File:Julia set for f(z) = z^2+0.355534 -0.337292*i.png
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Contents
Summary
DescriptionJulia set for f(z) = z^2+0.355534 -0.337292*i.png |
English: Julia set for f(z) = z^2+0.355534 -0.337292*i. it is inside the period 85 componnet with center c = 0.3555471250217805 -0.3372858052155804*i. Adress 1-(4/5)->5-(1/17)-> 85, Location by Chris Thomasson. |
Date | |
Source | Own work |
Author | Adam majewski |
Other versions |
|
steps
How to find c value on the parameter plane ?
- center ( nucleus) of period 1 component is c = 0.0
- go along internal ray 4/5 to the bifurcation point between period 1 and 5 components
- bifurcation point ( root point = bond) is c = 0.379513588015924 -0.334932305597498 i
- go along internal ray 0 to the center of period 5 component c = 0.379513588015924 -0.334932305597498 i
- go along internal ray 1/17 to bifurcation point
- bifurcation pointis aprox c = 0.355630631309901 -0.337285118827790 i ( real: 3.5563081682876507e-01 imag: -3.3729232913917928e-01
- go along internal ray 0 to the center
- center of period 85 componnet c = 0.355534 -0.337292*i
Tips for making better images by Paul Bourke
- do the fractal creation not in 8 bit, but use 16 bit or floating point for each pixel ( pfm file, kfb and exr ( OpenEXR) from Kalles Fraktaler or array with floating point values )
- apply antialiasing by supersampling each final pixel ( render a 30,000 pixel version of the image )
- do all colour external to the fractal creation using gradient maps...allows you to make appearance decisions independent to the creation
Summary
coefficients read from input file thomasson85.txt degree 2 coefficient = ( 1.0000000000000000 +0.0000000000000000*i) degree 0 coefficient = ( 0.3555340000000000 -0.3372920000000000*i) Input polynomial p(z)=(1+0i)*z^2+(0.35553400000000001668-0.33729199999999998072i) 1 critical points found cp#0: 0,0 . It's critical orbit is bounded and enters cycle #0 length=85 and it's stability = |multiplier|=0.1738 =attractive cycle = { 0.085363751140101362669,-0.13328474304205845113 ; 0.34505614728092159904,-0.36004737127162922405 ; 0.34496363521674267005,-0.58576511753922388959 ; 0.13141313669620913185,-0.74142732865898641759 ; -0.17691107118606408388,-0.53215858178273711676 ; 0.10363877094318652006,-0.14900251051191856777 ; 0.34407324670375988296,-0.36817687411380894957 ; 0.33836618846505089886,-0.59065162487515943113 ; 0.12115633552870030964,-0.73700507803939352414 ; -0.17296362741710957778,-0.51587766904259346745 ; 0.11932064699246511252,-0.15883585411781978025 ; 0.3445425882453631572,-0.37519679375787812203 ; 0.33347096106862217368,-0.59583454884540199004 ; 0.11171807227822680431,-0.73467903928273026182 ; -0.171738363087832413,-0.50144585202377256472 ; 0.13358012284424097826,-0.16505702039250771951 ; 0.34613382923822982873,-0.38138867412067112284 ; 0.32988530699559659443,-0.60131504440295846514 ; 0.10277853314624713077,-0.73402199604788154197 ; -0.17269086380642195699,-0.48817540810176329735 ; 0.14704090536688504964,-0.16868513417170785673 ; 0.34870035336058619047,-0.38689922970108481248 ; 0.32743492249050487608,-0.60711579622341371731 ; 0.094158038442373359622,-0.73487382735834927416 ; -0.17563980593299344246,-0.47568055617340299079 ; 0.16011134990674161238,-0.17019511885521021455 ; 0.3522032658868199384,-0.39179234045489208516 ; 0.32607990246221990605,-0.61327308371530764308 ; 0.085758227580088108866,-0.73724405464118469844 ; -0.18064032250609596275,-0.46374148683997123976 ; 0.17310875949855913936,-0.1697511765155422514 ; 0.35668518068671922761,-0.3960628311800529211 ; 0.32589255187915844303,-0.61983148500550155902 ; 0.077548885566184666551,-0.74128892876698238013 ; -0.18796144625994304356,-0.45226426061686064717 ; 0.18632054384881382991,-0.16727551096554213461 ; 0.36226824849133454354,-0.39962572835137605276 ; 0.3270715611046115523,-0.62683542532385372326 ; 0.06958715564247108798,-0.74733208223269198545 ; -0.19812886890384162619,-0.44130142784587667037 ; 0.20004209847430617919,-0.16242289451049232718 ; 0.3691696445008375016,-0.40227483331629948493 ; 0.32999518490121820813,-0.63430731441402388171 ; 0.062085052938857854254,-0.7559287190085018926 ; -0.21203967442341375982,-0.43115574907529163884 ; 0.2145995435689114772,-0.15444775068058413936 ; 