File:Fatou componenets4.jpg
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Summary
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| Description | Filled Julia set with marked components ( color) and centers of components ( black rectangle). Period = 4 . |
| Source | Own work |
| Author | Adam majewski |
Long desciption
[edit]
This image shows dynamical plane for f(z)=z*z + 0.281+0.533*i. There are 2 sets :
- Julia set ( boundary of filled Julia set)
- Fatou set, which consists of :
- basin of attraction of infinity ( light magenta points)
- basin of attraction of finite attractor ( points in 4 colors except white). This basin consists of infinitely many components
In this program period is set manually ( to do ).
Algorithm of coloring comonents is described by E Demidov:
color of component=last_iteration % period[1]
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. Use file viewer to see it.
/*
c console program:
1. draws components of Filled-in Julia set for Fc(z)=z*z +c
using escape time
it works except GivePeriod
-------------------------------
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;};
}
void GiveColorFromList6(int i ,unsigned char c[])
{ /* */
switch( i){
case 0: {
c[0] = 210;
c[1] = 0;
c[2] = 0;
break;
};
case 1: {
c[0] = 0 ;
c[1] = 0;
c[2] = 255;
break;
};
case 2: {
c[0] = 0;
c[1] = 255;
c[2] = 0;
break;
};
case 3: {
c[0] = 0;
c[1] = 255 ;
c[2] = 255;
break;
};
case 4: {
c[0] = 255;
c[1] = 0;
c[2] = 255;
break;
};
case 5: {
c[0] = 255;
c[1] = 0;
c[2] = 255 ;
break;
};
case 6: {
c[0] = 164;
c[1] = 136;
c[2] = 88 ;
break;
};
default: {
c[0] = 100;
c[1] = 100;
c[2] = 0;
break;
}
}
}
/* ----------------------*/
int main()
{ // 4: -0.15652016683376e+000 -1.03224710892283; 0.28227139076691e+000 -0.53006061757853
// 4: c=0.281+0.533, AR2=0.001, ER2=8*8,
//3: -0.12256116687665e+000 -0.74486176661974e
// 2: -1
// 1: 0.4*i
const double Cx=0.281,
Cy=0.533;
/* screen coordinate = coordinate of pixels */
int period=4,
p,
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 */
tempZx,
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,
ZAx,ZAy, /* attractor ZA = ZAx+ZAy*i */
Z_cr_x=0.0, Z_cr_y=0.0; /* critical point */
/* */
int Iteration,iIteration,
IterationMax=80,
IterationMaxBig=1000,
iTemp;
/* bail-out value , radius of circle ; */
const int EscapeRadius=8;
int ER2=EscapeRadius*EscapeRadius;
double //AR=PixelWidth, /* minimal distance from attractor = Attractor Radius */
AR2=0.001, /* proportional to period */
d,dX,dY; /* distance from attractor : d=sqrt(dx*dx+dy*dy) */
/* PPM file */
FILE * fp;
char *filename="fatou_4b.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;
static unsigned char color[3];
/* --------- find fixed points ---------------------------------*/
/* 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=0.0;//0.5-SqrtDeltaX/2;
AlphaY=0.0;//-SqrtDeltaY/2;
AbsLambdaA=2*sqrt(AlphaX*AlphaX+AlphaY*AlphaY);
AbsLambdaB=2*sqrt(BetaX*BetaX+BetaY*BetaY);
/* -- find attractor ZA = ZAx+ZAy*i ---------------*/
/* Z = 0 = critical point */
Zx=0.0;
Zy=0.0;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
/* */
for (Iteration=0;Iteration<IterationMaxBig && ((Zx2+Zy2)<ER2);Iteration++)
{
Zy=2*Zx*Zy + Cy;
Zx=Zx2-Zy2 +Cx;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
};
ZAx=Zx;
ZAy=Zy;
/*----*/
//period= GivePeriod( Cx,Cy,ZAx, ZAy,ER2,AR2 );
/* --------------- info -------------------------*/
printf(" C= (%f , %f) \n",Cx,Cy);
printf(" Fixed points : \n");
printf(" Beta= %f , %f\n",BetaX,BetaY);
printf(" Alpha= %f, %f\n",AlphaX,AlphaY);
printf(" abs(Lambda (Alpha))= %f\n",AbsLambdaA);
printf(" abs(lambda(Beta))= %f\n",AbsLambdaB);
