File:Mandelbrot set - multiplier map.png

Size of this preview: 600 × 600 pixels. Other resolutions: 240 × 240 pixels | 480 × 480 pixels | 1,000 × 1,000 pixels.

Original file (1,000 × 1,000 pixels, file size: 18 KB, MIME type: image/png)

This is a file from the Wikimedia Commons. Information from its description page there is shown below.
Commons is a freely licensed media file repository. You can help.

Summary

DescriptionMandelbrot set - multiplier map.png	English: Uniformization of the Mandelbrot set component's interior using multiplier map: internal rays , internal coordinate
Date	14 May 2017
Source	Own work. I use: the code and algorithm by Claude Heiland-Allen, algorithms and descriptions by Robert P. Munafo
Author	Adam majewski
Other versions	interior_coordinates_in_the_mandelbrot_set by Claude Heiland-Allen Displaying the Internal Structure of the Mandelbrot Set by A Cunningham Number Sequences in the Mandelbrot Set by the Mathemagicians' Guild on the dynamic plane : A Thompson Group for the Basilica by James Belk, Bradley Forrest, see fig 11 Mandelbrot set for with external and internal rays internal rays of Julia set Internal ray for angle 1/3 of main cardioid of Mandelbrot set as an image of internal ray of unit circle under conformal map $\gamma _{1}\,$ Uniformization of the exterior: External rays and equipotential lines of Mandelbrot set as an images of external rays and circles of unit circle under Jungreis function explicit exterior uniuformization of the dynamic plane

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.

compare with

Summary

choose rectangle of complex plane
for every point c of the rectangle
- find periodic point ( for periods from 1 to pMax)^[1]
  - if ther is no periodic ( = exterior or period > pMax or not enough precision or not enough iterations ??? ) then mark as the exterior
  - else compute it's multiplier = map it to standard plane ( multiplier map converts hyperbolic component to unit circle)

for (int p = 1; p < pMax; p++){
      
      if( cabs(c)<=2){
      w =  MultiplierMap(c,p);
      //if (p>2 ) printf (" w = %f ; %f p = %d \n", creal(w), cimag(w), p); 
      iRadius = cabs(w);
      if ( iRadius <= 1.0) {
           iAngle = GiveTurn(w);
           period=p;
           //if (p>2 ) printf (" c = %f ; %f p = %d \n", creal(c), cimag(c), p); 
           break;}
      }   }

multiplier map

Multiplier map $\lambda$ associated with hyperbolic component $\mathrm {H}$ gives an explicit uniformization of hyperbolic component $\mathrm {H}$ by the unit disk $\mathbb {D}$ :

$\lambda _{p}:\mathrm {H} \to \mathbb {D}$

In other words it maps hyperbolic component H to unit disk D.

It maps point c from parameter plane to point b from reference plane:

$\lambda _{p}(c)=w$

where:

c is a point in the parameter plane
w is a point in the reference plane. It is complex number
$\lambda$ is a multiplier map

Multiplier map is a conformal isomorphism.^[2]

For period:

1 has explicit form
2 has explicit form^[3]
3
- has explicit form but there are 3 components and each has it's own equation
- can be computed numericallly
for periods > 3 must be computed numerically ( AproximateMultiplierMap - Newton_method )

complex double MultiplierMap(complex double c, int period){

complex double w;
switch(period){
case 1: w = 1.0 - csqrt(1.0-4.0*c); 	break; // explicit
case 2: w = 4.0*c + 4; 			break; //explicit
default:w = AproximateMultiplierMap(c, period, EPS2, ER2); 	break; // 

}

return w;

}

Note that period=0 denotes exterior of Mandelbrot set. Here different map ( Riemann map) is used. It is called Boettcher map.

to do

add more precision ( one can see some erors on image : pots in place where should not be any hyperbolic omponents)
add Riemann map ( exterior to unit disc = Boettcher coordinate)
add boundary
- abs(multiplier (c)) = 1.0 +- eps
- DEM/M

c src code

/*

Multiplier map
https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set#internal_coordinate_and_multiplier_map

algorithm: 
https://mathr.co.uk/blog/2013-04-01_interior_coordinates_in_the_mandelbrot_set.html
by Claude Heiland-Allen

code mostly based on the : 
 http://mathr.co.uk/blog/2014-11-02_practical_interior_distance_rendering.html
 http://mathr.co.uk/blog/2013-04-01_interior_coordinates_in_the_mandelbrot_set.html
by 

 gcc -std=c99 -Wall -Wextra -pedantic -O3 -fopenmp m2.c -lm
 
 
 
