File:Julia set f(z)=1 over az5+z3+bz.png
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Summary
| Description |
English: Julia set f(z)=1/((0,15+0,15i)z5+z3+(-3+3i)z) on [-3;3]x[-3;3].
Location by Michael Becker[1]. Analysis by marcm200 and xenodreambuie[2]. Help of Claude Heiland-Allen. The Julia set (boundary of filled-in Julia set) itself is not drawn: we see it as the locus of points where the level curves are especially close to each other = a place with high density of level curves. {0, infinity} is superattractive period 2 cycle in the exterior. In the interior there are two period-4 cycles:
Deutsch: f(z)=1/((0,15+0,15i)z5+z3+(-3+3i)z), dargestellt auf [-3;3]x[-3;3]. |
| Date | |
| Source | Own work |
| Author | Adam majewski |
Licensing
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c source code
/*
https://web.archive.org/web/20161024194536/http://www.ijon.de/mathe/julia/some_julia_sets_3.html
a: (0,15+0,15i)
b: (-3+3i)
f: 1/(a*z^5+ b*z^3 + z);
dz = -(5*a*z^4+3*b*z^2+1)/(a*z^5+b*z^3+z)^2
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 = 0, IntLSM =1 , ExtLSM = 2, LSM = 3, DEM = 4, Unknown = 5 , BD = 6, MBD = 7 , SAC = 8, DLD = 9, ND = 10 , NP= 11, POT = 12 , Blend = 13, White= 14, Fatou_ab = 15, Fatou_abi = 16, LSM_m = 17
} 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 = 20000; //
// 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
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
// see SetPlane
double radius = 3.0;
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 AR; // bigger values do not works
double AR2;
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
*/
unsigned char iColorOfExterior = 245;
unsigned char iColorOfInterior = 99;
unsigned char iColorOfInterior1 = 100;
unsigned char iColorOfInterior2 = 183;
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
// critical points
complex double zcr1; //
complex double zcr2;// = -2.2351741790771484375e-08+9.4296410679817199707e-09*I;
complex double zcr3; //
complex double zcr4;//
const complex double z_cr[4]= {0.389374737779177*I-0.9400337727919567,
0.9400337727919567-0.389374737779177*I,
-1.816005797447402*I-0.7522142306508819,
1.816005797447402*I+0.7522142306508819};
const int period = 4;
// periodic points = attractors
//complex double z1 = 0.0 ; //fixed point (period 1) = attracting cycle
/*
attracting periodic points :
2 period 4 cycles found by marcm200
https://fractalforums.org/fractal-mathematics-and-new-theories/28/rational-function/4279/msg29227#msg29227
*/
const complex double zp4a[4]= { -0.753562725059588878195881989086-1.81926135093768714945383635495*I, 0.110964246025498911030204851613-0.0459628956422664519676501981849*I , 0.0951541874285335154137754898329-0.0394141549494900420014253938916*I, -0.877971698675046430260238139454-2.11961118232104128722426139575*I};
const complex double zp4b[4]= {0.753562725059588878195881989086+1.81926135093768714945383635495*I, -0.0951541874285335154137754898329+0.0394141549494900420014253938916*I, 0.877971698675046430260238139454+2.11961118232104128722426139575*I, -0.110964246025498911030204851613+0.0459628956422664519676501981849*I};
const complex double zp4a0 = -0.753562725059588878195881989086-1.81926135093768714945383635495*I;
const complex double zp4b0 = 0.753562725059588878195881989086+1.81926135093768714945383635495*I;
/* ------------------------------------------ functions -------------------------------------------------------------*/
/*
a: (0.15+0.15*%i);
b: (-3+3*%i);
f: 1/(a*z^5+ b*z^3 + z);
dz = -(5*a*z^4+3*b*z^2+1)/(a*z^5+b*z^3+z)^2
*/
const complex double a = 0.15 + 0.15*I;
const complex double b = -3.0 + 3.0*I;
// complex function
complex double f(const complex double z0) {
double complex z = z0;
complex double z2 = z*z;
complex double z3 = z2*z;
complex double z5 = z3*z2;
z = 1.0/(a*z5 + z3 + b*z);
return z;
}
//
complex double dfz(const complex double z0) {
// dz= -(5*a*z^4+3*z^2+b) / (a*z^5+z^3+b*z)^2
double complex z = z0;
complex double z2= z*z;
complex double z4 = z2*z2;
complex double numerator = -5.0*a*z4 - 3.0*z2 - b ;
complex double denom = a*z4*z + z2*z + b*z;
denom = denom*denom; // ^2
return numerator/denom;
}
// 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
*/
double GiveTunedAR(const int iter_Max){
fprintf(stdout, " GiveTunedAR\n");
complex double z = zcr1; // initial point z0 = criical point
int iter;
double r = 50 * PixelWidth; // initial value
double t;
// iterate critical point
for (iter=0; iter< iter_Max; ++iter ){
// check attractor from first basin
t = cabs(zp4a0 - z);
if ( t < r)
{
r = t;
break;
}
// check second basin
t = cabs(zp4b0 - z);
if (t < r)
{
r = t;
break;
}
z = f(z); // forward iteration
}
// check distance between zn = f^n(zcr) and periodic point zp
fprintf(stdout, " r = %f = %d * pixeWidth \n", r, (int) (r/PixelWidth));
// use it as a AR
return r;
}
// ****************** DYNAMICS = trap tests ( target sets) ****************************
// ???????
