File:Julia set for fc(z) = z*z+c where c = -0.749998153581339 +0.001569040474910*I; t = 0.49975027919634618290.png
Size of this preview: 600 × 600 pixels. Other resolutions: 240 × 240 pixels | 480 × 480 pixels | 768 × 768 pixels | 1,024 × 1,024 pixels | 2,000 × 2,000 pixels.
Original file (2,000 × 2,000 pixels, file size: 316 KB, MIME type: image/png)
 |
This is a file from the Wikimedia Commons |
Summary
| Description |
English: Julia set for fc(z) = z*z+c where c = -0.749998153581339 +0.001569040474910*I. Internal angle in turns is t = 0.49975027919634618290 |
| Date | |
| Source | Own work |
| Author | Adam majewski |
| Other versions |
|
_%3D_z*z%2Bc_where_c_%3D_-0.749998153581339_%2B0.001569040474910*I;_t_%3D_0.49975027919634618290.png)
_%3D_z*z%2Bc_where_c_%3D_-0.749998153581339_%2B0.001569040474910*I;_t_%3D_0.49975027919634618290.png){kind=link}
Algorithm
Program checks every pixel of image, iterates and divides them to 3 groups:
- escaping to infinity = exterior
- attracting to finite attractor = interior
- Unknown Pixels = not escaping and not attracting
Tests:
- simple bailout test : if (cabs(z) > 2.0)
- test for the interior : if (cabs(z-a) <= InternalSiegelDiscRadius) where a is a alfa fixed point which is a center of the Siegel disc
// ***************************************************************************************************************************
// ************************** binary Escape time *****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfBET(complex double z){
int nMax = iterMax;
//unsigned char iColor = iColorOfUnknown;
// double d;
int n;
for (n=0; n < nMax; n++){ //forward iteration
// test for the exterior: bailout test
if (cabs(z) > 2.0) {
NoOfExteriorPixels += 1;
return iColorOfExterior; // escaping set = exterior : LSM/J
}
// test for interior
if (cabs(z-a) <= InternalSiegelDiscRadius) {// falls into finite attractor = interior
NoOfInteriorPixels += 1;
return iColorOfInterior;
}
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
NoOfUnknownPixels += 1;
return iColorOfUnknown;
}
C source code
/*
Adam Majewski
adammaj1 aaattt o2 dot pl // o like oxygen not 0 like zero
console program in c programing language
==============================================
Structure of a program or how to analyze the program
============== Image X ========================
DrawImageOfX -> DrawPointOfX -> ComputeColorOfX
first 2 functions are identical for every X
check only last function = ComputeColorOfX
which computes color of one pixel !
==========================================
---------------------------------
indent d.c
default is gnu style
-------------------
c console progam
export OMP_DISPLAY_ENV="TRUE"
gcc d.c -lm -Wall -march=native -fopenmp
time ./a.out > b.txt
gcc d.c -lm -Wall -march=native -fopenmp
time ./a.out
time ./a.out >a.txt
----------------------
real 0m19,809s
user 2m26,763s
sys 0m0,161s
*/
#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> // OpenMP
/* --------------------------------- global variables and consts ------------------------------------------------------------ */
// 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 = 10000; //
// 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 char *edge2;
// 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 = -1.7; //-2.0; //-0.05;
static const double ZxMax = 1.7; // 2.0; //0.75;
static const double ZyMin = -1.7; //-2.0; //-0.1;
static const double ZyMax = 1.7; //2.0; //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
//int Period = 2;
static unsigned long int iterMax = 1000000; //iHeight*100;
static double ER = 1E60; // EscapeRadius for bailout test
double EscapeRadius=1000000; // = ER big !!!!