 */

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

const double pi = 3.141592653589793;
const double infinity = 1.0 / 0.0;
const double colour_modulus = 5.7581917135421046e-2; // (1.0 + 1.0 / (phi * phi)) / 24.0;
const double ER2 = 2.0 * 2.0; // ER*ER
const double EPS2 = 1e-100 * 1e-100; // EPS*EPS

static inline double cabs2(complex double z) {
  return creal(z) * creal(z) + cimag(z) * cimag(z);
}

static inline unsigned char *image_new(int width, int height) {
  return malloc(width * height * 3);
}

static inline void image_delete(unsigned char *image) {
  free(image);
}

static inline void image_save_ppm(unsigned char *image, int width, int height, const char *filename) {
  FILE *f = fopen(filename, "wb");
  if (f) {
    fprintf(f, "P6\n%d %d\n255\n", width, height);
    fwrite(image, width * height * 3, 1, f);
    fclose(f);
  } else {
    fprintf(stderr, "ERROR saving `%s'\n", filename);
  }
}

static inline void image_poke(unsigned char *image, int width, int i, int j, int r, int g, int b) {
  int k = (width * j + i) * 3;
  image[k++] = r;
  image[k++] = g;
  image[k  ] = b;
}

static inline void colour_hsv_to_rgb(double h, double s, double v, double *r, double *g, double *b) {
  double i, f, p, q, t;
  if (s == 0) { *r = *g = *b = v; } else {
    h = 6 * (h - floor(h));
    int 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;
    }
  }
}

static inline void colour_to_bytes(double r, double g, double b, int *r_out, int *g_out, int *b_out) {
  *r_out = fmin(fmax(255 * r, 0), 255);
  *g_out = fmin(fmax(255 * g, 0), 255);
  *b_out = fmin(fmax(255 * b, 0), 255);
}

static inline void colour_mandelbrot(unsigned char *image, int width, int i, int j, int period, double intRadius, double intAngle) {
  double r, g, b;
  
 /* if position > 1 then we have repetition of colors it maybe useful    */
      
  //if (intRadius > 1.0) intRadius -= 1.0; // normalize 
  
 
  
  
    
  colour_hsv_to_rgb(period * colour_modulus, 0.5, 1.0, &r, &g, &b); // changed b from tanh(distance )to 1.0 
  int ir, ig, ib;
  colour_to_bytes(r, g, b, &ir, &ig, &ib);
  
  //generate internal grid 
  if (intRadius <1.0){
    int rMax=10; /* number of color segments */
    double m=rMax* intRadius;
    int im=(int)m; // integer of m
 
    //if (intAngle > 1.0) intAngle -= 1.0; // normalize 
    int aMax=12; /* number of color segments */
    double k=aMax* intAngle;
    int ik =(int)k; // integer of m
    
     if ( im % 2 ) {ir -=40; }
     if ( ik % 2 ) {ig -=40; }
   }
   
   
   image_poke(image, width, i, j, ir, ig, ib);
   
}

/* newton function : N(z) = z - (fp(z)-z)/f'(z)) */

complex double N( complex double c, complex double zn , int pMax, double er2){

  
complex double z = zn;
complex double d = 1.0; /* d = first derivative with respect to z */
int p; 

for (p=0; p < pMax; p++){

   //printf ("p = %d ;  z = %f ; %f ;  d = %f ; %f \n", p, creal(z), cimag(z), creal(d), cimag(d)); 
   d = 2*z*d; /* first derivative with respect to z */
   z = z*z +c ; /* complex quadratic polynomial */
   //if (cabs(z) >er2) break;
    
}

 
 
 //printf (" next \n\n"); 
 //if ( cabs2(d) > 2) 
     z = zn - (z - zn)/(d - 1) ;
    
 

return z;
}

/* 
compute periodic point 
of complex quadratic polynomial
using Newton iteration = numerical method 

*/

complex double GivePeriodic(complex double c, complex double z0, int period, double eps2, double er2){

complex double z = z0;
complex double zPrev = z0; // prevoiuus value of z
int n ; // iteration
const int nMax = 64;

for (n=0; n<nMax; n++) {
     
    z = N( c, z, period, er2);
    if (cabs2(z - zPrev)< eps2) break;
    
    zPrev = z; }

return z;
}

complex double AproximateMultiplierMap(complex double c, int period, double eps2, double er2){
     
     complex double z;  // variable z 
     complex double zp ; // periodic point 
     complex double zcr = 0.0; // critical point
     complex double d = 1;
     complex double w;
     
     int p;
     
     zp =  GivePeriodic( c, zcr, period,  eps2, er2); // Find periodic point z0 such that f^p(z0,c)=z0 using Newton's method in one complex variable
     //zp = find(-1, 0, period, c); 
     //printf (" zp = %f ; %f p = %d \n", creal(zp), cimag(zp), period); 
     
