Title: alternative power interpolation
Post by: cKleinhuis on February 01, 2015, 01:40:35 AM
found this on reddit/shadertoy:
http://nl.reddit.com/r/fractals/comments/2u9uo4/figured_out_how_to_transition_between_powers/
and shadertoy
https://www.shadertoy.com/view/lls3D7
:)
http://nl.reddit.com/r/fractals/comments/2u9uo4/figured_out_how_to_transition_between_powers/
and shadertoy
https://www.shadertoy.com/view/lls3D7
:)
Title: Re: alternative power interpolation
Post by: Dinkydau on February 02, 2015, 10:42:51 PM
That's a nice discovery.
Title: Re: alternative power interpolation
Post by: DarkBeam on February 02, 2015, 11:09:36 PM
Very nice ;)
Title: Re: alternative power interpolation
Post by: cKleinhuis on February 02, 2015, 11:44:57 PM
what exactly has he done? removed the "mathematical correct" way to interpolate between powers to a more visual pleasing one ?!
Title: Re: alternative power interpolation
Post by: claude on February 03, 2015, 11:27:20 AM
Say you want to interpolate between the result sets from iterations of 
and 
.
One way would be to interpolate the power giving + 3t} + c)
with t from 0 to 1, but then you end up with fractional powers like 2.7123615245 which don't look so great (
with n integer wraps around a circle a whole number of times).
The other way presented in the article is to interpolate a sum giving z^2 + t z^3 + c)
, avoiding fractional powers entirely, while still starting and ending at the desired equations.
If it might help, a musical analogy: instead of a slide from C to G going through all microtonal pitches in between, it's like C fading out while G fades in.
Personally I think both methods are equally "mathematically correct", there are a lot of different ways to interpolate, and even with the same method there can be multiple routes (eg: on a sphere you could go the long way round).
One way would be to interpolate the power giving
The other way presented in the article is to interpolate a sum giving
If it might help, a musical analogy: instead of a slide from C to G going through all microtonal pitches in between, it's like C fading out while G fades in.
Personally I think both methods are equally "mathematically correct", there are a lot of different ways to interpolate, and even with the same method there can be multiple routes (eg: on a sphere you could go the long way round).
Title: Re: alternative power interpolation
Post by: cKleinhuis on February 03, 2015, 11:34:08 AM
ah, i see, its basically the "interpolation hybrid" which interpolates lineary between two outcomes of different formulas, like
 (z^2 +c_1) +t (z^3 + c_2))
but using just the exponential term to lineary interpolate
but using just the exponential term to lineary interpolate