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DescriptionImploded cauliflower LSM.png |
English: Imploded cauliflower LSM |
Date | |
Source | Own work |
Author | Adam majewski |
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c source code
/*
Adam Majewski
adammaj1 aaattt o2 dot pl // o like oxygen not 0 like zero
console program in c programing language
===============================================================
to do:
[[:File:Julia_IIM_6_basilica.png]]
Modified binary decomposition of dynamical plane
parabolic chesboard
[[:File:A_parabolic_checkerboard_for_rotation_number_1_over_2.png]]
https://arxiv.org/pdf/0802.3248.pdf
==============================================
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
./g.sh
----------------------
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 ------------------------------------------------------------ */
// see SetPlane
double radius = 1.4;
complex double center = 0.0;
double DisplayAspectRatio = 1.0; // https://en.wikipedia.org/wiki/Aspect_ratio_(image)
// 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
// z plane = dynamic plane
double ZxMin ; //-0.05;
double ZxMax ; //0.75;
double ZyMin ; //-0.1;
double ZyMax ; //0.7;
double PixelWidth; // =(ZxMax-ZxMin)/ixMax;
double PixelHeight; // =(ZyMax-ZyMin)/iyMax;
double ratio;
// w plane = 1/z plane
double WxMin = - 3; //-0.05;
double WxMax = 3; //0.75;
double WyMin = -3; //-0.1;
double WyMax = 3; //0.7;
double wPixelWidth; // =(WxMax-WxMin)/ixMax;
double wPixelHeight; // =(WyMax-WyMin)/iyMax;
// complex numbers of parametr plane
double complex c; // parameter of function fc(z)=z^2 + c
static unsigned long int iterMax = 1000000; //iHeight*100;
unsigned long int iterMax_LSM = 255;
double ER = 200.0; // EscapeRadius for bailout test
double EscapeRadius=1000000; // = ER big !!!!
double ER_LSM ; // see GiveER_LSM // 27.764 = manually find value such that level curves of escape time cross critical point and it's preimages
double ER_DLD ; // see GiveER_LSM // 27.764 = manually find value such that level curves of escape time cross critical point and it's preimages
// 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;
// ------------- DLD ----------------------
const int N = 20; // fixed number : maximal number of iterations
double p = 0.180; //0.01444322; //
// DLD colors
//double me = 1.0;
double mi = 0.9;
/* colors = shades of gray from 0 to 255 */
unsigned char iColorOfExterior = 250;
unsigned char iColorOfInterior = 200;
unsigned char iColorOfInterior1 = 210;
unsigned char iColorOfInterior2 = 180;
unsigned char iColorOfBoundary = 0;
unsigned char iColorOfUnknown = 30;
/* ------------------------------------------ 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;
}
// from screen to world coordinate ; linear mapping
// uses global cons
double GiveWx ( int ix)
{
return (WxMin + ix * wPixelWidth);
}
// uses globaal cons
double GiveWy (int iy) {
return (WyMax - iy * wPixelHeight);
} // reverse y axis
complex double GiveW( int ix, int iy){
double Wx = GiveWx(ix);
double Wy = GiveWy(iy);
return Wx + Wy*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;
}
/* ----------- 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
cabsz = cabs(z);
if (cabsz > 1e60 || cabs(dz)> 1e60) break; // big values
if (cabsz< 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
fprintf(stderr, "compute image DEM\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){
fprintf (stderr, " %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfDEMJ(A, ix, iy); //
}
return 0;
}
// ***************************************************************************************************************************
// ************************** only boundary by DEM/J*****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfDEMJ_boundary(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
cabsz = cabs(z);
if (cabsz > 1e60 || cabs(dz)> 1e60) break; // big values
if (cabsz< 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_boundary (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_boundary(z);
if (iColor == iColorOfBoundary)
{ A[i] = iColor ;} // draw only boundary without changing other parts
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfDEMJ_boundary (unsigned char A[])
{
unsigned int ix, iy; // pixel coordinate
fprintf(stderr, "compute image DEM boundary\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){
fprintf (stderr, " %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfDEMJ_boundary(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
cabsz = cabs(z);
if (cabsz > 10000000000*ER ) return iColorOfExterior; // big values
if (cabsz < (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
fprintf(stderr, "compute image 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){
fprintf (stderr, " %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_LSM;
double cabsz;
unsigned char iColor;
int n;
for (n=0; n < nMax; n++){ //forward iteration
cabsz = cabs(z);
if (cabsz > ER_LSM) break; // esacping
//if (cabsz< PixelWidth) break; // fails into finite attractor = interior, but not for disconnected Julia sets, then critical point and its preimages !!!!
