File:A parabolic checkerboard for rotation number 1 over 2.png

/*

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

  c console progam 

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


   gcc c.c -lm -Wall -march=native -fopenmp

  method of filling gaps in rays turns 
  should be the same as for drawing rays !!!!!!!!!!!!!

  !!!!!---------- algorithm ---------!!!!!!!!
  trap ( target set) for iteration :
  * of interior points = interior of circle centered at alfa and radius = dMaxDistance2Alfa, with iPeriodChild sectors described by periodic external rays landing on alfa
  * of exterior points = exterior of circle centered at origin ( z=0) and radius = ER
  Iterate point until it fails to one of 2 traps.
  If it it fail into interior trap then check in which sector is it. Color = number of sector

  Interior target set should be all inside Julia set 
  ( of course not counting external rays that cross it and exterior parts which are very thin = means width < pixelwidth)
  if it is a circle around alfa then it shrinks as iPeriodOfChild grows
  ( = time of creating image grows)
  -----------------------------------------
  Better is to take as a interior target set :
  part of this circle bordered by 2 external rays 
  (One can choose the longest one !!!)
  and color the interior proportionaly to 
  (i % iPeriodOfChild)




./a.out



without openmp
real	7m54.710s
user	7m54.861s
sys	0m0.008s

with openmp : 
real	1m47.034s
user	14m8.420s
sys	0m0.280s







*/

#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>

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

#define iPeriodChild 2 // Period of secondary component joined by root point with the parent component 
int iPeriodParent = 1; // main cardioid of Mandelbrot set

// radius of the target set ( circle around alfa fixed point ); it is related with iHeight
// so changing iHeight needs change of iMaxDistance2Alfa
int iMaxDistance2Alfa = 1; // distance point to alfa fixed point in pixels  150 when iHeight=1000; 280 when iHeight=2000

double dMaxDistance2Alfa2; // = (iMaxDistance2Alfa*PixelWidth)^2
double dMaxDistance2Alfa;

// 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 = 3000; //  
// 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 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=-1.5;
static  const double ZxMax=1.5;
static  const double ZyMin=-1.5;
static  const double ZyMax=1.5;
static  double PixelWidth; // =(ZxMax-ZxMin)/ixMax;
static  double PixelHeight; // =(ZyMax-ZyMin)/iyMax;
static  double ratio ;
 

// complex numbers of parametr plane 
double Cx; // c =Cx +Cy * i
double Cy;
double complex c; // parameter of function fc(z)=z^2 + c

double complex alfa; // alfa fixed point alfa=f(alfa)
double dAlfaX, dAlfaY;
int iAlfaX, iAlfaY;

static unsigned long int iterMax  = 10000000; //iHeight*100;

static double ER = 2.0; // 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


// Arrays are 0-indexed, so the first array element is at index = 0, and the highest is =(size_of_array – 1) 
unsigned char iColorInterior[2][iPeriodChild]={{255,231}, {123,99}}; /* shades of gray used in image */
unsigned char iColorsOfInterior[iPeriodChild]; //={255,230,180, 160,140,120,100}; // NumberOfPetal of colors = iPeriodChild
static unsigned char iColorOfExterior = 245;
unsigned char iColorOfUnknown = 133;


int iNumberOfUknknown = 0;



static double TwoPi=2*M_PI;


/* ------------------------------------------ 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 iPeriodOfTheParent)
{
  //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 ( iPeriodOfTheParent ) // 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 iPeriodOfTheParent  there are 2^(iPeriodOfTheParent-1) roots. 
    default: // higher periods : not works, 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;
}

// colors of components interior = shades of gray
int InitColors(int iMax, unsigned char iColors[])
{
  int i;
  //int iMax = iPeriodChild; iPeriodChild and iColorsOfInterior
  unsigned int iStep;

  iStep=  40 ;   //150/iMax;

  for (i = 1; i <= iMax; ++i)
    {
      iColors[i-1] = iColorOfExterior -i*iStep; 
      printf("i= %d color = %i  \n",i-1, iColors[i-1]); // debug
    }
  return 0;
}

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

// gives argument of complex number in turns 
// realted with alfa fixed point
// http://en.wikipedia.org/wiki/Turn_%28geometry%29

double GiveTurn(double zx, double zy)
{
  double argument;

  argument =atan2(zy, zx);// carg(zx +zy*I );   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
}

/* 
   principal square  root of complex number 
   http://en.wikipedia.org/wiki/Square_root

   z1= I;
   z2 = root(z1);
   printf("zx  = %f \n", creal(z2));
   printf("zy  = %f \n", cimag(z2));
*/
double complex root(double x, double y)
{ 
  
  double u;
  double v;
  double r = sqrt(x*x + y*y); 
  
  v = sqrt(0.5*(r - x));
  if (y < 0) v = -v; 
  u = sqrt(0.5*(r + x));
  return u + v*I;
}

