File:Quadratic Julia set with Internal binary decomposition for internal ray 1 over 3.ogv

 // divide interior into 3 components 
 angle =GiveTurn(dX +dY*I);
 if ((0.0<angle) && (angle<0.333333333)) return 0;
 if ((0.333333333<angle) && (angle<0.666666667)) return 1;
 return 2;

 while ( d>_AR2)
    {
      for(i=0;i<3;++i) // iMax = period !!!!
       { 
        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;
    };

/*

Adam Majewski
fraktal.republika.pl

c console progam using 
* symmetry
* openMP

 draw  julia sets

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

components of interior are invariant under f^period

*/

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

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

// virtual 2D array and integer ( screen) coordinate
// Indexes of array starts from 0 not 1 
unsigned int ix, iy; // var
unsigned int ixMin = 0; // Indexes of array starts from 0 not 1
unsigned int ixMax ; //
unsigned int iWidth ; // horizontal dimension of array
unsigned int ixAxisOfSymmetry  ; // 
unsigned int iyMin = 0; // Indexes of array starts from 0 not 1
unsigned int iyMax ; //
unsigned int iyAxisOfSymmetry  ; // 
unsigned int iyAbove ; // var, measured from 1 to (iyAboveAxisLength -1)
unsigned int iyAboveMin = 1 ; //
unsigned int iyAboveMax ; //
unsigned int iyAboveAxisLength ; //
unsigned int iyBelowAxisLength ; //
unsigned int iHeight = 1000; //  odd number !!!!!! = (iyMax -iyMin + 1) = iyAboveAxisLength + iyBelowAxisLength +1
// The size of array has to be a positive constant integer 
unsigned int iSize ; // = iWidth*iHeight; 

// memmory 1D array 
unsigned char *data;
// unsigned int i; // var = index of 1D array
unsigned int iMin = 0; // Indexes of array starts from 0 not 1
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 */
  const double ZxMin=-1.5;
  const double ZxMax=1.5;
  const double ZyMin=-1.5;
  const double ZyMax=1.5;
  double PixelWidth; // =(ZxMax-ZxMin)/iXmax;
   double PixelWidth2; // =  PixelWidth*PixelWidth;
  double PixelHeight; // =(ZyMax-ZyMin)/iYmax;
  double distanceMax;
  double ratio ;
  double TwoPi=2*M_PI;

// complex numbers of parametr plane 
double Cx; // c =Cx +Cy * i
double Cy;
double complex c; // 

double complex alfa; // alfa fixed point alfa=f(alfa)

unsigned long int iterMax  = 1000; //iHeight*100;
double ER = 2.0; // Escape Radius for bailout test 
double ER2;
double AR ; // radius around attractor
double AR2; // =AR*AR;

unsigned int period = 3; // period of secondary component joined by root point
  unsigned int denominator;
  double InternalAngle;

unsigned char colors[]={255,230,180};
/* ------------------------------------------ functions -------------------------------------------------------------*/

/* find c in component of Mandelbrot set 
 
 uses code by Wolf Jung from program Mandel
 see function mndlbrot::bifurcate from mandelbrot.cpp
 http://www.mndynamics.com/indexp.html

  */
double complex GiveC(double InternalAngleInTurns, double InternalRadius, unsigned int period)
{
  //0 <= InternalRay<= 1
  //0 <= InternalAngleInTurns <=1
  double t = InternalAngleInTurns *2*M_PI; // from turns to radians
  double R2 = InternalRadius * InternalRadius;
  //double Cx, Cy; /* C = Cx+Cy*i */
  switch ( period ) // of component 
  {
    case 1: // main cardioid
      Cx = (cos(t)*InternalRadius)/2-(cos(2*t)*R2)/4; 
      Cy = (sin(t)*InternalRadius)/2-(sin(2*t)*R2)/4; 
      break;
   case 2: // only one component 
      Cx = InternalRadius * 0.25*cos(t) - 1.0;
      Cy = InternalRadius * 0.25*sin(t); 
      break;
  // for each period  there are 2^(period-1) roots. 
  default: // higher periods : to do
      Cx = 0.0;
      Cy = 0.0; 
    break; }

  return Cx + Cy*I;
}

/*

http://en.wikipedia.org/wiki/Periodic_points_of_complex_quadratic_mappings
z^2 + c = z
z^2 - z + c = 0
ax^2 +bx + c =0 // ge3neral for  of quadratic equation
so :
a=1
b =-1
c = c
so :

