Author Topic: How many different ways are there to show such set ? (Read 1349 times)

Adam Majewski · « **on:** November 21, 2020, 07:20:59 PM »

Contest !!!!

Here are 22 different images of imploded cauliflower Julia set (edit : here c = 0.35): https://gitlab.com/adammajewski/implodedcauliflower and the source code for creating them ( c/bash)

Can you add new method/algoritm ?
Can make it better ? ( in any way )

Below one example ( BD/J inverted plane )

Linkback: https://fractalforums.org/programming/11/how-many-different-ways-are-there-to-show-such-set/3874/

Caleidoscope · « **Reply #1 on:** November 21, 2020, 11:56:54 PM »

This would have been a nice idea for the 'technical' challenge

gerrit · « **Reply #2 on:** November 22, 2020, 02:01:21 AM »

Orbit trap using ratio of smallest and next-to smallest |z|.

xenodreambuie · « **Reply #3 on:** November 22, 2020, 05:04:20 AM »

With a branching texture and vortex transform.

Adam Majewski · « **Reply #4 on:** November 22, 2020, 06:51:32 AM »

Quote from: xenodreambuie on November 22, 2020, 05:04:20 AM

With a branching texture and vortex transform.

Can you describe more ?
What program do you use ?
What is a branching texture ? ...

It looks similar to
http://www.juliasets.dk/RatioField-filer/ratiof28.jpg

Do you use similar technique ?

Adam Majewski · « **Reply #5 on:** November 22, 2020, 08:29:57 AM »

Here is normal shading https://gitlab.com/adammajewski/mandelbrot_wiki_ACh#normal-map

I'm not sure if it is good image because critical point and it's preimages are visible
compare http://www.math.titech.ac.jp/~kawahira/gallery/movies/0_1.avi
and http://www.math.titech.ac.jp/~kawahira/gallery/movies/0_1_050.png on page http://www.math.titech.ac.jp/~kawahira/gallery/movies.html

Code: [Select]




/* 
 The dot product of two vectors a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is defined as:[1]
 d = a1b1 + a2b2
  
*/
double cdot(double complex a, double complex b){
 
 
 return creal(a)*creal(b)+cimag(a)*cimag(b); 


}


// 
// output 
// 
double GiveReflection(double complex z )
  {
   int i=0; // iteration 
   int iMax = 2000;
   
   // https://en.wikipedia.org/wiki/Complex_quadratic_polynomial#First_derivative_with_respect_to_z
   double complex dz = 1.0; // derivative with respect to z 
   double reflection = 0.0; //  
   
   double h2 = 1.5 ; // height factor of the incoming light
   double angle = 45.0/360.0 ; // incoming direction of light in turns 
   double complex v = cexp(2.0*angle *M_PI* I); // = exp(1j*angle*2*pi/360)  // unit 2D vector in this direction
   // incoming light 3D vector = (v.re,v.im,h2)
  
  // https://en.wikipedia.org/wiki/Lambertian_reflectance

   
   double  complex u;
   
   
   z  = z*z+c; // 
   dz = 1.0;
   
   for(i=0;i<iMax;i++)
    {  
      dz = 2.0*dz*z ;
      z  = z*z+c; // 
      
      
      
      if(cabs(z) > ER_NP) 
      { // exterior
        u = z / dz;
        u = u / cabs(u);
        reflection =  cdot(u, v) + h2;  // use the simplest model for the shading: Lambert, which consists in using the dot product of (x,y,1) with a constant vector indicating the direction of the light.
        reflection = reflection/(1.0 + h2);  // rescale so that t does not get bigger than 1
        if (reflection<0.0) reflection =0.0;
        break;
      
      }
    }
    
   return reflection;  
 }





// Potential to color
unsigned char ComputeColorOfNP(complex double z){
//https://www.math.univ-toulouse.fr/~cheritat/wiki-draw/index.php/Mandelbrot_set#Normal_map_effect


  
  
  double reflection;
  unsigned char iColor;
   
   
   
  // compute 
   reflection = GiveReflection( z);
  
    
  // 
  //if (reflection <  )
    //{ /*  interior  */
    //  iColor = 0;}
  //else // exterior 
        
    { iColor = 255 * reflection;}
     
  return iColor;   
  
 
}

xenodreambuie · « **Reply #6 on:** November 22, 2020, 08:32:13 AM »

Quote from: Adam Majewski on November 22, 2020, 06:51:32 AM

Can you describe more ?
What program do you use ?
What is a branching texture ? ...

As field lines in exterior regions branch at each iteration boundary, one way to get a smooth branching pattern is by interpolating between the iterations, as in the triangle inequality average and related coloring methods. But for general procedural textures it is desirable that they can be calculated purely from the coordinates of a point (plus general information such as the degree of the formula.) The most obvious coordinates are continuous potential and field angle. Some patterns can be designed with a shape that branches within a unit square, so that the top edge matches with half of the bottom edge (or ideally, any moderate number of branches so it can work with higher degree formulas.) The pattern I used here is just an abs(sin), with interpolated frequency between 1 and 2 (or the degree, if greater than 2). So if I use a cubic formula it does a three to one branch.

The software is Jux, but a pre-release version for the swirly transformation.

