for an answer, but I think the first spherical coordinate system I suggested is one. It has only one... kind of double pole. If you construct the iso lines it looks like the attached pic. And I didnt try to support this mathematically, but it looks like all red and blue lines are even perpendicular. I hope I could brighten your view of the world a little :D Title: Re: Two further spherical coordinate systems Post by: hobold on March 24, 2010, 02:23:08 PM Links ... allright ...
the "monopolar" spherical coordinates: http://www.fractalforums.com/theory/alternate-co-ordinate-systems/msg11438/#msg11438 (http://www.fractalforums.com/theory/alternate-co-ordinate-systems/msg11438/#msg11438) the Juliabulb: http://www.fractalforums.com/theory/choosing-the-squaring-formula-by-location/msg14246/#msg14246 (http://www.fractalforums.com/theory/choosing-the-squaring-formula-by-location/msg14246/#msg14246) the Mandelbox, initially known as "The Amazing Fractal" http://www.fractalforums.com/3d-fractal-generation/amazing-fractal/ (http://www.fractalforums.com/3d-fractal-generation/amazing-fractal/) and finally some of the hybrid formulas: http://www.fractalforums.com/3d-fractal-generation/mandelbulb-and-mandelbox-mix/ (http://www.fractalforums.com/3d-fractal-generation/mandelbulb-and-mandelbox-mix/) http://www.fractalforums.com/images-showcase-(rate-my-fractal)/bulbbox-an-amazing-fractal-extension/ (http://www.fractalforums.com/images-showcase-(rate-my-fractal)/bulbbox-an-amazing-fractal-extension/) Title: Re: Two further spherical coordinate systems Post by: Mircode on March 25, 2010, 12:31:58 AM Thanks, hobold, for all the links.
And its nice to see ones exact idea already realised :angry: :P Here now the result of the transformation suggested in the second pic. Its frus-tra-ting! I also zoomed in a little, its just a mess. If someone is interested in the formula anyway, here it is. Code: R=sqrt(zx*zx+zy*zy+zz*zz); Title: Re: Two further spherical coordinate systems Post by: reesej2 on March 25, 2010, 01:55:32 AM As I recall, the standard order-2 Mandelbulb has a similar "unsatisfying" look at first. Have you tried higher orders? I'd like to see what the order-8 version of this looks like.
Title: Re: Two further spherical coordinate systems Post by: Mircode on March 25, 2010, 02:24:09 AM Well, I can handle setbacks. I can't expect to have a breakthrough after so few experiments. But seriously? Does it have to look THAT ugly?
Title: Re: Two further spherical coordinate systems Post by: reesej2 on March 25, 2010, 02:37:16 AM Interesting "claw"-like effect in the bulbs. Also, it's fascinating that, with the only change being the spherical coordinate system, the extra bulbs disappear entirely. All of the bulbs are on a single plane (unless there are others hidden on the other side). Also, the "whipping" effect resembles quaternions... hm...
Title: Re: Two further spherical coordinate systems Post by: jehovajah on March 25, 2010, 04:09:12 AM Not as disappointig as you think. You have a bit of tinkering to do and i would suggest starting in the step size and the bailout value. Your scheme seems to highlight a bilateral symmetry so you got some time savings there. However check atan2 to ensure right quadrant evaluation for your scheme, as it is non standard.
