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Welcome to Fractal Forums > Fractal Math, Chaos Theory & Research > Mandelbrot & Julia Set > The 3D Mandelbulb > 2d/3d conformal formulas for a tetrahedral projection Mandelbrot
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Author Topic: 2d/3d conformal formulas for a tetrahedral projection Mandelbrot  (Read 60844 times)
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cKleinhuis
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« Reply #15 on: February 16, 2015, 03:41:08 PM »
itercount2,3,4

* corrected_2.jpg (45.61 KB, 1024x768 - viewed 690 times.)

* corrected_3.jpg (44.5 KB, 1024x768 - viewed 678 times.)

* corrected_4.jpg (31.26 KB, 1024x768 - viewed 654 times.)
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cKleinhuis
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« Reply #16 on: February 16, 2015, 03:43:36 PM »
and elephant impression wink from itercount=3

* corrected_formula3elephants.jpg (238.27 KB, 1600x768 - viewed 676 times.)
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DarkBeam
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« Reply #17 on: February 16, 2015, 05:17:15 PM »
 snore I just have to report it in x & y form, that's the formula (hopefully...) smiley (just the mapping!!!)

Code:
// complex map; "sqrt(2)/4*z*(sqrt(8)-z^3)/(z^3+1/sqrt(8))" -> in x and y form

float x=z.x; float y = z.y;
float h = sqrt(8.); float k = 1./h;
float S = x*x*x - 3.* x*y*y;
float T = y*y*y - 3.* y*x*x; 
float B = (h + k) * T;
T = T*T;
float A = (S + k) * ( h - S) - T;
float C = -k / (T + (S + k)* (S + k));
z.x = (A* x - B* y)*C;
z.y = (A* y + B* x)*C;

WORKS woot woot grin
« Last Edit: February 16, 2015, 05:55:15 PM by DarkBeam » Logged
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Tglad
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« Reply #18 on: February 17, 2015, 10:40:49 AM »
Knighty:
Quote
why not use stereographic projection instead of spherical coordinates?
You're right, I wrote it quickly, but it should be possible to get rid of all the trigonometry, so make it run faster too.

I wrote a little blog post to present it with some more info:
http://tglad.blogspot.com.au/2015/02/tetrahedral-wrapping-function.html

The mandelbrot set is quite interesting,

Though this visualiser is not ideal, I need to write one which works better for transforms with poles in them.

I also realised that a similar transform can be made which is a bit more like the original mandelbulbs, it quadruple covers the sphere, and has bilateral symmetry, but is also continuous and smooth:

it looks a bit like a rocket-
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DarkBeam
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« Reply #19 on: February 17, 2015, 11:37:49 AM »
I think this formula is absolutely wonderful - never seen a so pretty power 2 Mandelbrot, it's the only contender of Aexion's find so far!



I am sorry that it renders slow, I hope that it will become very widely used! And OMG guys already a variant cheesy have some mercy!!!

-----------------------

(Aaand, what about the cosine variant? Please code code wink wink wink )
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DarkBeam
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« Reply #20 on: February 17, 2015, 12:19:08 PM »
Hum, while tempting naively to find the sine (or cosine) variant found this new buddy wink (dunno if it has any sense?)

Some features look semi-nice wink

Code:
void tetra(inout vec3 p, float r) {
     float angle = acos( p.z/r );
     float omega = atan( p.x, p.y );
     float zMag = tan(angle*0.5);
     vec2 z;
     z.y = sin(omega)*zMag;
     z.x = -cos(omega)*zMag;
     float x=z.x; float y = z.y;
     float h = sqrt(8.); float k = 1./h;
     float S = x*x*x - 3.* x*y*y;
     float T = y*y*y - 3.* y*x*x; 
     float B = (h + k) * T;
     T = T*T;
     float A = (S + k) * ( h - S) - T;
     float C = -k / (T + (S + k)* (S + k));
     z.y = (A* x - B* y)*C;
     z.x = (A* y + B* x)*C;
    
     zMag = sqrt(z.x*z.x + z.y*z.y);
     omega = atan(z.x,z.y);
     angle = 2.0*atan(zMag);

	p  = (r*r)*vec3( sin(omega)*sin(angle), cos(omega)*sin(angle), -cos(angle) );
}

Higher resolution...


Also now I will try to find "higher powers" cheesy

* variant.jpg (45.79 KB, 563x522 - viewed 668 times.)
« Last Edit: February 17, 2015, 07:42:12 PM by DarkBeam » Logged
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knighty
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« Reply #21 on: February 17, 2015, 12:44:27 PM »
Here is the stereographic projection variant. More "whipped cream" but the julias are interesting: for some C you have 2d julias encraved in.
To get some variations one can rotate z before applying tetra transform or,
Within tetra function, multiply z by some complex number just before transforming back to the sphere (a similarity transform).