// SAC/J
double lnER; // ln(ER)
int i_skip = 2; // exclude (i_skip+1) elements from average
unsigned char s = 7; // stripe density
double BoundaryWidth = 3.0; // % of image width
double distanceMax; //distanceMax = BoundaryWidth*PixelWidth;
double InternalSiegelDiscRadius; //https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/siegel
double ExternalSiegelDiscRadius;
/* colors = shades of gray from 0 to 255 */
unsigned char iColorOfExterior = 245;
unsigned char iColorOfInterior = 55;
unsigned char iColorOfInterior1 = 210;
unsigned char iColorOfInterior2 = 180;
unsigned char iColorOfBoundary = 0;
unsigned char iColorOfUnknown = 30;
int NoOfExteriorPixels = 0;
int NoOfInteriorPixels = 0;
int NoOfUnknownPixels = 0;
/* ------------------------------------------ functions -------------------------------------------------------------*/
//------------------complex numbers -----------------------------------------------------
// 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;
}
// ****************** DYNAMICS = trap tests ( target sets) ****************************
// bailout test
// z escapes when
// abs(z)> ER or cabs2(z)> ER2
// https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Julia_set#Boolean_Escape_time
int Escapes(complex double z){
// here target set (trap) is the exterior circle with radsius = ER ( EscapeRadius)
// with ceter = origin z= 0
// on the Riemann sphere it is a circle with point at infinity as a center
if (cabs(z)>ER) return 1;
return 0;
}
// compute alfa fixed point
// https://en.wikipedia.org/wiki/Periodic_points_of_complex_quadratic_mappings#Period-1_points_(fixed_points)
complex double GiveAlfa(complex double c)
{
// d=1-4c
// alfa = (1-sqrt(d))/2
return (1.0-csqrt(1.0 - 4.0*c))/2.0 ;
}
// https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/siegel
double GiveInternalSiegelDiscRadius(complex double c, complex double a)
{ /* compute critical orbit and finds smallest distance from fixed point */
int i; /* iteration */
double complex z =0.0; /* critical point */
/* center of Siegel disc = a */
double d; // distance
double dMin = 2.0;
for (i=0;i<=40000 ;i++) /* to small number of iMax gives bad result */
{
z = z*z + c;
/* */
d = cabs(z - a);
if (d < dMin) dMin = d; /* smallest distance */
}
return dMin;
}
double GiveExternalSiegelDiscRadius(complex double c, complex double a)
{ /* compute critical orbit and finds smallest distance from fixed point */
int i; /* iteration */
double complex z =0.0; /* critical point */
/* center of Siegel disc = a */
double d; // distance
double dMax = 0.0;
for (i=0;i<=40000 ;i++) /* to small number of iMax gives bad result */
{
z = z*z + c;
/* */
d = cabs(z - a);
if (d > dMax) dMax = d; /* smallest distance */
}
return dMax;
}
double GiveMaxDistanceFromCenter(complex double z, complex double c, complex double a)
{ /* compute critical orbit and finds smallest distance from fixed point */
int i; /* iteration */
/* center of Siegel disc = a */
double d; // distance
double dMax = 0.0;
for (i=0;i<=40000 ;i++) /* to small number of iMax gives bad result */
{
z = z*z + c;
/* */
d = cabs(z - a);
if (d > dMax) dMax = d; /* smallest distance */
}
return dMax;
}
/* ----------- 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;
}
// ***********************************************************************************************
// ********************** edge detection usung Sobel filter ***************************************
// ***************************************************************************************************
// 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)
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;
}
// ***************************************************************************************************************************
// ************************** DEM/J*****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfDEMJ(complex double z){
// https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Julia_set#DEM.2FJ
int nMax = iterMax;
complex double dz = 1.0; // is first derivative with respect to z.
double distance;
double cabsz;
int n;
for (n=0; n < nMax; n++){ //forward iteration
if (cabs(z) > 1e60 || cabs(dz)> 1e60) break; // big values
if (cabs(z-a)< PixelWidth) return iColorOfInterior; // falls into finite attractor = interior
dz = 2.0*z * dz;
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
distance = 2.0 * cabsz* log(cabsz)/ cabs(dz);
if (distance <distanceMax) return iColorOfBoundary; // distanceMax = BoundaryWidth*PixelWidth;
// else
return iColorOfExterior;
}
// plots raster point (ix,iy)
int DrawPointOfDEMJ (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 = ComputeColorOfDEMJ(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfDEMJ (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)
for (iy = iyMin; iy <= iyMax; ++iy){
printf ("DEM/J %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfDEMJ(A, ix, iy); //
}
return 0;
}
// ***************************************************************************************************************************
// ************************** Unknown: boundary and slow dynamics *****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfUnknown(complex double z){
int nMax = 20; // very low value
double cabsz;
int n;
for (n=0; n < nMax; n++){ //forward iteration
if (cabs(z) > 10000000000*ER ) return iColorOfExterior; // big values
if (cabs(z-a) < (PixelWidth/100)) return iColorOfInterior; // falls into finite attractor = interior
