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Automatically Generated, Perceptually Smooth Fractal Movie

IP.com Disclosure Number: IPCOM000123212D
Original Publication Date: 1998-Jul-01
Included in the Prior Art Database: 2005-Apr-04
Document File: 2 page(s) / 73K

Publishing Venue

IBM

Related People

Greer, TD: AUTHOR

Abstract

Disclosed is a method for generating a looping animation resulting in the appearance of smooth continuous magnification (or de-magnification) 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.

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Automatically Generated, Perceptually Smooth Fractal Movie

   Disclosed is a method for generating a looping animation
resulting in the appearance of smooth continuous magnification (or
de-magnification) 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.

   In(1), a method is described to automatically generate
a looping animation of zooming into or out of a fractal.  The
technique described there is satisfactory for relatively
low-resolution movies with not too many frames, up to 150 x 150
pixels and 16 frames, for example.  However, with more frames and at
higher resolutions an apparent flaw in the movie becomes noticeable.
The movie seems to speed up and slow down, speed up and slow down.
The "flaw" is not an error in calculation or presentation, however.
Rather, it is a psychological expectation of the way things should
grow.  The algorithm described(1) calls for a linear series of steps
in the transformation from the fractal at one scale to the fractal at
another scale, where the two scales differ by the rescaling inherent
in one of the mappings defining the fractal.  While linear steps may
seem a natural choice, a geometrically increasing step size is the
choice that results in smooth apparent growth.

   The linear series is in step 3 of (1) where a transformation
matrix k(n) is defined as the affine mapping whose matrix part takes
the (polar coordinates) point (1, 0) to the point (1-n(1-r(1))/N, n
theta(1)/N), and the point (1, pi/2) to the point (1-n(1-r(2))/N, n
theta(2)/N + o/2), and choosing the scalar part of k(n) so that k(n)
has the same fixed point as a particular mapping of the fractal being
scaled.  The parameters r(1), r(2), theta(1) and theta(2) come from
the rescaling mapping as described in (1), and n and N are the fr...