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
// manually udjusted series of ordered colors ( shades of gray )
iColor = 255 - 230.0*((double) n)/18.0; // nMax or lower values in denominator
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
fprintf(stderr, "compute image LSM\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){
fprintf (stderr, " %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfLSM(A, ix, iy); //
}
return 0;
}
// ---------------------- LSM inv ============================
// plots raster point (ix,iy)
int DrawPointOfLSM_inv (unsigned char A[], int ix, int iy)
{
int i; /* index of 1D array */
unsigned char iColor;
complex double w;
complex double z; // = 1/w
i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
w = GiveW(ix,iy);
z = 1/w;
iColor = ComputeColorOfLSM(z);
A[i] = iColor ;
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfLSM_inv (unsigned char A[])
{
unsigned int ix, iy; // pixel coordinate
fprintf(stderr, "compute image LSM inverted\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){
fprintf (stderr, " %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfLSM_inv(A, ix, iy); //
}
return 0;
}
// ***************************************************************************************************************************
// ************************** binary decomposition BD/J*****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfBD(complex double z){
int nMax = iterMax_LSM;
double cabsz;
unsigned char iColor;
int n;
for (n=0; n < nMax; n++){ //forward iteration
cabsz = cabs(z);
if (cabsz > ER_LSM) break; // esacping
//if (cabsz< PixelWidth) break; // fails into finite attractor = interior but not for disconnected Julia sets, then critical point and its preimages !!!!
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
if (cimag(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
fprintf(stderr, "compute image BD\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){
fprintf (stderr, " %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfBD(A, ix, iy); //
}
return 0;
}
// plots raster point (ix,iy)
int DrawPointOfBD_inv (unsigned char A[], int ix, int iy)
{
int i; /* index of 1D array */
unsigned char iColor;
complex double z; // z = 1/w
complex double w; // w = 1/z
i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
w = GiveW(ix,iy);
z = 1.0/w;
iColor = ComputeColorOfBD(z);
A[i] = iColor ; //
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfBD_inv (unsigned char A[])
{
unsigned int ix, iy; // pixel coordinate
fprintf(stderr, "compute image BD inverted \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){
fprintf (stderr, " %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfBD_inv(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
fprintf(stderr, "compute image MBD\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){
fprintf (stderr, " %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
fprintf(stderr, "compute image SAC\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){
fprintf (stderr, "SAC/J : %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfSAC(A, ix, iy); //
}
return 0;
}
// -------------------------- SAC inv ============================================
// plots raster point (ix,iy)
int DrawPointOfSAC_inv (unsigned char A[], int ix, int iy)
{
int i; /* index of 1D array */
unsigned char iColor;
complex double w;
complex double z; // z = 1/w
i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
w = GiveW(ix,iy);
z = 1/w;
iColor = ComputeColorOfSAC(z);
A[i] = iColor ; //
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOMfSAC_inv (unsigned char A[])
{
unsigned int ix, iy; // pixel coordinate
fprintf(stderr, "compute image SAC inv\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){
fprintf (stderr, "SAC/J inv: %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfSAC_inv(A, ix, iy); //
}
return 0;
}
// ***************************************************************************************************************************
// ************************** DLD/J*****************************************
// ****************************************************************************************************************************
/* partial pnorm
input: z , zn = f(z), p
output ppn
*/
double
ppnorm (complex double z, complex double zn, double p)
{
double s[2][3]; // array for 2 points on the Riemann sphere
int j;
double d; // denominator
double x;
double y;
double ds;
double ppn = 0.0;
// map from complex plane to riemann sphere
// z
x = creal (z);
y = cimag (z);
d = x * x + y * y + 1.0;
s[0][0] = (2.0 * x) / d;
s[0][1] = (2.0 * y) / d;
s[0][2] = (d - 2.0) / d; // (x^2 + y^2 - 1)/d
// zn
