/*

  Adam Majewski
  fraktal.republika.pl

  c console progam using 
  * symmetry
  * openMP

  draw  parabolic Julia set 

  and saves it to pgm file

 

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

*/

#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 constans ------------------------------------------------------------ */
//unsigned int period = 7; // period of secondary component joined by root point
#define period 7

//  unsigned int denominator;  denominator = period;
double InternalAngle;
unsigned char Colors[period]; //={255,230,180, 160,140,120,100}; // NumberOfPetal of colors >= period
unsigned char iExterior = 245;
unsigned char iPetal = 255;

// 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 = 2000; //  odd NumberOfPetal !!!!!! = (iyMax -iyMin + 1) = iyAboveAxisLength + iyBelowAxisLength +1
// The size of array has to be a positive constant integer 
unsigned int iSize ; // = iWidth*iHeight; 

// memmory 1D arrays 
unsigned char *data;
unsigned char *edge;

// 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 PixelHeight; // =(ZyMax-ZyMin)/iYmax;
 
double ratio ;

// 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)
double alfax,alfay;

unsigned long int iterMax  = 10000; //iHeight*100;
// target set for escaping points is a exterior of circle with center in origin
double ER = 2.0; // Escape Radius for bailout test 
double ER2;
/* trap triangle parameters */
double d1=-0.4; /* length */
double d2=0.6; /* width */

/* ------------------------------------------ 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 iPeriod)
{
  //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 ( iPeriod ) // 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 // general form  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 i;
  int iMax = period; // uses global var period and Colors
  unsigned int iStep;

  iStep=150/period;

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

/* -------------------------------------------------- SETUP --------------------------------------- */
int setup()
{
  /* 2D array ranges */
  if (!(iHeight % 2)) iHeight+=1; // it sholud be even NumberOfPetal (variable % 2) or (variable & 1)
  iWidth = iHeight;
  iSize = iWidth*iHeight; // size = NumberOfPetal 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].

  InternalAngle = 1.0/((double) period); // angle for 3/7 is different then angle for 2/7
  c = GiveC(InternalAngle, 1.0, 1) ;
  alfa = GiveAlfaFixedPoint(c);
  

  
  Cx=creal(c);
  Cy=cimag(c);
  alfax=creal(alfa);
  alfay=cimag(alfa);
  /* 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 
  ER2 = ER * ER;
  
   
  /* create dynamic 1D arrays for colors ( shades of gray ) */
  data = malloc( iSize * sizeof(unsigned char) );
  edge = malloc( iSize * sizeof(unsigned char) );
  if (edge == NULL || edge == NULL )
    {
      fprintf(stderr," Could not allocate memory\n");
      return 1;
    }
  else fprintf(stderr," memory is OK \n");

  
  InitColors();
  
 
  return 0;

}

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

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

/* 
   Checks to which petal ( = part of target set)  point of interior falls 
   bail-out test  = check if points escapes ) return 0
   if not escapes find what iteration it need to fall into trap ( target set inside one petal)

  
*/

 
/*

  uses code by Wolf Jung from program Mandel
  see function mndlbrot::pixcolor from ....cpp
  http://www.mndynamics.com/indexp.html

  uint mndlbrot::pixcolor(double x, double y)
  {  uint j; double a = A, b = b, x1, y1, u, v;
  for (j = 1; j <= 1000*maxiter; j++) //larger number !!!!!!!!!!
  {  u = x*x; v = y*y;
  if (u + v <= bailout) { y = 2*x*y + b; x = u - v + a; }
  else break;
  x1 = x - 0.311744900929367; y1 = y -0.390915741234015;
  if (x1 + y1 > -0.5 && y1 > x1 && y1 < 0.4*x1)
  return 9 + j % 7;
  }
  return 0;
  }
*/

unsigned char GiveColor(unsigned int ix, unsigned int iy)
{ 
  double Zx, Zy; //  Z= Zx+ZY*i;
  double Zx2, Zy2;
  double xt, yt;
  int j;
  