Tender hearted as you are you got to accept praise due! :dink: Title: Re: Two further spherical coordinate systems Post by: Mircode on March 25, 2010, 11:25:00 AM Quote starting in the step size Which step size? You mean along the view ray? That was already pretty small I think. Quote However check atan2 to ensure right quadrant evaluation for your scheme, as it is non standard I have a slight idea what this means, but I thought atan2 ensures right quadrant evaluation in contrast to atan. Maybe I get you wrong. Quote Tender hearted as you are No I'm not, I'm an orange! Title: Re: Two further spherical coordinate systems Post by: twinbee on March 25, 2010, 08:22:49 PM Since the new year, I haven't done much research into finding the true 3D brot, but it's great to see so much progress by everyone on this front. I'll probably get hooked in again myself at some point. ;)
Quote I would like to know if twinbee tried all the coordinate systems he posted here http://www.physicsforums.com/showthread.php?t=331883, at least the working ones. Are the results somewhere? I scrapped most of them for various reasons, but my best (though limited) hope now lies with system G. If someone could code that up (if it hasn't been tried already), that'd be awesome. Title: Re: Two further spherical coordinate systems Post by: Mircode on March 25, 2010, 10:07:42 PM Quote I scrapped most of them for various reasons, but my best (though limited) hope now lies with system G. If someone could code that up (if it hasn't been tried already), that'd be awesome. I'm afraid that system G also suffers from this: Quote While the lines of a coordinate system may intersect at almost any angle, they may NOT be parallel, because then they cease to distinguish different points. For example, your systems D and B suffer from this on the circle where the sphere intersects the y-z plane. (physics forums)Sorry ;) Title: Re: Two further spherical coordinate systems Post by: twinbee on March 25, 2010, 10:19:07 PM Yes, I remember that, although I still have a little hope because the numbers used for that part of the rotation may lead to a singularity 'join' in that part of the fractal (perhaps where two big spheres meet?), so it doesn't matter how the numbers come out. In theory anyway - I might be confused on that.
Any luck trying out the formula for the first pic in this thread? :) Title: Re: Two further spherical coordinate systems Post by: Timeroot on March 26, 2010, 12:21:50 AM I know that it doesn't represent the true Mset quite so accurately, but a while back I was considering spherical coordinates, where the radius coordinate actually measured the distance to some off-center point, say (0,1,0) or even (0,0.01,0). This leaves the angles on "coincident circles" determinable. Might be worth a shot, although with something like (0,0.01,0), I think the whipped cream would be taken to a whole new level. Either that, or pole-like "super distortion". :embarrass:
Title: Re: Two further spherical coordinate systems Post by: twinbee on March 26, 2010, 09:34:07 PM Sounds like an idea. After some rendering, I think whippy sounds about right ;)
I tried also offsetting the rotation origin point too. And with various permuations of numbers (x,y,z for each pic). Oh, small offsets like 0.01 didn't seem interesting (just tiny distortions mostly). http://www.skytopia.com/project/fractal/new/other/mr-whippy.jpg Btw, when I tried to post the pic here, Fractal Forums resized the pic to a tiny width (180 pixels). Does anyone know how to avoid the resizing altogether? Title: Re: Two further spherical coordinate systems Post by: cKleinhuis on March 26, 2010, 11:20:33 PM @twinbee max allowed attachments: 2
you have an extremely high image ... try to make it more rectangular, but attachment sizes are limited, because attachments are lost in space .... after time ! Title: Re: Two further spherical coordinate systems Post by: twinbee on March 26, 2010, 11:33:39 PM It wasn't an attachment to FF though, but a single tall picture link from my own site, so size (pixels and bytes) is theoretically infinite? (within reason :D ).
One could post separate pics, but sometimes it's easier to merge them together and upload just that one to save time. If taken separately, they're only small pics - around 600^2. What do you think? It seemed okay when I posted the 'tall' pic on this thread a couple of months ago btw: http://www.fractalforums.com/mandelbulb-renderings/higher-iterations/ Title: Re: Two further spherical coordinate systems Post by: cKleinhuis on March 26, 2010, 11:39:05 PM hmm, it was simply too high!
Title: Re: Two further spherical coordinate systems Post by: Timeroot on March 27, 2010, 12:37:11 AM angle offset (0,0,1) looked pretty cool! :D Was this was rendered with regular spherical coordinates, or the G system?
Title: Re: Two further spherical coordinate systems Post by: twinbee on March 27, 2010, 12:47:06 AM Quote Was this was rendered with regular spherical coordinates, or the G system? Regular. Title: Re: Two further spherical coordinate systems Post by: Timeroot on March 27, 2010, 12:49:15 AM Oh... the G system is what I think would get really interesting for very small offsets. |