The discriminant is the  d=b^2- 4ac 

d=1-4c = dx+dy*i
r(d)=sqrt(dx^2 + dy^2)
sqrt(d) = sqrt((r+dx)/2)+-sqrt((r-dx)/2)*i = sx +- sy*i

x1=(1+sqrt(d))/2 = beta = (1+sx+sy*i)/2

x2=(1-sqrt(d))/2 = alfa = (1-sx -sy*i)/2

alfa : attracting when c is in main cardioid of Mandelbrot set, then it is in interior of Filled-in Julia set, 
it means belongs to Fatou set ( strictly to basin of attraction of finite fixed point )

*/
// uses global variables : 
//  ax, ay (output = alfa(c)) 
double complex GiveAlfaFixedPoint(double complex c)
{
  double dx, dy; //The discriminant is the  d=b^2- 4ac = dx+dy*i
  double r; // r(d)=sqrt(dx^2 + dy^2)
  double sx, sy; // s = sqrt(d) = sqrt((r+dx)/2)+-sqrt((r-dx)/2)*i = sx + sy*i
  double ax, ay;
 
 // d=1-4c = dx+dy*i
 dx = 1 - 4*creal(c);
 dy = -4 * cimag(c);
 // r(d)=sqrt(dx^2 + dy^2)
 r = sqrt(dx*dx + dy*dy);
 //sqrt(d) = s =sx +sy*i
 sx = sqrt((r+dx)/2);
 sy = sqrt((r-dx)/2);
 // alfa = ax +ay*i = (1-sqrt(d))/2 = (1-sx + sy*i)/2
 ax = 0.5 - sx/2.0;
 ay =  sy/2.0;
 

return ax+ay*I;
}

int setup()
{

  
  

  denominator = period;
  InternalAngle = 1.0/((double) denominator);

 

  /* 2D array ranges */
  if (!(iHeight % 2)) iHeight+=1; // it sholud be even number (variable % 2) or (variable & 1)
  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].
  iyAboveAxisLength = (iHeight -1)/2;
  iyAboveMax = iyAboveAxisLength ; 
  iyBelowAxisLength = iyAboveAxisLength; // the same 
  iyAxisOfSymmetry = iyMin + iyBelowAxisLength ; 
  // 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 ...
  distanceMax = PixelWidth; 
  

  // for numerical optimisation 
  ER2 = ER * ER;
  
  PixelWidth2 =  PixelWidth*PixelWidth;
 
  /* create dynamic 1D arrays for colors ( shades of gray ) */
  
  data = malloc( iSize * sizeof(unsigned char) );
  if (data == NULL )
    {
      fprintf(stderr," Could not allocate memory\n");
      getchar();
      return 1;
    }
  else fprintf(stderr," memory is OK \n");

  
 
  return 0;

}

// from screen to world coordinate ; linear mapping
// uses global cons
double GiveZx(unsigned int ix)
{ return (ZxMin + ix*PixelWidth );}

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

// forward iteration of complex quadratic polynomial
// fc(z) = z*z +c
// initial point is a z0 = critical point 
// checks if points escapes to infinity
// uses global vars : ER2, Cx, Cy, iterMax
long int GiveLastIteration(double Zx, double Zy )
{ 
    long int iter;
    // z= Zx + Zy*i
    double Zx2, Zy2; /* Zx2=Zx*Zx;  Zy2=Zy*Zy  */
 
     /* initial value of orbit = critical point Z= 0 */
                       
                        Zx2=Zx*Zx;
                        Zy2=Zy*Zy;
                        // for iter from 0 to iterMax because of ++ after last loop
                        for (iter=0; iter<iterMax && ((Zx2+Zy2)<ER2); ++iter)
                        {
                            Zy=2*Zx*Zy + Cy;
                            Zx=Zx2-Zy2 + Cx;
                            Zx2=Zx*Zx;
                            Zy2=Zy*Zy;
                        };
  
   return iter;
}

double GiveTurn(double complex z)
{
 double argument;

 argument = carg(z); //   argument in radians from -pi to pi
 if (argument<0) argument=argument + TwoPi; //   argument in radians from 0 to 2*pi
 return argument/TwoPi ; // argument in turns from 0.0 to 1.0
}

