Print Page - New Mandelbulb-like formulas

Title: New Mandelbulb-like formulas
Post by: Buddhi on November 11, 2010, 02:21:12 PM

Because as Trifox said

Quote from: Trifox on November 10, 2010, 11:06:15 PM

people, this thread is growing too large, are you ok if i close it, with notice for next thread ?

"True 3D mandelbrot fractal (search for the holy grail continues)" is so long, I'm starting new thread for continuing this endless quest

Last days I decided to experiment with rotations in Mandelbulb formula. I tried to do some formula with no privileged axis. In standard Mandelbulb formula there are two rotations. First around Z axis and second around axis which is perpendicular to plane created by z axis and resultant vector of x and y vectors. So X axis is privileged. In my version of formula I used 3 simple rotations. First around Z axis, second around Y axis and third around X axis. There is a code is below:

Code:

double rp = pow(r, p-1);

double angZ = atan2(z.y, z.x);
double angY = atan2(z.z, z.x);
double angX = atan2(z.z, z.y);

CRotationMatrix rotM;
rotM.RotateX(angX*(p-1));
rotM.RotateY(angY*(p-1));
rotM.RotateZ(angZ*(p-1));

z = rotM.RotateVector(z) * rp + constant;
r = z.Length();

Results are not as beautiful as standard Mandelbulb but bulbs also exists in all directions.

Example of power 2 version:

(http://nocache-nocookies.digitalgott.com/gallery/4/640_11_11_10_2_16_27.jpeg)

Example of power 5 vesion with cross-sections:

(http://nocache-nocookies.digitalgott.com/gallery/4/640_11_11_10_1_57_50.jpeg)

Title: Re: New Mandelbulb-like formulas
Post by: cKleinhuis on November 11, 2010, 03:25:19 PM

it is like the definition of paolo bonzini, and i also think that the coordinate axis can be any orientation, in the base bulb functions there
is the y axis mirrored, so the rotation appears clockwise, instead of counter clockwise, i had an axis angle variant in my mind, you define 2 axis
to which the polar coordinates or calculated, and proceed as usual, the matrices gets a bit more complicated, but can be created
once for a whole image, also rotating around 3 axis makes sense, to squeeze out maximum transformation ;)
nice images

Title: Re: New Mandelbulb-like formulas
Post by: Buddhi on November 13, 2010, 05:41:56 PM

Some new formulas:

Two rotations: around Z and Y axis. Rotations are done using complex numbers. First the vector is normalized, next rotated and at the end multipled by inverted normalization factor. I have to use "signum" function, because fist part of formula loses sign of x value. Resultant fractal has 2 symmetries and there is visible 2D Mandelbrot set on slices XY and XZ.

Code:

double tempR;

double sign = 1.0;
double sign2 = 1.0;

//rotation around Z axis
if (z.x < 0) sign2 = -1.0;
tempR = sqrt(z.x * z.x + z.y * z.y);
z *= (1.0 / tempR);
temp = z.x * z.x - z.y * z.y;
z.y = 2.0 * z.x * z.y;
z.x = temp;
z *= tempR;

//rotation around X axis
if (z.x < 0) sign = -1.0;
tempR = sqrt(z.x * z.x + z.z * z.z);
z *= (1.0 / tempR);
temp = z.x * z.x - z.z * z.z;
z.z = 2.0 * z.x * z.z * sign2;
z.x = temp * sign;
z *= tempR;

z = z * r;
z += constant;
r = z.Length();

(http://nocache-nocookies.digitalgott.com/gallery/4/640_13_11_10_5_17_49.jpeg)

XY and XZ slices:

(http://nocache-nocookies.digitalgott.com/gallery/4/640_13_11_10_5_18_29.jpeg)

Second formula:

Three rotation, also done with complex numbers. Result is weird but some structures are like on xenodreambui's images http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/msg8246/#msg8246

Code:

double tempR;

