/*



here are:
*  3 critical points
* 2 periodic basins ( both have period 2 )








  https://web.archive.org/web/20161024194536/http://www.ijon.de/mathe/julia/some_julia_sets_3.html

  
f(z)=1/(z3+dz+c) mit c=0,37 und d=2,1, dargestellt auf [-2,1;2,1]x[-2,1;2,1].
https://web.archive.org/web/20161024194536/http://www.ijon.de/mathe/julia/some_julia_sets_3.html


https://fractalforums.org/fractal-mathematics-and-new-theories/28/rational-function/4279/45
f(z) := 1/{z^3 + 2.099609375*z  +0.349609375}

The parameters used here differ slightly from the ones on the site, as I prefer working with exactly double-representable numbers by using a near dyadic fraction, hoping the overall structure of the set remains the same (i.e. intersecting Jordan curves).


  Adam Majewski
  adammaj1 aaattt o2 dot pl  // o like oxygen not 0 like zero 
  
  
  
  Structure of a program or how to analyze the program 
  
  
  ============== Image X ========================
  
  DrawImageOf -> DrawPointOf -> ComputeColorOf ( FunctionTypeT FunctionType , complex double z) -> ComputeColor
  
  
  check only last function  which computes color of one pixel for given Function Type
  
  

   
  ==========================================

  
  ---------------------------------
  indent d.c 
  default is gnu style 
  -------------------



  c console progam 
  
  export  OMP_DISPLAY_ENV="TRUE"	
  gcc d.c -lm -Wall -march=native -fopenmp
  time ./a.out > b.txt


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


  time ./a.out

  time ./a.out >i.txt
  time ./a.out >e.txt
  
  
  
  
  
  
  convert -limit memory 1000mb -limit disk 1gb dd30010000_20_3_0.90.pgm -resize 2000x2000 10.png

  
  
  
*/

#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
#include <limits.h>		// Maximum value for an unsigned long long int



// https://sourceforge.net/p/predef/wiki/Standards/

#if defined(__STDC__)
#define PREDEF_STANDARD_C_1989
#if defined(__STDC_VERSION__)
#if (__STDC_VERSION__ >= 199409L)
#define PREDEF_STANDARD_C_1994
#endif
#if (__STDC_VERSION__ >= 199901L)
#define PREDEF_STANDARD_C_1999
#endif
#endif
#endif




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


//FunctionType
typedef enum  {Fatou_ab = 0, Fatou_abi = 2,  LSM = 3, LSM_m = 4, Unknown = 5 , BD = 6, MBD = 7 , SAC = 8, DLD = 9, ND = 10 , NP= 11, POT = 12 , Blend = 13, DEM = 14,
		
} FunctionTypeT; 
// FunctionTypeT FunctionType;

// 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 = 10000;	//  
// The size of array has to be a positive constant integer 
static unsigned long long int iSize;	// = iWidth*iHeight; 

// memmory 1D array 
unsigned char *data;
unsigned char *edge;
//unsigned char *edge2;

// unsigned int i; // var = index of 1D array
//static 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



// see SetPlane

double radius = 2.1; //1.0; //4.5; //2.1; //
complex double center = 0.0 ;
double  DisplayAspectRatio  = 1.0; // https://en.wikipedia.org/wiki/Aspect_ratio_(image)
// dx = dy compare setup : iWidth = iHeight;
double ZxMin; //= -1.3;	//-0.05;
double ZxMax;// = 1.3;	//0.75;
double ZyMin;// = -1.3;	//-0.1;
double ZyMax;// = 1.3;	//0.7;
double PixelWidth;	// =(ZxMax-ZxMin)/ixMax;
double PixelHeight;	// =(ZyMax-ZyMin)/iyMax;

// dem
double BoundaryWidth ; //= 1.0*iWidth/2000.0  ; //  measured in pixels ( when iWidth = 2000) 
double distanceMax ; //= BoundaryWidth*PixelWidth;


double ratio; 


/*
  ER = pow(10,ERe);
  AR = pow(10,-ARe);
*/
//int ARe ;			// increase ARe until black ( unknown) points disapear 
//int ERe ;
double ER;
double ER2;			//= 1e60;
double AR1; // bigger values do not works
double AR1_2;

double AR2; // bigger values do not works
double AR2_2;


double AR_max;
//double AR12;



int IterMax = 100000;
int IterMax_LSM = 1000;
int IterMax_DEM = 100000;

/* colors = shades of gray from 0 to 255 

   unsigned char colorArray[2][2]={{255,231},    {123,99}};
   color = 245;  exterior 
   
   here are two period 2 basins: basin1  and basin2
   each basin is a basin of attraction of period 2 cycle
   
   Each cycle has immediate basin of attraction which consist of 2 components ( and it's preimages)
   so we need 4 colors 
   
   also exterior is a component oof one basin , 
   it is not a basin of attraction to infiiniity
   
   
   
   
   
*/
unsigned char iColorOfBasin1 = 245;
unsigned char iColorOfBasin2 = 99;





unsigned char iColorOfBoundary = 0;
unsigned char iColorOfUnknown = 5;

// pixel counters
unsigned long long int uUnknown = 0;
unsigned long long int uInterior = 0;
unsigned long long int uExterior = 0;



/* critical points


> I have found 2 critical points :

Two finite ones and imo, infinity is a critical point too, as the 1st
derivative's denominator degree is strictly greater than the numerator's.