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
//printf("found \n");
return iColorOfUnknown;
}
// plots raster point (ix,iy)
int DrawPointOfUnknown (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 = ComputeColorOfUnknown(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfUnknown (unsigned char A[])
{
unsigned int ix, iy; // pixel coordinate
printf("compute image of Unknown\n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(A, ixMax , iyMax)
for (iy = iyMin; iy <= iyMax; ++iy){
printf ("Unknown %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfUnknown(A, ix, iy); //
}
return 0;
}
// ***************************************************************************************************************************
// ************************** LSM/J*****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfLSM(complex double z){
int nMax = iterMax;
unsigned char iColor = iColorOfUnknown;
double d;
int n;
for (n=0; n < nMax; n++){ //forward iteration
if (cabs(z) > 2.0) return 231+ 14*(n % 2); // escaping set = exterior : LSM/J
d = cabs(z-a);
if (d <= InternalSiegelDiscRadius) // falls into finite attractor = interior
{
//d = GiveMaxDistanceFromCenter(z,c,a);
//return 255-255.0 * d/ExternalSiegelDiscRadius;
return iColorOfInterior;
}
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
return iColor;
}
// plots raster point (ix,iy)
int DrawPointOfLSM (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 = ComputeColorOfLSM(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfLSM (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)
for (iy = iyMin; iy <= iyMax; ++iy){
printf ("LSM %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfLSM(A, ix, iy); //
}
return 0;
}
// ***************************************************************************************************************************
// ************************** Interior : orbits*****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfOrbits(complex double z){
int nMax = iterMax;
unsigned char iColor = iColorOfUnknown;
double d;
int n;
for (n=0; n < nMax; n++){ //forward iteration
if (cabs(z) > 2.0) return iColorOfExterior; // escaping set = exterior : LSM/J
d = cabs(z-a);
if (d <= InternalSiegelDiscRadius) // falls into finite attractor = interior
{
d = GiveMaxDistanceFromCenter(z,c,a);
return 255-255.0 * d/ExternalSiegelDiscRadius;
//return iColorOfInterior;
}
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
return iColor;
}
// plots raster point (ix,iy)
int DrawPointOfOrbits (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 = ComputeColorOfOrbits(z);
A[i] = iColor ; //
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImageOfOrbits (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)
for (iy = iyMin; iy <= iyMax; ++iy){
printf ("Orbits %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfOrbits(A, ix, iy); //
}
return 0;
}
// ***************************************************************************************************************************
// ************************** binary Escape time *****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfBET(complex double z){
int nMax = iterMax;
//unsigned char iColor = iColorOfUnknown;
// double d;
int n;
for (n=0; n < nMax; n++){ //forward iteration
// test for the exterior: bailout test
if (cabs(z) > 2.0) {
NoOfExteriorPixels += 1;
return iColorOfExterior; // escaping set = exterior : LSM/J
}
// test for interior
if (cabs(z-a) <= InternalSiegelDiscRadius) {// falls into finite attractor = interior
NoOfInteriorPixels += 1;
return iColorOfInterior;
}
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
NoOfUnknownPixels += 1;
return iColorOfUnknown;
}
// plots raster point (ix,iy)
int DrawPointOfBET (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 = ComputeColorOfBET(z);
A[i] = iColor ; //
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImageOfBET (unsigned char A[])
{
unsigned int ix, iy; // pixel coordinate
//
NoOfExteriorPixels = 0;
NoOfInteriorPixels = 0;
NoOfUnknownPixels = 0;
//printf("compute image \n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(A, ixMax , iyMax)
for (iy = iyMin; iy <= iyMax; ++iy){
printf ("BET %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfBET(A, ix, iy); //
}
// local info
// BET, number of pixels : exterior = 52 809 244 , interior = 13 749 273, unknown = 108 305 , all = 66 666 822
printf("BET, number of pixels : exterior = %d , interior = %d, unknown = %d , all = %d \n", NoOfExteriorPixels, NoOfInteriorPixels, NoOfUnknownPixels, NoOfExteriorPixels+NoOfInteriorPixels + NoOfUnknownPixels );
//
NoOfExteriorPixels = 0;
NoOfInteriorPixels = 0;
NoOfUnknownPixels = 0;
return 0;
}
// ***************************************************************************************************************************
// ************************** binary decomposition BD/J*****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfBD(complex double z){
int nMax = 255;
double cabsz;
unsigned char iColor;
int n;
for (n=0; n < nMax; n++){ //forward iteration
if (cabs(z) > ER) break; // esacping
if (cabs(z-a)< PixelWidth) break; // fails into finite attractor = interior
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
if (creal(z)>0.0)
iColor = 255;
else iColor = 0;
return iColor;
}
// plots raster point (ix,iy)
int DrawPointOfBD (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 = ComputeColorOfBD(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfBD (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)
