File:Binary decomposition.png

Size of this preview: 600 × 600 pixels. Other resolutions: 240 × 240 pixels | 480 × 480 pixels | 768 × 768 pixels | 1,024 × 1,024 pixels | 2,000 × 2,000 pixels.

Original file ‎(2,000 × 2,000 pixels, file size: 49 KB, MIME type: image/png)

This is a file from the Wikimedia Commons. The description on its description page there is shown below.
Commons is a freely licensed media file repository. You can help.

Summary

DescriptionBinary decomposition.png	English: Binary decomposition of unrolled circle plane. Relation between binary decomposition and binary numbers. It is a graphical explanation how to convert proper decimal fraction to binary fraction. A decomposition of a Carleson square into dyadic Whitney boxes
Date	15 April 2015
Source	Own work
Author	Adam majewski
Other versions	binary decomposition of Julia set exterior and interior for $f_{0}(z)=z*z$ binary decomposition of Mandelbrot set exterior unrolled cardioid Binary Tiling hue-Morse Sequence generated by an L-System Carleson dyadic square dyadic paritition shadertoy: binarytree

Licensing

I, the copyright holder of this work, hereby publish it under the following license:

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

You are free:

to share – to copy, distribute and transmit the work
to remix – to adapt the work

Under the following conditions:

attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

Summary

Relation between binary decomposition and external angles for quadratic polynomials^[1]

circle plane
unrolled circle plane

There are 3 planes :

extended complex plane ( parameter eplane or dynamic plane, but dynamic is easier to understand), for example plane for fc(z) = z*z +c where c= -2
extended complex plane for c= 0
above plane where circle is transformed int streight segment ( from 0 to 1 ). It can be called unrolled circle plane

Lines :

horizontal are boundaries of escape time level sets ( dwell bands )
vertical are dynamic external rays

Note that distance between horizontal lines is getting smaller when trace from up to down. Light gray band at the bottom of the image is a Juli a set.

how to convert proper decimal fraction to binary fraction

When fraction :

is greater then 1/2 then it starts with 0.1
is smaller then 1/2 then it starts with 0.0

( to do )

So when I trace external ray ( vertical line ) :

inwards ( from point at infinity towards Julia set ) I add bit at the end each time I cross level set boundary ( horizontal line )
outwards ( from boundary of Julia set toward point at inifinity) I add the bit at the beginning

C src code

"an image is basically a matrix" so program uses 2 arrays ( edge and data).

Program checks every pixel of the image ( see FillArray function ) :

// for all pixels of image 
  for(iy = iyMin; iy<=iyMax; ++iy) 
    { printf(" %d z %d\n", iy, iyMax); //info 
      for(ix= ixMin; ix<=ixMax; ++ix) PlotPoint(A, ix, iy, IterationMax, iMethod ) ; //  
    }

converts from c=0 plane to unrolled circle plane converting :

horizontal coordinate x to argument of complex number z
vertical coordinate y to radius of complex number z

It can be done using GiveZ function :

// from screen to world coordinate  mapping
complex double GiveZ(unsigned int ix, unsigned int iy)
{ 
  double Zx, Zy; // Z=Zx+Zy*I = radius*e^{turn*2*pi*I}
  double turn = ((double) ix)/((double ) ixMax); // arg(Z)
  double radius = ((double) (iyMax-iy))/((double) iyMax);  // abs(Z)
  Zx = ZxMax*radius*cos(turn*2.0*pi);
  Zy = ZyMax*radius*sin(turn*2.0*pi); 
 
 
  return (Zx+ I*Zy);
 
}

and computes it's color ( see ComputeColor function) :

if point escapes then it is exterior of the Julia set
if point do not escapes then it is interior of the Julia set

if (Zx2+Zy2<1.0) return  iColorOfInterior;
  if (Zx2+Zy2>ER2)  
    {  switch( iMethod )
	{
 
 
	case 1: // level set method
	  if (i%2 == 0) return 150; 
	  else return 190;
	  break;
 
	case 2: // binary decomposition method 
	  if (Zy>0) return iColorOfExteriorUp; 
	  else return iColorOfExteriorDown;
	  break;
 
 
	} // switch 
 
    }

Exterior can be colored using 2 methods :

level set method ( LSM )^[2]
binary decompositiom method ( BDM ) ^[3]

Program creates 6 pgm images.

