File:Parabolic Julia set from period 3 thru internal angle 1 over 3.png
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Summary
DescriptionParabolic Julia set from period 3 thru internal angle 1 over 3.png |
English: Parabolic Julia set for f(z) = z^2 + c from period 3 componenet thru internal angle 1 over 3. Parameter c is a root point between period 3 and period 9 components of Mandelbrot set: c = -0.040429288233396 +0.786653655622161*I |
Source | Own work |
Author | Adam majewski |
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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue |
C src code
Source code was formatted with Emacs. To compile :
gcc e.c -lm -Wall -march=native -fopenmp
To run :
time ./a.out
After 1828 miniutes ( on my i7-4770 CPU @ 3.40GHz × 8 ) it creates 5000x5000 pgm file in a program directory.
Convert pgm to png and downsize ( = supersampling) using Image Magic :
convert -resize 1500x1500 oar0.000933333.pgm a.png
/*
c console program
-----------------------------------------
1.ppm file code is based on the code of Claudio Rocchini
http://en.wikipedia.org/wiki/Image:Color_complex_plot.jpg
create 8 bit color graphic file , portable graymap file = PGM
see http://en.wikipedia.org/wiki/Portable_pixmap
to see the file use external application ( graphic viewer)
I think that creating graphic can't be simpler
---------------------------
2. first it creates data array which is used to store 1 byte color values of pixels,
fills tha array with data and after that writes the data (array) to binary pgm file in one step.
It alows free ( non sequential) acces to "pixels"
-------------------------------------------
Adam Majewski fraktal.republika.pl
Sobel filter
Gh = sum of six values ( 3 values of matrix are equal to 0 ). Each value is = pixel_color * filter_coefficients
gcc e.c -lm -Wall -march=native -fopenmp
time ./a.out
convert oar0.000933333.pgm a.png
----------------------------------
File oar0.000933333.pgm saved.
real 24m42.020s
c = -0.040429288233396 +0.786653655622161 i okres = 10000
==============================================
File oar0.000186667.pgm saved.
real 1828m56.372s
user 14512m23.201s
sys 2m15.393s
*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <complex.h>
#include <string.h>
#include <omp.h> // OpenMP; needs also -fopenmp
/* iXmax/iYmax = */
int iXmax = 5000; /* height of image in pixels */
int iYmax = 5000;
/* fc(z) = z*z + c */
int denominator =3; /* denominator of internal angle */
int periodOfParent=3;
int periodOfChild ; // = denominator*periodOfParent = period of child component , not parent
// it is impotran for quolity and time
double AR; // PixelWidth /* radius of circle around attractor ZA = target set for attracting points */
double AR2; // (AR*AR)
/* escape time to infinity */
int GiveExtLastIteration(double _Zx0, double _Zy0,double C_x, double C_y, int iMax, double _ER2)
{
int i;
double Zx, Zy;
double Zx2, Zy2; /* Zx2=Zx*Zx; Zy2=Zy*Zy */
Zx=_Zx0; /* initial value of orbit */
Zy=_Zy0;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
for (i=0;i<iMax && ((Zx2+Zy2)<_ER2);i++)
{
Zy=2*Zx*Zy + C_y;
Zx=Zx2-Zy2 +C_x;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
};
return i;
}
/* find attractor ZA using forward iteration of critical point Z = 0 */
/* if period is >1 gives one point from attracting cycle */
