Logo by mauxuam - Contribute your own Logo!
News: Visit us on facebook
 
*
Welcome, Guest. Please login or register. October 23, 2017, 01:37:53 PM


Login with username, password and session length


The All New FractalForums is now in Public Beta Testing! Visit FractalForums.org and check it out!


Pages: [1]   Go Down
  Print  
Share this topic on DiggShare this topic on FacebookShare this topic on GoogleShare this topic on RedditShare this topic on StumbleUponShare this topic on Twitter
Author Topic: Bounding circle of julia sets  (Read 2578 times)
Description: Who said it's radius is two?
0 Members and 1 Guest are viewing this topic.
knighty
Fractal Iambus
***
Posts: 818


« on: December 11, 2010, 08:47:52 PM »

Hello,
This is quite simple and you may already know it but I couldn't find any reference in the net:
The question is what is the radius of the smallest circle bounding a given julia set? I found a formula that gives that radius when c is real and is <= 0. For other c's it gives very good bounding circle. an interresting fact about the bounding circles found with this formula is that when their radius is used as the bailout, the leminscates are tangent.
The radius "r" is the positive solution of the equation: r^p-r-|c|=0
Where p is the power of the julia set (in z<-z^p+c).

Here are two evaldraw srcipts for illustration:
Code:
static n=2;
(x,y) {
   static cx=-1,cy=0.5;
   static bo=2;
   static frm=-1;
   if(numframes>frm){
      frm=numframes;
      cx=(mousx-0.5*xres)*0.005;
      cy=(0.5*yres-mousy)*0.005;
      bo=newton(1,sqrt(cx*cx+cy*cy))^2;
   }
   1-julia(x,y,cx,cy,32,bo)
}
julia(x,y,cx,cy,it,bo){
   r2=x*x+y*y;
   for(i=0;i<it && r2<bo;i++){
      r=sqrt(r2);
      t=atan2(y,x);
      r=r^n;t=n*t;
      x=r*cos(t)+cx;
      y=r*sin(t)+cy;
      /*x1=x*x-y*y+cx;
      y=2*x*y+cy;
      x=x1;*/
      r2=x*x+y*y;
   }
   i/it
}
newton(x,c){//Solves for the radius
   i=0;
   do{
      x1=x-(x^n-x-c)/(n*x^(n-1)-1);
      if(abs(x-x1)/x<2^-53) break;
      x=x1;
      i++;
   }while(i<5);
   x
}

Code:
static n=2;
(x,y) {
   cx=x;cy=y;
   bo=newton(1,sqrt(cx*cx+cy*cy))^2;
   1-julia(x,y,cx,cy,32,bo)//in fact this is the mandelbrot set :) replace "bo" by 2 and see the difference
}
julia(x,y,cx,cy,it,bo){
   r2=x*x+y*y;
   for(i=0;i<it && r2<bo;i++){
      r=sqrt(r2);
      t=atan2(y,x);
      r=r^n;t=n*t;
      x=r*cos(t)+cx;
      y=r*sin(t)+cy;
      /*x1=x*x-y*y+cx;
      y=2*x*y+cy;
      x=x1;*/
      r2=x*x+y*y;
   }
   i/it
}
newton(x,c){
   i=0;
   do{
      x1=x-(x^n-x-c)/(n*x^(n-1)-1);
      if(abs(x-x1)/x<2^-53) break;
      x=x1;
      i++;
   }while(i<5);
   x
}

I haven't tried it yet with quaternion julia and julia bulbs but I think it should work with them.


* EVAL0002.JPG (92.06 KB, 640x480 - viewed 260 times.)

* EVAL0001.JPG (104.68 KB, 640x480 - viewed 258 times.)
Logged
Adam Majewski
Fractal Lover
**
Posts: 220


WWW
« Reply #1 on: September 07, 2011, 03:42:44 PM »

Hi,
It is described in paper :
Julia Sets of Complex Polynomials and Their Implementation on the Computer by Christoph Martin Stroh

HTH
Ada
Logged
Fractal Ken
Fractal Lover
**
Posts: 246


Proud to be 2D


« Reply #2 on: September 07, 2011, 06:23:04 PM »

It is described in paper :
Julia Sets of Complex Polynomials and Their Implementation on the Computer by Christoph Martin Stroh

Here's the link.
Logged

Fortran will rise again
knighty
Fractal Iambus
***
Posts: 818


« Reply #3 on: September 07, 2011, 07:57:41 PM »

Thank you very much.
Very interesting document even if it is maybe too technical for my small brain. How did you find it?
Logged
Adam Majewski
Fractal Lover
**
Posts: 220


WWW
« Reply #4 on: September 07, 2011, 11:24:43 PM »

I have found it with google .
Please look also here :
http://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Julia_set

Adam
Logged
knighty
Fractal Iambus
***
Posts: 818


« Reply #5 on: September 08, 2011, 08:44:50 PM »

Well... I'll have to improve my "googling" skills.
So you are the author of that wikibook! Excellent! Thank you for citing this thread. Just one note: the formula I've found is exactly the same as Stroh's. Moreover he had mathematically justified and generalized it.
BTW welcome to FF.
Logged
Adam Majewski
Fractal Lover
**
Posts: 220


WWW
« Reply #6 on: September 11, 2011, 01:05:05 PM »

>So you are the author of that wikibook!
I started only some pages not the whole book.
Note that I use code made by other people so there are more authors.




>Excellent! Thank you for citing this thread. Just one note: the formula I've found is exactly the same as Stroh's
You can edit and change it or add new things.
There are many (new or old but hard to implement) algorithms even in case of http://en.wikipedia.org/wiki/Complex_quadratic_polynomial
Help is wellcome  smiley


Logged
knighty
Fractal Iambus
***
Posts: 818


« Reply #7 on: September 13, 2011, 07:54:28 PM »

Thank you for the invitation. grin
Logged
Pages: [1]   Go Down
  Print  
 
Jump to:  

Related Topics
Subject Started by Replies Views Last post
Virtual investigation to Mandelbrot sets and Julia sets Mandelbrot & Julia Set Jules Ruis 0 1862 Last post October 19, 2006, 06:36:54 PM
by Jules Ruis
Fun with Dynamic Spirograph: Circle Rolling around Rolling Circle Non-Fractal related Chit-Chat « 1 2 » benice 19 4318 Last post February 11, 2012, 01:20:47 AM
by Sockratease
Lattès Julia sets Mandelbrot & Julia Set s31415 1 807 Last post April 29, 2013, 12:33:52 AM
by Pauldelbrot
fractal dimension of Julia sets Mandelbrot & Julia Set claude 12 1100 Last post July 26, 2016, 03:35:34 AM
by valera_rozuvan
Julia-sets for continous functions (new) Theories & Research SamTiba 4 169 Last post February 05, 2017, 10:20:16 AM
by SamTiba

Powered by MySQL Powered by PHP Powered by SMF 1.1.21 | SMF © 2015, Simple Machines

Valid XHTML 1.0! Valid CSS! Dilber MC Theme by HarzeM
Page created in 0.285 seconds with 27 queries. (Pretty URLs adds 0.012s, 2q)