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From: wesley.loewer@dwarp.sccsi.com (Wesley Loewer) 
Newsgroups: sci.fractals
Subject: IFS examples
Message-ID: <217.523.uupcb@dwarp.sccsi.com>
Date: 6 Mar 93 18:38:00 GMT
Distribution: world
Organization: Data Warp Premium BBS - Spring/Houston, TX - 713-355-6107
Reply-To: wesley.loewer@dwarp.sccsi.com (Wesley Loewer) 
Lines: 79
 
AI>Could anyone give me some simple examples of IFS that could
AI>generate fractals in a 2d space
 
Concerning IFS examples, perhaps the following might be of interest to you
and others.  A couple of years ago I got really interested in polygons
formed by IFS definitions, in particular regular polygons that just touch
each other.  I don't know if the following is common knowledge or
something that I discovered, but here's what I came up with.  The proof for
this is a bit hard to describe via 7 bit ascii, but here are the results.
 
If you want an n_gon sided polygon (ie: n_gon=3 for Sierpinski Triangle),
For the IFS definition
 
 |x'| = |a b| * |x| + |e| with a probability p
 |y'|   |c d|   |y|   |f|
 
use   a = 1-r
      b = 0
      c = 0
      d = 1-r
      e = r * sin(k*x);
      f = r * cos(k*x);
      p = 1 / n_gon
 
where
  k = line number (ie: k=0, k=1, ..., k=n_gon-1)
  r = 1 - 1 / (2 * s)
  s = cos(n_angles*x/2) * sin((n_angles+1)*x/2) / sin(x/2)
  n_angles = integer part of (n_gon-1)/4
  x = 2 * pi / n_gon  (or 360/n_gon in degrees)
(also, if n_gon is odd try r = 1 + 1 / (2 * s) for an interesting
"inverted" polygon)
 
For example, using n_gon=3 gives the following 3 lines
ThreeMinus { ; Sierpinski Triangle
 0.5  0.0  0.0  0.5  0.0       0.5       0.333333
 0.5  0.0  0.0  0.5  0.433013 -0.25      0.333333
 0.5  0.0  0.0  0.5 -0.433013 -0.25      0.333333
}
 
ThreePlus { ; using r = 1 + 1 / (2 * s)
-0.5  0.0  0.0 -0.5  0.0       1.5       0.333333
-0.5  0.0  0.0 -0.5  1.299038 -0.75      0.333333
-0.5  0.0  0.0 -0.5 -1.299038 -0.75      0.333333
}
 
FiveMinus {
 0.381966  0.0  0.0  0.381966  0.0       0.618034  0.2
 0.381966  0.0  0.0  0.381966  0.587785  0.190983  0.2
 0.381966  0.0  0.0  0.381966  0.363271 -0.500000  0.2
 0.381966  0.0  0.0  0.381966 -0.363271 -0.500000  0.2
 0.381966  0.0  0.0  0.381966 -0.587785  0.190983  0.2
}
 
FivePlus {
-0.381970  0.0  0.0 -0.381970  0.0       1.381970  0.2
-0.381970  0.0  0.0 -0.381970  1.314332  0.427052  0.2
-0.381970  0.0  0.0 -0.381970  0.812302 -1.118037  0.2
-0.381970  0.0  0.0 -0.381970 -0.812302 -1.118037  0.2
-0.381970  0.0  0.0 -0.381970 -1.314332  0.427052  0.2
}
 
Six {
 0.333333  0.0  0.0  0.333333  0.0       0.666667  0.166667
 0.333333  0.0  0.0  0.333333  0.577351  0.333334  0.166667
 0.333333  0.0  0.0  0.333333  0.577351 -0.333333  0.166667
 0.333333  0.0  0.0  0.333333  0.000000 -0.666667  0.166667
 0.333333  0.0  0.0  0.333333 -0.577351 -0.333334  0.166667
 0.333333  0.0  0.0  0.333333 -0.577351  0.333333  0.166667
}
 
Hope someone finds this interesting,
-wes
                                                                                                                         
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