int IsInterior(complex double z){
if (
cabs2(zp4a0-z) < AR2 ||
cabs2(zp4b0-z) < AR2
)
{return 1;}
return 0;
}
/*
3 basins:
- exterior
- interior
- unknown ( possibly empty set )
*/
unsigned char ComputeColorOfFatou (complex double z)
{
double r2;
int i; // number of iteration
for (i = 0; i < IterMax; ++i)
{
r2 =cabs2(z);
if (r2 > ER2) // esaping = exterior
{
uExterior += 1;
return iColorOfExterior;
}
if (IsInterior(z)) //
{
return iColorOfInterior;
}
z = f(z); // iteration: z(n+1) = f(zn)
}
return iColorOfUnknown;
}
/*
3 basins
- exterior
- interior ( consist of 2 attracting basins)
- - basin 1
- - basin 2
- unknown ( possibly empty set )
*/
unsigned char ComputeColorOfFatou_ab (complex double z)
{
double r2;
int i; // number of iteration
for (i = 0; i < IterMax; ++i)
{
r2 =cabs2(z);
if (r2 > ER2) // esaping = exterior
{
uExterior += 1;
return iColorOfExterior;
}
//Interior = 2 Attraction basins
if ( cabs2(zp4a0-z) < AR2 ){ return iColorOfInterior1;}
if (cabs2(zp4b0-z) < AR2 ) { return iColorOfInterior2;}
z = f(z); // iteration: z(n+1) = f(zn)
}
return iColorOfUnknown;
}
/*
3 basins
- exterior
- interior ( consist of 2 attracting basins) each basin is colored according to position in the cycle
- - basin 1
- - basin 2
- unknown ( possibly empty set )
*/
unsigned char ComputeColorOfFatou_abi (complex double z)
{
double r2;
int i; // number of iteration
for (i = 0; i < IterMax; ++i)
{
r2 =cabs2(z);
if (r2 > ER2) // esaping = exterior
{
uExterior += 1;
return iColorOfExterior;
}
//Interior = 2 Attraction basins
if ( cabs2(zp4a0-z) < AR2 ){ return iColorOfInterior1 - (i % period)*20;}
if (cabs2(zp4b0-z) < AR2 ) { return iColorOfInterior2 - (i % period)*19;}
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)
{
// complex iteration f(z)=z^3 + c
r2 =cabs2(z);
if (r2 > ER2) // esaping = exterior
{
uExterior += 1;
return 255- (i % 255);
}
if (IsInterior(z)) //
{
return 255- ((2*i) % 255);
}
z = f(z);
}
return iColorOfUnknown;
}
unsigned char ComputeColorOfLSM_m (complex double z)
{
double r2;
int i; // number of iteration
for (i = 0; i < IterMax_LSM; ++i)
{
// complex iteration f(z)=z^3 + c
r2 =cabs2(z);
if (r2 > ER2) // esaping = exterior
{
uExterior += 1;
return 255- (i % 255);
}
if (IsInterior(z)) //
{
return (i % 255);
}
z = f(z);
}
return iColorOfUnknown; //iColorOfUnknown;
}
// ***************************************************************************************************************************
// ************************** DEM/J*****************************************
// ****************************************************************************************************************************
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 iColorOfInterior2;} // 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 iColorOfExterior;
}
/* ==================================================================================================
============================= Draw functions ===============================================================
=====================================================================================================
*/
unsigned char ComputeColor(FunctionTypeT FunctionType, complex double z){
unsigned char iColor;
switch(FunctionType){
case Fatou :{iColor = ComputeColorOfFatou(z); break;}
case Fatou_ab :{iColor = ComputeColorOfFatou_ab(z); break;}
case Fatou_abi :{iColor = ComputeColorOfFatou_abi(z); break;}
// case IntLSM :{iColor = ComputeColorOfIntLSM(z); break;}
// case ExtLSM :{iColor = ComputeColorOfExtLSM(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;}
*/
case White : {iColor = 255;}
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 */
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 */
A[i]= 0; //255-A[i]; // Mark point with inveres color
return 0;
}
int IsInside (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;
}
int PlotBigPoint(const complex double z, unsigned char A[]){
unsigned int ix_seed = (creal(z)-ZxMin)/PixelWidth;
unsigned int iy_seed = (ZyMax - cimag(z))/PixelHeight;
unsigned int i;
/* mark seed point by big pixel */
int iSide =4.0*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 (IsInside(iX, iY, ix_seed, iy_seed, iSide)) {
i= Give_i(iX,iY); /* index of _data array */
A[i]= 0; //255-A[i];
}
else {printf(" bad point \n");}
}}
return 0;
}
int PlotAllPoints(const complex double zz[], int kMax, 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], A);}
return 0;
}
int DrawForwardOrbit(complex double z, unsigned long long int iMax, unsigned char A[] )
{
unsigned long long int i; /* nr of point of critical orbit */
printf("draw forward orbit \n");
PlotBigPoint(z, A);
/* forward orbit of critical point */
for (i=1;i<iMax ; ++i)
{
z = f(z);
//if (cabs2(z - z2a) > 2.0) {return 1;} // escaping
PlotBigPoint(z, 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, iColorOfExterior, iSize*sizeof(*D)); // for heap-allocated arrays, where N is the number of elements = FillArrayWithColor(D , iColorOfExterior);
// 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;
}
// *******************************************************************************************
// ********************************** 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, "%s", n ); /* */
char *filename = strcat (name, ".pgm");
char long_comment[200];
sprintf (long_comment, "Julia set f(z)=1/((0,15+0,15i)z5+z3+(-3+3i)z) on [-3;3]x[-3;3]. 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) = z*c) \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 ...