for (iy = iyMin; iy <= iyMax; ++iy){
printf ("BSD %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfBD(A, ix, iy); //
}
return 0;
}
// ***************************************************************************************************************************
// ************************** modified binary decomposition BD/J*****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfMBD(complex double z){
// const number of iterations
int nMax = 7;
//double cabsz;
unsigned char iColor;
int n;
for (n=0; n < nMax; n++){ //forward iteration
//cabsz = cabs(z);
//if (cabsz > ER) break; // esacping
//if (cabsz< PixelWidth) break; // falls into finite attractor = interior
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
if (cabs(z) > 2.0)
{ // exterior
if (creal(z)>0.0)
iColor = 255;
else iColor = 0;
}
else iColor = iColorOfInterior;
return iColor;
}
// plots raster point (ix,iy)
int DrawPointOfMBD (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 = ComputeColorOfMBD(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOMfBD (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)
for (iy = iyMin; iy <= iyMax; ++iy){
printf ("MBD %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfMBD(A, ix, iy); //
}
return 0;
}
// ***********************************************************************************************
//*************************************** SAC/J **************************************************
// *****************************************************************************************
// https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/stripeAC
// SAC = Stripe Average Coloring
//
// the addend function
// input : complex number z
// output : double number t
double Give_t(double complex z){
return 0.5+0.5*sin(s*carg(z));
}
/*
input :
- complex number
- intege
output = average
*/
double Give_Arg(double complex z , int iMax)
{
int i=0; // iteration
//double complex Z= 0.0; // initial value for iteration Z0
double A = 0.0; // A(n)
double prevA = 0.0; // A(n-1)
double R; // =radius = cabs(Z)
double d; // smooth iteration count
double complex dz = 1.0; // first derivative with respect to z
double de; // Distance Estimation from DEM/J
// iteration = computing the orbit
for(i=0;i<iMax;i++)
{
dz = 2.0 * z * dz ;
z = z*z + c; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/qpolynomials
if (i>i_skip) A += Give_t(z); //
R = cabs(z);
// if(R > EscapeRadius) break; // exterior of M set
if (R > 1e60 || cabs(dz)> 1e60) break; // prevent NAN
prevA = A; // save value for interpolation
} // for(i=0
if (i == iMax)
A = -1.0; // interior
else { // exterior
de = 2 * R * log(R) / cabs(dz);
if (de < distanceMax) A = FP_ZERO; // boundary
else {
// computing interpolated average
A /= (i - i_skip) ; // A(n)
prevA /= (i - i_skip - 1) ; // A(n-1)
// smooth iteration count
d = i + 1 + log(lnER/log(R))/M_LN2;
d = d - (int)d; // only fractional part = interpolation coefficient
// linear interpolation
A = d*A + (1.0-d)*prevA;
}
}
return A;
}
unsigned char ComputeColorOfSAC(complex double z){
unsigned char iColor;
double arg;
arg = Give_Arg( z, 2500); // N in wiki
// color is proportional to arg
if (arg < 0.0)
iColor = 0; // interior
else //
{if (arg == FP_ZERO)
iColor = 255; // boundary
else iColor = (unsigned char) (255 - 255*arg );// exterior
}
return iColor;
}
// plots raster point (ix,iy)
int DrawPointOfSAC (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 = ComputeColorOfSAC(z);
A[i] = iColor ; //
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOMfSAC (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)
for (iy = iyMin; iy <= iyMax; ++iy){
printf ("SAC/J : %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfSAC(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, "%.1f", 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 PrintInfoAboutProgam()
{
// display info messages
printf ("Numerical approximation of 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 ("iSize = %d\n", iSize);
printf ("Image Width = %f in world coordinate\n", ZxMax - ZxMin);
printf ("PixelWidth = %f \n", PixelWidth);
printf("for DEM/J \n");
if ( distanceMax<0.0 || distanceMax > ER ) printf("bad distanceMax\n");
printf("Max distance from exterior to the boundary = distanceMax = %.16f = %f pixels\n", distanceMax, BoundaryWidth);
// 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("gcc version: %d.%d.%d\n",__GNUC__,__GNUC_MINOR__,__GNUC_PATCHLEVEL__); // https://stackoverflow.com/questions/20389193/how-do-i-check-my-gcc-c-compiler-version-for-my-eclipse
// OpenMP version is diplayed in the console
return 0;
}
int PrintInfoAboutPoint(complex double z){
//unsigned int ix, iy; // pixel coordinate
// to do
double arg;
unsigned char iColor;
arg = Give_Arg( z, 2500); // N in wiki
iColor = ComputeColorOfSAC(z);
printf ("parameter z = ( %.16f ; %.16f ) \n", creal(z), cimag(z));
printf ("SAC/J : arg = %.16f ; iColor = %d \n", arg, iColor);
return z;
}
// *****************************************************************************
//;;;;;;;;;;;;;;;;;;;;;; setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
// **************************************************************************************
int setup ()
{
printf ("setup start ");
c = -0.749998153581339 +0.001569040474910*I; // t = 0.49975027919634618290
a = GiveAlfa(c);
/* 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)) / ((double) iWidth / (double) iHeight); // it should be 1.000 ...
