Date: Sat, 10 Jan 1998 05:54:22 -0500 (EST)
Message-Id: <199801101054.FAA06276@freenet2.carleton.ca.carleton.ca>
From: ao950@freenet.carleton.ca (Paul Derbyshire)
To: fractint@lists.xmission.com
Subject: (fractint) Determining if Two Points Hit The Same Attractor...or Siegel Disks Too!
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If you iterate two z, how can you tell if they hit the same attractor?
If it's a cyclic attractor, the two z's might not be the same.


Simple trick: pick an upper cutoff maxit. Iterate z1 ... z_n
floor(maxit/2) times. Call the results b1 ... b_n.
With original z1 ... z_n again, iterate them again, until at most maxit
iterations in total (ceil(maxit/2) more iterations that is). After each
iteration, compare each z_k with all the b_i, i <> k. If it has reached
one of the b_i, remove z_k and b_k. i and k are the same attractor, out of
phase; the two b_i and b_k are redundant.
After the process is complete, the remaining b's, which can be renamed a1
through a_j, are points on j distinct attractors.

One problem remains: detecting the exceptional case of Siegel disks. A
point on a Siegel disk will eventually hit itself (using a "hit" function
returning true if two complex numbers differ by less than a tiny epsilon)
but if you have five points in a Siegel disk, chances are they will not
hit each other, only themselves, and you will obtain five would-be
"attractors" and large black areas in the image. This is Not Good.

To check for Siegel disks, after the above attractor detection, each of
the a_i is iterated along with say three random points in a triangle
about a_i and a small epsilon distance away; this is stopped when a_i has
hit itself.
A Siegel disk is assumed if: a_i has just hit itself, the three other
points all hit themselves, and they did not actually converge onto a_i. In
that case, we find the disk's centre as follows: starting from a_i,
iterate it once along with a smallish triangle of side r of surrounding
points. Note the distance traveled by each. The one that traveled the
shortest distance is closest to the disk's centre. Move a_i the distance r*1
in that direction, i.e. onto that point. Iterate the new a_i and a new
triangle (side r) and pick again the point closest to the center, move r*10
that way. Do one more iteration. Note as d1 the distance traveled by the
original a_i, d2 the subsequent a_i, and d3 this a_i. Choose
m=(2*d2-d3-d1)/9, compute d3/m, move a_i that far in the same direction.
Iterate it and note distance traveled. If nontrivial, repeat the whole
thing again. Repeat as necessary. At last, leave a_i fixed; it is the
Siegel disk center. Flag it as being such and not an attractor and record
the 'period'.

After doing this for all the a_k, eliminate redundant ones, ones differing
by less than an epsilon or so. These were in the same Siegel disk and
found the same centre.

Now during generation, check not only for points converging to any of the
a_i, but for a point that recurs with a period corresponding to (one of)
the Siegel disk(s), and for any such, use the distance from the original
pixel to the corresponding a_i Siegel center as m1 and the angle to a_i of
the original pixel as m2. (The m2 can be normalized readily enough but the
m1 for each/the disk will probably have to be normalized after the
generation, not during; for each Siegel disk keep track of an m1_max,
starts as 0 and after every pixel found to be in the Siegel disk, put
m1_max->max(m1_max,m1). Then go through the database of pixels and for all
the points in each disk, divide all the m1's by the appropriate disk's
m1_max to normalize them all to the interval [0,1].



--
    .*.  Friendship, companionship, love, and having fun are the reasons for
 -()  <  life. All else; sex, money, fame, etc.; are just to get/express these.
    `*'  Send any and all mail with attachments to the hotmail address please.
Paul Derbyshire ao950@freenet.carleton.ca pgd73@hotmail.com

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