double complex GiveAttractor(double complex C , double ER2, int _IterationMax)
{
int Iteration;
double Cx,Cy; /* C = Cx+Cy*I */
double Zx, Zy; /* z = zx+zy*i */
double Zx2, Zy2; /* Zx2=Zx*Zx; Zy2=Zy*Zy */
Cx = creal(C);
Cy = cimag(C);
/* -- find attractor ZA using forward iteration of critical point Z = 0 */
Zx=0.0;
Zy=0.0;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
for (Iteration=0;Iteration<_IterationMax && ((Zx2+Zy2)<ER2);Iteration++)
{
Zy=2*Zx*Zy + Cy;
Zx=Zx2-Zy2 + Cx;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
};
return Zx+Zy*I;
}
/* attracting time to finite attractor ZA */
int GiveIntLastIteration(double _Zx0, double _Zy0,double C_x, double C_y, int iMax, double _AR2, double _ZAx, double _ZAy )
{
int i;
double Zx, Zy; /* z = zx+zy*i */
double Zx2, Zy2; /* Zx2=Zx*Zx; Zy2=Zy*Zy */
double d, dX, dY; /* distance from z to Alpha */
Zx=_Zx0; /* initial value of orbit */
Zy=_Zy0;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
dX=Zx-_ZAx;
dY=Zy-_ZAy;
d=dX*dX+dY*dY;
for (i=0;i<iMax && (d>_AR2);i++)
{
Zy=2*Zx*Zy + C_y;
Zx=Zx2-Zy2 +C_x;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
dX=Zx-_ZAx;
dY=Zy-_ZAy;
d=dX*dX+dY*dY;
};
return i;
}
// gives c in hyperbolic component of Mandelbrot set
// #include <complex.h>
// turn is an internal angle in turns
// 0.0 <= radius <=1.0
double complex GiveC(int period, double radius, double turn)
{
double Cx, Cy; /* C = Cx + Cy*i */
double a = turn*(2*M_PI); // angle, from turns to radians
switch( period )
{
case 1 : Cx = radius*(0.5*cos(a) - 0.25*cos(2*a));
Cy = radius*(0.5*sin(a) - 0.25*sin(2*a));
break;
case 2 : Cx = radius*0.25*cos(a) - 1;
Cy = radius*0.25*sin(a) ;
break;
default : Cx=0; Cy=0; //
break;
}
return Cx+Cy*I;
}
/* gives position of point (iX,iY) in 1D array ; uses also global variables */
unsigned int f(unsigned int _iX, unsigned int _iY)
{return (_iX + (iYmax-_iY-1)*iXmax );}
/* --------------------------------------------------------------------------------------------------------- */
int main(){
unsigned int iX,iY, /* indices of 2D virtual array (image) = integer coordinate */
i, /* index of 1D array */
iLength = iXmax*iYmax;/* length of array in bytes = number of bytes = number of pixels of image * number of bytes of color */
/* world ( double) coordinate = parameter plane*/
const double ZxMin=-1.4;
const double ZxMax=1.4;
const double ZyMin=-1.4;
const double ZyMax=1.4;
double PixelWidth=(ZxMax-ZxMin)/iXmax;
double PixelHeight=(ZyMax-ZyMin)/iYmax;
/* */
double Zx, Zy; /* Z=Zx+Zy*i */
double complex ZA; /* atractor ZA = ZAx + ZAy*i */
double complex C; /* atractor C = Cx + Cy*i */
/* */
const double EscapeRadius=2.0; /* radius of circle around origin; its complement is a target set for escaping points */
double ER2=EscapeRadius*EscapeRadius;
const int IterationMax=60,
IterationMaxBig= 100000001;
int eLastIteration, iLastIteration;
periodOfChild = denominator*periodOfParent; //
/* sobel filter */
unsigned char G, Gh, Gv;
/* color; length of array should be >= periodOfChild !!!!! */
unsigned char color[9]={255,230,200,180,150,120, 90, 60,30}; /* shades of gray used in image */
const unsigned int MaxColorComponentValue=255; /* color component is coded from 0 to 255 ; it is 8 bit color file */
/* dynamic 1D arrays for colors ( shades of gray ) */
unsigned char *data, *edge;
data = malloc( iLength * sizeof(unsigned char) );
