File:Julia0bb.jpg
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DescriptionJulia0bb.jpg |
Fatou sets for |
Date | |
Source | self-made using C console program. |
Author | Adam majewski |
Permission (Reusing this file) |
This file is licensed under the Creative Commons Attribution 3.0 Unported license.
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Compare with
[edit]Binary fractal images by Mark Dow [1]
Comment
[edit]This image shows part of dynamical plane ( z-plane ) under iteration of points using function .
Here z-plane consists of 2 sets:
- Julia set ( circle of radius =1 and center = 0 ) which contain repeling fixed point ( red)
- Fatou set, which consist of 2 subsets:
- A(infinity)=basin of attration of infinity ( infinity is allways superattracting fixed point for polynomials )
- A(0) = basin of attraction of finite attractor ( here is z=0 which is also here superattracting finite fixed point ).
Made with binary decomposition of basins of both ( finite and infinite) attractors :
- Finite is
- infinite attractor is point at infinity.
Fixed points :
- Green point is a superattracting fixed point
,
- red point is repelling fixed point
.
Compare it with Fig.31 on page 42 of Peitgen, Richter "The beauty of fractals".
C source code
[edit]It is a console C program ( one file) It can be compiled under :
- windows ( gcc thru Dev-C++ )
- linux and mac using gcc :
gcc main.c -lm
it creates a.out file. Then run it :
./a.out
It creates ppm file in program directory. Conversion to jpg is made with convert from ImageMagic
/*
c program:
1. draws Fatou set for Fc(z)=z*z
using binary decomposition
-------------------------------
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)
---------------------------------
*/
#include <stdio.h>
#include <stdlib.h> /* for ISO C Random Number Functions */
#include <math.h>
/* gives sign of number */
double sign(double d)
{
if (d<0)
{return -1.0;}
else {return 1.0;};
};
/* ----------------------*/
int main()
{
const double Cx=0.0,Cy=0.0;
/* screen coordinate = coordinate of pixels */
int iX, iY,
iXmin=0, iXmax=2000,
iYmin=0, iYmax=2000,
iWidth=iXmax-iXmin+1,
iHeight=iYmax-iYmin+1,
/* 3D data : X , Y, color */
/* number of bytes = number of pixels of image * number of bytes of color */
iLength=iWidth*iHeight*3,/* 3 bytes of color */
index; /* of array */
int iXinc, iYinc,iIncMax=6;
/* world ( double) coordinate = parameter plane*/
const double ZxMin=-2.5;
const double ZxMax=2.5;
const double ZyMin=-2.5;
const double ZyMax=2.5;
/* */
double PixelWidth=(ZxMax-ZxMin)/iWidth;
double PixelHeight=(ZyMax-ZyMin)/iHeight;
double Zx, Zy, /* Z=Zx+Zy*i */
Z0x, Z0y, /* Z0 = Z0x + Z0y*i */
Zx2, Zy2, /* Zx2=Zx*Zx; Zy2=Zy*Zy */
NewZx, NewZy,
DeltaX, DeltaY,
SqrtDeltaX, SqrtDeltaY,
AlphaX, AlphaY,
BetaX,BetaY, /* repelling fixed point Beta */
AbsLambdaA,AbsLambdaB;
/* */
int Iteration,
IterationMax=100,
iTemp;
/* bail-out value , radius of circle ; */
const int EscapeRadius=400;
int ER2=EscapeRadius*EscapeRadius;
double AR=PixelWidth, /* minimal distance from attractor = Attractor Radius */
AR2=AR*AR,
d,dX,dY; /* distance from attractor : d=sqrt(dx*dx+dy*dy) */
/* PPM file */
FILE * fp;
char *filename="julia00bb_.ppm";
char *comment="# this is julia set for c= ";/* comment should start with # */
const int MaxColorComponentValue=255;/* color component ( R or G or B) is coded from 0 to 255 */
/* dynamic 1D array for 24-bit color values */
unsigned char *array;
/* --------- find repelling fixed point ---------------------------------*/
/* Delta=1-4*c */
DeltaX=1-4*Cx;
DeltaY=-4*Cy;
/* SqrtDelta = sqrt(Delta) */
/* sqrt of complex number algorithm from Peitgen, Jurgens, Saupe: Fractals for the classroom */
if (DeltaX>0)
{
SqrtDeltaX=sqrt((DeltaX+sqrt(DeltaX*DeltaX+DeltaY*DeltaY))/2);
SqrtDeltaY=DeltaY/(2*SqrtDeltaX);
}
else /* DeltaX <= 0 */
{
if (DeltaX<0)
{
SqrtDeltaY=sign(DeltaY)*sqrt((-DeltaX+sqrt(DeltaX*DeltaX+DeltaY*DeltaY))/2);
SqrtDeltaX=DeltaY/(2*SqrtDeltaY);
}
else /* DeltaX=0 */
{
SqrtDeltaX=sqrt(fabs(DeltaY)/2);
if (SqrtDeltaX>0) SqrtDeltaY=DeltaY/(2*SqrtDeltaX);
else SqrtDeltaY=0;
}
};
/* Beta=(1-sqrt(delta))/2 */
BetaX=0.5+SqrtDeltaX/2;
BetaY=SqrtDeltaY/2;
/* Alpha=(1+sqrt(delta))/2 */
AlphaX=0.5-SqrtDeltaX/2;
AlphaY=-SqrtDeltaY/2;
