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hi,
i would like to implement coloring function based on orbit period.
anyone know simple period detecting method in complex/quaternion series?

The simplest of which I'm aware is this:  iterate a bunch of times and save that final z value and its iteration number.  Then, for each subsequent iteration, compare that z with the stored value.  If the difference (typically, magnitude of z - zstored) is less than a threshold value, then the difference in iteration values is the (assumed) period.  Should work for any variable type.

Also see http://www.mrob.com/pub/muency/period.html (http://www.mrob.com/pub/muency/period.html).

One way to detect when a cycle has occurred is to iterate two variables at once, with one iterating half as fast as the other. If they ever become equal, then a cycle has occurred. The difference of their iteration counts when this happens will be a multiple of the period.

If you store the z[n] values in an array you do not need to iterate twice, you just need 2 index counters :)

I know this is an old topic, so this may be irrelevant, but with my recent work on a buddhabrot renderer, I wonder if it might not actually be faster to to iterate twice rather than storing the Z values because a memory access between each iteration might really hold the CPU up resulting in a lot of wasted cycles since CPU arithmetic is SO much faster than memory accesses.

Just a though, I don't know if it's true or not.

I don't know... I'd think it'd depend on the system you're using. Unless you're using a language with either really amazing support for complex numbers or really lousy memory procedures, I think storing it in an array would be faster. The only exception I can think of is if you're doing a tremendous number of iterations, so big that the array has to be put on the hard drive instead of the cache. THEN it'd be faster to just iterate twice.

For a very simple fractal like z^2+c then iterating twice may be more efficient but on a modern CPU I don't think you need to get much more complicated to get to a point where storing the values would be more efficient (simply due to improvements in memory access generally). Certainly if the calculations involved a divide or any transcendentals then storing would be faster.

maybe it would be better to use orbit radius, then you only need last 3 z's.

...why do you say that, makc?

What about period 5?

...or is this some kind of thing with radius of curvature??!?!!

his original intent was to color the thing using orbit period,and for that he seemingly needed to store lots of z-s. so I said the he could go with radius instead as the radius can be calculated from only 3 points. I am not saying using radius instead of period would make better picture or anything