> but it seems that all 2 fall into one period 2  cycle :
> What about second cycle ?
The two finite critical points go to the same cycle, infinity (or 0,
which I used numerically, as f(inf)=0)  goes into the other.



*/


const complex double z_cr[3]= { 0.8366600265340756*I, 0.0, -0.8366600265340756*I};

complex double zcr1 = 0.8366600265340756*I; //
complex double zcr2 = 0.0;// = -2.2351741790771484375e-08+9.4296410679817199707e-09*I;

// -0.8366600265340756*%i,0.8366600265340756*%i 



const int period = 2;

// periodic points = attractors
//complex double z1 =  0.0 ; //fixed point (period  1) =  attracting cycle

/*
  attracting periodic points : 
  2 period 2 cycles found by marcm200
  https://fractalforums.org/fractal-mathematics-and-new-theories/28/rational-function/4279/msg29227#msg29227
  
  z = +1.6890328811664648 +0.0000000000000000*I is in the probably attracting period 2 cyle  { +1.6890328811664648 +0.0000000000000000*I ,  +0.1147519899962205 +0.0000000000000000*I } 	
  z = +0.1147519899962201 +0.0000000000000000*I is in the probably attracting period 2 cyle  { +0.1147519899962201 +0.0000000000000000*I ,  +1.6890328811664670 +0.0000000000000000*I } 
  
  z = +0.4101296722285255 -0.5079485669960778*I is in the probably attracting period 2 cyle  { +0.4101296722285255 -0.5079485669960778*I ,  +0.4101296722285255 +0.5079485669960778*I } 		  
  z = +0.4101296722285255 +0.5079485669960778*I is in the probably attracting period 2 cyle  { +0.4101296722285255 +0.5079485669960778*I ,  +0.4101296722285255 -0.5079485669960778*I } 		  


  		  

  
*/

const complex double zp2a = 0.4101296722285255 +0.5079485669960778*I ;
const complex double zp2b = 0.1147519899962205 ;

const complex double zpa[2]= { 0.4101296722285255 +0.5079485669960778*I ,  0.4101296722285255 -0.5079485669960778*I };
const complex double zpb[2] = {   0.1147519899962205, 1.6890328811664648  };

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

/* 
  original
   f(z)=1/(z3+dz+c) mit c=0,37 und d=2,1, dargestellt auf [-2,1;2,1]x[-2,1;2,1].

modified 

*/
const complex double a =  2.099609375;
const complex double b = 0.349609375;

// complex function
complex double f(const complex double z0) {

  double complex z = z0;
  complex double z3 = z*z*z;
  
  z = 1.0/(z3 + a*z + b);
  return  z;
}
	

/* 

d(z):=-(3*z^2+2.1)/(z^3+2.1*z+0.37)^2

*/
complex double dfz(const complex double z0) {


  // dz= 
  double complex z = z0;
  complex double z2= z*z;
  complex double z3 = z*z2;
  complex double numerator = -3.0*z2 + 2.1 ;
  complex double denom = z3 + 2.1*z + 0.37;
  denom = denom*denom; // ^2
	
	
  return  numerator/denom;
	
}
		






int is_z_outside(complex double z){

  if (creal(z) >ZxMax ||
      creal(z) <ZxMin ||
      cimag(z) >ZyMax ||
      cimag(z) <ZyMin)
      	{return 1; } // is outside = true
      
    
  return 0; // is inside = false



}









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

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


complex double GiveZ (int ix, int iy)
{
  double Zx = GiveZx (ix);
  double Zy = GiveZy (iy);

  return Zx + Zy * I;




}







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

double cabs2(complex double z){

  return creal(z)*creal(z)+cimag(z)*cimag(z);


}






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












/* 

is it possible to adjust AR so that level curves in interior have figure 8?

find such AR for internal LCM/J and LSM that level curves croses critical point and it's preimages
for attracting ( also weakly attracting = parabolic) dynamics

it may fail 
* if one iteration is bigger then smallest distance between periodic point zp and Julia set
* if critical point is attracted by another cycye ( then change periodic point zp)

Made with help of Claude Heiland-Allen


  attracting radius of circle around finite attractor
  there are 2 basins so  
  
  
  It would have to be done separately in each basin.

