File:Julia set of rational function f(z)=z^2(3 − z^4 ) over 2.png
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Summary
DescriptionJulia set of rational function f(z)=z^2(3 − z^4 ) over 2.png |
English: Julia set of polynomial function f(z)=z^2(3 − z^4 )/2 |
Source | Own work |
Author | Adam majewski |
Other versions | see figure 4 on the page 19 in paper "ON THURSTON’S PULLBACK MAP" by XAVIER BUFF, ADAM EPSTEIN, SARAH KOCH, AND KEVIN PILGRIM |
Long description
basin | test | color |
---|---|---|
basin of attration to infinity | bailout or escape test for z=infinity : | iColorOfExterior = 245; |
basin to attraction to z=0 | test for falling into finite attractor z=1 : | iColorsOfInterior[1]=230; |
basin of attraction to z=1 | test for falling into finite attractor z=0 : | iColorsOfInterior[0]=200; |
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C src code
/*
Adam Majewski
adammaj1 aaattt o2 dot pl // o like oxygen not 0 like zero
fraktal.republika.pl
c console progam
ON THURSTON’S PULLBACK MAP by XAVIER BUFF, ADAM EPSTEIN, SARAH KOCH, AND KEVIN PILGRIM
see figure 4 on the page 19
How to compute iteration :
z:x+y*%i;
z1:z^2*(3-z^4)/2;
realpart(z1);
((x^2−y^2)*(−y^4+6*x^2*y^2−x^4+3)−2*x*y*(4*x*y^3−4*x^3*y))/2
imagpart(z1);
(2*x*y*(−y^4+6*x^2*y^2−x^4+3)+(x^2−y^2)*(4*x*y^3−4*x^3*y))/2
a= (x^2−y^2)
b=(−y^4+6*x^2*y^2−x^4+3)
c= 2*x*y
d= (4*x*y^3−4*x^3*y)
so
re(z1) = (a*b-c*d)/2
im(z1) = (c*b+a*d)/2
there are 2 finite superattracting fixed points :
za=0
zb=1
gcc r.c -lm -Wall -march=native
time ./a.out
m
*/
#include <stdio.h>
#include <stdlib.h> // malloc
#include <string.h> // strcat
#include <math.h> // M_PI; needs -lm also
#include <complex.h>
/* --------------------------------- global variables and consts ------------------------------------------------------------ */
// radius of the target set ( circle around alfa fixed point ); it is related with iHeight
// so changing iHeight needs change of iMaxDistance2fixed
#define iMaxDistance2fixed 10 // distance point to alfa fixed point in pixels 150 when iHeight=1000; 280 when iHeight=2000
int iMaxDistance2fixed2;
double dMaxDistance2fixed2; // = (iMaxDistance2fixed*PixelWidth)^2
double dMaxDistance2fixed;
// virtual 2D array and integer ( screen) coordinate
// Indexes of array starts from 0 not 1
//unsigned int ix, iy; // var
static unsigned int ixMin = 0; // Indexes of array starts from 0 not 1
static unsigned int ixMax ; //
static unsigned int iWidth ; // horizontal dimension of array
static unsigned int iyMin = 0; // Indexes of array starts from 0 not 1
static unsigned int iyMax ; //
static unsigned int iHeight = 4000; //
// The size of array has to be a positive constant integer
static unsigned int iSize ; // = iWidth*iHeight;
// memmory 1D array
unsigned char *data;
unsigned char *edge;
// unsigned int i; // var = index of 1D array
//static unsigned int iMin = 0; // Indexes of array starts from 0 not 1
static unsigned int iMax ; // = i2Dsize-1 =
// The size of array has to be a positive constant integer
// unsigned int i1Dsize ; // = i2Dsize = (iMax -iMin + 1) = ; 1D array with the same size as 2D array
/* world ( double) coordinate = dynamic plane */
