Fractals/Iterations in the complex plane/demj

This algorithm has 2 versions:


Compare it with version for parameter plane and Mandelbrot set : DEM/J

External distance estimationEdit

For distance estimate it has been proved that the computed value differs from the true distance at most by a factor of 4:

   Koebe Quarter Theorem. The image of an injective analytic function f : DC from the unit disk D onto a subset of the complex plane contains the disk whose center is f(0) and whose radius is |f′(0)|/4.[1]



Math formula :

 

where :

  is first derivative with respect to z.

This derivative can be found by iteration starting with

 

and then :

 

How to use distanceEdit

One can use distance for colouring  :

  • only Julia set ( boundary of filled Julia set)
  • boundary and exterior of filled Julia set.

Here is first example :

 if (LastIteration==IterationMax) 
     then { /*  interior of Julia set, distance = 0 , color black */ }
     else /* exterior or boundary of Filled-in Julia set  */
          {  double distance=give_distance(Z0,C,IterationMax);
             if (distance<distanceMax)
                 then { /*  Julia set : color = white */ }
                 else  { /*  exterior of Julia set : color = black */}
          }

Here is second example [2]

 if (LastIteration==IterationMax or distance < distanceMax) then ... // interior by ETM/J and boundary by DEM/J
 else .... // exterior by real escape time

ZoomEdit

DistanceMax is smaller than pixel size. During zooming pixel size is decreasing and DistanceMax should also be decreased to obtain good picture. It can be made by using formula :

 

where  

One can start with n=1 and increase n if picture is not good. Check also iMax !!

DistanceMax may also be proportional to zoom factor  :[3]

 

where thick is image width ( in world units) and mag is a zoom factor.

Examples of codeEdit

For cpp example see mndlbrot::dist from mndlbrot.cpp in src code of program mandel ver 5.3 by Wolf Jung.


C function using complex type :

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) {//printf(" distance = %f \n", distance); 
  		return iColorOfBoundary;}
  
  
  return iColorOfExterior;

 
}


C function using double type:

/*based on function  mndlbrot::dist  from  mndlbrot.cpp
 from program mandel by Wolf Jung (GNU GPL )
 http://www.mndynamics.com/indexp.html  */
double jdist(double Zx, double Zy, double Cx, double Cy ,  int iter_max)
 { 
 int i;
 double x = Zx, /* Z = x+y*i */
         y = Zy, 
         /* Zp = xp+yp*1 = 1  */
         xp = 1, 
         yp = 0, 
         /* temporary */
         nz,  
         nzp,
         /* a = abs(z) */
         a; 
 for (i = 1; i <= iter_max; i++)
  { /* first derivative   zp = 2*z*zp  = xp + yp*i; */
    nz = 2*(x*xp - y*yp) ; 
    yp = 2*(x*yp + y*xp); 
    xp = nz;
    /* z = z*z + c = x+y*i */
    nz = x*x - y*y + Cx; 
    y = 2*x*y + Cy; 
    x = nz; 
    /* */
    nz = x*x + y*y; 
    nzp = xp*xp + yp*yp;
    if (nzp > 1e60 || nz > 1e60) break;
  }
 a=sqrt(nz);
 /* distance = 2 * |Zn| * log|Zn| / |dZn| */
 return 2* a*log(a)/sqrt(nzp); 
 }

Delphi function :

function Give_eDistance(zx0,zy0,cx,cy,ER2:extended;iMax:integer):extended;

var zx,zy ,  // z=zx+zy*i
    dx,dy,  //d=dx+dy*i  derivative :  d(n+1)=  2 * zn * dn
    zx_temp,
    dx_temp,
    z2,  //
    d2, //
    a // abs(d2)
     :extended;
    i:integer;
begin
  //initial values
  // d0=1
  dx:=1;
  dy:=0;
  //
  zx:=zx0;
  zy:=zy0;
  // to remove warning : variables may be not initialised ?
  z2:=0;
  d2:=0;

  for i := 0 to iMax - 1 do
    begin
      // first derivative   d(n+1) = 2*zn*dn  = dx + dy*i;
      dx_temp := 2*(zx*dx - zy*dy) ;
      dy := 2*(zx*dy + zy*dx);
      dx := dx_temp;
      // z = z*z + c = zx+zy*i
      zx_temp := zx*zx - zy*zy + Cx;
      zy := 2*zx*zy + Cy;
      zx := zx_temp;
      //
      z2:=zx*zx+zy*zy;
      d2:=dx*dx+dy*dy;
      if ((z2>1e60) or (d2 > 1e60)) then  break;
      
    end; // for i
   if (d2 < 0.01) or (z2 < 0.1)  // when do not use escape time
    then  result := 10.0
    else
      begin
        a:=sqrt(z2);
        // distance = 2 * |Zn| * log|Zn| / |dZn|
        result := 2* a*log10(a)/sqrt(d2);
      end;

end;  //  function Give_eDistance

Matlab code by Jonas Lundgren[4]

function D = jdist(x0,y0,c,iter,D)
%JDIST Estimate distances to Julia set by potential function
% by Jonas Lundgren http://www.mathworks.ch/matlabcentral/fileexchange/27749-julia-sets
% Code covered by the BSD License http://www.mathworks.ch/matlabcentral/fileexchange/view_license?file_info_id=27749