A suggested method:

For each critical point, forward iterate to find an attractor and then thin out the critical point set to only one per basin by removing all but one that converge to a common attractor, for each attractor.
For each pixel, calculate a smoothed iteration value (e.g. using the methods in my GVC coloring ucl) and note which basin it is in.
For each critical point in the reduced set, calculate a smoothed iteration value using the same method as in step 2.
For each pixel, subtract from its smoothed iteration value the one found in step 3 for the critical point that shares its basin. Note that the critical point itself, if inside the image rectangle and in a pixel center, will end up with zero and some points may end up with negative values.
The level set boundaries you want will now be the boundaries where the sign or the integer part of the modified smoothed iteration value changes. In particular, the -0.something to +0.something transition will pass through the critical point, the n.something to (n+1).something transitions for nonnegative n will pass through its images, and the same for negative n will pass through its preimages.

  pauldebrot 
  https://fractalforums.org/programming/11/crtical-points-and-level-curves/4323/msg29514#new
  


*/
double GiveTunedAR1(const double iter_Max){

  fprintf(stdout, " GiveTunedAR1\n");

  complex double z = zcr1; // initial point z0 = criical point 
  double iter;
  double r ;//= 10 * PixelWidth; // initial value 
 // double t;
  
  // iterate critical point
	for (iter=0; iter< iter_Max; iter+=1.0 ){
		// check attractor from first basin  
		r = cabs(zp2a - z);
		//if ( t < r)
		//{
      		//	r = t;
		//	break;		//}
		
  	z = f(z); // forward iteration
  	z = f(z);
  
  }
  // check distance between zn = f^n(zcr) and periodic point zp
  
  fprintf(stdout, "AR1  = %f = %d * pixeWidth = %f %% of ImageWidth\n",  r, (int) (r/PixelWidth), r/ (ZxMax-ZxMin));
  
  
 // use it as a AR
 return r;
	
	
}




double GiveTunedAR2(const double iter_Max){

  fprintf(stdout, " GiveTunedAR2\n");

  complex double z = zcr2; // initial point z0 = criical point 
  double iter;
  double r;// = 10 * PixelWidth; // initial value 
  //double t;
  
  // iterate critical point
	for (iter=0; iter< iter_Max; iter+=1.0 ){
		// check attractor from first basin  
		
		// check second basin
		r = cabs(zp2b - z);
		//if (t < r)
		//{
		//	r = t;
		//	break;
		//}
  	
  	z = f(z); // forward iteration
  	z = f(z);
  
  }
  // check distance between zn = f^n(zcr) and periodic point zp
  
   fprintf(stdout, "AR2  = %f = %d * pixeWidth = %f %% of ImageWidth\n",  r, (int) (r/PixelWidth), r/ (ZxMax-ZxMin));
  
  
  
 // use it as a AR
 return r;
	
	
}





// ****************** DYNAMICS = trap tests ( target sets) ****************************


// ???????

int IsInterior(complex double z){

  if (
      cabs2(zp2a-z) < AR1_2 || 
      cabs2(zp2b-z) < AR2_2 
      )
    {return 1;}
  return 0;



}







/*
  2 basins 
  - basin 1 
  - basin 2
  - unknown ( possibly empty set ) 

*/

unsigned char ComputeColorOfFatou_ab (complex double z)
{



	
	
   int i;			// number of iteration
  for (i = 0; i < IterMax; ++i)
    {


		
	// infinity is not superattracting here !!!!!	
      
	
      // 2 Attraction basins 
      if ( cabs2(zp2a-z) < AR1_2 ){ return iColorOfBasin1;}
	 
      if (cabs2(zp2b-z) < AR2_2 ) { return iColorOfBasin2;}			
	
     
      z = f(z);		//  iteration: z(n+1) = f(zn)
	

    }

  
  return iColorOfUnknown;


}


/*
  2 basins 
  
  - - basin 1 
  - - basin 2
  - unknown ( possibly empty set ) 

*/

unsigned char ComputeColorOfFatou_abi (complex double z)
{



	
	
  


  int i;			// number of iteration
  for (i = 0; i < IterMax; ++i)
    {


	/// infinity is not superattracting here !!!!!	
	