static const double ZxMin=-1.8;
static const double ZxMax=1.8;
static const double ZyMin=-1.8;
static const double ZyMax=1.8;
static double PixelWidth; // =(ZxMax-ZxMin)/ixMax;
static double PixelHeight; // =(ZyMax-ZyMin)/iyMax;
static double ratio ;
// complex numbers of parametr plane
double Cx; // c =Cx +Cy * i
double Cy;
double complex c; // parameter of function fc(z)=z^2 + c
static unsigned long int iterMax = 1000; //iHeight*100;
static double ER = 2.0; // Escape Radius for bailout test
static double ER2;
/* colors = shades of gray from 0 to 255 */
// 8 bit color = int number from 0 to 255
unsigned char iColorsOfInterior[2]={200,230}; // NumberOfPetal of colors = iPeriodChild
static unsigned char iColorOfExterior = 245;
static unsigned char iColorOfUnknown = 100;
long int iUknownPixels=0;
/* ------------------------------------------ functions -------------------------------------------------------------*/
//------------------complex numbers -----------------------------------------------------
// from screen to world coordinate ; linear mapping
// uses global cons
double GiveZx(unsigned int ix)
{ return (ZxMin + ix*PixelWidth );}
// uses globaal cons
double GiveZy(unsigned int iy)
{ return (ZyMax - iy*PixelHeight);} // reverse y axis
/* ----------- array functions = drawing -------------- */
/* 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*iWidth; }
// plots raster point (ix,iy)
int iDrawPoint(unsigned char A[], unsigned int ix, unsigned int iy, unsigned char iColor)
{
/* i = Give_i(ix,iy) compute index of 1D array from indices of 2D array */
A[Give_i(ix,iy)] = iColor;
return 0;
}
// draws point to memmory array data
// uses complex type so #include <complex.h> and -lm
int dDrawPoint(unsigned char A[], complex double point,unsigned char iColor )
{
unsigned int ix, iy; // screen coordinate = indices of virtual 2D array
//unsigned int i; // index of 1D array
ix = (creal(point)- ZxMin)/PixelWidth;
iy = (ZyMax - cimag(point))/PixelHeight; // inverse Y axis
iDrawPoint(A, ix, iy, iColor);
return 0;
}
//;;;;;;;;;;;;;;;;;;;;;; setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
int setup()
{
printf("setup\n");
/* 2D array ranges */
iWidth = iHeight;
iSize = iWidth*iHeight; // size = number of points in array
// iy
iyMax = iHeight - 1 ; // Indexes of array starts from 0 not 1 so the highest elements of an array is = array_name[size-1].
//ix
ixMax = iWidth - 1;
/* 1D array ranges */
// i1Dsize = i2Dsize; // 1D array with the same size as 2D array
iMax = iSize-1; // Indexes of array starts from 0 not 1 so the highest elements of an array is = array_name[size-1].
/* Pixel sizes */
PixelWidth = (ZxMax-ZxMin)/ixMax; // ixMax = (iWidth-1) step between pixels in world coordinate
PixelHeight = (ZyMax-ZyMin)/iyMax;
ratio = ((ZxMax-ZxMin)/(ZyMax-ZyMin))/((float)iWidth/(float)iHeight); // it should be 1.000 ...
// for numerical optimisation in iteration
ER2 = ER * ER;
iMaxDistance2fixed2 =iMaxDistance2fixed * iMaxDistance2fixed;
dMaxDistance2fixed2 = iMaxDistance2fixed2*PixelWidth*PixelWidth; // dMaxDistance2fixed^2
dMaxDistance2fixed = sqrt(dMaxDistance2fixed2); // maybe it should be in reversed order ??
