File:Julia set for f(z) = z^4 - z.png
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Summary
DescriptionJulia set for f(z) = z^4 - z.png |
English: Julia set for f(z) = z^4 - z. It is described in Complex dynamics, Lennart Carleson, Theodore W. Gamelin, Springer, 1993, ISBN 978-0-387-97942-7. Page 40, Figure 2. Here λ is a 2nd root of unity and with and z = 0 is a parabolic fixed point with multiplicity 7 = 2 ∗ 3 1. There are thus 6 'attracting' directions.
Deutsch: Julia-Menge fur Hier ist λ eine 2-te Einheitswurzel und mit ist z = 0 ein parabolischer Fixpunkt mit Vielfachheit 7 = 2 ∗ 3 + 1. Es gibt somit 6 ’attracting’ -Richtungen.[1] |
Date | |
Source | Own work |
Author | Adam majewski |
C src code
/*
c program:
1. draws Julia set for z^4-z
using boolean escape time )
-------------------------------
2. technic of creating ppm file is based on the code of Claudio Rocchini
http://en.wikipedia.org/wiki/Image:Color_complex_plot.jpg
create 24 bit color graphic file , portable pixmap file = PPM
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
(%i2) z:x+y*%i;
(%o2) %i*y+x
(%i3) realpart(z^4-z);
(%o3) y^4-6*x^2*y^2+x^4-x
(%i4) imagpart(z^4-z);
(%o4) -4*x*y^3+4*x^3*y-y
gcc z.c -Wall -lm
*/
#include <stdio.h>
#include <stdlib.h> // malloc
#include <math.h>
#include <complex.h>
#include <string.h> // strcat
const int iXmax = 1000;
const int iYmax = 1000;
int iSize ; //= iXmax*iYmax; //
// target set for points falling into alfa fixed point
// is a circle around alfa fixed point
// with radius = AR
double AR ; // radius of target set around alfa fixed point in world coordinate AR = PixelWidth*TargetWidth;
double AR2; // =AR*AR;
double TargetRadiusInPixels; // radius of target set in pixels ;
// memmory 1D arrays
unsigned char *data;
unsigned char *edge;
int i; // index of array
double TwoPi=2.0*M_PI;
/* 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*iYmax; }
// save data array to pgm file
int SaveArray2PGMFile( unsigned char data[], double t)
{
FILE * fp;
const unsigned int MaxColorComponentValue=255; /* color component is coded from 0 to 255 ; it is 8 bit color file */
char name [10]; /* name of file */
sprintf(name,"%f", t); /* */
char *filename =strcat(name,".pgm");
char *comment="# ";/* 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 %u\n %u\n",comment,iXmax,iYmax,MaxColorComponentValue); /*write header to the file*/
fwrite(data,iSize,1,fp); /*write image data bytes to the file in one step */
printf("File %s saved. \n", filename);
fclose(fp);
return 0;
}
double GiveTurn(double complex z)
{
double argument;
argument = carg(z); // argument in radians from -pi to pi
if (argument<0) argument=argument + TwoPi; // argument in radians from 0 to 2*pi
return argument/TwoPi ; // argument in turns from 0.0 to 1.0
}
unsigned char GiveColorOfInterior(double x, double y)
{
double angle;
//double offset = 1.0/(2.0*6.0);
// all points tend to z=0 thru 6 petals
// check to which sector / petal fall
angle=GiveTurn(x+y*I);
if (angle<1.0/12.0 || angle > (11.0/12.0)) return 235; // (11/12 ; 1/12)
if (angle<3.0/12.0) return 225; // 1/12 ; 3/12
if (angle<5.0/12.0) return 215; // 3/12 ; 5/12
if (angle<7.0/12.0) return 205; // 5/12 ; 7/12
if (angle<9.0/12.0) return 195; // 7/12 ; 9/12
//if (angle<11.0/12.0)
return 175; // 9/12 ; 11/12
}
int AddBoundaries(unsigned char data[])
{
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;
printf(" find boundaries in data array using Sobel filter\n");
for(iY=1;iY<iYmax-1;++iY){
for(iX=1;iX<iXmax-1;++iX){
Gv= data[Give_i(iX-1,iY+1)] + 2*data[Give_i(iX,iY+1)] + data[Give_i(iX-1,iY+1)] - data[Give_i(iX-1,iY-1)] - 2*data[Give_i(iX-1,iY)] - data[Give_i(iX+1,iY-1)];
