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).
June 26, 2010, 05:53:02 PM
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