     // Find w by evaluating first derivative with respect to z of f^p at z0 
     if ( cabs2(zp)<er2) {
     
     //printf (" zp = %f ; %f p = %d \n", creal(zp), cimag(zp), period); 
     z = zp;
     for (p=0; p < period; p++){
        d = 2*z*d; /* first derivative with respect to z */
        z = z*z +c ; /* complex quadratic polynomial */
     
          }
          
          
     
     
     }
     else d= 10000;

return d;
}

complex double MultiplierMap(complex double c, int period){

complex double w;
switch(period){
case 1: w = 1.0 - csqrt(1.0-4.0*c); 	break; // explicit
case 2: w = 4.0*c + 4; 			break; //explicit
default:w = AproximateMultiplierMap(c, period, EPS2, ER2); 	break; // 

}

return w;

}

/*
carg_t(z):=
 block(
 [t],
 t:carg(z)/(2*%pi),  // now in turns 
 if t<0 then t:t+1, // map from (-1/2,1/2] to [0, 1) 
 return(t)
)$

*/

double GiveTurn( double complex z){
double t;

  t =  carg(z);
  t /= 2*pi; // now in turns
  if (t<0.0) t += 1.0; // map from (-1/2,1/2] to [0, 1) 
  return (t);
}

static inline void render(unsigned char *image,  int pMax, int width, int height, complex double center, double radius) {

  double pixel_spacing = radius / (height / 2.0);

   #pragma omp parallel for schedule(dynamic, 1)
    for (int j = 0; j < height; ++j) {
     for (int i = 0; i < width; ++i) {
      
      double x = i + 0.5 - width / 2.0;
      double y = height / 2.0 - j - 0.5;
      complex double c = center + pixel_spacing * (x + I * y);
      
      
      complex double w; // b(c):= (sqrt(1-4*c)+1)/4;
      double iRadius;
      double iAngle = 0.0;
      int period = 0;

      // find period and w 
      for (int p = 1; p < pMax; p++){
      
      if( cabs(c)<=2){
      w =  MultiplierMap(c,p);
      //if (p>2 ) printf (" w = %f ; %f p = %d \n", creal(w), cimag(w), p); 
      iRadius = cabs(w);
      if ( iRadius <= 1.0) {
           iAngle = GiveTurn(w);
           period=p;
           //if (p>2 ) printf (" c = %f ; %f p = %d \n", creal(c), cimag(c), p); 
           break;}
      }   } 
      // colour 
      colour_mandelbrot(image, width, i, j, period, iRadius, iAngle);
 
    }
  }
}

int main() {

  

  
 // int MaxIters = 100;
  int PeriodMax = 10;
  int width = 1000;
  int height = 1000;
  complex double center = -0.75 ;
  double radius = 1.5;
  const char *filename = "3.ppm";
 
  unsigned char *image = image_new(width, height);

  render(image,  PeriodMax, width, height, center, radius);
  image_save_ppm(image, width, height, filename);
  image_delete(image);
  
  
 /* 
  complex double z;
  z = GivePeriodic(-0.965 + 0.085*I , 0.0, 2, EPS2, ER2);
  printf (" z = %f ; %f \n",  creal(z), cimag(z)); 
  z = GivePeriodic(-0.965 + 0.085*I , 0.0, 2, EPS2, ER2);
  printf (" z = %f ; %f \n",  creal(z), cimag(z)); 
  */

  return 0;
}

references

File history

Click on a date/time to view the file as it appeared at that time.

	Date/Time	Thumbnail	Dimensions	User	Comment
current	18:53, 2 June 2017		1,000 × 1,000 (18 KB)	Soul windsurfer	User created page with UploadWizard

File usage

The following page uses this file:

Complex quadratic polynomial

Global file usage

The following other wikis use this file:

Usage on el.wikipedia.org
- Μιγαδικό τετραγωνικό πολυώνυμο
Usage on en.wikibooks.org

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.

If the file has been modified from its original state, some details may not fully reflect the modified file.

File change date and time	20:02, 13 May 2017

[1] terior_coordinates_in_the_mandelbrot_set by Claude Heiland-Allen

[2] Conformal Geometry and Dynamics of Quadratic Polynomials Mikhail Lyubich

[3] Exact Coordinates by Robert P. Munafo, 2003 Sep 22.

[1]

[2]

[3]

File:Mandelbrot set - multiplier map.png

Contents

Summary

Licensing

compare with

Summary

multiplier map

to do

c src code

references

Captions

Items portrayed in this file

depicts

creator

some value

copyright status

copyrighted

copyright license

Creative Commons Attribution-ShareAlike 4.0 International

inception

14 May 2017

media type

image/png

File history

File usage

Global file usage

Metadata