% Escape radius^2
R2 = 100^2;

% Parameters
N = numel(x0);
M = numel(y0);
cx = real(c);
cy = imag(c);
iter = round(1000*iter);

% Create waitbar
h = waitbar(0,'Please wait...','name','Julia Distance Estimation');
t1 = 1;

% Loop over pixels
for k = 1:N/2
    x0k = x0(k);
    for j = 1:M
        % Update distance?
        if D(j,k) == 0
            % Start values
            n = 0;
            x = x0k;
            y = y0(j);
            b2 = 1;                 % |dz0/dz0|^2
            a2 = x*x + y*y;         % |z0|^2
            % Iterate zn = zm^2 + c, m = n-1
            while n < iter && a2 <= R2
                n = n + 1;
                yn = 2*x*y + cy;
                x = x*x - y*y + cx;
                y = yn;
                b2 = 4*a2*b2;       % |dzn/dz0|^2
                a2 = x*x + y*y;     % |zn|^2
            end
            % Distance estimate
            if n < iter
                % log(|zn|)*|zn|/|dzn/dz0|
                D(j,k) = 0.5*log(a2)*sqrt(a2/b2);
            end
        end
    end
    % Lap time
    t = toc;
    % Update waitbar
    if t >= t1
        str = sprintf('%0.0f%% done in %0.0f sec',200*k/N,t);
        waitbar(2*k/N,h,str)
        t1 = t1 + 1;
    end
end

% Close waitbar
close(h)

Maxima function :

 GiveExtDistance(z0,c,e_r,i_max):=
 /* needs z in exterior of Kc */
 block(
 [z:z0,
 dz:1,
 cabsz:cabs(z),
 cabsdz:1, /* overflow limit */
 i:0],
 while  cabsdz < 10000000 and  i<i_max
  do 
   (
    dz:2*z*dz,
    z:z*z + c,
    cabsdz:cabs(dz),
    i:i+1
   ),
  cabsz:cabs(z), 
  return(2*cabsz*log(cabsz)/cabsdz)
 )$

Internal distance estimationEdit

Colouring the Julia set by Gert BuschmannEdit

 
Julia set drawn from distance estimation

In order to get a nice picture, we must also colour the Julia set, since otherwise the Julia set is only visible through the colouring of the Fatou domains, and this colouring changes vigorously near the Julia set, giving a muddy look (it is possible to avoid this by choosing the colour scale and the density carefully). A point z belongs to the Julia set if the iteration does not stop, that is, if we have reached the chosen maximum number of iterations, M. But as the Julia set is infinitely thin, and as we only perform calculations for regularly situated points, in practice we cannot colour the Julia set in this way. But happily there exists a formula that (up to a constant factor) estimates the distance from the points z outside the Julia set to the Julia set. This formula is associated to a Fatou domain, and the number given by the formula is the more correct the closer we come to the Julia set, so that the deviation is without significance.

The distance function is the function   (see the section Julia and Mandelbrot sets for non-complex functions), whose equipotential lines must lie approximately regularly. In the formula appears the derivative   of   with respect to z. But as   (the k-fold composition),   is the product of the numbers   (i = 0, 1, ..., k-1), and this sequence can be calculated recursively by   and   (before the calculation of the next iteration  ). In the three cases we have:

 limk→∞  (non-super-attraction)
 limk→∞  (super-attraction)
 limk→∞  (d ≥ 2 and z* = ∞)

If this number (calculated for the last iteration number kr - to be divided by r) is smaller that a given small number, we colour the point z dark-blue, for instance.

For more see Pictures_of_Julia_and_Mandelbrot_Sets

codeEdit

/*
gcc -std=c99 -Wall -Wextra -pedantic -O3 -o julia-de julia-de.c -lm
https://math.stackexchange.com/questions/1153052/interior-distance-estimate-for-julia-sets-getting-rid-of-spots
code by Claude Heiland-Allen
*/

