File:Level sets of attraction time to parabolic fixed point in the fat basilica Julia set.png
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Summary
DescriptionLevel sets of attraction time to parabolic fixed point in the fat basilica Julia set.png |
English: Level sets of attraction time to parabolic fixed point in the fat basilica Julia set. Target set is a circle inside Julia set with parabolic fixed point on it's boundary |
Date | |
Source | Own work |
Author | Adam majewski |
Other versions |
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Licensing
I, the copyright holder of this work, hereby publish it under the following license:
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Algorithm
How to choose size of attracting petal ( radius of a circle with parabolic point on it's boundary) such that level curves cross at z= 0 ?
// choose such value that level sets cross at z=0
// choose radius such a
double GivePetalRadius(complex double c, complex double fixed, int n){
complex double z = 0.0; // critical point
int k;
// best for n>1
int kMax = (n*ChildPeriod) - 1; // ????
for(k=0; k<kMax-1; ++k)
z = z*z + c; // forward iteration
return cabs(z-fixed)/2.0;
}
c source code
/*
Adam Majewski
adammaj1 aaattt o2 dot pl // o like oxygen not 0 like zero
https://plus.google.com/116648956837292097606/posts/b6J6z2u8soL
-------------------------------
cd existing_folder
git init
git remote add origin git@gitlab.com:adammajewski/SepalsOfCauliflower.git
git add .
git commit
git push -u origin master
---------------------------------
indent d.c
default is gnu style
-------------------
c console progam
gcc b.c -lm -Wall -march=native
time ./a.out
gcc b.c -lm -Wall -march=native -fopenmp
time ./a.out
time ./a.out >a.txt
----------------------
*/
#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 ------------------------------------------------------------ */
// https://mrob.com/pub/muency/child.html
int ChildPeriod = 2; // Period of secondary component joined by root point with the parent component
int ParentPeriod = 1; // 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 = 2000; //
// 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
static const double ZxMin = -1.6; //-0.05;
static const double ZxMax = 1.6; //0.75;
static const double ZyMin = -1.6; //-0.1;
static const double ZyMax = 1.6; //0.7;
static double PixelWidth; // =(ZxMax-ZxMin)/ixMax;
static double PixelHeight; // =(ZyMax-ZyMin)/iyMax;
static double ratio;
// complex numbers of parametr plane
double complex c; // parameter of function fc(z)=z^2 + c
double complex a; // alfa fixed point
static unsigned long int iterMax = 1000000; //iHeight*100;
static double ER = 2.0; // Escape Radius for bailout test
static double ER2;
double radius; //= 1.0-cabs(1.0-csqrt(1.0-4.0*c)) ; //0.1; // half of distance between critical point and fixed point
//double D2MaxGlobal; //= 0.0497920256372717 ;
//double DistanceMaxGlobal2 ;
/* colors = shades of gray from 0 to 255 */
static unsigned char iColorOfExterior = 250;
static unsigned char iColorOfInterior = 150;
unsigned char ColorStep; // (240- iColorOfInterior)/ChildPeriod
unsigned char iColorOfUnknown = 50;
int NoOfUnknownPoints = 0;
/* ------------------------------------------ functions -------------------------------------------------------------*/
//------------------complex numbers -----------------------------------------------------
/*
c functions using complex type numbers
computes c from component of Mandelbrot set */
complex double Give_c( int Period, int p, int q , double InternalRadius )
{
complex double w; // point of reference plane where image of the component is a unit disk
complex double c; // result
double t; // InternalAngleInTurns
t = (double) p/q;
t = t * M_PI * 2.0; // from turns to radians
w = InternalRadius*cexp(I*t); // map to the unit disk
switch ( Period ) // of component
{
case 1: // main cardioid = only one period 1 component
c = w/2 - w*w/4; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_1
break;
case 2: // only one period 2 component
c = (w-4)/4 ; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_2
break;
// period > 2
default:
printf("higher periods : to do, use newton method \n");
printf("for each q = Period of the Child component there are 2^(q-1) roots \n");
c = 10000.0; // bad value
break; }
return c;
}
// fast cabs
double cabs2(complex double z) {
return (creal(z) * creal(z) + cimag(z) * cimag(z));
}
// from screen to world coordinate ; linear mapping
// uses global cons
double
GiveZx ( int ix)
{
return (ZxMin + ix * PixelWidth);
}
// uses globaal cons
double GiveZy (int iy) {
return (ZyMax - iy * PixelHeight);
} // reverse y axis
complex double GiveZ( int ix, int iy){
double Zx = GiveZx(ix);
double Zy = GiveZy(iy);
return Zx + Zy*I;