  
      //2 Attraction basins 
      if ( cabs2(zp2a-z) < AR1_2 ){ return iColorOfBasin1 - (i % period)*50;}
	 
      if (cabs2(zp2b-z) < AR2_2 ) { return iColorOfBasin2 + (i % period)*50;}			
	
     
      z = f(z);		//  iteration: z(n+1) = f(zn)
	

    }

  
  return iColorOfUnknown;


}






unsigned char ComputeColorOfLSM (complex double z)
{



	
	
  //double r2;


  int i;			// number of iteration
  for (i = 0; i < IterMax_LSM; ++i)
    {


		

      // infinity is not superattracting here !!!!!
      //2 Attraction basins 
      if ( cabs2(zp2a-z) < AR1_2 ){ return iColorOfBasin1- ((2*i) % 255);} 
	 
      if (cabs2(zp2b-z) < AR2_2 ) { return iColorOfBasin2 + ((3*i) % 255);}		
	
      z = f(z);	

    }

  return iColorOfUnknown;


}






unsigned char ComputeColorOfLSM_m (complex double z)
{



	
	int p;
	int pMax = period;

  int i;			// number of iteration
  for (i = 0; i < IterMax_LSM; ++i)
    {


		

     // infinity is not superattracting here !!!!!
      
      
      	
	for (p=0; p < pMax; ++p){ //forward iteration
	
		 //2 Attraction basins 
      		if ( cabs2(zp2a-z) < AR1_2 ){ return iColorOfBasin2 - (2*(i-p) % 255);} 
      		if (cabs2(zp2b-z) < AR2_2 ) { return iColorOfBasin1 + (3*(i-p) % 255);}	
      		z = f(z);	
      		}

    }

  return iColorOfUnknown; 


}











// ***************************************************************************************************************************
// ************************** DEM/J*****************************************
// ****************************************************************************************************************************
/*

here infinity is critical point not a superattracting point
*/


unsigned char ComputeColorOfDEMJ(complex double z){
  // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Julia_set#DEM.2FJ


  
  int nMax = IterMax_DEM;
  complex double dz = 1.0; //  is first derivative with respect to z.
  double distance;
  double cabsz;
	
  int n;

  for (n=0; n < nMax; n++){ //forward iteration
    cabsz = cabs(z);
    //if (cabsz > 1e60 || cabs(dz)> 1e60) { break; }// big values 
    //if (IsInterior(z)) { return iColorOfBasin22;} // falls into finite attractor = interior
  			
    dz = dfz(z)*dz; 
    z = f(z) ; /* forward iteration : complex cubic polynomial */ 
  }
  
  
  distance = 2.0 * cabsz* log(cabsz)/ cabs(dz);
  if (distance <distanceMax) return iColorOfBoundary; // distanceMax = BoundaryWidth*PixelWidth;
  // else
  
  return iColorOfBasin1;

 
}






/* ==================================================================================================
   ============================= Draw functions ===============================================================
   =====================================================================================================
*/ 
unsigned char ComputeColor(FunctionTypeT FunctionType, complex double z){

  unsigned char iColor;
	
	
	
  switch(FunctionType){
  
  case Fatou_ab :{iColor = ComputeColorOfFatou_ab(z); break;}
  	
  case Fatou_abi :{iColor = ComputeColorOfFatou_abi(z); break;}
  
 
  
  case LSM :{iColor = ComputeColorOfLSM(z); break;}
  
  	
  case LSM_m :{iColor = ComputeColorOfLSM_m(z); break;}
  
  case DEM : {iColor = ComputeColorOfDEMJ(z); break;}
	
    /*	
  	case Unknown : {iColor = ComputeColorOfUnknown(z); break;}
		
  	case BD : {iColor = ComputeColorOfBD(z); break;}
		
  	case MBD : {iColor = ComputeColorOfMBD(z); break;}
		
  	case SAC : {iColor = ComputeColorOfSAC(z); break;}
  
  	case DLD : {iColor = ComputeColorOfDLD(z); break;}
		
  	case ND : {iColor = ComputeColorOfND(z); break;}
		
  	case NP : {iColor = ComputeColorOfNP(z); break;}
		
  	case POT : {iColor = ComputeColorOfPOT(z); break;}
		
  	case Blend : {iColor = ComputeColorOfBlend(z); break;}
    */	
  	
  
  	
  	
	
  default: {}
	
	
  }
	
  return iColor;



}


// plots raster point (ix,iy) 
int DrawPoint ( unsigned char A[], FunctionTypeT FunctionType, int ix, int iy)
{
  int i;			/* index of 1D array */
  unsigned char iColor;
  complex double z;


  i = Give_i (ix, iy);		/* compute index of 1D array from indices of 2D array */
  if(i<0 && i> iMax)
	{ return 1;}
  
  z = GiveZ(ix,iy);
  iColor = ComputeColor(FunctionType, z);
  A[i] = iColor ;		// 
  		
  	  
  return 0;
}




int DrawImage ( unsigned char A[], FunctionTypeT FunctionType)
{
  unsigned int ix, iy;		// pixel coordinate 