  // from screen to world coordinate 
  Zx = GiveZx(ix);
  Zy = GiveZy(iy);
  

  for (j = 1; j <= iterMax; j++) //larg number of iteration s
    {  Zx2 = Zx*Zx; 
      Zy2 = Zy*Zy;
       
      // bailout test 
      if (Zx2 + Zy2 > ER2) return iExterior; // if escaping stop iteration
       
      // if not escaping iterate
      // Z(n+1) = Zn * Zn  + C
      Zy = 2*Zx*Zy + Cy; 
      Zx = Zx2 - Zy2 + Cx;  
          
      // translation near fixed point alfa
      xt = Zx - alfax; 
      yt = Zy -alfay;
      // check if point z is inside triangle around arm of critical orbit 
      if (xt + yt > d1 && yt > xt && yt < d2*xt) return Colors[ j % period]; //  
    }
  return iPetal; // not escaping and not in attracting target set  
}

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

  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 */
    }
  }

  // copy boundaries from edge array to data array 
  //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;}}

  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;
}

int CopyBoundaries()
{
 
  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) data[i]=0;}
 
 
 
  return 0;
}

int MarkTrap()
{
 
  unsigned int ix, iy; // pixel coordinate 
  double Zx, Zy; //  Z= Zx+ZY*i;
  double xt, yt;
  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 */
	  // translation near fixed point alfa
          xt = Zx - alfax; 
          yt = Zy -alfay;
          // check if point z is inside triangle around arm of critical orbit 
          if (xt + yt > d1 && yt > xt && yt < d2*xt) data[i]=255-data[i];   // mark the trap

	}
    }

  return 0;
}

int DrawCriticalOrbit(unsigned int IterMax)
{
 
  unsigned int ix, iy; // pixel coordinate 
  double Zx=0.0;
  double Zy=0.0; //  Z= Zx+ZY*i;
  double Zx2=0.0;
  double Zy2=0.0;
  unsigned int i; /* index of 1D array */
  unsigned int j;

  // 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 */
  data[i]=255-data[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 iExterior; // 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 */
      data[i]=255-data[i];   // mark the trap

    }
  return 0;
}

// save data array to pgm file 

/*
Bug :
"... when trying to compile your C code on my Mac off of
the wiki-books page, I consistently find that I need to increase the
string size for the file name.  You always seem to allocate 10
characters for the file name and then use an sprintf to print a file
name to a string where the name is based on a float that you've
computed.  On my Mac, that float always has more than 10 digits so,
when I try to run the program, I get an error.  Increasing the buffer
size solved the problem." Mark McClure

*/
int SaveArray2PGMFile( unsigned char data[], double t)
{
  
  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", t); /*  */
  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("InternalAngle  = %f \n", InternalAngle);
  printf("Cx  = %f \n", Cx); 
  printf("Cy  = %f \n", Cy);
  // 
  printf("alfax  = %f \n", creal(alfa));
  printf("alfay  = %f \n", cimag(alfa));
  printf("iHeight  = %d \n", iHeight);
  printf("PixelWidth  = %f \n", PixelWidth);
  
  printf("distorsion of image  = %f \n", ratio);
  printf("iterMax = %lu \n", iterMax);
  return 0;
}

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

  
  setup(); 

  // here are procedures for creating image file
  
  //FillArray( data ); // no symmetry
  FillArraySymmetric(data);
  SaveArray2PGMFile(data , d2+0.000); // save edge array (only boundaries) to pgm file 
  AddBoundaries(data);
  //  CheckOrientation( data );
  SaveArray2PGMFile(edge ,d2+0.001); // save edge array (only boundaries) to pgm file 
  CopyBoundaries();
  SaveArray2PGMFile(data , d2+ 0.002); // save edge array (only boundaries) to pgm file 
  MarkTrap();
  DrawCriticalOrbit(1000000);
  SaveArray2PGMFile(data , d2+ 0.003); // save edge array (only boundaries) to pgm file
  //
  free(data);
  free(edge);
  
  //
  info();

  return 0;
}