//rotation around Z axis
tempR = sqrt(z.x * z.x + z.y * z.y);
z *= (1.0 / tempR);
temp = z.x * z.x - z.y * z.y;
z.y = 2.0 * z.x * z.y;
z.x = temp;
z *= tempR;

//rotation around X axis
tempR = sqrt(z.y * z.y + z.z * z.z);
z *= (1.0 / tempR);
temp = z.y * z.y - z.z * z.z;
z.z = 2.0 * z.y * z.z;
z.y = temp;
z *= tempR;

//rotation around Y axis
tempR = sqrt(z.x * z.x + z.z * z.z);
z *= (1.0 / tempR);
temp = z.x * z.x - z.z * z.z;
z.z = 2.0 * z.x * z.z;
z.x = temp;
z *= tempR;

z = z * r;
z += constant;
r = z.Length();

(http://nocache-nocookies.digitalgott.com/gallery/4/640_13_11_10_4_56_01.jpeg)

Some zoom:

(http://nocache-nocookies.digitalgott.com/gallery/4/640_13_11_10_4_59_22.jpeg)

I have already implemented these formulas in Mandelbulber 0.96

Title: Re: New Mandelbulb-like formulas
Post by: Collin237 on June 16, 2011, 12:28:38 PM

Unfortunately, I don't have a program to test this with.

A complex analytic function is a conformal mapping. So I was thinking, instead of looking for a 3D analog of complex numbers, what about simply looking for a 3D conformal mapping, regardless of whether it makes sense algebraically? I'm suggesting the following formula:

x → x² + 2x(y+z) - y² - z² + c1
y → y² + 2y(x+z) - x² - z² + c2
z → z² + 2z(x+y) - x² - y² + c3

If anyone can get an image of this, please let me know.
Thanks, Collin

Title: Re: New Mandelbulb-like formulas
Post by: Buddhi on June 16, 2011, 09:57:15 PM

Quote from: Collin237 on June 16, 2011, 12:28:38 PM

Here is the fractal rendered from your formula:

(http://nocache-nocookies.digitalgott.com/gallery/7/640_16_06_11_9_55_32.jpeg)

Title: Re: New Mandelbulb-like formulas
Post by: Tater on July 13, 2013, 08:31:00 PM

Quote from: Buddhi on November 11, 2010, 02:21:12 PM

Because as Trifox said "True 3D mandelbrot fractal (search for the holy grail continues)" is so long, I'm starting new thread for continuing this endless quest

Last days I decided to experiment with rotations in Mandelbulb formula. I tried to do some formula with no privileged axis. In standard Mandelbulb formula there are two rotations. First around Z axis and second around axis which is perpendicular to plane created by z axis and resultant vector of x and y vectors. So X axis is privileged. In my version of formula I used 3 simple rotations. First around Z axis, second around Y axis and third around X axis. There is a code is below:

Code:

double rp = pow(r, p-1);

double angZ = atan2(z.y, z.x);
double angY = atan2(z.z, z.x);
double angX = atan2(z.z, z.y);

CRotationMatrix rotM;
rotM.RotateX(angX*(p-1));
rotM.RotateY(angY*(p-1));
rotM.RotateZ(angZ*(p-1));

z = rotM.RotateVector(z) * rp + constant;
r = z.Length();

...

If I understand correctly, this is rotating the vector around each axis by a multiple of the current angle and using that same multiple for the power of the length. Have you tried rotating around each axis by a constant amount instead, so that, for instance, the vector is rotated by 30 degrees around the x, y and z axes each iterate? Perhaps a power of the length could be different from the constant of rotation too.

Title: Re: New Mandelbulb-like formulas
Post by: DarkBeam on November 26, 2016, 11:33:46 PM

Those formulas have an immense potential but they were simply forgot since years? :sad1:
Firstly I must point out that they should be simplified in the normalization part;

//rotation around Z axis
tempR = 1.0/sqrt(z.x * z.x + z.y * z.y);
temp = z.x * z.x - z.y * z.y;
z.y = 2.0 * z.x * z.y;
z.x = temp;
z.xy *= tempR;

Does the same as Buddhi's one but just two multiplication vs six.