x = creal (zn);
y = cimag (zn);
d = x * x + y * y + 1.0;
s[1][0] = (2.0 * x) / d;
s[1][1] = (2.0 * y) / d;
s[1][2] = (d - 2.0) / d; // (x^2 + y^2 - 1)/d
// sum
for (j = 0; j < 3; ++j)
{
ds = fabs (s[1][j] - s[0][j]);
// normal: neither zero, subnormal, infinite, nor NaN
//if (fpclassify (ds) !=FP_INFINITE)
//if (isnormal(ds))
// it is solved by if (cabs(z) > 1e60 ) break; procedure in parent function
ppn += pow (ds, p); // |ds|^p
// else {ppn = 10000.0; printf("ds = infty\t");} //
}
return ppn;
}
// DLD = Discret Lagrangian Descriptior
double
lagrangian (complex double z0, complex double c, int iMax, double p)
{
int i; // number of iteration
double d = 0.0; // DLD = sum
double ppn; // partial pnorm
complex double z = z0;
complex double zn; // next z
for (i = 0; i < iMax; ++i)
{
zn = z * z + c; // complex iteration
ppn = ppnorm (z, zn, p);
d += ppn; // sum
//
z = zn;
//if (! isnormal(d)) { return 0.0; } // not works
if (cabs (z) > ER_DLD ) //1e6)
break; // exterior : big values produces artifacts on the image
}
//if (d<0.0) {// interior
// d(z1a) - d(z21) = -0.0804163521959989
// d = - d;
// d = (db - d) /dd ; // normalize, see test_interior
//d = d*d;
//if (d>1.0) {printf("d int > 1.0\n");
/// }
// else {
d = d / ((double) i); // averaging not summation
//d = d*me;} // exterior
return d;
}
unsigned char
ComputeColor_DLD (complex double z, int FatouType)
{
//double cabsz;
int iColor;
double d;
if (FatouType == 1)
{ // interior
d = lagrangian (z, c, N, p);
// modify gradient position
//{d = d - (int)d;} // only fractional part
d = d * d * mi;
//if ( d< 1.0 ) d = 0.0;
} //
else
{
d = lagrangian (z, c, 10 * N, p);
}
iColor = (int) (d * 255) % 255; // nMax or lower walues in denominator
return (unsigned char) iColor;
}
// plots raster point (ix,iy)
int
DrawDLDPoint (unsigned char A[], int ix, int iy)
{
int i; /* index of 1D array */
unsigned char iColor;
complex double z;
int FatouType;
i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
z = GiveZ (ix, iy);
iColor = A[i]; // read color = read the information about Fatou component type ( interior/exterior)
if (iColor == iColorOfInterior)
{
FatouType = 1;
} // tru = interior
else
{
FatouType = 0;
}
iColor = ComputeColor_DLD (z, FatouType); // compute new color
A[i] = iColor; // save new colr to the array
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int
DrawDLDImage (unsigned char A[])
{
unsigned int ix, iy; // pixel coordinate
fprintf(stderr, "compute image DLD \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)
{
//fprintf (stderr,"%d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawDLDPoint (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, "%.0f", k ); /* */
char *filename = strcat (name, ".pgm");
char long_comment[200];
sprintf (long_comment, "fc(z)=z^2+ c where c = (%f %+f ); %s", creal(c), cimag(c),comment);
// save image array 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", long_comment, iWidth, iHeight, MaxColorComponentValue); // write header to the file
size_t rSize = fwrite (A, sizeof(A[0]), iSize, fp); // write whole array with image data bytes to the file in one step
fclose (fp);
// info
if ( rSize == iSize)
{
printf ("File %s saved ", filename);
if (long_comment == NULL || strlen (long_comment) == 0)
printf ("\n");
else { printf (". Comment = %s \n", long_comment); }
}
else {printf("wrote %zu elements out of %u requested\n", rSize, iSize);}
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 ("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 ("For LSM/J \n");
printf ("Maximal number of iterations = iterMax_LSM = %ld \n", iterMax_LSM);
printf ("Escape Radius = ER_LSM = %f \n", ER_LSM);
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;
}
// find such ER for LSM/J that level curves croses critical point and it's preimages
double GiveER(int i_Max){
complex double z= 0.0; // criical point
int i;
; // critical point escapes very fast here. Higher valus gives infinity
for (i=0; i< i_Max; ++i ){
z=z*z +c;
}
return cabs(z);
}
int SetPlane(complex double center, double radius, double a_ratio){
ZxMin = creal(center) - radius*a_ratio;
ZxMax = creal(center) + radius*a_ratio; //0.75;
ZyMin = cimag(center) - radius; // inv
ZyMax = cimag(center) + radius; //0.7;
return 0;
}
// *****************************************************************************
//;;;;;;;;;;;;;;;;;;;;;; setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
// **************************************************************************************
int setup ()
{
fprintf (stderr, "setup start\n");
c = 0.35; // parabolic parameter
/* 2D array ranges */
iWidth = iHeight* DisplayAspectRatio;
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].
