File:Basilica Julia set with binary decomposition.png
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[edit]DescriptionBasilica Julia set with binary decomposition.png |
English: Basilica Julia set with binary decomposition of both exterior and interior. Looks better when viewed in full size |
Date | |
Source | Own work |
Author | Adam majewski |
Other versions |
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c src code
[edit]/*
Adam Majewski
adammaj1 aaattt o2 dot pl // o like oxygen not 0 like zero
console program in c programing language
==============================================
Structure of a program or how to analyze the program
============== Image X ========================
DrawImageOfX -> DrawPointOfX -> ComputeColorOfX -> IsInTheYTrap
first 2 functions are identical for every X
check only last function = ComputeColorOfX
which computes color of one pixel !
==========================================
-------------------------------
cd existing_folder
git init
git remote add origin [email protected]:adammajewski/SepalsOfCauliflower.git
git add .
git commit
git push -u origin master
---------------------------------
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 >a.txt
----------------------
*/
#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 = 10000; //
// 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 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 = -2.0; //-0.05;
static const double ZxMax = 2.0; //0.75;
static const double ZyMin = -2.0; //-0.1;
static const double ZyMax = 2.0; //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; // parameter of function fc(z)=z^2 + c
static unsigned long int iterMax = 1000000; //iHeight*100;
static double ER = 200.0; // EscapeRadius for bailout test
double distanceMax;
//double D2MaxGlobal; //= 0.0497920256372717 ;
//double DistanceMaxGlobal2 ;
/* colors = shades of gray from 0 to 255 */
static unsigned char iColorOfExterior = 250;
static unsigned char iColorOfInterior = 200;
unsigned char iColorOfBoundary = 0;
double BoundaryWidth = 2.5;
/* ------------------------------------------ functions -------------------------------------------------------------*/
//------------------complex numbers -----------------------------------------------------
// fast cabs
double cabs2(complex double z) {
return (creal(z) * creal(z) + cimag(z) * cimag(z));
}
// 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) ****************************
// bailout test
// z escapes when
// abs(z)> ER or cabs2(z)> ER2
// https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Julia_set#Boolean_Escape_time
int Escapes(complex double z){
// here target set (trap) is the exterior circle with radsius = ER ( EscapeRadius)
// with ceter = origin z= 0
// on the Riemann sphere it is a circle with point at infinity as a center
if (cabs(z)>ER) return 1;
return 0;
}
/* ----------- 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;
}
// ***************************************************************************************************************************
// ************************** 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;
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 (cabsz< PixelWidth) return iColorOfInterior; // falls into finite attractor = interior
dz = 2.0*z * dz;
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
distance = 2.0 * cabsz* log(cabsz)/ cabs(dz);
if (distance <distanceMax) {//printf(" distance = %f \n", distance);
return iColorOfBoundary;}
return iColorOfExterior;
}
// plots raster point (ix,iy)
int DrawPointOfDEMJ (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 = ComputeColorOfDEMJ(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfDEMJ (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)
DrawPointOfDEMJ(A, ix, iy); //
}
return 0;
}
// ***************************************************************************************************************************
// ************************** bondary*****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfBoundary(complex double z){
// https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Julia_set#DEM.2FJ
int nMax = 20; // very low value
double cabsz;
int n;
for (n=0; n < nMax; n++){ //forward iteration
cabsz = cabs(z);
if (cabsz > 10000000000*ER ) return iColorOfExterior; // big values
if (cabsz < (PixelWidth/100)) return iColorOfInterior; // falls into finite attractor = interior
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
//printf("found \n");
return iColorOfBoundary;
}
// plots raster point (ix,iy)
int DrawPointOfBoundary (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 = ComputeColorOfBoundary(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfBoundary (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)
DrawPointOfBoundary(A, ix, iy); //
}
return 0;
}
// ***************************************************************************************************************************
// ************************** LSM/J*****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfLSM(complex double z){
int nMax = 255;
double cabsz;
unsigned char iColor;
int n;
for (n=0; n < nMax; n++){ //forward iteration
cabsz = cabs(z);
if (cabsz > ER) break; // esacping
if (cabsz< PixelWidth) break; // fails into finite attractor = interior
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
iColor = 255 - 255.0 * ((double) n)/20; // nMax or lower walues in denominator
return iColor;
}
// plots raster point (ix,iy)
int DrawPointOfLSM (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 = ComputeColorOfLSM(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfLSM (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)
DrawPointOfLSM(A, ix, iy); //
}
return 0;
}
// ***************************************************************************************************************************
// ************************** binary decomposition BD/J*****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfBD(complex double z){
int nMax = 255;
double cabsz;
unsigned char iColor;
int n;
for (n=0; n < nMax; n++){ //forward iteration
cabsz = cabs(z);
if (cabsz > ER) break; // esacping
if (cabsz< PixelWidth) break; // fails into finite attractor = interior
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
if (cimag(z)>0.0)
iColor = 255;
else iColor = 0;
return iColor;
}
// plots raster point (ix,iy)
int DrawPointOfBD (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 = ComputeColorOfBD(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfBD (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)
DrawPointOfBD(A, ix, iy); //
}
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, "%.10f", 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 info ()
{
// 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);
if ( distanceMax<0.0 || distanceMax > ER ) printf("bad distanceMax\n");
printf("distanceMax = %.16f\n", distanceMax);
// image corners in world coordinate
// center and radius
// center and zoom
// GradientRepetition
printf ("Maximal number of iterations = iterMax = %ld \n", iterMax);
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");
c = -1.0; //
/* 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)) / ((float) iWidth / (float) iHeight); // it should be 1.000 ...
//ER2 = ER * ER; // for numerical optimisation in iteration
/* 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;
}
distanceMax = BoundaryWidth*PixelWidth;
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);
info ();
return 0;
}
// ********************************************************************************************************************
/* ----------------------------------------- main -------------------------------------------------------------*/
// ********************************************************************************************************************
int main () {
setup ();
// ******************************** DEM/J **********************************************************
DrawImagerOfDEMJ(data);
SaveArray2PGMFile (data, iWidth+0.1, "boundary using DEM/J");
DrawImagerOfLSM(data);
SaveArray2PGMFile (data, iWidth+0.2, "LSM/J");
DrawImagerOfBD(data);
SaveArray2PGMFile (data, iWidth+0.3, "BD/J");
DrawImagerOfBoundary(data);
SaveArray2PGMFile (data, iWidth+0.4, "boundary using iteration");
end();
return 0;
}
test output:
setup start end of setup File 10000.1000000000.pgm saved . Comment = boundary using DEM/J File 10000.2000000000.pgm saved . Comment = LSM/J File 10000.3000000000.pgm saved . Comment = BD/J File 10000.4000000000.pgm saved . Comment = boundary using iteration allways free memory (deallocate ) to avoid memory leaks Numerical approximation of Julia set for fc(z)= z^2 + c parameter c = ( -1.0000000000000000 ; 0.0000000000000000 ) Image Width = 4.000000 in world coordinate PixelWidth = 0.000400 distanceMax = 0.0010001000100010 Maximal number of iterations = iterMax = 1000000 ratio of image = 1.000000 ; it should be 1.000 ... gcc version: 7.3.0
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