  fprintf (stderr, "compute image %d \n", FunctionType);
  // for all pixels of image 
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(A, ixMax , iyMax, uUnknown, uInterior, uExterior)
  for (iy = iyMin; iy <= iyMax; ++iy)
    {
      fprintf (stderr, " %d from %d \r", iy, iyMax);	//info 
      for (ix = ixMin; ix <= ixMax; ++ix)
	DrawPoint(A, FunctionType, ix, iy);	//  
    }
  fprintf (stderr, "\n");	//info 
  return 0;
}







int PlotPoint(const complex double z, unsigned char A[]){

	
  unsigned int ix = (creal(z)-ZxMin)/PixelWidth;
  unsigned int iy = (ZyMax - cimag(z))/PixelHeight;
  unsigned int i = Give_i(ix,iy); /* index of _data array */
	
	
  if(i>-1 && i< iMax)
		{A[i]= 0;} //255-A[i];
	
	
  return 0;
	
}






int IsInsideCircle (int x, int y, int xcenter, int ycenter, int r){

	
  double dx = x- xcenter;
  double dy = y - ycenter;
  double d = sqrt(dx*dx+dy*dy);
  if (d<r) {    return 1;}
  return 0;
	  

} 

// Big point = disk 
int PlotBigPoint(const complex double z, double p_size, unsigned char A[]){

	
  unsigned int ix_seed = (creal(z)-ZxMin)/PixelWidth;
  unsigned int iy_seed = (ZyMax - cimag(z))/PixelHeight;
  unsigned int i;
	
	
  if (  is_z_outside(z)) 
  	{fprintf (stdout,"PlotBigPoint :  z= %.16f %+.16f*I is outside\n", creal(z), cimag(z)); return 1;} // do not plot	
	
  /* mark seed point by big pixel */
  int iSide =p_size*iWidth/2000.0 ; /* half of width or height of big pixel */
  int iY;
  int iX;
  for(iY=iy_seed-iSide;iY<=iy_seed+iSide;++iY){ 
    for(iX=ix_seed-iSide;iX<=ix_seed+iSide;++iX){ 
      if (IsInsideCircle(iX, iY, ix_seed, iy_seed, iSide)) {
	i= Give_i(iX,iY); /* index of _data array */
	//if(i>-1 && i< iMax)
		{A[i]= 0;} //255-A[i];
      }
     // else {printf(" bad point \n");}
	
    }}
	
	
  return 0;
	
}



int PlotAllPoints(const complex double zz[], int kMax, double p_size,unsigned char A[]){

  int k;
	
	
  printf("kMax = %d \n",kMax);
	

  for (k = 0; k < kMax; ++k)
    {
      //fprintf(stderr, "z = %+f %+f \n", creal(zz[k]),cimag(zz[k]));
      PlotBigPoint(zz[k], p_size, A);}
  return 0;





}




int DrawForwardOrbit(const complex double z0, const unsigned long long int i_Max, double p_size, unsigned char A[]){
 

  
  unsigned long long int i; /* nr of point of critical orbit */
  complex double z = z0;
  printf("draw forward orbit \n");
 
  PlotBigPoint(z, p_size, A);
  
  /* forward orbit of critical point  */
  for (i=1;i<i_Max ; ++i)
    {
      z  = f(z);
      //if (cabs2(z - z2a) > 2.0) {return 1;} // escaping
      PlotBigPoint(z, p_size/2 , A);
    }
  
    fprintf (stdout,"first point of the orbit z0= %.16f %+.16f*I \n", creal(z0), cimag(z0));
    fprintf (stdout,"last point of the orbit z= %.16f %+.16f*I \n", creal(z), cimag(z));
   
  return 0;
 
}



// ***********************************************************************************************
// ********************** draw line segment ***************************************
// ***************************************************************************************************




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

 /* i =  Give_i(ix,iy) compute index of 1D array from indices of 2D array */
 if (ix >=ixMin && ix<=ixMax && iy >=iyMin && iy<=iyMax )
 	{A[Give_i(ix,iy)] = iColor;}
 	else {fprintf (stdout,"iDrawPoint :   (%d; %d) is outside\n", ix,iy); }

return 0;
}



/*
http://rosettacode.org/wiki/Bitmap/Bresenham%27s_line_algorithm
Instead of swaps in the initialisation use error calculation for both directions x and y simultaneously:
*/
void iDrawLine( int x0, int y0, int x1, int y1, unsigned char iColor, unsigned char A[]) 
{
  int x=x0; int y=y0;
  int dx = abs(x1-x0), sx = x0<x1 ? 1 : -1;
  int dy = abs(y1-y0), sy = y0<y1 ? 1 : -1; 
  int err = (dx>dy ? dx : -dy)/2, e2;

  for(;;){
    iDrawPoint(x, y, iColor, A);
    if (x==x1 && y==y1) break;
    e2 = err;
    if (e2 >-dx) { err -= dy; x += sx; }
    if (e2 < dy) { err += dx; y += sy; }
  }
}




int dDrawLineSegment(double complex Z0, double complex Z1, int color, unsigned char *array) 
{

	double Zx0 = creal(Z0);
	double Zy0 = cimag(Z0);
	double Zx1 = creal(Z1);
	double Zy1 = cimag(Z1);
 	unsigned int ix0, iy0; // screen coordinate = indices of virtual 2D array 
 	unsigned int ix1, iy1; // screen coordinate = indices of virtual 2D array

