File:Dynamic plane for parabolic parameter from period 2 thru internal angle 1 over 2.png
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[edit]DescriptionDynamic plane for parabolic parameter from period 2 thru internal angle 1 over 2.png |
English: Dynamic plane with parabolic Julia set for c = -1.25. It is the root point between period 2 component and period 4 component. Angled internal adress |
Date | |
Source | Own work |
Author | Adam majewski |
Compare with
[edit]-
Only Julia set
-
from period 1 to cusp
-
A parabolic checkerboard for rotation number 1 over 2
-
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C source code
[edit]/*
Adam Majewski
adammaj1 aaattt o2 dot pl // o like oxygen not 0 like zero
fraktal.republika.pl
c console progam
gcc c.c -lm -Wall -march=native
time ./a.out
gcc c.c -lm -Wall -march=native -fopenmp
method of filling gaps in rays turns
should be the same as for drawing rays !!!!!!!!!!!!!
!!!!!---------- algorithm ---------!!!!!!!!
trap ( target set) for iteration :
* of interior points = interior of circle centered at alfa and radius = dMaxDistance2Alfa, with iPeriodChild sectors described by periodic external rays landing on alfa
* of exterior points = exterior of circle centered at origin ( z=0) and radius = ER
Iterate point until it fails to one of 2 traps.
If it it fail into interior trap then check in which sector is it. Color = number of sector
Interior target set should be all inside Julia set
( of course not counting external rays that cross it and exterior parts which are very thin = means width < pixelwidth)
if it is a circle around alfa then it shrinks as iPeriodOfChild grows
( = time of creating image grows)
-----------------------------------------
Better is to take as a interior target set :
part of this circle bordered by 2 external rays
(One can choose the longest one !!!)
and color the interior proportionaly to
(i % iPeriodOfChild)
gcc c.c -lm -Wall -march=native
./a.out
without openmp
real 7m54.710s
user 7m54.861s
sys 0m0.008s
with openmp :
real 1m47.034s
user 14m8.420s
sys 0m0.280s
*/
#include <stdio.h>
#include <stdlib.h> // malloc
#include <string.h> // strcat
#include <math.h> // M_PI; needs -lm also
#include <complex.h>
#include <omp.h>
/* --------------------------------- global variables and consts ------------------------------------------------------------ */
#define iPeriodChild 4 // Period of secondary component joined by root point with the parent component
int iPeriodParent = 2; // main cardioid of Mandelbrot set
// 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 = 5000; //
// The size of array has to be a positive constant integer
static unsigned int iSize ; // = iWidth*iHeight;
// memmory 1D array
unsigned char *data; // fatou components
unsigned char *julia; // julia set = common edge between fatou components
unsigned char *edge; // pther edges
unsigned char *zero; // zero algorithm
// 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=-0.9;
static const double ZyMax=0.9;
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 = 10000; //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
// Arrays are 0-indexed, so the first array element is at index = 0, and the highest is =(size_of_array – 1)
//unsigned char iColorInterior[2][iPeriodChild]={{255,231}, {123,99}}; /* shades of gray used in image */
unsigned char iColors[4]={240,200,160, 120}; //
static unsigned char iColorOfExterior = 245;
// static unsigned char iColorOfInterior = 200;
unsigned char iColorOfUnknown = 100;
int iNumberOfUknknown = 0;
// period 2 cycle = {Zx2a, Zx2b} , Zy = 0
double Zx2a = -1.207106781186548;
double Zx2b = 0.207106781186547;
// limits : Zx2aMinus < Zx2a < Zx2aPlus < Zx2bMinus < Zx2b < Zx2bPlus
double Zx2aMinus;
double Zx2bMinus;
