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Summary

Description
English: Imploaded Rabbit Julia set : Julia set for fc(z)=z*z + c and c=-0.1+0.651. Made using DEM/J. C parameter for this file is from program by Jonas Lundgren
Date
Source Own work
Author Adam majewski
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Long description

program

This is console c program. It makes ppm file. It uses DEM/J method[1]

c parameter

C parameter for this file is from program by Jonas Lundgren[2]

it is :

  • outside Mandelbrot set thus Julia set is disconnected
  • between period 1 and 3
  • on the right to the root of period 3 component
  • in the 1/3-wake ( wake of the period-3 component) = above ray 1/7=0.(001) =
  • in the (1/3, 1/25)-subwake bounded by rays 5396990266136737387082/37778931862957161709567 and 5396990266136737387089/37778931862957161709567

dynamic rays and periodic points

Rays : 1/7, 2/7, 4/7 land on period one point :

 z = -0.237943688964178  +0.441090566757054 i

which is a center of 3-arm spiral

So here period of landing point is different then period of points on the external rays that lands on it


Period 3 points :

  • z = -0.437544375897169 +0.596837602749673 i
  • z = -0.264770043176733 +0.128714127185863 i
  • z = -0.046464150773409 +0.582840709975087 i

are centers of 3 one-arm spirals

What rays land on these points ?

Not (checked manually ):

  • period 6 wher angle = p/63
  • not period 9 where angle = p/511

First angle is :

10/63 = 0.158 < angle < 1/7 = 0.1428

Note that 1/7 = 9/63 = 73/511

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported 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.


C source code

gcc d.c -lm -Wall 
./a.out

Postprocessing using ImageMagic library :

convert f.ppm -resize 2000x2000 f.png
/* 
   c console  program:
   1. draws Julia setfor Fc(z)=z*z +c
   using DEM/J algorithm ( Distance Esthimation Method for Julia set )
   -------------------------------         
   2. technic of creating ppm file is  based on the code of Claudio Rocchini
   http://en.wikipedia.org/wiki/Image:Color_complex_plot.jpg
   create 24 bit color graphic file ,  portable pixmap file = PPM 
   see http://en.wikipedia.org/wiki/Portable_pixmap
   to see the file use external application ( graphic viewer)
   ---------------------------------
   I think that creating graphic can't be simpler
   comments : Adam Majewski 

   gcc d.c -lm -Wall

   it creates a.out file. Then run it :

   ./a.out




*/



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


/*
  estimates distance from point c to nearest point in Julia  set 
  for Fc(z)= z*z + c
  z(n+1) = Fc(zn)  
  this function is based on function  mndlbrot::dist  from  mndlbrot.cpp
  from program mandel by Wolf Jung (GNU GPL )
  http://www.mndynamics.com/indexp.html 

  Hyunsuk Kim  : 
  For Julia sets, z is the variable and c is a constant. Therefore df[n+1](z)/dz = 2*f[n]*f'[n] -- you don't add 1.

  For the Mandelbrot set on the parameter plane, you start at z=0 and c becomes the variable. df[n+1](c)/dc = 2*f[n]*f'[n] + 1. 


*/
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); 
}
/* ------------------------------------------------------*/
int main(void)
{
  const double Cx=-0.74543;
  const double Cy=0.11301;
  /* screen ( integer) coordinate */
  int iX,iY;
  const int iXmax = 10000; 
  const int iYmax = 10000;
  /* world ( double) coordinate = parameter plane*/
  const double ZxMin=-2.;
  const double ZxMax=2.0;
  const double ZyMin=-2.0;
  const double ZyMax=2.0;
  /* */
  double PixelWidth=(ZxMax-ZxMin)/iXmax;
  double PixelHeight=(ZyMax-ZyMin)/iYmax;
  /* color component ( R or G or B) is coded from 0 to 255 */
  /* it is 24 bit color RGB file */
  const int MaxColorComponentValue=255; 
  FILE * fp;
  char *filename="demj-.ppm";
  char *comment="# ";/* comment should start with # */
  static unsigned char color[3];
  double Zx, Zy,    /* Z=Zx+Zy*i   */
    Z0x, Z0y,  /* Z0 = Z0x + Z0y*i */
    Zx2, Zy2; /* Zx2=Zx*Zx;  Zy2=Zy*Zy  */
  /*  */
  int Iteration;
  const int IterationMax=2000;
  /* bail-out value , radius of circle ;  */
  const int EscapeRadius=400;
  int ER2=EscapeRadius*EscapeRadius;
  double distanceMax=PixelWidth/15; /*jdist( 0,0,Cx,Cy, IterationMax);*/
  /*create new file,give it a name and open it in binary mode  */
  fp= fopen(filename,"wb"); /* b -  binary mode */
  /*write ASCII header to the file*/
  fprintf(fp,"P6\n %s\n %d\n %d\n %d\n",comment,iXmax,iYmax,MaxColorComponentValue);
  /* compute and write image data bytes to the file*/
  for(iY=0;iY<iYmax;++iY)
    {
      Z0y=ZyMax - iY*PixelHeight; /* reverse Y  axis */
      if (fabs(Z0y)<PixelHeight/2) Z0y=0.0; /*  */    
      for(iX=0;iX<iXmax;++iX)
	{    /* initial value of orbit Z0 */
	  Z0x=ZxMin + iX*PixelWidth;
	  /* Z = Z0 */
	  Zx=Z0x;
	  Zy=Z0y;
	  Zx2=Zx*Zx;
	  Zy2=Zy*Zy;
	  /* */
	  for (Iteration=0;Iteration<IterationMax && ((Zx2+Zy2)<ER2);Iteration++)
	    {
	      Zy=2*Zx*Zy + Cy;
	      Zx=Zx2-Zy2 +Cx;
	      Zx2=Zx*Zx;
	      Zy2=Zy*Zy;
	    };
	  /* compute  pixel color (24 bit = 3 bytes) */
	  if (Iteration==IterationMax)
	    { /*  interior of Julia set  = black */
	      color[0]=0;
	      color[1]=0;
	      color[2]=0;                           
	    }
	  else /* exterior of Filled-in Julia set  =  */
	    {  double distance=jdist(Z0x,Z0y,Cx,Cy,IterationMax);
	      if (distance<distanceMax)
		{ /*  Julia set  = white */
		  color[0]=255; /* Red*/
		  color[1]=255;  /* Green */ 
		  color[2]=255;/* Blue */
		}
	      else 
		{ /*  exterior of Julia set  = black */
		  color[0]=0;
		  color[1]=0;
		  color[2]=0;                           
		};
	    }
	  /* check the orientation of Z-plane */
	  /* mark first quadrant of cartesian plane*/     
	  /*     if (Z0x>0 && Z0y>0) color[0]=255-color[0];  */
	  /*write color to the file*/
	  fwrite(color,1,3,fp);
	}
    }
  fclose(fp);
  printf("file saved ");
  getchar();
  return 0;
}

References

  1. DEM/J in wikibooks
  2. Julia sets in Matlab using DEM by Jonas Lundgren

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26 June 2011

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Date/TimeThumbnailDimensionsUserComment
current13:21, 26 June 2011Thumbnail for version as of 13:21, 26 June 20112,000 × 2,000 (454 KB)Soul windsurfer

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