File:Cauliflower Julia set DLD.png
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Summary
editDescriptionCauliflower Julia set DLD.png |
English: Cauliflower Julia set DLD. Algorithm : Discrete Lagrangian Descriptors (DLD) by Víctor J. García-Garrido[1] |
Date | |
Source | Own work with help of pauldelbrot[2] |
Author | Adam majewski |
Other versions |
|
Licensing
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c src code
editNote that this code producec 2 files , each has : 20000x20000 pixels = 400 MB !!! If one wants smaller files change line 87 :
static unsigned int iHeight = 20000; //
to smaller values like:
static unsigned int iHeight = 1000; //
/*
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 ========================
DrawImageOfX -> DrawPointOfX -> ComputeColorOfX
first 2 functions are identical for every X
check only last function = ComputeColorOfX
which computes color of one pixel !
==========================================
---------------------------------
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
=======================
# gnuplot "i.plt"
set terminal svg enhanced background rgb 'white'
set xlabel "re(z)"
set ylabel "DLD"
set title "Relation between z and DLD in the interior of Julia set for c = -1"
set output "interior.svg"
plot "i.txt" with lines
----------------------
*/
#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
/* --------------------------------- global variables and consts ------------------------------------------------------------ */
// 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 int iSize; // = iWidth*iHeight;
// memmory 1D array
unsigned char *data;
unsigned char *edge;
unsigned char *bin;
// 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
static const double ZxMin = -1.2; //-0.05;
static const double ZxMax = 1.2; //0.75;
static const double ZyMin = -1.2; //-0.1;
static const double ZyMax = 1.2; //0.7;
static double PixelWidth; // =(ZxMax-ZxMin)/ixMax;
static double PixelHeight; // =(ZyMax-ZyMin)/iyMax;
static double ratio;
// complex numbers of parametr plane
double complex c = 0.25; // parameter of function fc(z)=z^2 + c
double complex zf = 0.5;
double ER = 1e60;
double AR = 1e-20; //1e-0;
const int N = 1000; // fixed number : maximal number of iterations
double p = 0.015625; // 1/64 // 0.25;
double m = 8.0;
int iInterior = 0;
int iExterior = 0;
/* colors = shades of gray from 0 to 255 */
unsigned char iColorOfExterior = 250;
unsigned char iColorOfInterior = 200;
unsigned char iColorOfBoundary = 0;
/* ------------------------------------------ functions -------------------------------------------------------------*/
//------------------complex numbers -----------------------------------------------------
// 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;
}
// ****************** DYNAMICS = trap tests ( target sets) ****************************
/* ----------- 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;
}
// ***********************************************************************************************
// ********************** 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;
}
// ***************************************************************************************************************************
// ************************** DLD/J*****************************************
// ****************************************************************************************************************************
/* partial pnorm
input: z , zn = f(z), p
output ppn
*/
double ppnorm( complex double z, complex double zn, double p){
double s[2][3]; // array for 2 points on the Riemann sphere
int j;
double d; // denominator
double x;
double y;
double ds;
double ppn = 0.0;
// map from complex plane to riemann sphere
// z
x = creal(z);
y = cimag(z);
d = x*x + y*y + 1.0;
s[0][0] = (2.0*x)/d;
s[0][1] = (2.0*y)/d;
s[0][2] = (d-2.0)/d; // (x^2 + y^2 - 1)/d
// zn
x = creal(zn);
y = cimag(zn);
d = x*x + y*y + 1.0;
s[1][0] = (2.0*x)/d;
s[1][1] = (2.0*y)/d;
s[1][2] = (d-2.0)/d; // (x^2 + y^2 - 1)/d
// sum
for (j=0; j <3; ++j){
ds = fabs(s[1][j] - s[0][j]);
// normal: neither zero, subnormal, infinite, nor NaN
//if (fpclassify (ds) !=FP_INFINITE)
//if (isnormal(ds))
// it is solved by if (cabs(z) > 1e60 ) break; procedure in parent function
ppn += pow(ds,p); // |ds|^p
// else {ppn = 10000.0; printf("ds = infty\t");} //
}
return ppn;
}
// DLD = Discret Lagrangian Descriptior
double lagrangian( complex double z0, complex double c, int iMax, double p ){
int i; // number of iteration
double d = 0.0; // DLD = sum
double ppn; // partial pnorm
complex double z = z0;
complex double zn; // next z
//if (cabs(z) < AR || cabs(z +1)< AR) return 5.0; // for z= 0.0 d = inf
for (i=0; i<iMax; ++i){
zn = z*z +c; // complex iteration
ppn = ppnorm(z, zn, p);
d += ppn; // sum
//
z = zn;
//if (! isnormal(d)) { return 0.0; } // not works
if (cabs(z) > ER ) {
iExterior +=1;
d = -d;
break; // exterior : big values produces NAN error in ppnorm computing
}
// it not works ????