double Zx2aPlus;
double Zx2bPlus;
//static double TwoPi=2*M_PI;
/* ------------------------------------------ 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(int ParentPeriod, int ChildPeriod)
{
printf("setup\n");
ratio = (ZxMax-ZxMin)/(ZyMax-ZyMin);
Cx=-1.25;
Cy=0.0;
c=Cx+Cy*I;
/* 2D array ranges */
if (!(iHeight % 2)) iHeight+=1; // it sholud be even number (variable % 2) or (variable & 1)
iWidth = ratio*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;
//
Zx2bMinus = Zx2b - 0.2;
Zx2aMinus = Zx2a - 0.2;
Zx2bPlus = Zx2b + 0.2;
Zx2aPlus = Zx2a + 0.2;
/* create dynamic 1D arrays for colors ( shades of gray ) */
data = malloc( iSize * sizeof(unsigned char) );
edge = malloc( iSize * sizeof(unsigned char) );
julia = malloc( iSize * sizeof(unsigned char) );
zero = malloc( iSize * sizeof(unsigned char) );
if (edge== NULL || data == NULL || zero==NULL || julia ==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 fc(z)=z^2+c
// using 1 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
double Zx2, Zy2;
int i=0; // number of the iteration = fc(z)
double Zx, Zy;
// from screen to world coordinate
Zx = GiveZx(ix);
Zy = GiveZy(iy);
// if not inside target set around attractor ( alfa fixed point )
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 : Z(n+1) = Zn * Zn + C
Zy = 2*Zx*Zy + Cy;
Zx = Zx2 - Zy2 + Cx;
//
i+=1;
if (i > IterationMax) break;
}
if (Zx2aMinus < Zx && Zx < Zx2a ) return iColors[0];
if (Zx2a < Zx && Zx < Zx2aPlus ) return iColors[1];
if (Zx2bMinus < Zx && Zx < Zx2b ) return iColors[2];
if (Zx2b < Zx && Zx < Zx2bPlus ) return iColors[3];
iNumberOfUknknown +=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;
}
int FillArrayWithColor(unsigned char A[], unsigned char color )
{
int i;
for(i = 0; i <= iSize; ++i) A[i]=color;
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
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(ixMax , iyMax, IterationMax)
for(iy = iyMin; iy<=iyMax; ++iy)
{ printf(" %d z %d \r", iy, iyMax); //info
for(ix= ixMin; ix<=ixMax; ++ix) PlotPoint(A, ix, iy, IterationMax ) ; //
}
return 0;
}
// from Source to Destination
int ComputeBoundaries(unsigned char S[], unsigned char D[])
{
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 D array ( global var )
// clear D array
FillArrayWithColor(D , iColorOfExterior);
// printf(" find boundaries in S 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= S[Give_i(iX-1,iY+1)] + 2*S[Give_i(iX,iY+1)] + S[Give_i(iX-1,iY+1)] - S[Give_i(iX-1,iY-1)] - 2*S[Give_i(iX-1,iY)] - S[Give_i(iX+1,iY-1)];
Gh= S[Give_i(iX+1,iY+1)] + 2*S[Give_i(iX+1,iY)] + S[Give_i(iX-1,iY-1)] - S[Give_i(iX+1,iY-1)] - 2*S[Give_i(iX-1,iY)] - S[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) {D[i]=255;} /* background */
else {D[i]=0;} /* boundary */
}
}
return 0;
}
// copy from Source to Destination
int CopyBoundaries(unsigned char S[], unsigned char D[])
{
unsigned int iX,iY; /* indices of 2D virtual array (image) = integer coordinate */
unsigned int i; /* index of 1D array */
//printf("copy boundaries from S array to D array \n");
for(iY=1;iY<iyMax-1;++iY)
for(iX=1;iX<ixMax-1;++iX)
{i= Give_i(iX,iY); if (S[i]==0) D[i]=0;}
return 0;
}
int DrawCriticalOrbit(unsigned char A[], unsigned int IterMax)
{
// integer = pixel coordinate
unsigned int ix, iy;
// double = world coordinate
// critical point Z= Zx+ZY*i;
double Zx=0.0;
double Zy=0.0;
double Zx2=0.0;
double Zy2=0.0;
unsigned int i; /* index of 1D array */
unsigned int j; // number of iteration
// draw critical point
ix = (int)((Zx-ZxMin)/PixelWidth);
iy = (int)((ZyMax-Zy)/PixelHeight); // reverse y axis
i = Give_i(ix, iy); /* compute index of 1D array from indices of 2D array */
A[i]=255-A[i];
// iterate
for (j = 1; j <= IterMax; j++) //larg number of iteration s
{ Zx2 = Zx*Zx;
