Logo by Pauldelbrot - Contribute your own Logo!

END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

it was a great time but no longer maintainable by c.Kleinhuis contact him for any data retrieval,
thanks and see you perhaps in 10 years again

this forum will stay online for reference
News: Visit the official fractalforums.com Youtube Channel
 
*
Welcome, Guest. Please login or register. March 29, 2024, 03:54:29 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] 2   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: IFS fractals from Mobius transforms  (Read 17494 times)
0 Members and 1 Guest are viewing this topic.
fractalrebel
Fractal Lover
**
Posts: 211



WWW
« on: December 16, 2009, 08:43:28 PM »

Iterated Function System (IFS) fractals are normally generated using affine transforms, which are linear transforms of the type:

xn+1 = a*xn + b*yn + e
yn+1 = c*xn + d*yn + f

Several affine transforms are used to generate an IFS fractal, with the choice at each iteration being made randomly.  Apophysis is an example of software for generating IFS fractals using affine transforms.

Another type of transform, the Mobius transform

T(z) = (a*z + b)/(c*z + d)     where z is complex

can also be used to generate IFS fractals. Mobius transforms are also known as fractional linear transforms. Several Mobius transforms are selected randomly at each iteration, analogous to affine transforms, to generate a fractal. The following image is a simple demonstration of this approach.


* F3M1I0_3LxGn_058.jpg (249.93 KB, 900x900 - viewed 719 times.)
Logged

fractalrebel
Fractal Lover
**
Posts: 211



WWW
« Reply #1 on: December 16, 2009, 08:53:26 PM »

Here is an IFS Mobius example which has been mapped to a Riemann sphere.


* Sieripinskish.jpg (104.31 KB, 900x900 - viewed 693 times.)
Logged

kram1032
Fractal Senior
******
Posts: 1863


« Reply #2 on: December 16, 2009, 09:02:50 PM »

very nice but the jpg compression kills them quite much...
Logged
BradC
Safarist
******
Posts: 85



« Reply #3 on: December 16, 2009, 09:03:11 PM »

Wow, nice! I can almost see spiders crawling all over that first one. smiley

I'm curious how many transforms are involved and how you chose them.
Logged
fractalrebel
Fractal Lover
**
Posts: 211



WWW
« Reply #4 on: December 16, 2009, 10:09:19 PM »

Wow, nice! I can almost see spiders crawling all over that first one. smiley

I'm curious how many transforms are involved and how you chose them.

The images were created with IFS Mobius which is in the Ultrafractal database. The number of Mobius transforms can vary from 2 to 6. The first image uses two transforms and the second one uses 4 transforms. Rather than acting on points, as is the case with affine transforms. the Mobius transforms act upon circles and lines. In the first image the transforms operate repeatedly on 4 starting circles which were chosen randomly as long as they met the Kissing Schottky criteria (see Indra's Pearls). In the second they operate on 3 starting circles chosen randomly which met the criteria of no overlap.
Logged

fractalrebel
Fractal Lover
**
Posts: 211



WWW
« Reply #5 on: December 16, 2009, 10:22:17 PM »

Here is one of my favorites, called Cosmic Chicken:


* CosmicChicken.jpg (158.11 KB, 900x720 - viewed 657 times.)
Logged

fractalrebel
Fractal Lover
**
Posts: 211



WWW
« Reply #6 on: December 16, 2009, 10:23:46 PM »

The Cosmic Chicken (or any other IFS fractal) can easily be used as an orbit trap. Here is an example:


* cosmicchickenorbittrap.jpg (208.78 KB, 900x720 - viewed 684 times.)
Logged

kram1032
Fractal Senior
******
Posts: 1863


« Reply #7 on: December 16, 2009, 10:36:07 PM »

cool cheesy
Logged
Nahee_Enterprises
World Renowned
Fractal Senior
******
Posts: 2250


use email to contact


nahee_enterprises Nahee.Enterprises NaheeEnterprise
WWW
« Reply #8 on: December 30, 2009, 01:11:54 AM »

Interesting !!    smiley
Logged

stijnw
Guest
« Reply #9 on: January 23, 2010, 11:40:44 PM »


This looks real nice! The first image reminds me of a popular flame-fractal I used as background image for quite some time...

