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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
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
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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