printf(" Limit distances : \n");
printf(" AR2= %f\n",AR2);
printf(" ER2= %d\n",ER2);
printf(" Attractor : \n");
printf(" ZA= %f , %f\n",ZAx,ZAy);
printf(" period= %d\n",period);
/*--------------------------------------------------------*/
array = malloc( iLength * sizeof(unsigned char) );
if (array == NULL)
{
fprintf(stderr,"Could not allocate memory");
getchar();
return 1;
}
else
{
printf(" I'm working. Wait \n");
/* 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 (iIteration=0;iIteration<IterationMax && (d>AR2);iIteration++)
{
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;
};
/* iLSM */
if (iIteration==IterationMax)
{ // points of interior which did not reached attractor = solid red
array[iTemp]=255; /* Red*/
array[iTemp+1]=0; /* Green */
array[iTemp+2]=0;/* Blue */
}
else
{ /* components of filled Julia set */
iIteration%=period;// modulo period
GiveColorFromList6(iIteration ,color);
array[iTemp]=color[0]; /* Red*/
array[iTemp+1]=color[1]; /* Green */
array[iTemp+2]=color[2];/* Blue */
}
}
else /* exterior of Filled-in Julia set */
{ /* LSM */
//GiveColorFromList6(Iteration ,color);
int i=Iteration*5;
{
array[iTemp]=(255-i)%255; /* Red*/
array[iTemp+1]=(215-i)%255; /* Green */
array[iTemp+2]=(200-i)%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 special points ----------------------------*/
/* --- fixed points : Alpha and Beta ---------*/
iX=(AlphaX-ZxMin)/PixelWidth; /*translate from world to screen coordinate */
iY=(AlphaY-ZxMin)/PixelHeight; /* */
/* plot big green pixel = 2*iIncMax 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 red pixel = 2*iIncMax 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;
}
}
/* -------------- critical point ------------*/
/* translate from world to screen coordinate */
iX=(Z_cr_x-ZxMin)/PixelWidth;
iY=(Z_cr_y-ZyMin)/PixelHeight; /* */
/* plot big blue pixel = 2*iIncMax 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]=0;
array[iTemp+2]=255;
}
}
/*----------- attracting cycle ---------------------*/
Zx=ZAx;
Zy=ZAy;
for(p=0;p<period;p++)
{
/* translate from world to screen coordinate */
iX=(Zx-ZxMin)/PixelWidth;
iY=(Zy-ZyMin)/PixelHeight; /* */
/* plot big white pixel = 2*iIncMax 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]=0;
array[iTemp+2]=0;
}
}
/* compute new point */
tempZx= Zx*Zx - Zy*Zy + Cx;
Zy = 2*Zx*Zy + Cy;
Zx = tempZx;
}
/* ----- 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 ... */
}
Licensing
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I, the copyright holder of this work, hereby publish it under the following licenses:
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
|
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. |
You may select the license of your choice.
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 15:34, 30 June 2010 | | 2,000 × 2,000 (233 KB) | Soul windsurfer (talk | contribs) | smaller size |
| 11:02, 31 May 2008 |  | 10,000 × 10,000 (3.53 MB) | Soul windsurfer (talk | contribs) | earlier version had errors | |
| 10:24, 31 May 2008 |  | 15,000 × 15,000 (10.33 MB) | Soul windsurfer (talk | contribs) | bigger size | |
| 21:21, 26 May 2008 |  | 4,000 × 4,000 (1.52 MB) | Soul windsurfer (talk | contribs) | better quality | |
| 10:57, 25 May 2008 |  | 2,000 × 2,000 (343 KB) | Soul windsurfer (talk | contribs) | better quality | |
| 10:48, 25 May 2008 |  | 2,000 × 2,000 (152 KB) | Soul windsurfer (talk | contribs) | {{Information |Description=Filled Julia set with marked components |Source=self-made |Date= |Author= Adam majewski |Permission= |other_versions= }} |
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