0.37773285640969328503,-0.40358083360259677796 ; 0.33533862156005905808,-0.64218348213782805445 ; 0.055586366379134388715,-0.76798984737747610918 ; -0.23118456154764366328,-0.42267153006355973055 ; 0.23032907917170558632,-0.14186173528716850001 ; 0.38846073277339915331,-0.40264176571678755856 ; 0.34431534940731378658,-0.65011303071103720974 ; 0.051440107137190604902,-0.78497979064703704566 ; -0.25801318710198040129,-0.41805088906282605876 ; 0.24733825887230226348,-0.12156671548416758677 ; 0.40193174798837350004,-0.3974281994893570924 ; 0.35913395629163719081,-0.65677002184125865547 ; 0.053164336972316261765,-0.80902883263519242973 ; -0.29616720530935625177,-0.42331496295707404798 ; 0.26405345563740539871,-0.086547980910739397142 ; 0.41776767443432955007,-0.38299858687584187322 ; 0.38337591225337619427,-0.65730085790150993841 ; 0.070466672298047483558,-0.84127863204583630541 ; -0.34725018483215319698,-0.45585621135144721583 ; 0.26831180543826926854,-0.020699692702649707954 ; 0.42709674765955951692,-0.34839994384213057721 ; 0.41656311099217374627,-0.6348929657994942799 ; 0.12596974741780025964,-0.86623797796097012913 ; -0.37896585719740571996,-0.55553155857489888714 ; 0.19053380834870822791,0.083762986591094767963 ; 0.38482089420120235701,-0.30537263833227384913 ; 0.41036867237179930346,-0.57231954349521196868 ; 0.19638678739762535641,-0.80701602247312875793 ; -0.25717309026398937899,-0.65426656806381511267 ; -0.0063927437300727585345,-0.00077248966922782713596 ; 0.35557427043210931839,-0.33728212334302098974 ; 0.36820783106654997319,-0.5771496898749748361 ; 0.15800924233595339818,-0.76231407101919290703 ; -0.20062182221027302642,-0.57819733756755764542 ; 0.061470954376758202287,-0.10529399308013645542 ; 0.34822585325322968908,-0.35023704448955150781 ; 0.35412925754116375554,-0.58121518731652688228 ; 0.14313043707927045056,-0.74894260551210012355 ; -0.18489470433275023709,-0.55168496494846896994 ; }
old c src code
/*
https://www.facebook.com/photo.php?fbid=111908386634835&set=a.110008616824812&type=3&theater
center of period 85 componnet. Adress 1->5 -> 85
4K antialiased version. Max iterations 5000, and it was reached!
Paul Bourke I rendered a 30,000 pixel version
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 git@gitlab.com: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
----------------------
a@zelman:~/c/julia/demj/85$ gcc d.c -lm -Wall -march=native -fopenmp
a@zelman:~/c/julia/demj/85$ time ./a.out
setup start
end of setup
File 30001.0000000000.pgm saved . Comment = DEM/J
allways free memory (deallocate ) to avoid memory leaks
Numerical approximation of Julia set for fc(z)= z^2 + c
parameter c = ( 0.3555340000000000 ; -0.3372920000000000 )
Image Width = 2.400000 in world coordinate
PixelWidth = 0.000080
distanceMax = 0.0000800026667556
Maximal number of iterations = iterMax = 50000
ratio of image = 1.000000 ; it should be 1.000 ...
gcc version: 7.3.0
real 2021m48,597s
user 16100m25,607s
sys 1m52,759s
*/
#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 = 30000; //
// 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 = -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; // parameter of function fc(z)=z^2 + c
static unsigned long int iterMax = 50000; //iHeight*100;
static double ER = 1e60; // EscapeRadius for bailout test , minimal = 2.0
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 = 255;
unsigned char iColorOfBoundary = 0;
double BoundaryWidth = 1.0; // distanceMax = BoundaryWidth*PixelWidth; // here boundary is changing with resolution, maybe % of ImageWidth would be better ?
/* ------------------------------------------ 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
if (cabs(z)> ER || cabs(dz)> 1e60) break; // big values
dz = 2.0*z * dz;
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
if (n==nMax) return iColorOfInterior;
cabsz = cabs(z);
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[])
{
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;
}
// *******************************************************************************************
// ********************************** 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 = 0.355534-I*0.337292; //
/* 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; // here boundary is changing with resolution, maybe % of ImageWidth would be better ?