// 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

/* -----------  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;
}

// draws point to memmory array data
// uses complex type so #include <complex.h> and -lm 
int dDrawPoint(unsigned char A[], complex double point,unsigned char iColor )
{

  unsigned int ix, iy; // screen coordinate = indices of virtual 2D array
  //unsigned int i; // index of 1D array
  
  ix = (creal(point)- ZxMin)/PixelWidth; 
  iy = (ZyMax - cimag(point))/PixelHeight; // inverse Y axis 
  iDrawPoint(A, ix, iy, iColor);
  return 0;
}



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

int setup(int ParentPeriod, int ChildPeriod)
{

  
  
  unsigned int denominator;
  double InternalAngle;
  
  printf("setup\n");

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

  c = GiveC(InternalAngle, 1.0, ParentPeriod) ; // internal radius= 1.0 gives root point = parabolic parameter 
  Cx=creal(c);
  Cy=cimag(c);
  alfa = GiveAlfaFixedPoint(c);
  dAlfaX = creal(alfa);
  dAlfaY = cimag(alfa);
  iAlfaX = (dAlfaX- ZxMin)/PixelWidth; 
  iAlfaY = (ZyMax - dAlfaY)/PixelHeight; // inverse Y axis 
  
  

  /* 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].
  //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;
  dMaxDistance2Alfa = PixelWidth*iMaxDistance2Alfa; // trap around fixed point 
  dMaxDistance2Alfa2 = dMaxDistance2Alfa* dMaxDistance2Alfa; //  
  
  /* create dynamic 1D arrays for colors ( shades of gray ) */
  data = malloc( iSize * sizeof(unsigned char) );
  edge = malloc( iSize * sizeof(unsigned char) );

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

  
  // fill array iColorsOfInterior with iPeriodChild colors ( shades of gray )
  InitColors(iPeriodChild, iColorsOfInterior);

 
  

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

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

unsigned char ComputeColor(unsigned int ix, unsigned int iy, int IterationMax)
{ 
  // 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 dMaxDistance2Alfa
  // as a target set for points of interior of Julia set 
  //  Z= Zx+ZY*i;

  double Zx2, Zy2;
  int i=0;  // number of the iteration = fc(z)
  double d2 ; /* d2= (distance from z to Alpha)^2   */
  double Zxt,Zyt ; // 
  double Zx, Zy;
  int m,n;
 
 
  
  
  // from screen to world coordinate 
  Zx = GiveZx(ix);
  Zy = GiveZy(iy);
  /* distance from z to Alpha  */
  Zxt=Zx-dAlfaX;
  Zyt=Zy-dAlfaY;
  d2=Zxt*Zxt +Zyt*Zyt;
  if (d2<dMaxDistance2Alfa2) 
    {
         return  iColorsOfInterior[i % iPeriodChild];
	
    }

  // if not inside target set around attractor ( alfa fixed point )
  while (1 )
    { // then iterate 
      
      Zx2 = Zx*Zx; 
      Zy2 = Zy*Zy;
       
      // bailout test 
      if (Zx2 + Zy2 > ER2) return iColorOfExterior; // if escaping stop iteration
       
      // if not escaping or not attracting then iterate = check behaviour
      // new z : Z(n+1) = Zn * Zn  + C
      Zy = 2*Zx*Zy + Cy; 
      Zx = Zx2 - Zy2 + Cx; 
      //
      i+=1;
      if (i > IterationMax) break; 
      
      /* distance from z to Alpha  */
      Zxt=Zx-dAlfaX;
      Zyt=Zy-dAlfaY;
      d2=Zxt*Zxt +Zyt*Zyt;
      // check if fall into internal target set = interior 
      if (d2<dMaxDistance2Alfa2) 
	{
	          m = (Zyt > 0 ? 0 : 1); // part of the target set
                  n = (i % iPeriodChild); // attraction time 
	        
                  return  iColorInterior[m][n];
                  
	  
	}
    
      
      
      
    }
   iNumberOfUknknown +=1;
  return   iColorOfUnknown;   //
}

// plots raster point (ix,iy) 
int PlotPoint(unsigned char A[] , unsigned int ix, unsigned int iy, int IterationMax)
{
  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);
  A[i] = iColor;

  return 0;
}

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

  //printf("compute image \n");
  // for all pixels of image 
  #pragma omp parallel for schedule(dynamic) private(ix,iy) shared(ixMax , iyMax, IterationMax)
  for(iy = iyMin; iy<=iyMax; ++iy) 
    { printf(" %d z %d \r", iy, iyMax); //info 
      for(ix= ixMin; ix<=ixMax; ++ix) PlotPoint(A, ix, iy, IterationMax ) ; //  
    } 
   
  return 0;
}








int ComputeBoundariesIn(unsigned char A[])
{
 
  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) {edge[i]=255;} /* background */
      else {edge[i]=0;}  /* boundary */
    }
  }
 
