Auto-Generating Fractal Movie
Original Publication Date: 1997-Mar-01
Included in the Prior Art Database: 2005-Apr-01
Disclosed is a method for generating a looping animation, resulting in the appearance of continuous magnification (or demagnification) of a fractal. The method generates an animation based on a self-similarity relation intrinsic to the fractal itself and, therefore, avoids the need for special calculation and planning on the part of the user.
Auto-Generating Fractal Movie
a method for generating a looping animation,
resulting in the appearance of continuous magnification (or
demagnification) of a fractal. The method generates an animation
based on a self-similarity relation intrinsic to the fractal itself
and, therefore, avoids the need for special calculation and planning
on the part of the user.
creates a looping series of N digital images which,
shown in sequence, display the effect of flying ever deeper into a
fractal, specifically a fractal from the class representable by
iterated function systems. The exposition that follows assumes that
the fractal is defined as an iterated function system, that the
range of the fractal is two-dimensional, and that the particular
function the user chooses to scale with is an affine mapping.
However, relaxing these assumptions -- to scale three-dimensional
fractals, for example -- is generally straightforward, and the method
described, suitably augmented, is still useful. A requirement which
cannot be relaxed, however, is that the particular scaling function
chosen must be invertible.
Start with an
iterated function system fractal defined with the
J mappings f sub 1, f sub 2, ..., f sub j. Select one of these
mappings, say the ith mapping, to use for scaling. As stated above,
the mapping selected must be invertible. By the (fractal)
properties of iterated function systems, the image of the entire
fractal is contained in the range of f sub i and, likewise, in the
range of f sub i overmark 2 (the composition of f sub i with itself).
These images of the fractal are distorted according to the functions
f sub i and f sub i overmark 2. The desired series of images by
transforming the range of f sub i into that of f sub i overmark 2 in
a series of smooth steps and mapping these regions into the desired
display window to produce the actual pictures.
1. Define a display window (e.g., (100,100) x (144,144)
for a 45x45 pixel window) and scale the fractal to
it. This scaling can be done automatically. The
re-scaled fractal is now defined by J mappings g sub j,
each corresponding to an original mapping f sub j.
2. Re-scale to fit just the range of the ith mapping in
the window by using the inverse of the ith mapping
(of the scaled-to-the-window fractal) as the new scaling
factor. That is, for each mapping g sub j of the
window-scaled fractal, re-scale to get a fractal defined
by the J mappings:
h sub j = g sub i sup -1 g sub j g sub i
where g sub i is the window-scaled version of the original
chosen mapping f sub i. Alternatively, one may wish to
re-scale with g sub i overmark -2 or even higher powers
in order to ensure that no extraneous points from other
mappings protrude into the display window. ...