File:Quadratic Julia set with Internal level sets for internal ray 0.ogv

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Summary

Description	English: Quadratic Julia set with Internal level sets for c values along internal ray 0 of main cardioid of Mandelbrot set
Date	21 February 2012
Source	Own work
Author	Adam majewski

This file was selected as the media of the day for 06 March 2012. It was captioned as follows:

English: Quadratic Julia set with Internal level sets for c values along internal ray 0 of main cardioid of Mandelbrot set

Other languages

English: Quadratic Julia set with Internal level sets for c values along internal ray 0 of main cardioid of Mandelbrot set

Македонски: Квадратно Жулиино множество со множества на внатрешно ниво за вредностите на c, заедно со внатрешен зрак 0 на главната кардоида од Манделбротово множество.

中文（简体）：朱利亚集合

Compare with

How the target set is changing along intrnal ray 0
Binary decomposition of interior of filled Julia set
Full tile = binary decomposition and internal level sets

Long description

This video shows dynamical planes for complex quadratic polynomial ^[1]. Here parameter c is changing along the internal ray of main cardioid of Mandelbrot set^[2]. In other words c is changing from 0 to 0.25 ( real number because imaginary part is allways 0 ).

On every image there is filled Julia set with it's interior coloured with internal level sets around fixed point. Here are 3 cases :

hyperbolic ( when c= 0.0). Here fixed point is in the center of Julia set
attracting ( when 0.0 < c <0.25 ). Here fixed point moves to right
parabolic ^[3]( c= 0.25 ). Here fixed point is on boundary = Julia set.

See how fixed point moves from center of interior to its boundary.

Above video is inspired by these images by T Kawahira^{[dead link]}

Algorithm for one image

For every point of z plane do :

check if point escapes to infinity under iteration of quadratic polynomial. It is done in GiveExtLastIteration function.
- if point escapes it is exterior point = is in basin of attraction to infinity
- if point not escapes it is ( not precisely but ...) interior point so it should be attracted to finite attractor. Check speed of attraction ( using function GiveIntLastIteration) and colour it proportionaly to it. So :

 if ( IterationMax != eLastIteration ) 
       {data[i]=245;} /* exterior */
     else /* interior */
       { iLastIteration = GiveIntLastIteration(Zx, Zy, Cx, Cy, IterationMaxBig, AR2, creal(ZA), cimag(ZA));
         data[i]=color[iLastIteration % 2];/*  level sets of attraction time */
       }

Finite attractor ZA is found using forward iteration of critical point z=0

C source code

This C code creates nMax pgm files for c values from carray :

 for(n=0;n<=nMax;++n)
  {

    Cx = carray[n];
    // create pgm file for Cx
  }

/*
 
  c console program
  -----------------------------------------
  1.ppm file code is  based on the code of Claudio Rocchini
  http://en.wikipedia.org/wiki/Image:Color_complex_plot.jpg
  create 24 bit color graphic file ,  portable pixmap file = PPM 
  see http://en.wikipedia.org/wiki/Portable_pixmap
  to see the file use external application ( graphic viewer)
  I think that creating graphic can't be simpler
  ---------------------------
  2. first it creates data array which is used to store rgb color values of pixels,
  fills tha array with data and after that writes the data from array to pgm file.
  It alows free ( non sequential) acces to "pixels"
 
  -------------------------------------------
  Adam Majewski   fraktal.republika.pl 
 
  Sobel filter 
  Gh = sum of six values ( 3 values of matrix are equal to 0 ). Each value is = pixel_color * filter_coefficients 
 
 
  gcc ilsv.c -lm -Wall -o2
  gcc ilsv.c -lm -Wall -march=native
  time ./a.out

 
 
 
*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <complex.h>
#include <string.h>
 
/* iXmax/iYmax = 1 */
unsigned int iXmax = 1000; /* height of image in pixels */
unsigned int iYmax = 1000;
unsigned int iLength; 
/* fc(z) = z*z + c */
#define denominator 1 /* denominator of internal angle */

double  AR = 0.0014998955;  /* PixelWidth*1.5   radius of circle around attractor ZA = target set for attracting points */