SetPlane( center, radius, DisplayAspectRatio );
/* 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 ...
wPixelWidth = (WxMax-WxMin)/ixMax;
wPixelHeight =(WyMax-WyMin)/iyMax;
//ER2 = ER * ER; // for numerical optimisation in iteration
lnER = log(EscapeRadius); // ln(ER)
ER_LSM = GiveER(10); // find such ER for LSM/J that level curves croses critical point and it's preimages
ER_DLD = GiveER(15);
/* 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)
distanceMax = BoundaryWidth*PixelWidth;
fprintf (stderr," end of setup \n");
return 0;
} // ;;;;;;;;;;;;;;;;;;;;;;;;; end of the setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
int end(){
fprintf (stderr," 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+1, "boundary using DEM/J");
DrawImagerOfBD(data);
SaveArray2PGMFile (data, iWidth+2, "BD/J");
ComputeBoundaries(data, edge);
SaveArray2PGMFile (edge, iWidth+3, "boundaries of BD/J");
DrawImagerOMfBD(data);
SaveArray2PGMFile (data, iWidth+4, "MBD/J");
ComputeBoundaries(data, edge2);
SaveArray2PGMFile (edge2, iWidth+5, "boundaries of MBD/J");
DrawImagerOfLSM(data);
SaveArray2PGMFile (data, iWidth+6, "LSM/J");
ComputeBoundaries(data, edge);
SaveArray2PGMFile (edge, iWidth+7, "boundaries of LSM/J");
CopyBoundaries(edge, data);
SaveArray2PGMFile (data, iWidth+8, "LSM + boundaries of LSM/J");
CopyBoundaries(edge, edge2);
SaveArray2PGMFile (edge2, iWidth+9, "boundaries of LSM/J and MBD");
DrawImagerOfUnknown(data);
SaveArray2PGMFile (data, iWidth+10, "Unknown : boundary and slow dynamics");
DrawImagerOMfSAC(data);
SaveArray2PGMFile (data, iWidth+11, "SAC/J + DEM/J");
DrawDLDImage(data);
DrawImagerOfDEMJ_boundary(data);
SaveArray2PGMFile (data, iWidth+12, "DLD/J + boundary by DEM");
// inverterd plane = wplane = 1/z plane
DrawImagerOfBD_inv(data);
SaveArray2PGMFile (data, iWidth+13, "BD/J inverted ");
ComputeBoundaries(data, edge);
SaveArray2PGMFile (edge, iWidth+14, "boundaries of BD/J inv");
DrawImagerOMfSAC_inv(data);
SaveArray2PGMFile (data, iWidth+15, "SAC/J + DEM/J inverted");
DrawImagerOfLSM_inv(data);
SaveArray2PGMFile (data, iWidth+16, "LSM/J inv");
ComputeBoundaries(data, edge);
SaveArray2PGMFile (edge, iWidth+17, "boundaries of LSM/J inv");
CopyBoundaries(edge, data);
SaveArray2PGMFile (data, iWidth+1.8, "LSM + boundaries of LSM/J inv");
//PrintInfoAboutPoint(ZxMin+ZyMax*I);
end();
return 0;
}
text output
File 10000.100.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); boundary using DEM/J File 10000.200.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); BD/J File 10000.300.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); boundaries of BD/J File 10000.400.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); MBD/J File 10000.500.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); boundaries of MBD/J File 10000.600.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); LSM/J File 10000.700.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); boundaries of LSM/J File 10000.800.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); LSM + boundaries of LSM/J File 10000.900.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); boundaries of LSM/J and MBD File 10001.000.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); Unknown : boundary and slow dynamics File 10001.100.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); SAC/J + DEM/J File 10001.200.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); DLD/J + boundary by DEM File 10001.300.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); BD/J inverted File 10001.400.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); boundaries of BD/J inv File 10001.500.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); SAC/J + DEM/J inverted File 10001.600.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); LSM/J inv File 10001.700.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); boundaries of LSM/J inv File 10001.800.pgm saved . Comment = fc(z)=z^2+ c where c = (0.350000 +0.000000 ); LSM + boundaries of LSM/J inv Numerical approximation of Julia set for fc(z)= z^2 + c parameter c = ( 0.3500000000000000 ; 0.0000000000000000 ) Image Width = 2.800000 in world coordinate PixelWidth = 0.000280 for DEM/J Max distance from exterior to the boundary = distanceMax = 0.0084008400840084 = 30.000000 pixels Maximal number of iterations = iterMax = 1000000 For LSM/J Maximal number of iterations = iterMax_LSM = 255 Escape Radius = ER_LSM = 27.763998 ratio of image = 1.000000 ; it should be 1.000 ... gcc version: 9.3.0 real 2m11,910s
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