edge = malloc( iLength * sizeof(unsigned char) );
if (data == NULL || edge==NULL)
{
fprintf(stderr," Could not allocate memory. End of the program. ");
getchar();
return 1;
}
else printf(" memory is OK\n");
// computed with program Mandel by Wolf Jung
C = -0.040429288233396 +0.786653655622161*I; //GiveC(periodOfParent, 1.0, 1.0/denominator);
printf(" Cx = %f \n", creal(C));
printf(" Cy = %f \n", cimag(C));
ZA = GiveAttractor( C, ER2, IterationMaxBig); /* find attractor ZA using forward iteration of critical point Z = 0 */
printf(" ZAx = %f \n", creal(ZA));
printf(" ZAy = %f \n", cimag(ZA));
AR = PixelWidth/3.0;
AR2=AR*AR;
printf(" fill the data array \n");
#pragma omp parallel for schedule(dynamic) private(i,iX,iY,Zy, Zx, eLastIteration,iLastIteration) shared(iYmax,iXmax, ER2)
for(iY=0;iY<iYmax;++iY){
Zy=ZyMin + iY*PixelHeight; /* */
if (fabs(Zy)<PixelHeight/2) Zy=0.0; /* */
printf(" row %u from %u \n",iY, iYmax);
for(iX=0;iX<iXmax;++iX){
Zx=ZxMin + iX*PixelWidth;
eLastIteration = GiveExtLastIteration(Zx, Zy, creal(C), cimag(C), IterationMax, ER2 );
i= f(iX,iY); /* compute index of 1D array from indices of 2D array */
if ( IterationMax != eLastIteration )
{data[i]=245;} /* exterior */
else /* interior */
{ iLastIteration = GiveIntLastIteration(Zx, Zy, creal(C), cimag(C), IterationMaxBig, AR2, creal(ZA), cimag(ZA));
data[i]=color[iLastIteration % periodOfChild];} /* level sets of attraction time */
/* if (Zx>0 && Zy>0) data[i]=255-data[i]; check the orientation of Z-plane by marking first quadrant */
}
}
printf(" find boundaries in data array using Sobel filter\n");
#pragma omp parallel for schedule(dynamic) private(i,iX,iY,Gv,Gh,G) shared(iYmax,iXmax, ER2)
for(iY=1;iY<iYmax-1;++iY){
for(iX=1;iX<iXmax-1;++iX){
Gv= data[f(iX-1,iY+1)] + 2*data[f(iX,iY+1)] + data[f(iX-1,iY+1)] - data[f(iX-1,iY-1)] - 2*data[f(iX-1,iY)] - data[f(iX+1,iY-1)];
Gh= data[f(iX+1,iY+1)] + 2*data[f(iX+1,iY)] + data[f(iX-1,iY-1)] - data[f(iX+1,iY-1)] - 2*data[f(iX-1,iY)] - data[f(iX-1,iY-1)];
G = sqrt(Gh*Gh + Gv*Gv);
i= f(iX,iY); /* compute index of 1D array from indices of 2D array */
if (G==0) {edge[i]=255;} /* background */
else {edge[i]=0;} /* boundary */
}
}
/* ---------- file -------------------------------------*/
printf(" save data array to the file \n");
FILE * fp;
char name [10]; /* name of file */
i = sprintf(name,"oar%2.9f",AR); /* result (is saved in i) but is not used */
char *filename =strcat(name,".pgm");
char *comment="# C=";/* comment should start with # */
/* 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\n %u\n %u\n",comment,iXmax,iYmax,MaxColorComponentValue); /*write header to the file*/
fwrite(edge,iLength,1,fp); /*write image data bytes to the file in one step */
printf("File %s saved. \n", filename);
fclose(fp);
/* --------------free memory ---------------------*/
free(data);
free(edge);
return 0;
}
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 06:25, 3 August 2014 | 1,500 × 1,500 (194 KB) | Soul windsurfer | better quality | |
09:31, 13 October 2012 | 1,000 × 1,000 (16 KB) | Soul windsurfer | {{Information |Description ={{en|1=Parabolic Julia set from period 3 thru internal angle 1 over 3. Parameter c is a root point between period 3 and period 9 components of Mandelbrot set}} |Source ={{own}} |Author =[[User:Adam majewsk... |
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