/*  */
int GiveNumber(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  */
  double angle;
 
  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;
 
  while ( d>_AR2)
    {
      for(i=0;i<3;++i) // iMax = period !!!!
       { 
        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;
    };

 // divide interior into 3 components 
 angle =GiveTurn(dX +dY*I);
 if ((0.0<angle) && (angle<0.333333333)) return 0;
 if ((0.333333333<angle) && (angle<0.666666667)) return 1;
 return 2;
}

 

unsigned char GiveColor(unsigned int ix, unsigned int iy)
{ 
   double Zx, Zy; //  Z= Zx+ZY*i;
  int iter;
  unsigned char color; // gray from 0 to 255 
// unsigned char ColorList[]={255,230,180}; /* shades of gray used in imaage
  //color=0;
  // from screen to world coordinate 
  Zx = GiveZx(ix);
  Zy = GiveZy(iy);
  
 iter  =GiveLastIteration(Zx, Zy );
   if (iter< iterMax ) // point escapes   
   { color = 245; } // exterior 
   else // Filled Julia Set
      { int number =GiveNumber(Zx,Zy, Cx, Cy, period, AR2, creal(alfa), cimag(alfa));
          color = colors[number]; } 

  return color;
}

/* -----------  array functions -------------- */

/* 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; }
//  ix = i % iWidth;
//  iy = (i- ix) / iWidth;
//  i  = Give_i(ix, iy);

// plots raster point (ix,iy) 
int PlotPoint(unsigned int ix, unsigned int iy, unsigned char iColor)
{
 unsigned i; /* index of 1D array */
 i = Give_i(ix,iy); /* compute index of 1D array from indices of 2D array */
 data[i] = iColor;

return 0;
}

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

// 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(ix, iy, GiveColor(ix, iy) ); //  
   } 
   
 return 0;
}

// fill array using symmetry of image 
// uses global var :  ...
int FillArraySymmetric(unsigned char data[] )
{
   
 unsigned char Color; // gray from 0 to 255 

//printf("axis of symmetry \n"); 
iy = iyAxisOfSymmetry; 
#pragma omp parallel for schedule(dynamic) private(ix,Color) shared(ixMin,ixMax, iyAxisOfSymmetry)
for(ix=ixMin;ix<=ixMax;++ix) {//printf(" %d from %d\n", ix, ixMax); //info  
                              PlotPoint(ix, iy, GiveColor(ix, iy));
}

/*
The use of ‘shared(variable, variable2) specifies that these variables should be shared among all the threads.
The use of ‘private(variable, variable2)’ specifies that these variables should have a seperate instance in each thread.
*/

#pragma omp parallel for schedule(dynamic) private(iyAbove,ix,iy,Color) shared(iyAboveMin, iyAboveMax,ixMin,ixMax, iyAxisOfSymmetry)

// above and below axis 
for(iyAbove = iyAboveMin; iyAbove<=iyAboveMax; ++iyAbove) 
  {//printf(" %d from %d\r", iyAbove, iyAboveMax); //info 
  for(ix=ixMin; ix<=ixMax; ++ix) 

  { // above axis compute color and save it to the array
    iy = iyAxisOfSymmetry + iyAbove;
    Color = GiveColor(ix, iy);
    PlotPoint(ix, iy, Color ); 
    // below the axis only copy Color the same as above without computing it 
    PlotPoint(ixMax-ix, iyAxisOfSymmetry - iyAbove , Color ); 
   } 
}  
 return 0;
}

int AddBoundaries(unsigned char data[])
{

  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; 
  // memmory 1D array 
  unsigned char *edge;
  /* create dynamic 1D arrays for colors ( shades of gray ) */
  
  edge = malloc( iSize * sizeof(unsigned char) );
  if (edge == NULL )
    {
      fprintf(stderr," Could not allocate memory\n");
      return 1;
    }
  