Regards,
Stijn Wolters.
Logged
paxinum
Guest
« Reply #10 on: July 08, 2010, 05:23:58 PM »

All fractals here can be generated by Möbius mappings:
http://en.wikipedia.org/wiki/Apollonian_gasket

There is a book called Indras Pearls using only Möbius mappings.
Logged
bib
Global Moderator
Fractal Senior
******
Posts: 2070


At the borders...


100008697663777 @bib993
WWW
« Reply #11 on: July 08, 2010, 06:13:25 PM »

Thanks paxinum for upping this thread.

The Cosmic Chicken (or any other IFS fractal) can easily be used as an orbit trap. Here is an example:

Very interesting fractalrebel. I did not find any IFS or Mobius trap shape or formula in reb.ulb or in the public classes. Would you mind sharing the UF parameter set or give some explanations ?
Logged

Between order and disorder reigns a delicious moment. (Paul Valéry)
KRAFTWERK
Global Moderator
Fractal Senior
******
Posts: 1439


Virtual Surreality


WWW
« Reply #12 on: July 09, 2010, 10:21:44 AM »

Whoah!  afro

Can we lift this to the 3D KIFS?

Think I will skip my vacation and hang here...  smiley
Logged

kram1032
Fractal Senior
******
Posts: 1863


« Reply #13 on: July 09, 2010, 01:54:21 PM »

A 3D variant would be from just adding another variable...

I wonder about a mixed variation:

(a*x+b*y[+c*z]+d)/(e*x+f*y[+g*z])+h)

Or even further extended... You could do that from a polynomial of any degree, basically...

{a_1 x^2 + b_1 y^2 + c_1 z^2 + a_2 x + b_2 y + c_2 z + d1}\over{a_3 x^2 + b_3 y^2 + c_3 z^2 + a_4 x + b_4 y + c_4 z + d2}

That would be a mix of a Möbius transform and a general 3-cone section. smiley

You could extend that even further by allowing permutations....

x*x=x²     y*x=x*y     z*x=x*z
x*y         y*y=y²       z*y=y*z
x*z         y*z            z*z=z²

So... a*x²,b*y²,c*z²,d*2xy,e*2xz,f*2yz
« Last Edit: July 09, 2010, 02:00:26 PM by kram1032 » Logged
KRAFTWERK
Global Moderator
Fractal Senior
******
Posts: 1439


Virtual Surreality


WWW
« Reply #14 on: July 09, 2010, 02:26:33 PM »

Wahooooo here we go!!! 8-)

Sorry, holiday...  afro
Logged

Pages: [1] 2   Go Down
  Print  
 
Jump to:  

Related Topics
Subject Started by Replies Views Last post
Custom formulas & transforms (old post) Mandelbulb 3d DarkBeam 0 37836 Last post February 07, 2011, 12:38:22 PM
by DarkBeam
Cayley transforms Mandelbulb3D Gallery DarkBeam 0 968 Last post February 02, 2012, 01:00:37 PM
by DarkBeam
Fractals and Fourier Transforms Programming David Makin 5 6019 Last post August 05, 2012, 11:46:36 AM
by xueer
The Purpose of 'Mobius' & 'Mobius Inverse' Pre/Post Transforms in Chaotica General Discussion Fractalcathedral 0 3521 Last post September 04, 2015, 08:48:31 PM
by Fractalcathedral
Benesi Mag Transforms (new) Theories & Research mclarekin 5 456 Last post September 08, 2016, 04:22:17 AM
by mclarekin

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 0.234 seconds with 27 queries. (Pretty URLs adds 0.012s, 2q)