The interesting part?

Try to insert some fabs around, anywhere. I dunno how many possible variations you can get but some are really wonderful.

Images will come soon O0

Title: Re: New Mandelbulb-like formulas
Post by: DarkBeam on November 27, 2016, 11:38:34 AM

I have "generalized" the rotating function like this;

Code:

void Rho(double* u, double* v, int* op)
{
  double Rtemp, tempR, temp;
  
  //rotation around any axis; pass them to this function
  Rtemp = Hypot(*u,*v); // unused, hypot = sqrt(y^2+x^2)
  tempR = recip(Rtemp); // 1/arg
  double uu = *u, vv = *v;
  
  if (*op & 8) uu = fabs(uu);
  if (*op & 16) vv = fabs(vv);
  temp = uu * *u - vv * *v;
  if (*op & 4) temp = fabs(temp);
  
  uu = *u; vv = *v;
  if (*op & 1) uu = fabs(uu);
  if (*op & 2) vv = fabs(vv);
  *v = uu * vv; *v += *v;

  *u = temp; 
  *u *= tempR; *v *= tempR; 
  return;   
}

Where you let the user choose an integer as op, op can go 0 to 31 for each rotation; total of 32^3 = 32768 :o variations, some very nice some very distorted. :D
If all user params are zero you get the normal thingy. :D
It gives burning ship, celtic and normal mandelbrot for the xy slice, but the xz slice is always fuzzy at least for the triple rotation version.

Pictures now. :dink:

Title: Re: New Mandelbulb-like formulas
Post by: DarkBeam on November 27, 2016, 12:28:59 PM

...

(Here I disabled the YZ rotation, as it incredibly increases fuzziness & seems to piss off Mandel) :'(

Title: Re: New Mandelbulb-like formulas
Post by: Sabine on November 27, 2016, 01:04:49 PM

:o There really is too much to try and experiment with! :yes:

Title: Re: New Mandelbulb-like formulas
Post by: DarkBeam on November 27, 2016, 07:03:54 PM

A grailish Julia set... ;) Variant (2; NO; 12) (No YZ rot ;D )

(http://nocache-nocookies.digitalgott.com/gallery/19/4162_27_11_16_7_02_09.jpeg)

Title: Re: New Mandelbulb-like formulas
Post by: M Benesi on November 27, 2016, 08:04:28 PM

combine with:

Code:

//sr12= sqrt (1/2)  sr13= sqrt(1/3).....

tx=z.x*sr23-z.z*sr13;     //rotate
	z.z=z.x*sr13 + z.z*sr23;   
	z.x=tx*sr12-z.y*sr12;               
	z.y=tx*sr12+z.y*sr12; 
	
	z=abs(z)*T1Scale;   //do thingy and scale    t1scale=2
	dr*=T1Scale;

	tx=z.x*sr12+z.y*sr12;    //rotate back and subtract offset from x...
	z.y=-z.x*sr12+z.y*sr12; 
	z.x=tx*sr23+z.z*sr13-T1offset;    //t1offset~2??  maybe sqrt(3) if it's after... forgot  :D
	z.z=-tx*sr13+z.z*sr23;

Title: Re: New Mandelbulb-like formulas
Post by: DarkBeam on November 27, 2016, 08:48:14 PM

Matthew images plsss :)

Title: Re: New Mandelbulb-like formulas
Post by: Sabine on November 27, 2016, 11:07:33 PM

A little preview (very slow render!) with XYrot on ;)

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Fractal Math, Chaos Theory & Research => The 3D Mandelbulb => Topic started by: Buddhi on November 11, 2010, 02:21:12 PM