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Method for Finding Geometrically Similar Fractal Confined to Fixed Region

IP.com Disclosure Number: IPCOM000122543D
Original Publication Date: 1991-Dec-01
Included in the Prior Art Database: 2005-Apr-04
Document File: 2 page(s) / 81K

Publishing Venue

IBM

Related People

Greer, TD: AUTHOR

Abstract

Disclosed is a method for determining the parameters of an iterated function system which generates a scaled fractal geometrically similar to the fractal generated by an original iterated function system. The terminology used is for fractals imbedded in two dimensions, but the extension to higher dimensions is straightforward. Likewise, stretching or compressing, to correct for aspect ratios, for example, is easy to incorporate into the procedure.

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Method for Finding Geometrically Similar Fractal Confined to Fixed
Region

      Disclosed is a method for determining the parameters of
an iterated function system which generates a scaled fractal
geometrically similar to the fractal generated by an original
iterated function system.  The terminology used is for fractals
imbedded in two dimensions, but the extension to higher dimensions is
straightforward.  Likewise, stretching or compressing, to correct for
aspect ratios, for example, is easy to incorporate into the
procedure.

      An Iterated Function System definition of a fractal may consist
of a set of affine contraction mappings [*].  Affine mappings are
vector functions of the form f(x) = Ax + b, where x, f(x), and b
are n-dimensional vectors and A is an nxn matrix.  The fixed point of
each mapping is the vector y such that y = Ay + b.  So the
fixed point is
                 y = (I - A)-1b
where I is the nxn identity matrix and the -1 denotes inverse.  The
"image of a circle" under this mapping f means the points that are
obtained when every point on the circle is subjected to the mapping
f.  Mathematically, this is the range of f when the domain of f is
restricted to the circle.

      The following procedure produces the mappings which define a
new fractal, geometrically similar to the original, which lies within
a pre-defined circle, say, C(s,S).  In the following, C(c,R) denotes
a circle centered at c, with radius R.
1.   Calculate the fixed points of the mappings making up the IFS,
and then determine their average (the centroid, giving each point
equal weight).  Call the average c.  It will be the center of a
circle which will be guaranteed to contain the fractal.
2.   Find the most distant fixed point from c and choose an initial
radius R of the containing circle to be the distance between c and
this most distant fixed point.
3.   For each mapping in turn, increase R if necessary so that the
image of C(c,R) just lies within C(c,R).  This can always be done
because the mappings are contraction mappings.  An iterative method
to bracket the necessary R is to success...