   	// first step of clipping
   	//if (  Zx0 < ZxMax &&  Zx0 > ZxMin && Zy0 > ZyMin && Zy0 <ZyMax 
    	 // && Zx1 < ZxMax &&  Zx1 > ZxMin && Zy1 > ZyMin && Zy1 <ZyMax )
   	
   	ix0= (Zx0- ZxMin)/PixelWidth; 
   	iy0 = (ZyMax - Zy0)/PixelHeight; // inverse Y axis 
   	ix1= (Zx1- ZxMin)/PixelWidth; 
   	iy1= (ZyMax - Zy1)/PixelHeight; // inverse Y axis 
   	
   	// second step of clipping
   	if (ix0 >=ixMin && ix0<=ixMax && ix0 >=ixMin && ix0<=ixMax && iy0 >=iyMin && iy0<=iyMax && iy1 >=iyMin && iy1<=iyMax )
   		iDrawLine(ix0,iy0,ix1,iy1,color, array) ;

return 0;
}




int DrawAttractors(const complex double zpa[], const complex double zpb[], int kMax, double p_size, unsigned char A[]){

	
	PlotAllPoints(zpa, period, p_size,A);
	dDrawLineSegment(zpa[0], zpa[1],0,A);
	
	PlotAllPoints(zpb, period, p_size, A);
	dDrawLineSegment(zpb[0], zpb[1],0,A);
	

	return 0;

}






// ***********************************************************************************************
// ********************** edge detection usung Sobel filter ***************************************
// ***************************************************************************************************

// from Source to Destination
int ComputeBoundaries(unsigned char S[], unsigned char D[])
{
 
  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 D  array ( global var )
 
  // clear D array
  memset(D, iColorOfBasin1, iSize*sizeof(*D)); // for heap-allocated arrays, where N is the number of elements = FillArrayWithColor(D , iColorOfBasin1);
 
  // printf(" find boundaries in S array using  Sobel filter\n");   
#pragma omp parallel for schedule(dynamic) private(i,iY,iX,Gv,Gh,G) shared(iyMax,ixMax)
  for(iY=1;iY<iyMax-1;++iY){ 
    for(iX=1;iX<ixMax-1;++iX){ 
      Gv= S[Give_i(iX-1,iY+1)] + 2*S[Give_i(iX,iY+1)] + S[Give_i(iX-1,iY+1)] - S[Give_i(iX-1,iY-1)] - 2*S[Give_i(iX-1,iY)] - S[Give_i(iX+1,iY-1)];
      Gh= S[Give_i(iX+1,iY+1)] + 2*S[Give_i(iX+1,iY)] + S[Give_i(iX-1,iY-1)] - S[Give_i(iX+1,iY-1)] - 2*S[Give_i(iX-1,iY)] - S[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) {D[i]=255;} /* background */
      else {D[i]=0;}  /* boundary */
    }
  }
 
   
 
  return 0;
}



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







// FillAllArrayWithColor
//memset (data, 255, sizeof (unsigned char ) * iSize);








// *******************************************************************************************
// ********************************** save A array to pgm file ****************************
// *********************************************************************************************

int SaveArray2PGMFile (unsigned char A[],  char * n, char *comment)
{

  FILE *fp;
  const unsigned int MaxColorComponentValue = 255;	/* color component is coded from 0 to 255 ;  it is 8 bit color file */
  char name[100];		/* name of file */
  snprintf (name, sizeof name, "%.1f_%d_%s", radius, iHeight, n );	/* radius and iHeght are global variables */
  char *filename = strcat (name, ".pgm");
  char long_comment[200];
  sprintf (long_comment, "Julia set f(z) := 1/(z^3 + a*z + b)  Location by Michael Becker %s", comment);





  // save image array 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", long_comment, iWidth, iHeight, MaxColorComponentValue);	// write header to the file
  size_t rSize = fwrite (A, sizeof(A[0]), iSize, fp);	// write whole array with image data bytes to the file in one step 
  fclose (fp);