It seems quite difficult to get a nearly conformal transform In 3D. It is like there is not enought space/room. Maybe in hyperbolic space...? Or maybe a continuous but non smooth transform? (my 2ct thought)
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DarkBeam
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« Reply #22 on: February 17, 2015, 01:04:48 PM »
Quote from: knighty on February 17, 2015, 12:44:27 PM
Here is the stereographic projection variant. More "whipped cream" but the julias are interesting: for some C you have 2d julias encraved in.

On the "normal" formula some J-Sets have... 3D spirals! Beautiful shocked
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cKleinhuis
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« Reply #23 on: February 17, 2015, 01:23:06 PM »
Quote from: DarkBeam on February 17, 2015, 01:04:48 PM
On the "normal" formula some J-Sets have... 3D spirals! Beautiful shocked

lol, how nice, although i wonder what is special about the new formula, it definately produces cool pictures in 2d wink show those nice 3d spirals cheesy !!!
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KRAFTWERK
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« Reply #24 on: February 17, 2015, 03:03:50 PM »

 Repeating Zooming Self-Silimilar Thumb Up, by Craig
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knighty
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« Reply #25 on: February 17, 2015, 04:55:03 PM »
Tglad:
In your formula(s) there are always points where the derivative vanishes (=0).(tere are 6 here). those have 180° rotation. Is it possible to modify the formula in order to transform those points from parabolic (dervative=0) to conical (derivative!=0)? I think that would reduce a lot the stretching.
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LMarkoya
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« Reply #26 on: February 17, 2015, 05:24:31 PM »
outstanding, is the formula available in Mandelbulb?
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DarkBeam
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« Reply #27 on: February 17, 2015, 06:06:27 PM »
Quote from: LMarkoya on February 17, 2015, 05:24:31 PM
outstanding, is the formula available in Mandelbulb?

Yes! Look in my archive! smiley Luca
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DarkBeam
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« Reply #28 on: February 17, 2015, 06:30:34 PM »
Quote from: Tglad on February 17, 2015, 10:40:49 AM
I also realised that a similar transform can be made which is a bit more like the original mandelbulbs, it quadruple covers the sphere, and has bilateral symmetry, but is also continuous and smooth:
<Quoted Image Removed>
it looks a bit like a rocket-
<Quoted Image Removed>

Equivalent in flat x-y code;

Code:
     float x=z.x; float y = z.y;
     float j = x*x+y*y;
     float Ni = -y*(j+1.)*(j-2.*x-1.)*(j+2.*x-1.);
     float Nr= x*(j-1.)*(j*j+2.*j+4.*y*y+1.);
     float C= (j-2.*y+1.)*(j+2.*y+1.);
     C = 2. / (C*C);
     z.x =Nr*C;
     z.y = Ni*C;

 A Beer Cup
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DarkBeam
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Fragments of the fractal -like the tip of it
« Reply #29 on: February 17, 2015, 07:31:44 PM »
Quote from: cKleinhuis on February 17, 2015, 01:23:06 PM
lol, how nice, although i wonder what is special about the new formula, it definately produces cool pictures in 2d wink show those nice 3d spirals cheesy !!!

Christian I can easily find some marine creatures and curious swirls in my variation, not so lucky with Tglad's original! smiley

This is a normal Julia with param included;

Mandelbulb3Dv18{
g.....w....o0...w....I....E4B.AWyA9xz8ZVUKKjhD0Edf9/3tv3pz18h6/N5h6qzOCJmpNYl4xj
................................3.KBnjFSD.2........A./..................y.2...wD
...Uz6.....8..../Mk0/.....Uw/...y/....E3.....kue/6uw28oD/..........G0dkpXm1....U
z.UaNadD12../2Uqja1y5rmtzqEOaSa6ysvj3Xs5ELEgOz9...........U0.....y1...sD...../..
.z1...sD25dhsNgqNw1ty8vO2cXTy62/nkdrm7pDGByRL8nqcqfTsD5cnHcIzYwQn.oYLtbjWHnjRuhQ
GxftdXY0dTGFy.JbAOblOblDU.....2BE.............sD.6....sD..G.0...................
.............oAnAt1...sD....z...........................................T....k1.
.....Ksulz1.......kz.cNmO1.U..6.P....Q2....L....m....c3...kB....u....I1.....SJ52
Sk3U.ubFP5lb2zj7....p.....6.pc..zXCc..kvrEtMc7xD6ocyFE0ujz1..........2.28.kFrA0.
.Ub96aAIVz9.1se7Umvxz0........../EU0.wzzz1...........s/...................E.2c..
zzzz.............0...................2./8.kzzzD............8....................
/EU0.wzzz1...................................wAVk..Etyfvxa0qbQ0..Zvji1SCohAB..Ii
zuCXyIYir/.EtyfrJkoO5V9..Zvji1UJwCoa..IizSynY3nhC1.Etyfve3rn201..ZvjiTfRDH6A..Ii
zuC.wxAVk..Etyfv..EsUa3feeWCNqGQIJ36wk8EwyLsUa3f................................
E....2..F2E.....I....2.....JblKMYFJNo7LMm.UQ.........................6..........
...................../........zj................................................
................................................................................
........................}
{Titel: Pretty seahorses}




 afro
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