Logo by DsyneGrafix - 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: Check out the originating "3d Mandelbulb" thread here
 
*
Welcome, Guest. Please login or register. August 08, 2022, 03:51:20 AM


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: Any function you like....  (Read 1692 times)
0 Members and 1 Guest are viewing this topic.
David Makin
Global Moderator
Fractal Senior
******
Posts: 2286



Makin' Magic Fractals
WWW
« on: December 18, 2009, 01:58:02 PM »

Hi all, after this:

http://www.fractalforums.com/mandelbulb-renderings/a-new-class-of-bulb/msg9988/#msg9988

And something Jos Leys pointed out to me, i.e. that for the Mandelbulb Julias because the rotations do not affect the magnitude then for the analytical distance estimate for z^p+c (Julias where p is real) because we only need the magnitude of the final derivative in the end then the running derivative can be done using just the magnitude rather than the full triplex value, i.e. to get the derivative we simply iterate:

dz = p*dz*m^(p-1)

where dz is the (both running and final) magnitude of the derivative, p is the power/degree (real) and m is the current magnitude of the main iteration.

Putting the above ideas together I came up with the following for the main iterate:

Convert the triplex (or any multi-dimensional number) to the magnitude and the relevant unit vector, if the magnitude is 0 return the constant as the new value else...
New magnitude=magnitude^p
Apply *any* function f() you like to the unit vector.
If the function used is such that the result may not maintain unity then normalise the result throwing away the magnitude - special case a zero result e.g. return 1 (real) if zero generally (or another appropriate unit vector depending on your function e.g. based on numeric or geometric limits).
Now simply multiply the new unit vector by the new magnitude and add the constant.

When doing the above then the optimisation of using just the magnitude for the analytical derivative for Julia Sets still applies for any function f using power p.

Note that using the optimisation of just using the magnitude for the derivative on the Mandelbrot versions does work to some extent but the calculations really require the full vector derivative because of the "+1" added on each iteration.
« Last Edit: December 18, 2009, 03:09:38 PM by David Makin » Logged

The meaning and purpose of life is to give life purpose and meaning.

http://www.fractalgallery.co.uk/
"Makin' Magic Music" on Jango
David Makin
Global Moderator
Fractal Senior
******
Posts: 2286



Makin' Magic Fractals
WWW
« Reply #1 on: December 18, 2009, 03:00:47 PM »

I should add that Paul also suggested a method using *any* function, here:

http://www.fractalforums.com/theory/the-christmas-tree-3d-mandelbrot-set/msg10213/#msg10213

That method also allows the possibility of the optimised (magnitude only) Julia derivative for z^p+c where p is real.
Logged

The meaning and purpose of life is to give life purpose and meaning.

http://www.fractalgallery.co.uk/
"Makin' Magic Music" on Jango
kram1032
Fractal Senior
******
Posts: 1863


« Reply #2 on: December 18, 2009, 04:33:05 PM »

any pics of that sofar? smiley
Logged
David Makin
Global Moderator
Fractal Senior
******
Posts: 2286



Makin' Magic Fractals
WWW
« Reply #3 on: December 18, 2009, 05:32:15 PM »

any pics of that sofar? smiley

Well not for *any* function but for rotational functions using angles other than the normal Mandelbulb ones.
I have formulas in my updated 3D wip formula for using the "simultaeneous" rotation around 2 or all 3 2D angles, these formulae do not preserve the unity of the vector so one version produced this:

"Hairy Ball"


http://makinmagic.deviantart.com/art/Hairy-Ball-144652915

But renormalising as I described gets rid of the "hairiness" smiley

Others using different angles, in some cases also with the described renormalising:

http://makinmagic.deviantart.com/art/Terraformed-Mars-144722355
http://makinmagic.deviantart.com/art/A-Martian-Delicacy-144901891
http://makinmagic.deviantart.com/art/Under-Red-Skies-145191644
http://makinmagic.deviantart.com/art/Centauri-Prime-145239769
http://makinmagic.deviantart.com/art/The-Lost-Isle-145246194
http://makinmagic.deviantart.com/art/Temple-of-the-Anointed-145450385
http://makinmagic.deviantart.com/art/Summer-on-the-Tundra-145532566
http://makinmagic.deviantart.com/art/Shropshire-Crags-146372767
http://makinmagic.deviantart.com/art/Limestone-pavement-146400211

If I remember correctly (without checking) I think "Summer on the Tundra" was the same formula as "Hairy Ball" except with renormalisation added.
Logged

The meaning and purpose of life is to give life purpose and meaning.

http://www.fractalgallery.co.uk/
"Makin' Magic Music" on Jango
kram1032
Fractal Senior
******
Posts: 1863


« Reply #4 on: December 18, 2009, 05:44:17 PM »

I see smiley

well in that case you know already that they look great cheesy
Logged
David Makin
Global Moderator
Fractal Senior
******
Posts: 2286



Makin' Magic Fractals
WWW
« Reply #5 on: December 18, 2009, 11:16:48 PM »

I should add that the optimisation of using the magnitude only for the derivative will only work for formulas with the basic form a*z^p + c.
As soon as you do something like a*z^p + b*z^p1 + c then you have to use the full vector derivative.
Logged

The meaning and purpose of life is to give life purpose and meaning.

http://www.fractalgallery.co.uk/
"Makin' Magic Music" on Jango
Pages: [1]   Go Down
  Print  
 
Jump to:  

Related Topics
Subject Started by Replies Views Last post
Please help recursive function Discuss Fractal Forums kevinmorais 4 356 Last post January 25, 2013, 11:59:39 AM
by AtomicZagnut
Velvet function Images Showcase (Rate My Fractal) Chaos_Ink 0 584 Last post January 07, 2013, 09:48:31 PM
by Chaos_Ink
Continuous Boxtiling Function (Accordion+cycleshift) Mandelbulb 3d M Benesi 6 1462 Last post December 21, 2016, 04:58:47 AM
by M Benesi
the paradox of complex function by elliptical integral (new) Theories & Research hgjf2 1 586 Last post September 03, 2017, 11:25:59 AM
by hgjf2
Separating the power function into 2 parts General Discussion Khashishi 3 1077 Last post September 08, 2017, 12:59:09 AM
by vinecius

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 0.165 seconds with 29 queries. (Pretty URLs adds 0.006s, 2q)