Adam Majewski · « **Reply #7 on:** November 22, 2020, 10:03:23 AM »

Thx. Image is cool and Jux is a great software http://www.xenodream.com/jux.htm

Quote from: xenodreambuie on November 22, 2020, 08:32:13 AM

As field lines in exterior regions branch at each iteration boundary ....

What do you mean by field lines? I have found in chm :

Quote

Field Lines are divisions of potential in a direction perpendicular to the potential gradient

Should it be : (external) rays are simply the curves that run perpendicular to the equipotential curves = along potential gradient ??
Seems similar to the external rays = field lines of Douady-Hubbard potential
https://en.wikipedia.org/wiki/External_ray

Sho

Here some external rays branch at critical point and it's preimages
https://commons.wikimedia.org/wiki/File:Dynamical_plane_with_branched_periodic_external_ray_0_for_map_f(z)_=_z*z_+_0.35.png#file

xenodreambuie · « **Reply #8 on:** November 22, 2020, 10:02:58 PM »

Quote from: Adam Majewski on November 22, 2020, 10:03:23 AM

What do you mean by field lines? I have found in chm :
"Field Lines are divisions of potential in a direction perpendicular to the potential gradient"

Should it be : (external) rays are simply the curves that run perpendicular to the equipotential curves = along potential gradient ??
Seems similar to the external rays = field lines of Douady-Hubbard potential
https://en.wikipedia.org/wiki/External_ray

Yes, in a potential field the definition of field lines is lines that are perpendicular to equipotential curves. External rays are field lines in an exterior region that are continuous.

But the most convenient way to get a metric perpendicular to equipotentials is to take the arctan2 of the point at bailout. That gives field lines (or lines of longitude) that branch at each iteration boundary in exterior regions. In interior regions they do not branch, so it's easy to apply continuous textures.

Adam Majewski · « **Reply #9 on:** November 23, 2020, 10:23:06 PM »

Cool. Masterpiece. I like everything. Nice color gradient.

Adam Majewski · « **Reply #10 on:** November 23, 2020, 10:25:30 PM »

Quote from: xenodreambuie on November 22, 2020, 08:32:13 AM

As field lines in exterior regions branch at each iteration boundary

Do you mean boundaries of binary decomposition ?

Adam Majewski · « **Reply #11 on:** November 23, 2020, 10:41:08 PM »

I have tried to recreate your cool image using
* boundaries of binary decomposition ( using Sobel filter) = bdb
* boundaries of level sets ( of escape time) = lc ( level curves)

so file name is lcbdb.png

Please note that I "normalize " parameters to have level curves cross at critical point ( z=0)

Code: [Select]

ER_LSM = GiveER(20); // find such ER for LSM/J that level curves croses critical point and it's preimages
where

Code: [Select]


// find such ER for LSM/J that level curves croses critical point and it's preimages
double GiveER(int i_Max){

	complex double z= 0.0; // criical point
	double ER;
	int i;
	 ; // critical point escapes very fast here. Higher valus gives infinity
	for (i=0; i< i_Max; ++i ){
		z=z*z +c; 
	 
	 }
	 
	 ER = cabs(z);
	 if (ER < 2 ) 
	 	{printf("errror :  ER < 2.0 !!!!!\n");}
	 
	 return cabs(z);
	
	
}

Of course your image is much better but the skeleton seem the same. Am I right ?

Procedural texture : https://github.com/adammaj1/Procedural-texture

It seems similar to https://gitlab.com/adammajewski/mandelbrot-book_book#final-angle but you use smth like https://code2d.wordpress.com/2020/07/21/island-gradient/

xenodreambuie · « **Reply #12 on:** November 24, 2020, 01:02:02 AM »

Thanks Adam; yes you are correct. The boundaries are the same as for binary decomposition. I was going to show an image with just the black lines, but decided to add a star to show how any pattern can be used.

The density and alignment of the grid depends on the high bailout, so I just adjust that visually for the desired result.

Adam Majewski · « **Reply #13 on:** November 24, 2020, 06:20:38 PM »

Cartograpic techniques:

superheal · « **Reply #14 on:** November 25, 2020, 01:01:51 PM »

Xeno, can you elaborate a bit more on how you created this spiral like triangle inequality looking pattern? Do you mind providing some code?

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Author Topic: How many different ways are there to show such set ? (Read 1349 times)

Adam Majewski

How many different ways are there to show such set ?

Caleidoscope

Re: How many different ways are there to show such set ?

gerrit

Re: How many different ways are there to show such set ?

xenodreambuie

Re: How many different ways are there to show such set ?

Adam Majewski

Re: How many different ways are there to show such set ?

Adam Majewski

Re: How many different ways are there to show such set ?

xenodreambuie

Re: How many different ways are there to show such set ?

Adam Majewski

Re: How many different ways are there to show such set ?

xenodreambuie

Re: How many different ways are there to show such set ?

Adam Majewski

Re: How many different ways are there to show such set ?

Adam Majewski

Re: How many different ways are there to show such set ?

Adam Majewski

Re: How many different ways are there to show such set ?

xenodreambuie

Re: How many different ways are there to show such set ?

Adam Majewski

Re: How many different ways are there to show such set ?

superheal

Re: How many different ways are there to show such set ?

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