  // info 
  if ( rSize == iSize) 
    {
      printf ("File %s saved ", filename);
      if (long_comment == NULL || strlen (long_comment) == 0)
	printf ("\n");
      else { printf (". Comment = %s \n", long_comment); }
    }
  else {printf("wrote %zu elements out of %llu requested\n", rSize,  iSize);}

  return 0;
}




int PrintCInfo ()
{

  printf ("gcc version: %d.%d.%d\n", __GNUC__, __GNUC_MINOR__, __GNUC_PATCHLEVEL__);	// https://stackoverflow.com/questions/20389193/how-do-i-check-my-gcc-c-compiler-version-for-my-eclipse
  // OpenMP version is displayed in the console : export  OMP_DISPLAY_ENV="TRUE"

  printf ("__STDC__ = %d\n", __STDC__);
  printf ("__STDC_VERSION__ = %ld\n", __STDC_VERSION__);
  printf ("c dialect = ");
  switch (__STDC_VERSION__)
    {				// the format YYYYMM 
    case 199409L:
      printf ("C94\n");
      break;
    case 199901L:
      printf ("C99\n");
      break;
    case 201112L:
      printf ("C11\n");
      break;
    case 201710L:
      printf ("C18\n");
      break;
      //default : /* Optional */

    }

  return 0;
}


int
PrintProgramInfo ()
{


  // display info messages
  printf ("Numerical approximation of Julia set for F(z) =  ) \n");
  //printf ("parameter C = ( %.16f ; %.16f ) \n", creal (C), cimag (C));
  

  printf ("Image Width = %f in world coordinate\n", ZxMax - ZxMin);
  printf ("PixelWidth = %.16f \n", PixelWidth);
  //printf ("AR = %.16f = %f *PixelWidth = %f %% of ImageWidth \n", AR, AR / PixelWidth, AR / ZxMax - ZxMin);



  // image corners in world coordinate
  // center and radius
  // center and zoom
  // GradientRepetition
  printf ("Maximal number of iterations = iterMax = %d \n", IterMax);
  printf ("ratio of image  = %f ; it should be 1.000 ...\n", ratio);
  //




  return 0;
}



int SetPlane(complex double center, double radius, double a_ratio){

  ZxMin = creal(center) - radius*a_ratio;	
  ZxMax = creal(center) + radius*a_ratio;	//0.75;
  ZyMin = cimag(center) - radius;	// inv
  ZyMax = cimag(center) + radius;	//0.7;
  return 0;

}



// Check Orientation of z-plane image : mark first quadrant of complex plane 
// it should be in the upper right position
// uses global var :  ...
int CheckZPlaneOrientation(unsigned char A[] )
{
 
  double Zx, Zy; //  Z= Zx+ZY*i;
  unsigned i; /* index of 1D array */
  unsigned int ix, iy;		// pixel coordinate 
	
  fprintf(stderr, "compute image CheckOrientation\n");
  // for all pixels of image 
#pragma omp parallel for schedule(dynamic) private(ix,iy, i, Zx, Zy) shared(A, ixMax , iyMax) 
  for (iy = iyMin; iy <= iyMax; ++iy){
    //fprintf (stderr, " %d from %d \r", iy, iyMax);	//info 
    for (ix = ixMin; ix <= ixMax; ++ix){
      // from screen to world coordinate 
      Zy = GiveZy(iy);
      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;
}







// *****************************************************************************
//;;;;;;;;;;;;;;;;;;;;;;  setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
// **************************************************************************************

int setup ()
{

  fprintf (stderr, "setup start\n");






  /* 2D array ranges */

  iWidth = iHeight* DisplayAspectRatio ;
  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].

  
  SetPlane( center, radius,  DisplayAspectRatio );	
  /* Pixel sizes */
  PixelWidth = (ZxMax - ZxMin) / ixMax;	//  ixMax = (iWidth-1)  step between pixels in world coordinate 
  PixelHeight = (ZyMax - ZyMin) / iyMax;
  ratio = ((ZxMax - ZxMin) / (ZyMax - ZyMin)) / ((double) iWidth / (double) iHeight);	// it should be 1.000 ...
  
    
  
  // LSM  
  // escape radius ( of circle around infinity 
 // ER = 200.0; // 
  //ER2 = ER*ER;
  
  
  /* 
  
  attracting radius of circle arounf finite attractor
  there are 2 basins so 2 
  
  
  It would have to be done separately in each basin.