/* create dynamic 1D arrays for colors ( shades of gray ) */
data = malloc( iSize * sizeof(unsigned char) );
edge = malloc( iSize * sizeof(unsigned char) );
if (edge== NULL || data == NULL)
{
fprintf(stderr," Could not allocate memory");
getchar();
return 1;
}
printf(" end of setup \n");
return 0;
} // ;;;;;;;;;;;;;;;;;;;;;;;;; end of the setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
unsigned char ComputeColor(unsigned int ix, unsigned int iy, int IterationMax)
{
// check behavour of z under f(z)=z^2+c
// using 3 target set:
// 1. exterior or circle (center at origin and radius ER )
// as a target set containing infinity = for escaping points ( bailout test)
// for points of exterior of julia set
// 2. interior of circle with center z=0 and radius=dMaxDistance2fixed
// 3. interior of the circle with center z=1 and radius = dMaxDistance2fixed
// Z= Zx+ZY*i;
double Zx2, Zy2;
int i=0;
//int j; // iteration = fc(z)
double d2 ; /* d2= (distance from z to zb)^2 */
double Zxt,Zyt ; //
double Zx, Zy;
double a,b,c,d; // temporary variables
int iDistance;
// from screen to world coordinate
Zx = GiveZx(ix);
Zy = GiveZy(iy);
/* distance from z to zb=1 */
Zxt=Zx-1.0;
Zyt=Zy;
d2=Zxt*Zxt +Zyt*Zyt;
if (d2<dMaxDistance2fixed2)
{
iDistance = (int)(sqrt(d2)/PixelWidth);
if (iDistance<iMaxDistance2fixed)
{
return iColorsOfInterior[1];
}
}
/* distance from z to za=0 */
d2=Zx*Zx +Zy*Zy;
if (d2<dMaxDistance2fixed2)
{
iDistance = (int)(sqrt(d2)/PixelWidth);
if (iDistance<iMaxDistance2fixed)
{
return iColorsOfInterior[0];
}
}
// if not inside target set around
while (1 )
{ // then iterate
Zx2 = Zx*Zx;
Zy2 = Zy*Zy;
// bailout test
if (Zx2 + Zy2 > ER2) return iColorOfExterior; // if escaping stop iteration
// if not escaping or not attracting then iterate = check behaviour
// new_z = f(z) = z^2*(3-z^4)/2
a= Zx*Zx-(Zy*Zy);
b=-(Zy*Zy*Zy*Zy)+6*Zx*Zx*Zy*Zy-(Zx*Zx*Zx*Zx)+3;
c= 2*Zx*Zy;
d= 4*Zx*Zy*Zy*Zy-4*Zx*Zx*Zx*Zy;
Zx = (a*b-c*d)/2;
Zy = (c*b+a*d)/2;
//
i+=1;
/* distance from z to zb=1 */
Zxt=Zx-1.0;
Zyt=Zy;
d2=Zxt*Zxt +Zyt*Zyt;
if (d2<dMaxDistance2fixed2)
{
iDistance = (int)(sqrt(d2)/PixelWidth);
if (iDistance<iMaxDistance2fixed)
{
return iColorsOfInterior[1];
}
}
/* distance from z to za=0 */
d2=Zx*Zx +Zy*Zy;
if (d2<dMaxDistance2fixed2)
{
iDistance = (int)(sqrt(d2)/PixelWidth);
if (iDistance<iMaxDistance2fixed)
{
return iColorsOfInterior[0];
}
}
if (i > IterationMax) break;
}
// pixel is not escaping to infinity or not attracting to fixed attractore :
// change parameters : iterMax, distance ...
iUknownPixels+=1;
return iColorOfUnknown ; //
}
// plots raster point (ix,iy)
int PlotPoint(unsigned char A[] , unsigned int ix, unsigned int iy, int IterationMax)
{
unsigned i; /* index of 1D array */
unsigned char iColor;
i = Give_i(ix,iy); /* compute index of 1D array from indices of 2D array */
iColor = ComputeColor(ix, iy, IterationMax);
A[i] = iColor;
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int ComputeFatouComponents(unsigned char A[], int IterationMax )
{
unsigned int ix, iy; // pixel coordinate
printf("compute image \n");
// for all pixels of image
for(iy = iyMin; iy<=iyMax; ++iy)
{ printf(" %d z %d\n", iy, iyMax); //info
for(ix= ixMin; ix<=ixMax; ++ix) PlotPoint(A, ix, iy, IterationMax ) ; //
}
return 0;
}
int ComputeBoundariesIn(unsigned char A[])
{
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;
// boundaries are in edge array ( global var )
printf(" find boundaries in A array using Sobel filter\n");
// #pragma omp parallel for schedule(dynamic) private(i,iY,iX,Gv,Gh,G) shared(iyMax,ixMax, ER2)
for(iY=1;iY<iyMax-1;++iY){
for(iX=1;iX<ixMax-1;++iX){
Gv= A[Give_i(iX-1,iY+1)] + 2*A[Give_i(iX,iY+1)] + A[Give_i(iX-1,iY+1)] - A[Give_i(iX-1,iY-1)] - 2*A[Give_i(iX-1,iY)] - A[Give_i(iX+1,iY-1)];
Gh= A[Give_i(iX+1,iY+1)] + 2*A[Give_i(iX+1,iY)] + A[Give_i(iX-1,iY-1)] - A[Give_i(iX+1,iY-1)] - 2*A[Give_i(iX-1,iY)] - A[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]=255;} /* background */
else {edge[i]=0;} /* boundary */
}
}
return 0;
}
int CopyBoundariesTo(unsigned char A[])
{
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) A[i]=0;}
return 0;
}
// Check Orientation of image : mark first quadrant
// it should be in the upper right position
// uses global var : ...