#include <complex.h>
#include <math.h>
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>

void hsv2rgb(double h, double s, double v, int *rp, int *gp, int *bp) {
  double i, f, p, q, t, r, g, b;
  int ii;
  if (s == 0.0) { r = g = b = v; } else {
    h = 6 * (h - floor(h));
    ii = i = floor(h);
    f = h - i;
    p = v * (1 - s);
    q = v * (1 - (s * f));
    t = v * (1 - (s * (1 - f)));
    switch(ii) {
      case 0: r = v; g = t; b = p; break;
      case 1: r = q; g = v; b = p; break;
      case 2: r = p; g = v; b = t; break;
      case 3: r = p; g = q; b = v; break;
      case 4: r = t; g = p; b = v; break;
      default:r = v; g = p; b = q; break;
    }
  }
  *rp = fmin(fmax(round(r * 255), 0), 255);
  *gp = fmin(fmax(round(g * 255), 0), 255);
  *bp = fmin(fmax(round(b * 255), 0), 255);
}

complex double julia_attractor(complex double c, int maxiters, int *period) {
  double epsilon = nextafter(2, 4) - 2;
  complex double z = c;
  double mzp = 1.0/0.0;
  int p = 0;
  for (int n = 1; n < maxiters; ++n) {
    double mzn = cabs(z);
    if (mzn < mzp) {
      mzp = mzn;
      p = n;
      complex double z0 = z;
      for (int i = 0; i < 64; ++i) {
        complex double f = z0;
        complex double df = 1;
        for (int j = 0; j < p; ++j) {
          df = 2 * f * df;
          f = f * f + c;
        }
        complex double z1 = z0 - (f - z0) / (df - 1);
        if (cabs(z1 - z0) <= epsilon) {
          z0 = z1;
          break;
        }
        if (isinf(creal(z1)) || isinf(cimag(z1)) || isnan(creal(z1)) || isnan(cimag(z1))) {
          break;
        }
        z0 = z1;
      }
      complex double w = z0;
      complex double dw = 1;
      for (int i = 0; i < p; ++i) {
        dw = 2 * w * dw;
        w = w * w + c;
      }
      if (cabs(dw) <= 1) {
        *period = p;
        return z0;
      }
    }
    z = z * z + c;
  }
  *period = 0;
  return 0;
}

double julia_exterior_de(complex double c, complex double z, int maxiters, double escape_radius) {
  complex double dz = 1;
  for (int n = 0; n < maxiters; ++n) {
    if (cabs(z) >= escape_radius) {
      return cabs(z) * log(cabs(z)) / cabs(dz);
    }
    dz = 2 * z * dz;
    z = z * z + c;
  }
  return 0;
}

double julia_interior_de(complex double c, complex double z, int maxiters, double escape_radius, double pixel_size, complex double z0, int period, bool superattracting, int *fatou) {
  complex double dz = 1;
  for (int n = 0; n < maxiters; ++n) {
    if (cabs(z) >= escape_radius) {
      *fatou = -1;
      return cabs(z) * log(cabs(z)) / cabs(dz);
    }
    if (cabs(z - z0) <= pixel_size) {
      *fatou = n % period;
      if (superattracting) {
        return cabs(z - z0) * fabs(log(cabs(z - z0))) / cabs(dz);
      } else {
        return cabs(z - z0) / cabs(dz);
      }
    }
    dz = 2 * z * dz;
    z = z * z + c;
  }
  *fatou = -2;
  return 0;
}

int main(int argc, char **argv) {
  int size = 512;
  double radius = 2;
  double escape_radius = 1 << 10;
  int maxiters = 1 << 13;
  if (! (argc > 2)) { return 1; }
  complex double c = atof(argv[1]) + I * atof(argv[2]);

  int period = 0;
  bool superattracting = false;
  complex double z0 = julia_attractor(c, maxiters, &period);
  if (period > 0) {
    double epsilon = nextafter(1, 2) - 1;
    complex double z = z0;
    complex double dz = 1;
    for (int i = 0; i < period; ++i) {
      dz = 2 * z * dz;
      z = z * z + c;
    }
    superattracting = cabs(dz) <= epsilon;
  }

  double pixel_size = 2 * radius / size;
  printf("P6\n%d %d\n255\n", size, size);
  for (int j = 0; j < size; ++j) {
    for (int i = 0; i < size; ++i) {
      double x = 2 * radius * ((i + 0.5) / size - 0.5);
      double y = 2 * radius * (0.5 - (j + 0.5) / size);
      complex double z = x + I * y;
      double de = 0;
      int fatou = -1;
      if (period > 0) {
        de = julia_interior_de(c, z, maxiters, escape_radius, pixel_size, z0, period, superattracting, &fatou);
      } else {
        de = julia_exterior_de(c, z, maxiters, escape_radius);
      }
      int r, g, b;
      hsv2rgb(fatou / (double) period, 0.25 * (0 <= fatou), tanh(de / pixel_size), &r, &g, &b);
      putchar(r);
      putchar(g);
      putchar(b);
    }
  }
  return 0;
}


referencesEdit

  1. Koebe quarter theorem in wikipedia
  2. Pictures of Julia and Mandelbrot sets by Gert Buschmann
  3. Pictures of Julia and Mandelbrot sets by Gert Buschmann
  4. Julia sets by Jonas Lundgren in Matlab
Fractals/Iterations in the complex plane
demj Fractals/Iterations_in_the_complex_plane/def_cqp#Riemann_map →