// critical points
zcr1 = z_cr[0];
zcr2 = z_cr[1];
zcr3 = z_cr[2];
zcr4 = z_cr[3];
// LSM
// escape radius ( of circle around infinity
ER = 200.0; //
ER2 = ER*ER;
// attracting radius of ciurcel arounf finite attractor
//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 = 50*PixelWidth; // 0.03; // 10*0.0006 = 0.006
AR = GiveTunedAR(10000);
AR2 = AR * AR;
//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);
SaveArray2PGMFile (data, "Fatou" , "Fatou ");
ComputeBoundaries(data,edge);
SaveArray2PGMFile (edge, "LCM_Fatou" , "LCM of Fatou ");
DrawImage (data, Fatou_ab);
SaveArray2PGMFile (data, "Fatou_ab" , "Fatou_ab ");
DrawImage (data, Fatou_abi);
SaveArray2PGMFile (data, "Fatou_abi" , "Fatou_abi ");
*/
DrawImage (data, LSM); // first
//SaveArray2PGMFile (data, "LSM" , "LSM");
ComputeBoundaries(data,edge);
SaveArray2PGMFile (edge, "LCM" , "LCM ");
DrawForwardOrbit(zcr1, 200, edge);
SaveArray2PGMFile (edge, "LCM_cr" , "LCM and critical orbit");
/*
CopyBoundaries(edge, data);
SaveArray2PGMFile (data, "LSCM" , "LSCM");
PlotAllPoints(z_cr, 4, data);
SaveArray2PGMFile (data, "Fatou_cr" , "Fatou_cr");
DrawImage (data, Fatou);
PlotBigPoint(zp4a0, data);
PlotBigPoint(zp4b0, data);
SaveArray2PGMFile (data, "Fatou_zp4" , "Fatou_zp4");
DrawImage (data, LSM_m); // first
SaveArray2PGMFile (data, "LSM_m2" , "LSM_m ");
ComputeBoundaries(data,edge);
SaveArray2PGMFile (edge, "LCM_m2_LS" , "LCM_m ");
CopyBoundaries(edge, data);
SaveArray2PGMFile (data, "LSCM_mm2" , "LSCM mm");
DrawImage (data, DEM); // first
SaveArray2PGMFile (data, "DEM" , "DEM ");
*/
end ();
return 0;
}
bash source code
#!/bin/bash
# script file for BASH
# which bash
# save this file as d.sh
# chmod +x d.sh
# ./d.sh
# checked in https://www.shellcheck.net/
printf "make pgm files \n"
gcc d.c -lm -Wall -march=native -fopenmp
if [ $? -ne 0 ]
then
echo ERROR: compilation failed !!!!!!
exit 1
fi
export OMP_DISPLAY_ENV="TRUE"
printf "display OMP info \n"
time ./a.out > a.txt
export OMP_DISPLAY_ENV="FALSE"
printf "convert all pgm files to png using Image Magic convert \n"
# for all pgm files in this directory
for file in *.pgm ; do
# b is name of file without extension
b=$(basename "$file" .pgm)
# convert using ImageMagic
convert "${b}".pgm -resize 2000x2000 "${b}".png
echo "$file"
done
printf "delete all pgm files \n"
rm ./*.pgm
echo OK
# end
makefile
all:
chmod +x d.sh
./d.sh
Tu run the program simply
make
One can also uncomment some procedures in main functions to see more images
references
- ↑ Some Julia sets 3 by Michael Becker, 8/2003. Last modification: 8/2003.
- ↑ fractalforums.org : rational-function
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 18:50, 25 June 2021 | | 2,000 × 2,000 (844 KB) | Soul windsurfer | level curves cross at critical point |
| 20:03, 24 June 2021 |  | 2,000 × 2,000 (843 KB) | Soul windsurfer | comment | |
| 19:32, 24 June 2021 |  | 2,000 × 2,000 (843 KB) | Soul windsurfer | Uploaded own work with UploadWizard |
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