File:Interior of cauliflower with sepals.png
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[edit]DescriptionInterior of cauliflower with sepals.png |
English: Interior of cauliflower showing parabolic sepals ( Invariant lamination). Inspired by parabolic bifurcation image by Vladlen Timorin from paper Smart Criticality for Cubic Laminations by Alexander Blokh, Lex Oversteegen, Ross Ptacek, Vladlen Timorin. See also fractalforums |
Date | |
Source | Own work |
Author | Adam majewski |
Other versions |
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c src code
[edit]/*
Adam Majewski
adammaj1 aaattt o2 dot pl // o like oxygen not 0 like zero
https://plus.google.com/116648956837292097606/posts/b6J6z2u8soL
how to show sepals inside main box of parabolic chessboard ?
- compute full orbit ( forward and backward of every point)
- for each whole orbit ( not point) compute maximal distance from orbit to fixed point alfa
- normalize distance ( dustance/ distance max ) so it will have value from 0 to 1.0
_ use such normalized distance for coloring
- then one can see orbits
==========================================
-------------------------------
cd existing_folder
git init
git remote add origin [email protected]:adammajewski/SepalsOfCauliflower.git
git add .
git commit
git push -u origin master
---------------------------------
indent d.c
default is gnu style
-------------------
c console progam
gcc c.c -lm -Wall -march=native
time ./a.out
gcc c.c -lm -Wall -march=native -fopenmp
time ./a.out
time ./a.out >a.txt
----------------------
period 2 cycle :
z = -0.500000000000000 +1.000000000000000 i
z = -0.500000000000000 -1.000000000000000 i
*/
#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 0 // Period of secondary component joined by root point with the parent component
//int iPeriodParent = 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 = 4000; //
// The size of array has to be a positive constant integer
static unsigned int iSize; // = iWidth*iHeight;
// memmory 1D array
unsigned char *data;
// unsigned 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 = -1.15; //-0.05;
static const double ZxMax = 1.15; //0.75;
static const double ZyMin = -1.15; //-0.1;
static const double ZyMax = 1.15; //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
static unsigned long int iterMax = 1000000; //iHeight*100;
static double ER = 2.0; // Escape Radius for bailout test
static double ER2;
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 iColorOfUnknown = 50;
int NoOfUnknownPoints = 0;
/* ------------------------------------------ functions -------------------------------------------------------------*/
//------------------complex numbers -----------------------------------------------------
// 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;
}
int IsInTarget(complex double z){
// here target set is a square with side r
// with fixed point on it's boundary
double s = 0.25;
double zx = creal(z);
//double dx; // = fabs(creal(z) - r - 0.5);
double dy; // = fabs(cimag(z));
// fixed point is z=0.5
// target set is any
if (zx > 0.50) return 0; //
//dx = fabs(zx - r - 0.5);
dy = fabs(cimag(z));
if ( zx <= 0.5 &&
zx > 0.0 &&
dy< s) return 1;
return 0;
}
double GiveDistance2ToAlfa(complex double z){
return (creal(z) - 0.5) * (creal(z) - 0.5) + cimag(z) * cimag(z);
}
double GiveDistance2MaxLocal(complex double z){
double d2 = GiveDistance2ToAlfa(z); // d2 = d*d = d without sqrt
double d2max = 0.0;
while (d2>d2max) {
d2max = d2; // save new local max
z = csqrt(z - 0.25); // inverse iteration : complex quadratic polynomial
d2 = GiveDistance2ToAlfa(z);
}
return d2max;
}
unsigned char GiveColorOfInterior(complex double z){
double D2MaxLocal = GiveDistance2MaxLocal(z);
unsigned char iColor = 255.0*D2MaxLocal/D2MaxGlobal;
//printf(" z = %.16f ; %.16f \t D2MaxLocal = %.16f \tcolor = %.16f = %d \n ", creal(z), cimag(z), D2MaxLocal, 255.0*D2MaxLocal/D2MaxGlobal, iColor);
return iColor;
}
unsigned char ComputeColorSepal(complex double z){
int nMax = iterMax;
int n;
for (n=0; n < nMax; n++){
if (cabs2(z)>ER2) return iColorOfExterior;
if (IsInTarget(z)) return GiveColorOfInterior(z);
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
printf("unknown point ( probably periodic) : z = %.16f; %.16f\n", creal(z), cimag(z));
NoOfUnknownPoints +=1;
return iColorOfUnknown;
}
// plots raster point (ix,iy)
int PlotPointSepal (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 = ComputeColorSepal (z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawSepals (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)
PlotPointSepal (A, ix, iy); //
}
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 ()
{
// 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 = ( %f ; %f ) \n", creal(c), cimag(c));
printf ("Maximal Distance to alfa = %.16f \n", D2MaxGlobal);
printf ("Image Width = %f \n", ZxMax - ZxMin);
printf ("PixelWidth = %f \n", PixelWidth);
// 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\n");
c = 0.25;
/* 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(0.5*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));
if (data == NULL )
{
fprintf (stderr, " Could not allocate memory");
getchar ();
return 1;
}
printf (" end of setup \n");
return 0;
} // ;;;;;;;;;;;;;;;;;;;;;;;;; end of the setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
int end(){
printf (" allways free memory to avoid buffer overflow \n");
free (data);
info ();
return 0;
}
/* ----------------------------------------- main -------------------------------------------------------------*/
int main () {
setup ();
//
DrawSepals(data);
SaveArray2PGMFile (data, iHeight);
//
end();
return 0;
}
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