/*

  Adam Majewski
  adammaj1 aaattt o2 dot pl  // o like oxygen not 0 like zero 
  fraktal.republika.pl

  c console progam 


  How to compute iteration : 

  gcc r.c -lm -Wall -march=native 
  time ./a.out

  m
*/

#include <stdio.h>
#include <stdlib.h> // malloc
#include <string.h> // strcat
#include <math.h> // M_PI; needs -lm also 
#include <complex.h>

/* --------------------------------- global variables and consts ------------------------------------------------------------ */





// virtual 2D array and integer ( screen) coordinate
// Indexes of array starts from 0 not 1 
//unsigned int ix, iy; // var
static unsigned int ixMin = 0; // Indexes of array starts from 0 not 1
static unsigned int ixMax ; //
static unsigned int iWidth ; // horizontal dimension of array

static unsigned int iyMin = 0; // Indexes of array starts from 0 not 1
static unsigned int iyMax ; //

static unsigned int iHeight = 2000; //  
// The size of array has to be a positive constant integer 
static unsigned int iSize ; // = iWidth*iHeight; 

// memmory 1D array 

unsigned char *data;
unsigned char *edge;
unsigned char *edge1;

// unsigned int i; // var = index of 1D array
//static unsigned int iMin = 0; // Indexes of array starts from 0 not 1
static unsigned int iMax ; // = i2Dsize-1  = 
// The size of array has to be a positive constant integer 
// unsigned int i1Dsize ; // = i2Dsize  = (iMax -iMin + 1) =  ;  1D array with the same size as 2D array

/* world ( double) coordinate = dynamic plane */
static   const double ZxMin=-10.0;
static  const double ZxMax=10.0;
static  const double ZyMin=-10.0;
static  const double ZyMax=10.0;
static  double PixelWidth; // =(ZxMax-ZxMin)/ixMax;
static  double PixelHeight; // =(ZyMax-ZyMin)/iyMax;
static  double ratio ;
 




static unsigned long int iterMax  = 1000; //iHeight*100;

static double ER = 9.0; // Escape Radius for bailout test 
static double ER2;

/* colors = shades of gray from 0 to 255 */
// 8 bit color = int number from 0 to 255
unsigned char iColorOfInterior=200; // 
static unsigned char iColorOfExteriorUp = 125;
static unsigned char iColorOfExteriorDown = 245;

static unsigned char iColorOfUnknown = 100;

long int iUknownPixels=0;


const double pi = 3.141592653589793;
/* ------------------------------------------ functions -------------------------------------------------------------*/



      

//------------------complex numbers -----------------------------------------------------



// from screen to world coordinate mapping
// uses global cons
complex double GiveZ(unsigned int ix, unsigned int iy)
{ 
  double Zx, Zy; // Z=Zx+Zy*I = radius*e^{turn*2*pi*I}
  double turn = ((double) ix)/((double ) ixMax); // arg(Z)
  double radius = ((double) (iyMax-iy))/((double) iyMax);  // abs(Z)
  Zx = ZxMax*radius*cos(turn*2.0*pi);
  Zy = ZyMax*radius*sin(turn*2.0*pi); 
 

  return (Zx+ I*Zy);

}

// uses globaal cons
//double GiveZy(unsigned int ix, unsigned int iy)
// { return (ZyMax - iy*PixelHeight);} // reverse y axis

/* -----------  array functions = drawing -------------- */

/* gives position of 2D point (ix,iy) in 1D array  ; uses also global variable iWidth */
unsigned int Give_i(unsigned int ix, unsigned int iy)
{ return ix + iy*iWidth; }