if (cabs(z -zf) < AR )
{ // interior
iInterior +=1;
d = -d;
break;
}
}
d = d/((double)i); // averaging
if (d<0.0) {// exterior
d = -d;
}
else {d = m* d;}
d = d - (int)d; // fractional part}
return d;
}
unsigned char ComputeColorOfDLD(complex double z){
//double cabsz;
int iColor;
double d;
d = lagrangian(z,c, N,p);
iColor = (int)(d*255.0) % 255; // nMax or lower walues in denominator
return (unsigned char) iColor;
}
// plots raster point (ix,iy)
int DrawPointOfDLD (unsigned char A[], 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 = ComputeColorOfDLD(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfDLD (unsigned char A[])
{
unsigned int ix, iy; // pixel coordinate
//printf("compute image \n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(A, ixMax , iyMax)
for (iy = iyMin; iy <= iyMax; ++iy){
printf (" %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfDLD(A, ix, iy); //
}
return 0;
}
// test how values changes to tune color
int test_interior(){
// choose 2 points such that color is changing the most
complex double z = zf;
complex double z2 = 0.5*I;
int iMax = 20;
complex double dz = (zf- z2)/iMax;
printf("dz = %.16f ; %.16f\n", creal(dz), cimag(dz));
int i;
printf("z = %.16f ; %.16f\n", creal(z), cimag(z));
printf("# z d\n"); // gnuplot
for (i=0; i<iMax; ++i){
double d = lagrangian(z, c, N, p);
//int iColor = ComputeColorOfDLD(z);
// printf(" z = %.16f d = %.16f color = %d \n",creal(z), d, iColor);
printf("%d %.16f %.16f %.16f\t %d\n",i , creal(z), cimag(z),d, (int)(d*255.0) % 255); // gnuplot
z = z - dz;
}
//
double d0 = lagrangian(zf, c, N, p);
double db = lagrangian(z2, c, N, p);
double dd = d0 - db;
printf("d0 - db = %.16f - %.16f = %.16f\n",d0, db, dd);
return 0;
}
// test how values changes to tune color
int test_exterior(){
complex double z;
complex double z0 = zf;
complex double z1 = 3.0;
complex double dz = cabs(z1 - z0)/20;
z = z0;
printf("# z d color\n"); // gnuplot
while (creal(z) < creal(z1)){
double d = lagrangian(z, c, N, p);
//int iColor = ComputeColorOfDLD(z);
// printf(" z = %.16f d = %.16f color = %d \n",creal(z), d, iColor);
printf(" %.16f\t %.16f \t%d\n",creal(z), d, (int)(d*255.0) % 255); // gnuplot
z += dz;
}
//
double d0 = lagrangian(z0, c, N, p);
double d1 = lagrangian(z1, c, N, p);
double dd = d0 - d1;
printf("d0 - d1 = %.16f - %.16f = %.16f\n",d0, d1, dd);
return 0;
}
// *******************************************************************************************
// ********************************** save A array to pgm file ****************************