Zy2 = Zy*Zy;
// bailout test
if (Zx2 + Zy2 > ER2) return iColorOfExterior; // if escaping stop iteration
// if not escaping iterate
// Z(n+1) = Zn * Zn + C
Zy = 2*Zx*Zy + Cy;
Zx = Zx2 - Zy2 + Cx;
//compute integer coordinate
ix = (int)((Zx-ZxMin)/PixelWidth);
iy = (int)((ZyMax-Zy)/PixelHeight); // reverse y axis
i = Give_i(ix, iy); /* compute index of 1D array from indices of 2D array */
A[i]=0; //255-A[i]; // mark the critical orbit
}
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;
}
unsigned char ComputeColorZero(unsigned int ix, unsigned int iy, int iMax)
{
// check behavour of z under fc(z)=z^2+c
// using 1 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
double Zx2, Zy2;
int i; // number of the iteration = fc(z)
unsigned char iColor;
double Zx, Zy;
i = 0;
// from screen to world coordinate
Zx = GiveZx(ix);
Zy = GiveZy(iy);
while (i <iMax)
{ // then iterate
Zx2 = Zx*Zx;
Zy2 = Zy*Zy;
// if not escaping or not attracting then iterate = check behaviour
// new z : Z(n+1) = Zn * Zn + C
Zy = 2*Zx*Zy + Cy;
Zx = Zx2 - Zy2 + Cx;
//
i+=1;
}
//
if ( Zy>0) iColor = 150; //re(z_n) > 0 and im(z_n) > 0 (first quadrant)
// if ( Zx<0 && Zy>0) iColor = 170; //re(z_n) < 0 and im(z_n) > 0 (second)
//if ( Zx<0 && Zy<0) iColor = 190; //re(z_n) < 0 and im(z_n) < 0 (third)
if ( Zy<0) iColor = 200; //re(z_n) > 0 and im(z_n) < 0 (fourth).
//
return iColor;
}
// plots raster point (ix,iy)
int PlotPointZero(unsigned char A[] , unsigned int ix, unsigned int iy, int n)
{
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 = ComputeColorZero(ix, iy, n);
// mark Fatou components, first interior
if (data[i]==iColors[0] || data[i]==iColors[1] || data[i]==iColors[2] || data[i]==iColors[3] )
A[i] = iColor+15; // !!!
else // exterior
{if (n<10) A[i]= iColor; else A[i]= iColorOfExterior;} // exterior , only for low n
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int ComputeZerosOfQnc(unsigned char A[], int n)
{
unsigned int ix, iy; // pixel coordinate
//printf("compute image \n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(ixMax , iyMax, n)
for(iy = iyMin; iy<=iyMax; ++iy)
{ printf(" %d z %d \r", iy, iyMax); //info
for(ix= ixMin; ix<=ixMax; ++ix) PlotPointZero(A, ix, iy, n ) ; //
}
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="# ";/* 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 parabolic Julia set for fc(z)= z^2 + c \n");
printf("iPeriodParent = %d \n", iPeriodParent);
printf("iPeriodOfChild = %d \n", iPeriodChild);
printf("parameter c = ( %f ; %f ) \n", Cx, Cy);
printf("Image Width = %f \n", ZxMax-ZxMin);
printf("PixelWidth = %f \n", PixelWidth);
printf("Maximal number of iterations = iterMax = %ld \n", iterMax);
printf("iNumberOfUknknown = %d \n", iNumberOfUknknown);
printf("ratio of image = %f \n", ratio);
return 0;
}
/* ----------------------------------------- main -------------------------------------------------------------*/
int main()
{
setup(iPeriodParent, iPeriodChild);
int n;
printf("components of Fatou set : \n");
ComputeFatouComponents(data, iterMax);
SaveArray2PGMFile( data, 10000);
printf("done \n");
ComputeBoundaries(data, julia);
SaveArray2PGMFile( julia, 20000);
printf(" Julia set \n");
CopyBoundaries(julia, data);
SaveArray2PGMFile( data, 30000);
for(n = 0; n<=25; ++n)
{
ComputeZerosOfQnc(zero, n);
ComputeBoundaries(zero,edge);
CopyBoundaries(julia, edge); // julia is computed more precisely then edge
CopyBoundaries(edge, zero);
SaveArray2PGMFile( zero, n);
}
printf(" allways free memory to avoid buffer overflow \n");
free(data);
free(edge);
free(zero);
free(julia);
info();
return 0;
}
Image Magic src code
[edit]This big image is downsized with Image Magic console tool :
convert d.pgm -resize 2000x1000 d.png
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