Gh= data[Give_i(iX+1,iY+1)] + 2*data[Give_i(iX+1,iY)] + data[Give_i(iX-1,iY-1)] - data[Give_i(iX+1,iY-1)] - 2*data[Give_i(iX-1,iY)] - data[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) {edge[i]=245;} /* background */
else {edge[i]=0;} /* boundary */
}
}
// copy boundaries from edge array to data array
//for(iY=1;iY<iyMax-1;++iY){
// for(iX=1;iX<ixMax-1;++iX){i= Give_i(iX,iY); if (edge[i]==0) data[i]=0;}}
return 0;
}
int CopyBoundaries()
{
unsigned int iX,iY; /* indices of 2D virtual array (image) = integer coordinate */
unsigned int i; /* index of 1D array */
printf("copy boundaries from edge array to data array \n");
for(iY=1;iY<iYmax-1;++iY)
for(iX=1;iX<iXmax-1;++iX)
{i= Give_i(iX,iY); if (edge[i]==0) data[i]=0;}
return 0;
}
int main()
{
/* screen ( integer) coordinate */
int iX,iY;
iSize = iXmax*iYmax; //
/* world ( double) coordinate = parameter plane*/
const double xMin=-1.0;
const double xMax=1.4;
const double yMin=-1.2;
const double yMax=1.2;
/* */
double PixelWidth=(xMax-xMin)/iXmax;
double PixelHeight=(yMax-yMin)/iYmax;
//
/* color component ( R or G or B) is coded from 0 to 255 */
/* it is 24 bit color RGB file */
unsigned char color;
//
double x, y, tempx, /* Z=x+y*i */
x0, y0, /* Z0 = x0 + y0*i */
x2, y2; /* x2=x*x; y2=y*y */
double x2y2; //, y4, x4, x2my2;
/* */
int Iteration;
const int IterationMax=2000;
/* bail-out value , radius of circle ; */
const int EscapeRadius=2;
int ER2=EscapeRadius*EscapeRadius;
TargetRadiusInPixels = iXmax/100.0; // = 15.0; // radius of target set in pixels ; Maybe increase to 20 = 1000/50
AR = PixelWidth*TargetRadiusInPixels; // !!!! important value ( and precision and time of creation of the pgm image )
AR2= AR*AR;
/* create dynamic 1D arrays for colors ( shades of gray ) */
data = malloc( iSize * sizeof(unsigned char) );
edge = malloc( iSize * sizeof(unsigned char) );
if (edge == NULL || edge == NULL )
{
fprintf(stderr," Could not allocate memory\n");
return 1;
}
else fprintf(stderr," memory is OK \n");
/* compute and write image data bytes to the file*/
for(iY=0;iY<iYmax;++iY)
{
y0=yMax - iY*PixelHeight; /* reverse Y axis */
if (fabs(y0)<PixelHeight/2) y0=0.0; /* */
printf(" iy = %d from %d\r", iY, iYmax); //info
for(iX=0;iX<iXmax;++iX)
{ /* initial value of orbit Z0 */
x0=xMin + iX*PixelWidth;
/* Z = Z0 */
x=x0;
y=y0;
x2=x*x;
y2=y*y;
x2y2= x2+y2;
/* */
for (Iteration=0;Iteration<IterationMax ;Iteration++)
{
if ((x2y2)>ER2) { color=255; break;}/* exterior of Filled-in Julia set */
if ((x2y2)<AR2) { color=GiveColorOfInterior(x,y); break;}/* interior of Filled-in Julia set */
// z^4- z
tempx = y*y*y*y -6*y2*x2 + x*x*x*x - x ; // y^4-6*x^2*y^2+x^4-x
y = -4*x*y*y*y +4*x*x*x*y - y ; // -4*x*y^3+4*x^3*y-y
x=tempx;
x2=x*x;
y2=y*y;
x2y2= x2+y2;
};
if (Iteration==IterationMax) color=GiveColorOfInterior(x,y); ;
i=Give_i(iX,iY);
data[i]=color;
}
}
SaveArray2PGMFile(data,0.101);
AddBoundaries(data);
SaveArray2PGMFile(edge,0.102);
CopyBoundaries();
SaveArray2PGMFile(data,0.103);
//
free(data);
free(edge);
return 0;
}
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I, the copyright holder of this work, hereby publish it under the following licenses:
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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 |
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 19:14, 8 February 2013 | 1,000 × 1,000 (22 KB) | Soul windsurfer | {{Information |Description ={{en|1=Julia set for f(z) = z^4 - z}} |Source ={{own}} |Author =Adam majewski |Date =2013-02-08 |Permission = |other_versions = }} |
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