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Author Topic: Mandelbulb, spiral structures & orbit traps  (Read 4982 times)
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cytotox
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« on: December 07, 2010, 01:10:29 PM »

Hi

I've got the impression that the quest for the "true" 3D version of the Mandelbrot Set was inspired mainly by the intricate and mostly spiral-containing nature of the colorful renderings obtained using this fractal formula, and that there was (and to some extent still is) a certain disappointment that the byzantine structural richness of those images could not be reproduced - in a similar fashion - in 3 dimensions.

Exploration of the power 8 Mandelbulb (from the outside as well as the inside) has shown unlimited detail, but not yielded the variaty of different shapes that a lot of people had probably expected or at least hoped for. But as far as I understand it, the Mandelbulb just presents the (approximated) volume defined by the points (voxels?) that actually belong to the M.bulb set (the 'inside'). The patterns (e.g. spirals) that we associate with the M-set, however, are actually (mainly) represented by points that are 'outside' the set and colored according to their escaping/orbit characteristics.

As it would not make sense to encase the Mandelbulb set in a solid mass of outside points (even colored differently, spirals would not be visible from the outside), I am wondering if the use of orbit traps could reveal spiral details outside the M.bulb set without the view being obscured.

I am aware of some attempts in this direction (mostly using a 3D version of a pickover stalk trap shape) by Paul Nylander, I˝igo Quilez & David Makin

http://www.fractalforums.com/3d-fractal-generation/3d-pickover-stalks/
http://www.bugman123.com/Hypercomplex/PickoverStalks-Tricorn-large.jpg
http://iquilezles.org/www/articles/orbittraps3d/orbittraps3d.htm

but I have the impression that these were not designed to zoom-in close to the surface of the 3D fractal where spiraling might become visible (like you have to zoom into the Seahorse or Elephant valleys to observe spiral structures in the M-set).

So - could it be that the use of e.g. spherical orbit traps (appropriately sized and placed) would make it possible to trace / visualize convoluted three-dimensional patterns close to the M.bulb set, e.g. in the form of beads-on-a-string spirals?
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David Makin
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« Reply #1 on: December 07, 2010, 06:02:40 PM »

You are correct, one of the main issues with viewing in projected 3D rather than in plain 2D is revealing the true detail.
IMHO the Mandelbulbs for instance do actually contain 3D detail that is as rich and varied as the 2D detail in 2D Mandelbrots (though not so self-similar or consistently "fractal" for the lower powers) but the main problem is how to show this in what is finally a 2D image.
Of course one way is to simply show the object sliced through so at least a surface shows full detail, another is maybe to have multiple "solid thresholds" each level of which is treated as partially transparent (maybe as different materials) and using much more complicated lighting including diffusion/diffraction etc.
Using alternative "solid conditions" such as rendering from the inside-out or other "colouring" calculations such as orbit traps can certainly show detail that is not visible from the "normal" methods using either DE or plain iteration count.
The main problem is that the calculations required to do "efficient" rendering using other methods are considerably more complicated and generally I do not think any will ever be as efficient as say analytical DE for a standard Mandelbulb or similar.
Even rendering inside-out (properly with the viewpoint "inside" and scanning from there outwards rather than the cheating method of scanning from a far point towards the viewpoint) is not easy because the equivalent maths for using distance estimation to step through the "inside" is considerably more complicated than stepping through the "outside" (it has to allow for different periodic attractors and no matter the period still return consistent DE values) and other colourings used to give a "solid" threshold give even bigger headaches smiley
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cytotox
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« Reply #2 on: December 09, 2010, 03:27:25 PM »

Hi David


..... Using alternative "solid conditions" such as rendering from the inside-out or other "colouring" calculations such as orbit traps can certainly show detail that is not visible from the "normal" methods using either DE or plain iteration count.
The main problem is that the calculations required to do "efficient" rendering using other methods are considerably more complicated and generally I do not think any will ever be as efficient as say analytical DE for a standard Mandelbulb or similar ....


So (as I understand it), the DE calculation for the M.-bulb is done in quite a special way which is not based on a combiniation / integration of the raymarching algorithm with the 'standard' iteration process, where the fractal's boundary is found when raymarching hits a solid spot (an 'inside' point that belongs to the M.-bulb set, e.g. for which the max. iteration condition is fullfilled while the bailout value has not been reached)?

Since there is actually not much difference in calculation effort required when switching from iteration coloring to orbit trap coloring with the 2-dimensional M-set, I had assumed that a similar "switch" between these two options for 3D Mandelbulb calculation could be implemented fairly easily ...
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David Makin
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« Reply #3 on: December 10, 2010, 06:17:38 AM »

Well, sort of, the issue is that there are both analytical and non-analytcal ways of estimating the distance to the "inside" of a fractal from a given point on the outside and all work reasonably well because the behaviour of basic information is well-known.
To use the equivalent algorithms to render from the inside such that the outside is solid is somewhat more complicated mathematically making both analytical and non-analytical distance estimation much more difficult.
Using distance estimation to estimate a distance to a given threshold of *any* colouring formula you choose to use is another matter again - say using a given small value of point orbit trap (nearest) as "solid".
The problem is AFAIK there is no analytical method (for orbit trap nearest or any other colouring) and the non-analytical "delta" methods ae very "tricky" to use because the colouring contours are basically still fractal in nature but far less predictable or understood than standard distace estimation.
I have done some investigation into finding an efficient method for orbit trap (nearest) but haven't spent anywhere near enough time on it to pin down a decent way of scaling the data using deltas or whatever to get a decemt estimate to a given orbit trap nearest threshold.
If you can find an analytical method then that would be great.
Obviously the same goes for any other colouring to be used as a solid condition - either an analytical method needs to be found or a delta method needs to be tailored for the colouring concerned - anothet possibility would be say triangle inequality average.
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