   
 
  return 0;
}

int CopyBoundariesTo(unsigned char A[])
{
 
  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 (edge[i]==0) A[i]=0;}
 
 
 
  return 0;
}


 
 
int DrawCriticalOrbit(unsigned char A[], unsigned int IterMax)
{
  // integer = pixel coordinate
  unsigned int ix, iy; 
  // double = world coordinate
  // critical point Z= Zx+ZY*i;
  double Zx=0.0;
  double Zy=0.0; 
  double Zx2=0.0;
  double Zy2=0.0;
  
  unsigned int i; /* index of 1D array */
  unsigned int j; // number of iteration
 
 
  // draw critical point  
  ix = (int)((Zx-ZxMin)/PixelWidth);
  iy = (int)((ZyMax-Zy)/PixelHeight); // reverse y axis
  i = Give_i(ix, iy); /* compute index of 1D array from indices of 2D array */
  A[i]=255-A[i]; 
 
  // iterate
  for (j = 1; j <= IterMax; j++) //larg number of iteration s
    {  Zx2 = Zx*Zx; 
      Zy2 = Zy*Zy;
 
      // bailout test 
      if (Zx2 + Zy2 > ER2) return iColorOfExterior; // if escaping stop iteration
 
      // if not escaping iterate
      // Z(n+1) = Zn * Zn  + C
      Zy = 2*Zx*Zy + Cy; 
      Zx = Zx2 - Zy2 + Cx;
      //compute integer coordinate  
      ix = (int)((Zx-ZxMin)/PixelWidth);
      iy = (int)((ZyMax-Zy)/PixelHeight); // reverse y axis
      i = Give_i(ix, iy); /* compute index of 1D array from indices of 2D array */
      A[i]=0; //255-A[i];   // mark the critical orbit
 
    }
  return 0;
}

// Check Orientation of image : mark first quadrant 
// it should be in the upper right position
// uses global var :  ...
int CheckOrientation(unsigned char A[] )
{
  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) A[i]=255-A[i];   // check the orientation of Z-plane by marking first quadrant */

	}
    }
   
  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", k); /*  */
  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(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 parabolic Julia set for fc(z)= z^2 + c \n");
  printf("iPeriodParent = %d \n", iPeriodParent);
  printf("iPeriodOfChild  = %d \n", iPeriodChild);
  printf("parameter c = ( %f ; %f ) \n", Cx, Cy); 
  printf("alfa fixed point z = ( %f ; %f )  \n", dAlfaX, dAlfaY);
  printf("Image Width = %f \n", ZxMax-ZxMin);
  printf("PixelWidth = %f \n", PixelWidth);
  printf("size of target set in screen units = iMaxDistance2Alfa  = %d pixels \n", iMaxDistance2Alfa); 
  printf("size of target set  = dMaxDistance2Alfa  = %f [world units] =  %f [percent of image width] \n", dMaxDistance2Alfa, dMaxDistance2Alfa/(ZxMax-ZxMin));
   printf("iNumberOfUknknown = %d ; it should be 0 ...\n", iNumberOfUknknown);
  printf("Maximal number of iterations = iterMax = %ld \n", iterMax);
  printf("ratio of image  = %f ; it should be 1.000 ...\n", ratio);
  return 0;
}

/* -----------------------------------------  main   -------------------------------------------------------------*/
int main()
{
  setup(iPeriodParent, iPeriodChild);
 

 
   
    
 
  printf("components of Fatou set  : \n");
  ComputeFatouComponents(data, iterMax);
  SaveArray2PGMFile( data, iWidth+iMaxDistance2Alfa+0.0); 
  printf("done \n");

  ComputeBoundariesIn(data);
  SaveArray2PGMFile( edge, iWidth+iMaxDistance2Alfa+0.1); 
 printf(" Julia set \n");

  


  CopyBoundariesTo(data);
  SaveArray2PGMFile( data, iWidth+iMaxDistance2Alfa+0.2); 
  printf("Julia set and components \n") ;

  

  
 
  DrawCriticalOrbit(data, 100000);
  SaveArray2PGMFile(data, iWidth+iMaxDistance2Alfa+0.3); 
  printf(" critical orbit \n");

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

  
  info();

  return 0;
}