// plots raster point (ix,iy) 
int iDrawPoint(unsigned char A[], unsigned int ix, unsigned int iy, unsigned char iColor)
{ 

  /* i =  Give_i(ix,iy) compute index of 1D array from indices of 2D array */
  A[Give_i(ix,iy)] = iColor;

  return 0;
}


 





//;;;;;;;;;;;;;;;;;;;;;;  setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

int setup()
{

  
 
  
  
  printf("setup\n");
 
  /* 2D array ranges */
  
  iWidth = iHeight;
  iSize = iWidth*iHeight; // size = number of points in array 
  // iy
  iyMax = iHeight - 1 ; // Indexes of array starts from 0 not 1 so the highest elements of an array is = array_name[size-1].
  //ix
  
  ixMax = iWidth - 1;

  /* 1D array ranges */
  // i1Dsize = i2Dsize; // 1D array with the same size as 2D array
  iMax = iSize-1; // Indexes of array starts from 0 not 1 so the highest elements of an array is = array_name[size-1].

  /* Pixel sizes */
  PixelWidth = (ZxMax-ZxMin)/ixMax; //  ixMax = (iWidth-1)  step between pixels in world coordinate 
  PixelHeight = (ZyMax-ZyMin)/iyMax;
  ratio = ((ZxMax-ZxMin)/(ZyMax-ZyMin))/((float)iWidth/(float)iHeight); // it should be 1.000 ...
  
  

  // for numerical optimisation in iteration
  ER2 = ER * ER;
  
  
  /* create dynamic 1D arrays for colors ( shades of gray ) */
  data = malloc( iSize * sizeof(unsigned char) );
  edge = malloc( iSize * sizeof(unsigned char) );
  edge1 = malloc( iSize * sizeof(unsigned char) );

  if (edge1==NULL || edge== NULL || data == NULL)
    {
      fprintf(stderr," Could not allocate memory");
      getchar(); 
      return 1;
    }

  

 
  

  
   
  
  printf(" end of setup \n");
  
  return 0;

} // ;;;;;;;;;;;;;;;;;;;;;;;;; end of the setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

unsigned char ComputeColor(unsigned int ix, unsigned int iy, int IterationMax, int iMethod)
{ 
  // check behavour of z under fc(z)=z^2+c
  // using 2 target set:
  // 1. exterior or circle (center at origin and radius ER ) 
  // as a target set containing infinity = for escaping points ( bailout test)
  // for points of exterior of julia set
  // 2. interior of circle with center = alfa and radius dMaxDistance2fixed
  // as a target set for points of interior of Julia set 
  //  Z= Zx+ZY*i;

 
  int i=0; // number of iteration 
  
   
  complex double Z; // Z= Zx + Zy*I
  double Zx, Zy; 
  double Zx2, Zy2; // Zx2 = Zx* Zx
  
  
  // from screen to world coordinate 
  Z = GiveZ(ix,iy);
  Zx = creal(Z);
  Zy = cimag(Z);
  //
  Zx2=Zx*Zx;
  Zy2=Zy*Zy;
  
  
  

  if (Zx2+Zy2<1.0) return  iColorOfInterior;
  if (Zx2+Zy2>ER2)  
    {  switch( iMethod )
	{
            
    
	case 1: // level set method
	  if (i%2 == 0) return 150; 
	  else return 190;
	  break;

	case 2: // binary decomposition method 
	  if (Zy>0) return iColorOfExteriorUp; 
	  else return iColorOfExteriorDown;
	  break;


	} // switch 
           
    }
  	
   
    
  // if not inside target set around 
  while (i< IterationMax)
    { // then iterate 
      Zy=2*Zx*Zy ;      
      Zx=Zx2-Zy2 ;
      Zx2=Zx*Zx;
      Zy2=Zy*Zy;
            
      // escaping test
      if (Zx2+Zy2>ER2)  
	{  switch( iMethod )
	    {
            case 1: // level set method
              if (i%2 == 0) return 100; 
	      else return 200;
	      break;

            case 2: // binary decomposition method 
	      if (Zy>0) return iColorOfExteriorUp; 
	      else return iColorOfExteriorDown;
	      break;

