File:Julia set for f(z)= z^14-z.png
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editDescriptionJulia set for f(z)= z^14-z.png | |
Source | Own work |
Author | Adam majewski |
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C src code
editThis program creates 3 pgm files.
Convert one image to png using Image Magic :
convert 0.420000.pgm p.png
/*
c program:
1. draws Julia set for z^14-z
http://www.ijon.de/mathe/julia/node6.html
using escape time
aproximates julia set near fixed point z=0 using 13-arm star
http://www.ijon.de/mathe/julia/node6.html
-------------------------------
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
Maxima CAS session :
(%i1) z:x+y*%i;
(%o1) %i*y+x
(%i2) z1:z^14-z;
(%o2) (%i*y+x)^14-%i*y-x
(%i3) realpart(z1);
(%o3) -y^14+91*x^2*y^12-1001*x^4*y^10+3003*x^6*y^8-3003*x^8*y^6+1001*x^10*y^4-91*x^12*y^2+x^14-x
(%i4) imagpart(z1);
(%o4) 14*x*y^13-364*x^3*y^11+2002*x^5*y^9-3432*x^7*y^7+2002*x^9*y^5-364*x^11*y^3+14*x^13*y-y
gcc z.c -Wall -lm
author : Adam Majewski
fraktal.republika.pl
*/
#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; }
int NumberOfPetals=26;
// 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
}
// all points of interior fall into fixed point z=0
// thru 2*13 petals
unsigned char GiveColorOfInterior(double x, double y)
{
int i;
int iMax=NumberOfPetals;
double angle;
double offset = 1.0/(2.0*iMax);
// works !!!!!!
angle=GiveTurn(x+y*I);
if (angle< offset || (1.0-offset)<angle) return 231; //
for(i=0;i<iMax;++i) if (angle< offset+ (1.0*i)/iMax) return ((unsigned char)(230-i));
return (230-iMax - 1); //
}
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=-1.2;
const double xMax=1.2;
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;
double x10,y10;
/* */
int Iteration;
const int IterationMax=2000;
/* bail-out value , radius of circle ; */
const int EscapeRadius=2;
int ER2=EscapeRadius*EscapeRadius;
TargetRadiusInPixels = 280.0; // = 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;
// color=100;
/* */
for (Iteration=0;Iteration<IterationMax ;Iteration++)
{
if (x2y2>ER2) { color=255; break;}/* exterior of Filled-in Julia set */
// check test every 26 iteration
if (!(Iteration % NumberOfPetals) && (x2y2<AR2)) { color=GiveColorOfInterior(x,y); break;}/* interior of Filled-in Julia set */
// z^14- z
y10= y2*y2*y2*y2*y2;
x10= x2*x2*x2*x2*x2;
//realpart(z1);
// -y^14 +91*x^2*y^12 -1001*x^4*y^10 +3003*x^6*y^8 -3003*x^8*y^6 +1001*x^10*y^4 -91*x^12*y^2 +x^14 -x
tempx = -y10*y2*y2 + 91.0*x2*y10*y2 -1001.0*x2*x2*y10 +3003.0*x2*x2*x2*y2*y2*y2*y2 -3003.0*x2*x2*x2*x2*y2*y2*y2 +1001.0*x10*y2*y2 -91.0*x10*x2*y2 +x10*x2*x2 -x ;
// 14*x*y^13 -364*x^3*y^11 +2002*x^5*y^9 -3432*x^7*y^7 +2002*x^9*y^5 -364*x^11*y^3 +14*x^13*y -y
y = 14.0*x*y10*y2*y -364.0*x2*x*y10*y +2002.0*x*x2*x2*y2*y2*y2*y2*y -3432.0*x2*x2*x2*x*y2*y2*y2*y +2002.0*x2*x2*x2*x2*x*y2*y2*y -364.0*x10*x*y*y2 +14.0*x10*x2*x*y -y;
x=tempx;
x2=x*x;
y2=y*y;
x2y2= x2+y2;
};
i=Give_i(iX,iY);
data[i]=color;
}
}
SaveArray2PGMFile(data,TargetRadiusInPixels+0.0100);
AddBoundaries(data);
SaveArray2PGMFile(edge,TargetRadiusInPixels+0.0200);
CopyBoundaries();
SaveArray2PGMFile(data,TargetRadiusInPixels+0.0300);
//
free(data);
free(edge);
return 0;
}
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current | 20:13, 4 March 2013 | 1,000 × 1,000 (29 KB) | Soul windsurfer (talk | contribs) | {{Information |Description ={{en|1=Julia set for f(z)= z^15-z}} |Source ={{own}} |Author =Adam majewski |Date = |Permission = |other_versions = }} |
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