//ER2 = ER * ER; // for numerical optimisation in iteration
lnER = log(EscapeRadius); // ln(ER)
/* create dynamic 1D arrays for colors ( shades of gray ) */
data = malloc (iSize * sizeof (unsigned char));
edge = malloc (iSize * sizeof (unsigned char));
edge2 = malloc (iSize * sizeof (unsigned char));
if (data == NULL || edge == NULL || edge2 == NULL){
fprintf (stderr, " Could not allocate memory");
return 1;
}
BoundaryWidth = 6.0*iWidth/2000.0 ; // measured in pixels ( when iWidth = 2000) ; such function is stable when iWidth is changing
distanceMax = BoundaryWidth*PixelWidth; // distance to the boundary from exterior
InternalSiegelDiscRadius = GiveInternalSiegelDiscRadius(c,a);
ExternalSiegelDiscRadius = GiveExternalSiegelDiscRadius(c,a);
printf ("and end\n");
return 0;
} // ;;;;;;;;;;;;;;;;;;;;;;;;; end of the setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
int end(){
printf (" allways free memory (deallocate ) to avoid memory leaks \n"); // https://en.wikipedia.org/wiki/C_dynamic_memory_allocation
free (data);
free(edge);
free(edge2);
PrintInfoAboutProgam();
return 0;
}
// ********************************************************************************************************************
/* ----------------------------------------- main -------------------------------------------------------------*/
// ********************************************************************************************************************
int main () {
setup ();
/*
DrawImagerOfDEMJ(data);
SaveArray2PGMFile (data, iWidth+0.1, "boundary using DEM/J");
DrawImagerOfBD(data);
SaveArray2PGMFile (data, iWidth+0.2, "BD/J");
ComputeBoundaries(data, edge);
SaveArray2PGMFile (edge, iWidth+0.3, "boundaries of BD/J");
DrawImagerOMfBD(data);
SaveArray2PGMFile (data, iWidth+0.4, "MBD/J");
ComputeBoundaries(data, edge2);
SaveArray2PGMFile (edge2, iWidth+0.5, "boundaries of MBD/J");
DrawImagerOfLSM(data);
SaveArray2PGMFile (data, iWidth+0.6, "LSM/J");
ComputeBoundaries(data, edge);
SaveArray2PGMFile (edge, iWidth+0.7, "boundaries of LSM/J");
DrawImageOfOrbits(data);
SaveArray2PGMFile (data, iWidth+0.8, "Orbits");
CopyBoundaries(edge, data);
SaveArray2PGMFile (data, iWidth+0.8, "orbits + boundaries of LSM/J ");
*/
DrawImageOfBET(data);
SaveArray2PGMFile (data, iWidth+0.9, "BET = binary escape time");
/*
CopyBoundaries(edge, data);
SaveArray2PGMFile (data, iWidth+1.0, "BET + boundaries of LSM/J ");
CopyBoundaries(edge, edge2);
SaveArray2PGMFile (edge2, iWidth+0.8, "boundaries of LSM/J and MBD");
DrawImagerOfUnknown(data);
SaveArray2PGMFile (data, iWidth+0.9, "Unknown : boundary and slow dynamics");
DrawImagerOMfSAC(data);
SaveArray2PGMFile (data, iWidth+1.0, "SAC/J + DEM/J");
*/
//PrintInfoAboutPoint(ZxMin+ZyMax*I);
end();
return 0;
}
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International 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.
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 16:59, 20 January 2019 | | 2,000 × 2,000 (316 KB) | Soul windsurfer | User created page with UploadWizard |
File usage
The following page uses this file:
_%3D_z*z%2Bc_where_c_%3D_-0.749998153581339_%2B0.001569040474910*I;_t_%3D_0.49975027919634618290.png){kind=link}