}
/* ----------- 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;
}
// bailout test
// escapes = abs(z)> ER
int Escapes(complex double z){
if (cabs2(z)>ER2) return 1;
return 0;
}
int IsInTarget(complex double z){
// here target set is a circle inside immediate basin component containing critical point
// with fixed point on it's boundary
// attracting petal
complex double center = a+radius;
if (cabs(z-center) <= radius) return 1;
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
memset(D, iColorOfExterior, iSize*sizeof(*D)); // for heap-allocated arrays, where N is the number of elements = 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;
}
// Interior: Level Sets of Attraction time
unsigned char ComputeColorOfInteriorLevelSets(complex double z){
int nMax = iterMax;
int n;
int p;
int pMax = ChildPeriod;
for (n=0; n < nMax; n++){ //forward iteration
if (Escapes(z)) return iColorOfExterior;
for (p=0; p < pMax; p++){ //forward iteration
if (IsInTarget(z)) return iColorOfInterior + (n % ChildPeriod)* ColorStep; // immediate basin of attraction and it's preimages
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
}
printf("unknown point : z = %.16f; %.16f\n", creal(z), cimag(z));
NoOfUnknownPoints +=1;
return iColorOfUnknown;
}
// plots raster point (ix,iy)
int DrawPointOfInteriorLevelSets (unsigned char A[], int ix, int iy)
{
int i; /* index of 1D array */
unsigned char iColor;
complex double z;
i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
z = GiveZ(ix,iy);
iColor = ComputeColorOfInteriorLevelSets(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImageOfInteriorLevelSets (unsigned char A[])
{
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(A, ixMax , iyMax, iColorOfUnknown)
for (iy = iyMin; iy <= iyMax; ++iy){
//printf (" %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfInteriorLevelSets(A, ix, iy); //
}
return 0;
}
// INterior : components of Immediate Basin of Attraction
unsigned char ComputeColorOfImmediateBasin(complex double z){
int nMax = iterMax;
int n;
for (n=0; n < nMax; n++){ //forward iteration
if (Escapes(z)) return iColorOfExterior;
if (IsInTarget(z)) return iColorOfInterior + (n % ChildPeriod)* ColorStep; // immediate basin of attraction and it's preimages
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
printf("unknown point : z = %.16f; %.16f\n", creal(z), cimag(z));
NoOfUnknownPoints +=1;
return iColorOfUnknown;
}
// plots raster point (ix,iy)
int DrawPointOfImmediateBasin (unsigned char A[], int ix, int iy)
{
int i; /* index of 1D array */
unsigned char iColor;
complex double z;
i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
z = GiveZ(ix,iy);
iColor = ComputeColorOfImmediateBasin(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfImmediateBasin (unsigned char A[])
{
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(A, ixMax , iyMax, iColorOfUnknown)
for (iy = iyMin; iy <= iyMax; ++iy){
//printf (" %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfImmediateBasin(A, ix, iy); //
}
return 0;
}
// save A array to pgm file
int SaveArray2PGMFile( unsigned char A[], double k, char* comment )
{
FILE * fp;
const unsigned int MaxColorComponentValue=255; /* color component is coded from 0 to 255 ; it is 8 bit color file */
char name [100]; /* name of file */
snprintf(name, sizeof name, "%.3f", k); /* */
char *filename =strncat(name,".pgm", 4);
// 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 array with image data bytes to the file in one step
fclose(fp);
// info
printf("File %s saved ", filename);
if (comment == NULL || strlen(comment) ==0)
printf("\n");
else printf (". Comment = %s \n", comment);
return 0;
}
int info ()
{
// display 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 = ( %.16f ; %.16f ) \n", creal(c), cimag(c));
printf ("is a root point between period %d and %d components \n", ChildPeriod, ParentPeriod);
printf ("Image Width = %f in world coordinate\n", ZxMax - ZxMin);
printf ("PixelWidth = %f \n", PixelWidth);
printf("radius of attracting circular petal = %.16f\n", radius);
// image corners in world coordinate
// center and radius
// center and zoom
// GradientRepetition
printf ("Maximal number of iterations = iterMax = %ld \n", iterMax);
printf ("ratio of image = %f ; it should be 1.000 ...\n", ratio);
printf("NoOfUnknownPoints = %d NoOfAllPoints = %d so ratio unknown/all = %f \n", NoOfUnknownPoints, iSize, (double) NoOfUnknownPoints/ iSize);
return 0;
}
//;;;;;;;;;;;;;;;;;;;;;; setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
int setup ()
{
printf ("setup start\n");
c = Give_c(ParentPeriod, 1, ChildPeriod, 1.0);
a = -0.5; // alfa fixed point
radius = cabs(c)/4.8 ; // choose such value that level sets cross at z=0
/* 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 ...