int CheckOrientation(unsigned char A[] )
{
unsigned int ix, iy; // pixel coordinate
double Zx, Zy; // Z= Zx+ZY*i;
unsigned i; /* index of 1D array */
for(iy=iyMin;iy<=iyMax;++iy)
{
Zy = GiveZy(iy);
for(ix=ixMin;ix<=ixMax;++ix)
{
// from screen to world coordinate
Zx = GiveZx(ix);
i = Give_i(ix, iy); /* compute index of 1D array from indices of 2D array */
if (Zx>0 && Zy>0) A[i]=255-A[i]; // check the orientation of Z-plane by marking first quadrant */
}
}
return 0;
}
// save "A" array to pgm file
int SaveArray2PGMFile( unsigned char A[], double k)
{
FILE * fp;
const unsigned int MaxColorComponentValue=255; /* color component is coded from 0 to 255 ; it is 8 bit color file */
char name [30]; /* name of file */
sprintf(name,"%.0f", k); /* */
char *filename =strcat(name,".pgm");
char *comment="# Numerical approximation of Julia set for f(z)= z^2*(3-z^4)/2; Adam Majewski";/* 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,iWidth,iHeight,MaxColorComponentValue); /*write header to the file*/
fwrite(A,iSize,1,fp); /*write A array to the file in one step */
printf("File %s saved. \n", filename);
fclose(fp);
return 0;
}
int info()
{
// diplay info messages
printf("Numerical approximation of Julia set for f(z)= z^2*(3-z^4)/2; \n");
printf("Image Width = %f \n", ZxMax-ZxMin);
printf("PixelWidth = %f \n", PixelWidth);
printf("size of target set in screen units = iMaxDistance2fixed = %d pixels \n", iMaxDistance2fixed);
printf("size of target set in world units = dMaxDistance2fixed = %f ; \n", dMaxDistance2fixed);
printf("Maximal number of iterations = iterMax = %ld \n", iterMax);
printf("ratio of image = %f ; it should be 1.000 ...\n", ratio);
printf("Unknown pixels = %ld ; it should be 0 ...\n", iUknownPixels);
return 0;
}
/* ----------------------------------------- main -------------------------------------------------------------*/
int main()
{
setup();
ComputeFatouComponents(data, iterMax);
SaveArray2PGMFile( data, iHeight+0); // save array data (components of Fatou set ) to pgm file
ComputeBoundariesIn(data);
SaveArray2PGMFile( edge, iHeight+1); // save array edge (Julia set ) to pgm file
CopyBoundariesTo(data);
SaveArray2PGMFile( data, iHeight+2); // save array data (Julia set and components ) to pgm file
printf(" allways free memory to avoid buffer overflow \n");
free(data);
free(edge);
info();
return 0;
}
Text output :
File 4000.pgm saved. find boundaries in A array using Sobel filter File 4001.pgm saved. copy boundaries from edge array to data array File 4002.pgm saved. File 4003.pgm saved. allways free memory to avoid buffer overflow Numerical approximation of Julia set for f(z)= z^2*(3-z^4)/2; Image Width = 3.600000 PixelWidth = 0.000900 size of target set in screen units = iMaxDistance2fixed = 10 pixels size of target set in world units = dMaxDistance2fixed = 0.009002 ; Maximal number of iterations = iterMax = 1000 ratio of image = 1.000000 ; it should be 1.000 ... Unknown pixels = 0 ; it should be 0 ...
Image Magic src code
convert 4003.pgm -resize 1000x1000 s4003.png
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