	    } // switch 
           
	}
      // if escaping stop iteration
      i+=1;
    }

  // pixel is not escaping to infinity or not attracting to fixed attractore : 
  // change parameters : iterMax, distance ...
  iUknownPixels+=1;
  return  iColorOfUnknown ;   //
}

// plots raster point (ix,iy) 
int PlotPoint(unsigned char A[] , unsigned int ix, unsigned int iy, int IterationMax, int iMethod)
{
  unsigned i; /* index of 1D array */
  unsigned char iColor;
  

  i = Give_i(ix,iy); /* compute index of 1D array from indices of 2D array */
  iColor = ComputeColor(ix, iy, IterationMax, iMethod);
  A[i] = iColor;

  return 0;
}

// fill array 
// uses global var :  ...
// scanning complex plane 
int FillArray(unsigned char A[], int IterationMax, int iMethod )
{
  unsigned int ix, iy; // pixel coordinate 

  printf("compute image \n");
  // for all pixels of image 
  for(iy = iyMin; iy<=iyMax; ++iy) 
    { printf(" %d z %d\n", iy, iyMax); //info 
      for(ix= ixMin; ix<=ixMax; ++ix) PlotPoint(A, ix, iy, IterationMax, iMethod ) ; //  
    } 
   
  return 0;
}

int ComputeBoundariesFromA2B(unsigned char A[], unsigned char B[])
{
 
  unsigned int iX,iY; /* indices of 2D virtual array (image) = integer coordinate */
  unsigned int i; /* index of 1D array  */
  /* sobel filter */
  unsigned char G, Gh, Gv; 
  // boundaries are in edge array ( global var )
 
 
 
 
  printf(" find boundaries in A array using  Sobel filter\n");   
  // #pragma omp parallel for schedule(dynamic) private(i,iY,iX,Gv,Gh,G) shared(iyMax,ixMax, ER2)
  for(iY=1;iY<iyMax-1;++iY){ 
    for(iX=1;iX<ixMax-1;++iX){ 
      Gv= A[Give_i(iX-1,iY+1)] + 2*A[Give_i(iX,iY+1)] + A[Give_i(iX-1,iY+1)] - A[Give_i(iX-1,iY-1)] - 2*A[Give_i(iX-1,iY)] - A[Give_i(iX+1,iY-1)];
      Gh= A[Give_i(iX+1,iY+1)] + 2*A[Give_i(iX+1,iY)] + A[Give_i(iX-1,iY-1)] - A[Give_i(iX+1,iY-1)] - 2*A[Give_i(iX-1,iY)] - A[Give_i(iX-1,iY-1)];
      G = sqrt(Gh*Gh + Gv*Gv);
      i= Give_i(iX,iY); /* compute index of 1D array from indices of 2D array */
      if (G==0) {B[i]=255;} /* background */
      else {B[i]=0;}  /* boundary */
    }
  }
 
   
 
  return 0;
}

int CopyBoundariesFromA2B(unsigned char A[], unsigned char B[])
{
 
  unsigned int iX,iY; /* indices of 2D virtual array (image) = integer coordinate */
  unsigned int i; /* index of 1D array  */
 
 
  printf("copy boundaries from edge array to data array \n");
  for(iY=1;iY<iyMax-1;++iY)
    for(iX=1;iX<ixMax-1;++iX)
      {i= Give_i(iX,iY); if (A[i]==0) B[i]=0;}
 
 
 
  return 0;
}



 
 


 

// save "A" array to pgm file 
int SaveArray2PGMFile( unsigned char A[], double k)
{
  
  FILE * fp;
  const unsigned int MaxColorComponentValue=255; /* color component is coded from 0 to 255 ;  it is 8 bit color file */
  char name [30]; /* name of file */
  sprintf(name,"f%.0f", k); /*  */
  char *filename =strcat(name,".pgm");
  char *comment="# Numerical approximation of  Julia set for f(z)= z^2 after plane transformation; Adam Majewski";/* comment should start with # */

