Vary Exponent

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Offline E8Guy

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« on: August 12, 2018, 06:16:44 AM »
I saw a zoom vid with an exponent of 10, all of the spokes and nubs were 10-fold. Instead of zooming in 101000000 times, what if we fixed the viewing plane and varied the exponent z2.0000000z2.0000001z2.0000002...z9.9999999. How finely do you have to chop before it looks smooth?

Offline gerrit

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« Reply #1 on: August 12, 2018, 07:02:09 AM »
Not sure what you have in mind exactly, increasing power  of \( z^p+c \) iterations would just morph it into a circle with the interesting stuff getting smaller and smaller.

I tried some related ideas here and here

Here's a basic animation of M-set with power going from 2 to 10. Power is not really defined for nonintegers so it looks discontinuous:
« Last Edit: August 12, 2018, 07:28:14 AM by gerrit »

Offline E8Guy

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« Reply #2 on: August 12, 2018, 05:01:54 PM »

Here's a basic animation of M-set with power going from 2 to 10. Power is not really defined for nonintegers so it looks discontinuous:
That's it, I was thinking of a "zoomed in" area of that. I can see now that the frame would have to pan as the exponent increases -thanks!

Online pauldelbrot

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« Reply #3 on: August 12, 2018, 08:09:20 PM »
Panning the view could be done automatically.

Say we have a family of formulae, parametrized by something -- in this case, exponent.

We pick a feature we want to observe as this parameter is varied -- say, a seahorse. This has a Misiurewicz point where its arms join together, which we can pick out approximate coordinates for, and then make it exact by using Newton's method. We need a second feature -- perhaps the center of the bulb the thing attaches to, or the Misiurewicz point at the tip of its curled-up tail. We can nail either of those precisely with Newton's method as well. Call the first, arm-joining point A and the other point selected B.

Now here's how to keep the thing centered in the view in a parameter morph video. For the first frame, set the view center on A and the magnification so that B is a fixed radius from A in pixels. For every subsequent frame: the parameter is tweaked; the previous locations of A and B are used as initial guesses in Newton's method to find where those features have moved to; the view is recentered on A; and the magnification is adjusted to put B the desired pixel distance away from A. Then the frame is rendered. And so on and so forth.

If the parameter is not changed too quickly, and the involved points don't completely cease to exist at some threshold, the seahorse, or whatever, should remain centered in the view and of unchanging size, even as it morphs and evolves. Failure modes could include: it shrinks fast enough to hit the precision limit at some point and cause glitches or a massive slowdown (fix: have perturbation available to use if necessary); the key Misiurewicz points, component centers, or etc. cease to exist, or one of them does; one of these moves too quickly at some point and causes Newton's method to fly off to some other target.

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