AbsLambdaA=2*sqrt(AlphaX*AlphaX+AlphaY*AlphaY);
AbsLambdaB=2*sqrt(BetaX*BetaX+BetaY*BetaY);
printf(" Cx= %f\n",Cx);
printf(" Cy= %f\n",Cy);
printf(" Beta= %f , %f\n",BetaX,BetaY);
//printf(" BetaY= %f\n",BetaY);
printf(" Alpha= %f, %f\n",AlphaX,AlphaY);
//printf(" AlphaY= %f\n",AlphaY);
printf(" abs(Lambda (Alpha))= %f\n",AbsLambdaA);
printf(" abs(lambda(Beta))= %f\n",AbsLambdaB);
/* initial value of orbit Z=Z0 is repelling fixed point */
Zy=BetaY; /* */
Zx=BetaX;
/*-------------------------------------------------------------------*/
array = malloc( iLength * sizeof(unsigned char) );
if (array == NULL)
{
fprintf(stderr,"Could not allocate memory");
getchar();
return 1;
}
else
{
/* fill the data array with white points */
for(index=0;index<iLength-1;++index) array[index]=255;
/* ---------------------------------------------------------------*/
for(iY=0;iY<iYmax;++iY)
{
Z0y=ZyMin + iY*PixelHeight; /* reverse Y axis */
if (fabs(Z0y)<PixelHeight/2) Z0y=0.0; /* */
for(iX=0;iX<iXmax;++iX)
{ /* initial value of orbit Z0 */
Z0x=ZxMin + iX*PixelWidth;
/* Z = Z0 */
Zx=Z0x;
Zy=Z0y;
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;
};
iTemp=((iYmax-iY-1)*iXmax+iX)*3;
/* compute pixel color (24 bit = 3 bajts) */
if (Iteration==IterationMax)
{ /* interior of Filled-in Julia set = */
/* Z = Z0 */
Zx=Z0x;
Zy=Z0y;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
dX=Zx-AlphaX;
dY=Zy-AlphaY;
d=dX*dX+dY*dY;
for (Iteration=0;Iteration<IterationMax && (d>AR2);Iteration++)
{
Zy=2*Zx*Zy + Cy;
Zx=Zx2-Zy2 +Cx;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
dX=Zx-AlphaX;
dY=Zy-AlphaY;
d=dX*dX+dY*dY;
};
/* */
if (Zy>AlphaY)
{
array[iTemp]=0; /* Red*/
array[iTemp+1]=0; /* Green */
array[iTemp+2]=0;/* Blue */
}
else
{
array[iTemp]=255; /* Red*/
array[iTemp+1]=255; /* Green */
array[iTemp+2]=255;/* Blue */
};
}
else
/* exterior of Filled-in Julia set */
/* */
if (Zy>0)
{
array[iTemp]=0; /* Red*/
array[iTemp+1]=0; /* Green */
array[iTemp+2]=0;/* Blue */
}
else
{
array[iTemp]=255; /* Red*/
array[iTemp+1]=255; /* Green */
array[iTemp+2]=255;/* Blue */
};
/* check the orientation of Z-plane */
/* mark first quadrant of cartesian plane*/
// if (Z0x>0 && Z0y>0) array[((iYmax-iY-1)*iXmax+iX)*3]=255-array[((iYmax-iY-1)*iXmax+iX)*3];
}
}
/* draw fixed points ----------------------------------------------------*/
/* translate from world to screen coordinate */
iX=(AlphaX-ZxMin)/PixelWidth;
iY=(AlphaY-ZxMin)/PixelHeight; /* */
/* plot big green pixel = 6 pixel wide */
for(iYinc=-iIncMax;iYinc<iIncMax;++iYinc){
for(iXinc=-iIncMax;iXinc<iIncMax;++iXinc)
{
iTemp=((iYmax-iY-1+iYinc)*iXmax+iX+iXinc)*3;
array[iTemp]=0;
array[iTemp+1]=255;
array[iTemp+2]=0;
}
}
/* translate from world to screen coordinate */
iX=(BetaX-ZxMin)/PixelWidth;
iY=(BetaY-ZyMin)/PixelHeight; /* */
/* plot big green pixel = 6 pixel wide */
for(iYinc=-iIncMax;iYinc<iIncMax;++iYinc){
for(iXinc=-iIncMax;iXinc<iIncMax;++iXinc)
{
iTemp=((iYmax-iY-1+iYinc)*iXmax+iX+iXinc)*3;
array[iTemp]=255;
array[iTemp+1]=0;
array[iTemp+2]=0;
}
}
/* write the whole data array to ppm file in one step ----------------------- */
/*create new file,give it a name and open it in binary mode */
fp= fopen(filename,"wb"); /* b - binary mode */
if (fp == NULL){ fprintf(stderr,"file error"); }
else
{
/*write ASCII header to the file*/
fprintf(fp,"P6\n %s\n %d\n %d\n %d\n",comment,iXmax,iYmax,MaxColorComponentValue);
/*write image data bytes to the file*/
fwrite(array,iLength ,1,fp);
fclose(fp);
fprintf(stderr,"file saved");
getchar();
}
free(array);
return 0;
} /* if (array .. else ... */
}
References
[edit]File history
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 09:26, 11 June 2011 | 1,000 × 1,000 (161 KB) | Soul windsurfer (talk | contribs) | I have made 10 000 x 10 000 ppm image and resized with Image Magic : convert b1.ppm -resize 1000x1000 b.jpg | |
15:39, 24 January 2008 | 2,000 × 2,000 (618 KB) | Soul windsurfer (talk | contribs) | {{Information |Description= Fatou set for F(z)=z*z. Made with binary decomposition of basins of both ( finite and infinite) attractors. Finite is z=Alpha=0, infinite attractor is point at infinity. |Source=self-made using C console program. |Date=24.11.2 |
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