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+BoundaryWidth, "DEM/J");
end();
return 0;
}
new c src code
/*
https://www.facebook.com/photo.php?fbid=111908386634835&set=a.110008616824812&type=3&theater
center of period 85 componnet. Adress 1->5 -> 85
4K antialiased version. Max iterations 5000, and it was reached!
Paul Bourke I rendered a 30,000 pixel version
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 !
Vec3f(color[2], color[1], color[0]);
==========================================
---------------------------------
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
----------------------
ls -l *
=====
*/
#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 = 1000; // // The size of array has to be a positive constant integer
static unsigned int iSize; // = iWidth*iHeight;
static unsigned int iSizeC; // = iWidth*iHeight*iSizeC
int iColorSize = 3 ; // RGB = 3*(unsigned char)
// memmory 1D arrays
double *dData; // double for dem ( double = 8 bytes per pixel)
unsigned char * ucColor; //
// 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; // parameter of function fc(z)=z^2 + c
static unsigned long int iterMax = 50000; //iHeight*100;
static double ER = 160; // EscapeRadius for bailout test , minimal = 2.0
//double distanceMax;
double distanceMaxImage = 0.0;
double BoundaryWidth = 0.0;
//double D2MaxGlobal; //= 0.0497920256372717 ;
//double DistanceMaxGlobal2 ;
// sum of points = iSize;
int iExterior = 0;
int iInterior = 0;
int iBoundary = 0;
int iAll =0;
double PixelWidthRatio = 1.0; // BoundaryWidth= PixelWidthRatio*PixelWidth; // here boundary is changing with resolution, maybe % of ImageWidth would be better ?
/* ------------------------------------------ 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 global constants
double GiveZy (int iy) {
//return (ZyMin + iy * PixelHeight);
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;
}
// ***************************************************************************************************************************
// ************************** DEM/J*****************************************
// ****************************************************************************************************************************
double ComputeDistance(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 = 0.0;
double cabsz;
int n;
// compute distance from point z to the boundary of Julia set
for (n=0; n < nMax; n++){ //forward iteration
if (cabs(z)> ER || cabs(dz)> 1e60) break; // big values
dz = 2.0*z * dz;
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
if (n==nMax)
{distance = -1.0;} //
else {
cabsz = cabs(z);
distance = 2.0 * cabsz* log(cabsz)/ cabs(dz);
}
return distance;
}
// compute and save raster point (ix,iy) data
int ComputePointData (double A[], int ix, int iy)
{
int i; /* index of 1D array */
double distance;
complex double z;
i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
z = GiveZ(ix,iy);
distance = ComputeDistance(z);
if (distance >distanceMaxImage ) distanceMaxImage = distance; // compute also max distance of the image
A[i] = distance ; //
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int FillDataArray (double A[])
{
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)
ComputePointData(A, ix, iy); //
}
return 0;
}
/*
normalize values in array D
normalize = change range to [0,1.0]
using transformation : d = d/dMax
to do it one muust know dMax = max (D)
uses global var : ...
*/
int NormalizeDataArray (double D[])
{
int i=0; // array index
//printf("compute image \n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(i) shared(iAll, iBoundary, iExterior, iInterior, D, iSize)
for (i = 0; i < iSize; ++i){
D[i] = D[i]/distanceMaxImage;
}
//printf(" i = %d\n", i);
return 0;
}
int SaveDataFile(double A[], double k, char* comment ){
FILE * fp;
char name [100]; /* name of file */
snprintf(name, sizeof name, "%.0f", k); /* */
char *filename =strncat(name,".dat", 4);
// save image to the pgm file
fp= fopen(filename,"wb"); // create new file,give it a name and open it in binary mode
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;
}
// *******************************************************************************************
// ********************************** pfm file ****************************
// *********************************************************************************************
/*
input :
* int i
* array D of double numbers ( distance)
output : array of rgb colors
*/
int ComputePointColor(const int i, const double D[], unsigned char C[] ){
int iC = i*iColorSize; // compute index of F array
// color channels from 0 to 255
unsigned char R;
unsigned char G;
unsigned char B;
double Position = D[i]; // read distance form // distance/distanceMax;
if (Position<0.0)// interior = white
{
R = 255;
G = 255;
B = 255;
iInterior +=1;
}
else {
if ( Position > BoundaryWidth) // exterior = color gradient
{ Position = sqrt(Position); // Position*Position*Position;
R = 0; // red
G = 255.0*Position; //x^3 = non linear gradient
B = 0; // blue
iExterior +=1;
}
else { // boundary = black
R = 0;
G = 0;
B = 0;
iBoundary +=1;
} // boundary
}
//printf("R = %u \t G = %u \t B = %u \n", R, G, B); // info
// save color to the array
C[iC] = R;
C[iC+1] = G;
C[iC+2] = B;
iAll +=1;
return 0;
};
// fill array f using data from d array
// uses global var : ...