// *********************************************************************************************
int SaveArray2PGMFile( unsigned char A[], double k, 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, "%.3f", k); /* */
char *filename =strncat(name,".pgm", 4);
// save image to the pgm file
fp= fopen(filename,"wb"); // create new file,give it a name and open it in binary mode
fprintf(fp,"P5\n # %s\n %u %u\n %u\n", comment, iWidth, iHeight, MaxColorComponentValue); // write header to the file
fwrite(A,iSize,1,fp); // write array with image data bytes to the file in one step
fclose(fp);
// info
printf("File %s saved ", filename);
if (comment == NULL || strlen(comment) ==0)
printf("\n");
else printf (". Comment = %s \n", comment);
return 0;
}
int PrintInfoAboutProgam()
{
// display info messages
printf ("Numerical approximation of Julia set for fc(z)= z^2 + c \n");
//printf ("iPeriodParent = %d \n", iPeriodParent);
//printf ("iPeriodOfChild = %d \n", iPeriodChild);
printf ("parameter c = ( %.16f ; %.16f ) \n", creal(c), cimag(c));
printf ("Image Width = %f in world coordinate\n", ZxMax - ZxMin);
printf ("PixelWidth = %f \n", PixelWidth);
printf("iInterior = %d \n", iInterior);
printf("iExterior = %d \n", iExterior);
// image corners in world coordinate
// center and radius
// center and zoom
// GradientRepetition
printf ("Maximal number of iterations = iterMax = %d \n", N);
printf ("ratio of image = %f ; it should be 1.000 ...\n", ratio);
//
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 diplayed in the console
return 0;
}
// *****************************************************************************
//;;;;;;;;;;;;;;;;;;;;;; setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
// **************************************************************************************
int setup ()
{
printf ("setup start\n");
/* 2D array ranges */
iWidth = iHeight;
iSize = iWidth * iHeight; // size = number of points in array
// iy
iyMax = iHeight - 1; // Indexes of array starts from 0 not 1 so the highest elements of an array is = array_name[size-1].
//ix
ixMax = iWidth - 1;
/* 1D array ranges */
// i1Dsize = i2Dsize; // 1D array with the same size as 2D array
iMax = iSize - 1; // Indexes of array starts from 0 not 1 so the highest elements of an array is = array_name[size-1].
/* Pixel sizes */
PixelWidth = (ZxMax - ZxMin) / ixMax; // ixMax = (iWidth-1) step between pixels in world coordinate
PixelHeight = (ZyMax - ZyMin) / iyMax;
ratio = ((ZxMax - ZxMin) / (ZyMax - ZyMin)) / ((double) iWidth / (double) iHeight); // it should be 1.000 ...