A suggested method:

For each critical point, forward iterate to find an attractor and then thin out the critical point set to only one per basin by removing all but one that converge to a common attractor, for each attractor.
For each pixel, calculate a smoothed iteration value (e.g. using the methods in my GVC coloring ucl) and note which basin it is in.
For each critical point in the reduced set, calculate a smoothed iteration value using the same method as in step 2.
For each pixel, subtract from its smoothed iteration value the one found in step 3 for the critical point that shares its basin. Note that the critical point itself, if inside the image rectangle and in a pixel center, will end up with zero and some points may end up with negative values.
The level set boundaries you want will now be the boundaries where the sign or the integer part of the modified smoothed iteration value changes. In particular, the -0.something to +0.something transition will pass through the critical point, the n.something to (n+1).something transitions for nonnegative n will pass through its images, and the same for negative n will pass through its preimages.

  pauldebrot 
  https://fractalforums.org/programming/11/crtical-points-and-level-curves/4323/msg29514#new
  
  
  AR_max = 5*PixelWidth*iWidth/2000.0 ; // adjust first number 
   GiveTunedAR(const int i_Max, const complex double zcr, const double c, const double zp){
  AR = 0.5*PixelWidth; // 0.03; // 10*0.0006 = 0.006
  */
  AR1 = GiveTunedAR1(10); 
  AR1_2 = AR1 * AR1;
  //
  AR2 = GiveTunedAR2(10); 
  AR2_2 = AR2 * AR2;
  
  //AR12 = AR/2.0;
  
  
  
  
  
  // DEM
  BoundaryWidth = 0.5*iWidth/2000.0  ; //  measured in pixels ( when iWidth = 2000) 
  distanceMax = BoundaryWidth*PixelWidth;



  /* create dynamic 1D arrays for colors ( shades of gray ) */
  data = malloc (iSize * sizeof (unsigned char));

  edge = malloc (iSize * sizeof (unsigned char));
  if (data == NULL || edge == NULL)
    {
      fprintf (stderr, " Could not allocate memory");
      return 1;
    }
  




 


  fprintf (stderr, " end of setup \n");

  return 0;

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




int end ()
{


  fprintf (stderr, " allways free memory (deallocate )  to avoid memory leaks \n");	// https://en.wikipedia.org/wiki/C_dynamic_memory_allocation
  free (data);
  free(edge);


  PrintProgramInfo ();
  PrintCInfo ();
  return 0;

}

// ********************************************************************************************************************
/* -----------------------------------------  main   -------------------------------------------------------------*/
// ********************************************************************************************************************

int main ()
{
  setup ();
  
  
   

   
    DrawImage (data, Fatou_ab);	 
    SaveArray2PGMFile (data,  "Fatou_ab" , "Fatou_ab ");
  
    DrawImage (data, Fatou_abi);	 
    SaveArray2PGMFile (data,  "Fatou_abi" , "Fatou_abi ");
    
    
    ComputeBoundaries(data,edge);
    SaveArray2PGMFile (edge,  "Fatou_abi_LCM" , "Fatou_abi_LCM ");
    
    CopyBoundaries(edge, data);
    SaveArray2PGMFile (data,  "Fatou_abi_LSCM" , "Fatou_abi_LSCM");
    
    
    DrawAttractors(zpa, zpb, 2, 14, data);
    SaveArray2PGMFile (data,  "Fatou_abi_LSCM_zp" , "Fatou_abi_LSCM_zp");
    
    DrawForwardOrbit(zcr1, 100, 8, data);
    DrawForwardOrbit(zcr2, 100, 8, data);
    SaveArray2PGMFile (data,  "Fatou_abi_LSCM_zp_cr" , "Fatou_abi_LSCM_zp_cr");
    
   
    
   
    
  
  
    DrawImage (data, LSM);	
    SaveArray2PGMFile (data,  "LSM" , "LSM");
  
    ComputeBoundaries(data,edge);
    SaveArray2PGMFile (edge,  "LCM" , "LCM ");
    
    CopyBoundaries(edge, data);
    SaveArray2PGMFile (data,  "LSCM" , "LSCM");
    
    
    PlotBigPoint(zcr1,8,edge);
    PlotBigPoint(zcr2,8, edge);
    SaveArray2PGMFile (edge,  "LCM_cr" , "LCM + critical points");
    
    DrawAttractors(zpa, zpb, 2, 14, edge);
    SaveArray2PGMFile (edge,  "LCM_zp_cr" , "LCM + critical points");
    
    DrawForwardOrbit(zcr1, 200, 8, edge);
    DrawForwardOrbit(zcr2, 200, 8,  edge);
    SaveArray2PGMFile (edge,  "LCM_zp_cr_cro" , "LCM + critical orbit + periodic points");
    
       
    /*  
    DrawImage (data, LSM_m);	 
    SaveArray2PGMFile (data,  "LSM_m" , "LSM_m ");
  
    ComputeBoundaries(data,edge);
    SaveArray2PGMFile (edge,  "LCM_m" , "LCM_m ");
  
    CopyBoundaries(edge, data);
    SaveArray2PGMFile (data,  "LSCM_m" , "LSCM m");

  
   DrawImage (data, DEM);	// first 
    SaveArray2PGMFile (data,  "DEM" , "DEM ");
    */

  
  end ();

  return 0;
}