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VIZUÁLIS MATEMATIKA ÉS FIZIKA / VIZUALNA MATEMATIKA I FIZIKA / VISUAL MATHEMATICS AND PHYSICS
Anne Burns Professor of Mathematics, Long Island University - C.W. Post Campus, USA aburns@liu / anneburns.net
Iterált Möbiusz A komplex síkon alkalmazott Möbiusz-transzformáció egyik tulajdonsága, hogy kör képe kör lesz. A transzformációt többször egymás után alkalmazva a mellékelt színpompás körökből álló képeket kapjuk. A transzformáció paramétereinek már egészen kis változtatása is jelentős módosulást hoz létre a képeken.
Iterated Möbius A Möbius Transformation is a function from the complex plane to itself that maps circles (and lines) to circles (and lines). Two or more contracting Möbius Transformations can be composed to form an iterated function system. Starting with an initial circle and n Möbius Transformations that map an initial circle into n smaller circles inside the initial circle, after k iterations there will be nk circles. Where the original circles are tangent, the images will be tangent. If the original circles overlap, the images will overlap. The Unit Circle Group is a subgroup of the group of Möbius Transformations. A transformation from this group maps the unit circle onto itself and the interior onto the interior; such a transformation can be written in terms of three real parameters. If the transformation is not the identity, a geometric figure (that is not a circle) inside the unit circle will be mapped to a distorted image somewhere else inside the unit circle. Starting with an initial set of n circles inside the unit circle and n Möbius Transformations that map the unit circle into the n circles, then composing the n transformations with a transformation from the unit circle group to form an iterated function system, some artistic images are created. Each new iteration causes a compounding of the distortion yielding interesting patterns. Changing one of more of the three parameters by even a small amount can drastically change the picture.
More about this subject can be found at http://www.anneburns.net/circles/unitcircle.html and at http://myweb.cwpost.liu.edu/aburns/flash/evmthart.htm