 // printf(" find boundaries in data 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= data[Give_i(iX-1,iY+1)] + 2*data[Give_i(iX,iY+1)] + data[Give_i(iX-1,iY+1)] - data[Give_i(iX-1,iY-1)] - 2*data[Give_i(iX-1,iY)] - data[Give_i(iX+1,iY-1)];
      Gh= data[Give_i(iX+1,iY+1)] + 2*data[Give_i(iX+1,iY)] + data[Give_i(iX-1,iY-1)] - data[Give_i(iX+1,iY-1)] - 2*data[Give_i(iX-1,iY)] - data[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) {edge[i]=255;} /* background */
      else {edge[i]=0;}  /* boundary */
    }
  }

for(iY=1;iY<iyMax-1;++iY){ 
    for(iX=1;iX<ixMax-1;++iX){i= Give_i(iX,iY); if (edge[i]==0) data[i]=0;}}

free(edge);

return 0;
}

// Check Orientation of image : first quadrant in upper right position
// uses global var :  ...
int CheckOrientation(unsigned char data[] )
{
   unsigned int ix, iy; // pixel coordinate 
   double Zx, Zy; //  Z= Zx+ZY*i;
   unsigned i; /* index of 1D array */
   for(iy=iyMin;iy<=iyMax;++iy) 
   {
     Zy = GiveZy(iy);
    for(ix=ixMin;ix<=ixMax;++ix) 
   {

    // from screen to world coordinate 
    Zx = GiveZx(ix);
     i = Give_i(ix, iy); /* compute index of 1D array from indices of 2D array */
     if (Zx>0 && Zy>0) data[i]=255-data[i];   // check the orientation of Z-plane by marking first quadrant */

    }
   }
   
  return 0;
}

// save data array to pgm file 
int SaveArray2PGMFile( unsigned char data[],double Radius)
{
  
  FILE * fp;
  const unsigned int MaxColorComponentValue=255; /* color component is coded from 0 to 255 ;  it is 8 bit color file */
  char name [10]; /* name of file */
  sprintf(name,"%f", Radius); /*  */
  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,iWidth,iHeight,MaxColorComponentValue);  /*write header to the file*/
  fwrite(data,iSize,1,fp);  /*write image data bytes to the file in one step */
  printf("File %s saved. \n", filename);
  fclose(fp);

  return 0;
}

int info()
{
 // diplay info messages
  printf("Cx  = %f \n", Cx); 
  printf("Cy  = %f \n", Cy);
  // 
 
  printf("alfax  = %f \n", creal(alfa));
  printf("alfay  = %f \n", cimag(alfa));
  printf("distorsion of image  = %f \n", ratio);
return 0;
}

double GiveAR2(double R, int i, int iMax, double PixelWidth )
{

if (R>=0.09) 
  AR = PixelWidth*15;// AR is const 
  else AR = PixelWidth*(i/(0.06*iMax)+1);// AR is related with internal radius 
return (AR*AR);

}

/* -----------------------------------------  main   -------------------------------------------------------------*/
int main()
{
// here are procedures for creating image file
  int i;
  int iMax=100;
  double stepR =1.0/iMax;
  double R= 0.0;

  setup();

  for(i=0;i<=iMax;++i)
   {
    //printf("R= %f \n", R);
    R=i*stepR;
  c = GiveC(InternalAngle, R, 1) ;
  Cx=creal(c);
  Cy=cimag(c);
  alfa = GiveAlfaFixedPoint(c);
 // AR = PixelWidth*15;// AR is const 
  AR2= GiveAR2(R, i, iMax,PixelWidth);
  // compute colors of pixels = image
  //FillArray( data ); // no symmetry
  FillArraySymmetric(data); 
  AddBoundaries(data);
  //  CheckOrientation( data );
  SaveArray2PGMFile( data,R); // save array (image) to pgm file 
 }
  free(data);
  //info();
  return 0;
}

#!/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 video file
# ffmpeg2theora %d.gif --framerate 5 --videoquality 9 -f webm --artist "Adam Majewski" -o o${i}.webm 
ffmpeg2theora %d.gif --framerate 10 --videoquality 9 -f ogv --artist "Adam Majewski" -o o${i}.ogv 

 
echo o${i} OK
# end