//#define alfa (1-sqrt(1-4*Cx))/2 /* attracting or parabolic fixed point z = alfa */
//#define beta (1+sqrt(1-4*Cx))/2 /* repelling or parabolic fixed point z = beta */

/* color */
unsigned char color[]={255,231,123,99}; /* shades of gray used in image */
const unsigned int MaxColorComponentValue=255; /* color component is coded from 0 to 255 ;  it is 8 bit color file */

 
/* escape time to infinity */
int GiveExtLastIteration(double _Zx0, double _Zy0,double C_x, double C_y, int iMax, double _ER2)
{ 
  int i;
  double Zx, Zy;
  double Zx2, Zy2; /* Zx2=Zx*Zx;  Zy2=Zy*Zy  */
  Zx=_Zx0; /* initial value of orbit  */
  Zy=_Zy0;
  Zx2=Zx*Zx;
  Zy2=Zy*Zy;
  for (i=0;i<iMax && ((Zx2+Zy2)<_ER2);i++)
    {
      Zy=2*Zx*Zy + C_y;
      Zx=Zx2-Zy2 +C_x;
      Zx2=Zx*Zx;
      Zy2=Zy*Zy;
    };
  return i;
}
 
 
/* find attractor ZA  using forward iteration of critical point Z = 0  */
/* if period is >1 gives one point from attracting cycle */
double complex GiveAttractor(double _Cx, double _Cy, double ER2, int _IterationMax)
{
  int Iteration;
  double Zx, Zy; /* z = zx+zy*i */
  double Zx2, Zy2; /* Zx2=Zx*Zx;  Zy2=Zy*Zy  */
  /* -- find attractor ZA  using forward iteration of critical point Z = 0  */
  Zx=0.0;
  Zy=0.0;
  Zx2=Zx*Zx;
  Zy2=Zy*Zy;
  for (Iteration=0;Iteration<_IterationMax && ((Zx2+Zy2)<ER2);Iteration++)
    {
      Zy=2*Zx*Zy + _Cy;
      Zx=Zx2-Zy2 + _Cx;
      Zx2=Zx*Zx;
      Zy2=Zy*Zy;
    };
  return Zx+Zy*I;
}
 
/* attracting time to finite attractor ZA */
int GiveIntLastIteration(double _Zx0, double _Zy0,double C_x, double C_y, int iMax, double _AR2, double _ZAx, double _ZAy )
{ 
  int i;
  double Zx, Zy; /* z = zx+zy*i */
  double Zx2, Zy2; /* Zx2=Zx*Zx;  Zy2=Zy*Zy  */
  double d, dX, dY; /* distance from z to Alpha  */
  Zx=_Zx0; /* initial value of orbit  */
  Zy=_Zy0;
  Zx2=Zx*Zx;
  Zy2=Zy*Zy;
  dX=Zx-_ZAx;
  dY=Zy-_ZAy;
  d=dX*dX+dY*dY;
  for (i=0;i<iMax && (d>_AR2);i++)
    {
      Zy=2*Zx*Zy + C_y;
      Zx=Zx2-Zy2 +C_x;
      Zx2=Zx*Zx;
      Zy2=Zy*Zy;
      dX=Zx-_ZAx;
      dY=Zy-_ZAy;
      d=dX*dX+dY*dY;
    };
  return i;
}

 
/* gives position of point (iX,iY) in 1D array  ; uses also global variables */
unsigned int f(unsigned int _iX, unsigned int _iY)
{return (_iX + (iYmax-_iY-1)*iXmax );}

// save data array to pgm file 
int SavePGMFile(double Cx, unsigned char data[])
{
FILE * fp;
  char name [15]; /* name of file */
  sprintf(name,"%11.10f", Cx); /*  basename = file name without extension*/
  char *filename =strcat(name,".pgm");
  char *comment="# ";/* 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,iXmax,iYmax,MaxColorComponentValue);  /*write header to the file*/
  fwrite(data,iLength,1,fp);  /*write image data bytes to the file in one step */
  printf("File %s saved. \n", filename);
  fclose(fp);
  return 0;
}

/*

 
/* --------------------------------------------------------------------------------------------------------- */
 
int main(){
 
 unsigned int nMax ; /* number of steps = number of images */
 unsigned int n;