/* create dynamic 1D arrays for colors ( shades of gray ) */
data = malloc (iSize * sizeof (unsigned char));
edge = malloc (iSize * sizeof (unsigned char));
bin = malloc (iSize * sizeof (unsigned char));
if (data == NULL || edge == NULL || bin == NULL){
fprintf (stderr, " Could not allocate memory");
return 1;
}
printf (" end of setup \n");
return 0;
} // ;;;;;;;;;;;;;;;;;;;;;;;;; end of the setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
int end(){
printf (" allways free memory (deallocate ) to avoid memory leaks \n"); // https://en.wikipedia.org/wiki/C_dynamic_memory_allocation
free (data);
free(edge);
free(bin);
PrintInfoAboutProgam();
return 0;
}
// ********************************************************************************************************************
/* ----------------------------------------- main -------------------------------------------------------------*/
// ********************************************************************************************************************
int main () {
setup ();
DrawImagerOfDLD(data);
SaveArray2PGMFile (data, iWidth+m+p, "DLD/J");
ComputeBoundaries(data, edge);
SaveArray2PGMFile (edge, iWidth+100+m+p, "boundaries of DLD/J");
test_exterior();
test_interior();
end();
return 0;
}
txt output
editgcc d.c -lm -Wall -march=native -fopenmp ./a.out setup start end of setup File 20008.016.pgm saved . Comment = DLD/J File 20108.016.pgm saved . Comment = boundaries of DLD/J # z d color 0.5000000000000000 -nan -128 0.6250000000000000 0.6956723222621100 177 0.7500000000000000 0.6548270978041459 166 0.8750000000000000 0.6442809897988127 164 1.0000000000000000 0.6265944697132493 159 1.1250000000000000 0.5471862349697154 139 1.2500000000000000 0.5898922393465287 150 1.3750000000000000 0.5431682790690553 138 1.5000000000000000 0.5027246360756843 128 1.6250000000000000 0.4675603911239450 119 1.7500000000000000 0.5503411844773443 140 1.8750000000000000 0.5269507911846940 134 2.0000000000000000 0.5057177276273157 128 2.1250000000000000 0.4863605298130780 124 2.2500000000000000 0.4686360587290348 119 2.3750000000000000 0.4523370349494329 115 2.5000000000000000 0.4372871860310705 111 2.6250000000000000 0.4233383125753438 107 2.7500000000000000 0.4103701503419785 104 2.8750000000000000 0.3982571942958144 101 d0 - d1 = -nan - 0.5998479213011405 = -nan dz = 0.0250000000000000 ; -0.0250000000000000 z = 0.5000000000000000 ; 0.0000000000000000 # z d 0 0.5000000000000000 0.0000000000000000 -nan -128 1 0.4750000000000000 0.0250000000000000 0.7121325813233135 181 2 0.4500000000000000 0.0500000000000000 0.6771641863530853 172 3 0.4249999999999999 0.0750000000000000 0.6461577065251660 164 4 0.3999999999999999 0.1000000000000000 0.6197877204696240 158 5 0.3749999999999999 0.1250000000000000 0.5965414459168343 152 6 0.3499999999999999 0.1500000000000000 0.5753191789442660 146 7 0.3249999999999998 0.1750000000000000 0.5553461177196439 141 8 0.2999999999999998 0.2000000000000000 0.5360320675183665 136 9 0.2749999999999998 0.2250000000000000 0.5168792739832853 131 10 0.2499999999999998 0.2500000000000000 0.4974192989426989 126 11 0.2249999999999998 0.2750000000000000 0.4771609068859917 121 12 0.1999999999999998 0.3000000000000000 0.4555349628489580 116 13 0.1749999999999998 0.3250000000000000 0.4318210274734753 110 14 0.1499999999999998 0.3500000000000000 0.4050307792444734 103 15 0.1249999999999998 0.3750000000000001 0.3736957950628046 95 16 0.0999999999999998 0.4000000000000001 0.3354274035426386 85 17 0.0749999999999998 0.4250000000000001 0.2858481394390004 72 18 0.0499999999999998 0.4500000000000001 0.2153260815300087 54 19 0.0249999999999998 0.4750000000000001 0.0937798292447631 23 d0 - db = -nan - 0.3205178129943942 = -nan allways free memory (deallocate ) to avoid memory leaks Numerical approximation of Julia set for fc(z)= z^2 + c parameter c = ( 0.2500000000000000 ; 0.0000000000000000 ) Image Width = 2.400000 in world coordinate PixelWidth = 0.000120 iInterior = 4 iExterior = 195263944 Maximal number of iterations = iterMax = 1000 ratio of image = 1.000000 ; it should be 1.000 ... gcc version: 7.5.0
postprocessing
editconvert 20008.016.pgm -resize 2000x2000 2.png
references
edit- ↑ Unveiling the Fractal Structure of Julia Sets with Lagrangian Descriptors by Víctor J. García-Garrido
- ↑ fractalforums.org: unveiling-the-fractal-structure-of-julia-sets-with-lagrangian-descriptors
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