//D2MaxGlobal = GiveDistance2ToAlfa(0.0*I); // manually chooosen
// for numerical optimisation in iteration
ER2 = ER * ER;
/* create dynamic 1D arrays for colors ( shades of gray ) */
data = malloc (iSize * sizeof (unsigned char));
edge = malloc (iSize * sizeof (unsigned char));
if (data == NULL || edge == NULL){
fprintf (stderr, " Could not allocate memory");
return 1;
}
ColorStep = (230 - iColorOfInterior)/ChildPeriod;
if (ColorStep <1) {printf("error from setup : ColorStep < 0 ; It should be greater\n"); return 1; } // check
printf (" end of setup \n");
return 0;
} // ;;;;;;;;;;;;;;;;;;;;;;;;; end of the setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
int end(){
printf (" allways free memory (deallocate ) to avoid memory leaks \n"); // https://wiki.riteme.site/wiki/C_dynamic_memory_allocation
free (data);
free(edge);
info ();
return 0;
}
/* ----------------------------------------- main -------------------------------------------------------------*/
int main () {
setup ();
//
DrawImagerOfImmediateBasin(data);
SaveArray2PGMFile (data, iHeight, "components of immediate basin of attraction (IBA) and it's preimages");
//
ComputeBoundaries(data,edge);
SaveArray2PGMFile (edge, iHeight+1.0, "only boundary of components");
//
CopyBoundaries(edge,data);
SaveArray2PGMFile (data, iHeight+2.0, "components with boundaries");
//
DrawImageOfInteriorLevelSets (data);
SaveArray2PGMFile (data, iHeight+3.0+radius, "Interior: level sets of attraction time to the parabolic fixed point");
//
ComputeBoundaries(data,edge);
SaveArray2PGMFile (edge, iHeight+4.0+radius, "only boundaries of level sets");
//
CopyBoundaries(edge,data);
SaveArray2PGMFile (data, iHeight+5.0+radius, "level sets with boundaries");
//
end();
return 0;
}
Run:
gcc b.c -lm -Wall -march=native -fopenmp
./a.out
convert 10004.pgm -resize 2000x2000 1.png
In case of ImageMagic errr see here
Text output
setup
end of setup
File 2000.000.pgm saved
File 2001.000.pgm saved
File 2002.000.pgm saved
File 2003.156.pgm saved
File 2004.156.pgm saved
File 2005.156.pgm saved
allways free memory to avoid buffer overflow
Numerical approximation of parabolic Julia set for fc(z)= z^2 + c
parameter c = ( -0.7500000000000000 ; 0.0000000000000001 )
Image Width = 3.200000
PixelWidth = 0.001601
radius of attracting circular petal = 0.1562500000000000
Maximal number of iterations = iterMax = 1000000
ratio of image = 1.000000 ; it should be 1.000 ...
NoOfUnknownPoints = 0 NoOfAllPoints = 4000000 so ratio unknown/all = 0.000000
Items portrayed in this file
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