// manually choosed values to show crossing of internal curves
double carray[]={
0.0, 0.01, 0.02, 0.03, 0.035, 0.03722, 0.04, 0.045, 0.05, 0.06, 
0.07, 0.077, 0.08378, 0.09, 0.10, 0.11, 0.12214, 0.13, 0.14,
0.150656, 0.155, 0.16, 0.165, 0.171528, 0.175, 0.18, 0.183,
0.186948, 0.19, 0.193, 0.197, 0.207388, 0.203, 0.206, 0.209, 0.2142728,
0.215, 0.2165, 0.2175, 0.21971144, 0.221, 0.222, 0.223, 0.22406680,
0.226, 0.228, 0.23050176,  0.232, 0.2335, 0.23492, 0.2353, 0.2356,
0.2366192000, 0.24, 0.245, 0.25};

nMax=sizeof(carray)/sizeof(double);

double Cx;
//double stepCx;
double Cy =  0.0;
// stepCx = (CxMax - CxMin)/ nMax;
 
  unsigned int iX,iY, /* indices of 2D virtual array (image) = integer coordinate */
    i; /* index of 1D array  */
    iLength = iXmax*iYmax;/* length of array in bytes = number of bytes = number of pixels of image * number of bytes of color */
  /* world ( double) coordinate = dynamic plane = z-plane */
  const double dSide = 1.5;
  const double ZxMin=-dSide;
  const double ZxMax=dSide;
  const double ZyMin=-dSide;
  const double ZyMax=dSide;
  double PixelWidth=(ZxMax-ZxMin)/iXmax;
  double PixelHeight=(ZyMax-ZyMin)/iYmax;
  /* */
  double Zx, Zy;    /* Z=Zx+Zy*i   */
 // double alfa; // define alfa (1-sqrt(1-4*Cx))/2 /* attracting or parabolic fixed point z = alfa */
  double complex ZA;  /* atractor ZA = ZAx + ZAy*i */
  double AR2 =  AR*AR;
  /* */
 
  const double EscapeRadius=2.0; /* radius of circle around origin; its complement is a target set for escaping points */
  double ER2=EscapeRadius*EscapeRadius;
 
  const int IterationMax=60,
    IterationMaxBig= 1000001;
  int eLastIteration, iLastIteration;
  //int InternalTile;
  /* sobel filter */
  unsigned char G, Gh, Gv; 
  
 
 
  /* dynamic 1D arrays for colors ( shades of gray ) */
  unsigned char *data, *edge;
  data = malloc( iLength * sizeof(unsigned char) );
  edge = malloc( iLength * sizeof(unsigned char) );
  if (data == NULL || edge==NULL)
    {
      fprintf(stderr," Could not allocate memory");
      getchar(); 
      return 1;
    }
  else printf(" memory is OK\n");
 
 
 
  for(n=0;n<=nMax;++n)
   {

   Cx = carray[n];
   //alfa = (1-sqrt(1-4*Cx))/2 ; /* attracting or parabolic fixed point z = alfa */
 
  ZA = GiveAttractor( Cx, Cy, ER2, IterationMaxBig); /* find attractor ZA  using forward iteration of critical point Z = 0  */
 
 
 
 // printf(" fill the data array \n");
  for(iY=0;iY<iYmax;++iY){ 
    Zy=ZyMin + iY*PixelHeight; /*  */
    if (fabs(Zy)<PixelHeight/2) Zy=0.0; /*  */
   //printf(" row %u from %u \n",iY, iYmax); /* info */   
    for(iX=0;iX<iXmax;++iX){ 
      Zx=ZxMin + iX*PixelWidth;
      eLastIteration = GiveExtLastIteration(Zx, Zy, Cx, Cy, IterationMax, ER2 );
      i= f(iX,iY); /* compute index of 1D array from indices of 2D array */
      if ( IterationMax != eLastIteration ) 
        {data[i]=245;} /* exterior */
      else /* interior */
        { iLastIteration = GiveIntLastIteration(Zx, Zy, Cx, Cy, IterationMaxBig, AR2, creal(ZA), cimag(ZA));
	  //InternalTile = GiveIntTile( Zx, Zy, Cx, Cy, IterationMaxBig );
          data[i]=color[iLastIteration % 2];/*  level sets of attraction time */