int FillColorArray (double D[], unsigned char C[])
{
int i=0; // array index
//printf("compute image \n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(i) shared(iAll, iBoundary, iExterior, iInterior, D, C, iSize)
for (i = 0; i < iSize; ++i){
//printf (" %d from %d \n", i, iSize); //info
ComputePointColor(i, D, C); //
}
//printf(" i = %d\n", i);
return 0;
}
int Z_IsInFirstQuadrant(const int ix, const int iy){
double zx = GiveZx(ix);
double zy = GiveZy(iy);
int r = 0;
if (zx >0.0 && zy > 0.0) { r = 1;}
return r;
}
int InverseColor(const int i, unsigned char C[]){
C[i] = 255 - C[i];
C[i+1] = 255 - C[i+1];
C[i+2] = 255 - C[i+2];
return 0;
}
// scanning complex plane
int CheckArrayOrientation (unsigned char C[])
{
int ix, iy; // pixel coordinate
int i;
//printf("compute image \n");
// for all pixels of image
//#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(F, ixMax , iyMax)
for (iy = iyMin; iy <= iyMax; ++iy){
for (ix = ixMin; ix <= ixMax; ++ix){
if ( Z_IsInFirstQuadrant(ix, iy)) {
i = Give_i(ix,iy);
InverseColor(i*iColorSize, C);
}
}
//
}
return 0;
}
int Save_PPM( unsigned char Cl[], double k, char* comment )
{
FILE * fp;
char name [100]; /* name of file */
snprintf(name, sizeof name, "%.0f", k); /* */
char *filename =strncat(name,".ppm", 4);
// save image to the pgm file
fp= fopen(filename,"wb"); // create new file,give it a name and open it in binary mode
if (fp ) {
fprintf(fp,"P6\n%d %d\n255\n", iWidth, iHeight); // write header to the file
fwrite(Cl, iSizeC, 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);
}
else printf( "ERROR saving `%s'\n", filename);
return 0;
}
int info ()
{
int iSum = iExterior+iBoundary+iInterior;
// 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);
printf("distanceMaxImage = %.16f\n", distanceMaxImage);
printf("BoundaryWidth = %f = %f *distanceMaxImage = %f * PixelWidth\n" , BoundaryWidth , BoundaryWidth/((double) distanceMaxImage), PixelWidthRatio);
printf("iSize = number of points in the dData array = %d\n", iSize );
//
printf("iSizeC = number of points in the ucColor array = %d\n", iSizeC);
printf(" iExterior = %d \t iBoundary = %d \t iInterior = %d \t iSum = %d\n", iExterior, iBoundary,iInterior, iSum);
printf("iAll = %d \n", iAll);
if (iSum != iSize) { printf("Error : iSum != iSize \n");}
printf("estimated ppm file size = %d B\n ", iSizeC*((int) sizeof(unsigned char)) +16); // header ( 16 bytes ) + series of three 4-byte IEEE 754 single precision floating point numbers for each pixel
// 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 = 0.355534-I*0.337292; //
/* 2D array ranges */
iWidth = iHeight;// square image
iSize = iWidth * iHeight; // size = number of points in the dData array
iSizeC = iSize * iColorSize ; // size of the fRGB 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 data ( dData) and colors ( ucColor ) */
dData = malloc (iSize * sizeof (double));
ucColor = malloc (iSizeC * sizeof (unsigned char));
if (dData == NULL || ucColor == NULL){
printf (" Could not allocate memory");
return 1;
}
BoundaryWidth = PixelWidthRatio*PixelWidth; // here boundary is changing with resolution, maybe % of ImageWidth would be better ?
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 (dData);
free(ucColor);
info ();
return 0;
}
// ********************************************************************************************************************
/* ----------------------------------------- main -------------------------------------------------------------*/
// ********************************************************************************************************************
int main () {
setup ();
// DEM/J
FillDataArray(dData); // compute distasnce of each pixel and also find max distance of the image
NormalizeDataArray(dData); // convert distance normalized distance = distance/distanceMaxImage
SaveDataFile(dData, iHeight+PixelWidthRatio, "") ;
//
FillColorArray(dData, ucColor);
Save_PPM(ucColor, iHeight+PixelWidthRatio, "");
//
// CheckArrayOrientation(fRGB);
// Save_fRGB_Array_2_PFM(fRGB, iHeight, "");
end();
return 0;
}
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