alternative power interpolation

[cKleinhuis]

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

[Dinkydau]

That's a nice discovery.

[DarkBeam]

Very nice

[cKleinhuis]

what exactly has he done? removed the "mathematical correct" way to interpolate between powers to a more visual pleasing one ?!

[claude]

Say you want to interpolate between the result sets from iterations of and .

One way would be to interpolate the power giving 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 , 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).

[cKleinhuis]

ah, i see, its basically the "interpolation hybrid" which interpolates lineary between two outcomes of different formulas, like

 

but using just the exponential term to lineary interpolate