        //if (Zx>=0 && Zx <= 0.5 && (Zy > 0 ? Zy : -Zy) <= 0.5 - Zx) data[i]=255-data[i]; // show petal

        } 
      /*  if (Zx>0 && Zy>0) data[i]=255-data[i];    check the orientation of Z-plane by marking first quadrant */

    }
  }
 
 
 // printf(" find boundaries in data array using  Sobel filter\n");   
 
  for(iY=1;iY<iYmax-1;++iY){ 
    for(iX=1;iX<iXmax-1;++iX){ 
      Gv= data[f(iX-1,iY+1)] + 2*data[f(iX,iY+1)] + data[f(iX-1,iY+1)] - data[f(iX-1,iY-1)] - 2*data[f(iX-1,iY)] - data[f(iX+1,iY-1)];
      Gh= data[f(iX+1,iY+1)] + 2*data[f(iX+1,iY)] + data[f(iX-1,iY-1)] - data[f(iX+1,iY-1)] - 2*data[f(iX-1,iY)] - data[f(iX-1,iY-1)];
      G = sqrt(Gh*Gh + Gv*Gv);
      i= f(iX,iY); /* compute index of 1D array from indices of 2D array */
      if (G==0) {edge[i]=255;} /* background */
      else {edge[i]=0;}  /* boundary */
    }
  }
 
    //printf(" copy boundaries from edge to data array \n");
    for(iY=1;iY<iYmax-1;++iY){ 
     for(iX=1;iX<iXmax-1;++iX)
      {i= f(iX,iY); /* compute index of 1D array from indices of 2D array */
    if (edge[i]==0) data[i]=0;}}
 
 
  /* ---------- file  -------------------------------------*/
  //printf(" save  data array to the file \n");
  SavePGMFile( Cx, data);

  } // for n ....
 
  /* --------------free memory ---------------------*/
  free(data);
  free(edge);
 
 
 
  return 0;
}

Bash source code

#!/bin/bash
 
# script file for BASH 
# which bash
# save this file as g
# chmod +x g
# ./g

i=0
# for all pgm files in this directory
for file in *.pgm ; do
  # b is name of file without extension
  b=$(basename $file .pgm)
  # change file name to integers and count files
  ((i= i+1))
  # convert from pgm to gif and add text ( level ) using ImageMagic
  convert $file -pointsize 50 -annotate +10+100 $b ${i}.gif
  echo $file
done
 
echo convert all gif files to one ogv_file
# convert -delay 50   -loop 1 %d.gif[1-$i] aa${i}.gif
# convert -delay 50 -loop 0 %d.gif[1-$i] b${i}.mpg
# convert -delay 50   -loop 1 %d.gif[1-$i] aa${i}.ogv
ffmpeg2theora %d.gif --framerate 5 --videoquality 9  -o output5.ogv

 
echo b${i} OK
# end

References

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 3.0 Unported 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.

File history

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

	Date/Time	Thumbnail	Dimensions	User	Comment
current	17:02, 21 February 2012		12 s, 1,000 × 1,000 (5.67 MB)	Soul windsurfer

File usage

The following 3 pages use this file:

Global file usage

The following other wikis use this file:

Usage on el.wikipedia.org
- Μιγαδικό τετραγωνικό πολυώνυμο
Usage on en.wikipedia.org
- Complex quadratic polynomial

Metadata

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

Software used	Xiph.Org libtheora 1.1 20090822 (Thusnelda) ffmpeg2theora-0.24

[1] wikipedia : Complex quadratic polynomial

[2] Internal ray for angle 1/3 of main cardioid of Mandlebrot set

[3] wikibooks : parabolic basin

[1]

[2]

[3]

File:Quadratic Julia set with Internal level sets for internal ray 0.ogv

Contents

Summary

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Long description

Algorithm for one image

C source code

Bash source code

References

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copyright license

Creative Commons Attribution-ShareAlike 3.0 Unported

source of file

original creation by uploader

inception

21 February 2012

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