  /* save image to the pgm file  */      
  fp= fopen(filename,"wb"); /*create new file,give it a name and open it in binary mode  */
  fprintf(fp,"P5\n %s\n %u %u\n %u\n",comment,iWidth,iHeight,MaxColorComponentValue);  /*write header to the file*/
  fwrite(A,iSize,1,fp);  /*write A array to the file in one step */
  printf("File %s saved. \n", filename);
  fclose(fp);

  return 0;
}

int info()
{
  // diplay info messages
  printf("Numerical approximation of   \n");
  printf("Image Width = %f \n", ZxMax-ZxMin);
  printf("PixelWidth = %f \n", PixelWidth);
  printf("Maximal number of iterations = iterMax = %ld \n", iterMax);
  printf("ratio of image  = %f ; it should be 1.000 ...\n", ratio);
  printf("Unknown pixels = %ld ; it should be 0 ...\n", iUknownPixels);
  return 0;
}

/* -----------------------------------------  main   -------------------------------------------------------------*/
int main()
{

 
  setup();
 

 
   
 
  FillArray(data, iterMax, 1 ); // level set method
  SaveArray2PGMFile( data, iHeight+0); // save array data (components of Fatou set ) to pgm file

  ComputeBoundariesFromA2B(data, edge1);
  SaveArray2PGMFile( edge1, iHeight+1); // save array edge (Julia set ) to pgm file

  FillArray(data, iterMax, 2 ); // binary decomposition method
  SaveArray2PGMFile( data, iHeight+2); // save array data (components of Fatou set ) to pgm file

  ComputeBoundariesFromA2B(data, edge);
  SaveArray2PGMFile( edge, iHeight+3); // save array edge (Julia set ) to pgm file

  

  CopyBoundariesFromA2B(edge1, edge); // boundary = boundary from LSM + boundary from BDM
  SaveArray2PGMFile( edge, iHeight+4); // save array data (Julia set and components ) to pgm file
  CopyBoundariesFromA2B(edge, data);
  SaveArray2PGMFile( data, iHeight+5); // save array data (Julia set and components ) to pgm file

  

  
  

  

  printf(" allways free memory  to avoid buffer overflow \n");
  free(data);
  free(edge);
  free(edge1);
 

  
  info();

 

  return 0;
}

Image magic code

convert f2005.pgm f.png

GIMP

Gimp was used for manually adding the numbers to the image

References

File history

Click on a date/time to view the file as it appeared at that time.

	Date/Time	Thumbnail	Dimensions	User	Comment
current	13:59, 15 April 2015		2,000 × 2,000 (49 KB)	Soul windsurfer	User created page with UploadWizard

File usage

The following 2 pages use this file:

Global file usage

The following other wikis use this file:

Usage on el.wikipedia.org
- Εξωτερική ακτίνα
Usage on en.wikipedia.org
- External ray

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.

If the file has been modified from its original state, some details may not fully reflect the modified file.

Horizontal resolution	28.35 dpc
Vertical resolution	28.35 dpc
File change date and time	13:53, 15 April 2015

[1] ry decomposition and external angles by claude

[2] wikiobokks : LSM

[3] wikibooks : BDM

[1]

[2]

[3]

File:Binary decomposition.png

Contents

Summary

Licensing

Summary

how to convert proper decimal fraction to binary fraction

C src code

Image magic code

GIMP

References

Captions

Items portrayed in this file

depicts

creator

some value

copyright status

copyrighted

copyright license

Creative Commons Attribution-ShareAlike 4.0 International

inception

15 April 2015

source of file

original creation by uploader

MIME type

image/png

checksum

9a9cf5d30655705e45d283f2fb5ce6d3d6a10456

data size

49,922 byte

height

2,000 pixel

width

2,000 pixel

File history

File usage

Global file usage

Metadata