Moebius Transformation

print

Möbius Transformation


Mathematically, perhaps the most interesting transformation available in Cinderella is the Möbius transformation. This is a transformation with the property that circles and lines are again mapped to circles and lines. Any rotation, translation, or similarity can be considered a special case of a Möbius transformation, whereas neither an affine transformation nor a projective transformation is a Möbius transformation.

A Möbius transformation is defined by three preimage/image pairs of points. Defining a Möbius transformation in Cinderella is analogous to the creation of a similarity, though with six required points instead of four.

The following two pictures demonstrate how a circle is mapped under a Möbius transformation. Since the Möbius transformation in the picture maps A → D, B → E, C → F, a circle through A, B, C is mapped to a circle through D, E, F.


Defining a Möbius transformation Mapping a circle


If one iterates a Möbius transformation, the sequence of mapped pictures forms a mathematically and aesthetically very interesting pattern that can be considered a generalization of a logarithmic spiral. The following picture shows the image of an iterated Möbius transformation applied to a single circle and its center.

Iterating a Möbius transformation


A Möbius transformation usually has two distinct fixed points. In the above drawing these fixed points have been constructed explicitly by making sure that the transformation maps A to itself and B to itself.



See Also





Contributors to this page: Kortenkamp , Richter , Kohler and Kramer .
Page last modified on Friday 26 of August, 2011 [16:49:27 UTC] by Kortenkamp.

The content on this page is licensed under the terms of the License.


Menu
free viagra samplehentai girlsmature pussymilfseeker cialis samplescam clip movie nude webcammother incest order viagra online cialis ukanime rapeparis hilton phone numbers viagra alternatives cialis forum cialis free samplehot girls in pantiesmonster of cocks discount cialis cilias free viagra samplesfree chat rooms cilia structurefree cartoon sex comics buy cialis order viagrafree adult videosplump girl gallerypantyhose gallerycum on her face cheapest cialisbisexual moviestampa bukakehuge black cock thumbnails buy cialis onlineporn star cialis drugwomen having sex cheap generic viagra alternative to viagra natural viagra cheap viagraoral sexteen webcam strip videosnude spanish girlserotic sex cams movies viagra side effectscartoon adultdisney sex animenude blonde hairygang bang swinger cialis viagrabisexual free moviesgay twinkswebcam chat live xxxyoung teens order cialislatina girls thongscum loversjapanese girl viagra cheapyoung japanese girlsmr chews asian beavergangbang squadshoshone indiansmature wiveslive webcam chat girlsfree ebony viagra on lineasian ladyboysteen boys viagra pillsself bondage techniques cailisincest familyfree ebony cheap cialisgay amateur cialis genericbusty asian viagra onlinemature breasts viagra for women free viagrabig boobies cialis generic viagragloryhole gaylatinas in thongs female viagraindian tits viagra 6 free samplesamateur upskirt viagra alternativefree xxx video cialis online discountgalleries of teen girls cialis dosage cheap generic cialisparis hilton pornopussy cat dollsbrutal sexgay peopleblack milfsno tits discount viagrablonde hairy pussyshemale animefree hardcore moviesmom strips for sonfat titscelebrity legsdouble anal levitra vs cialis cialis tadalafil cialis cheapgay bdsmcelebrities exposed viagra generic alternatives to viagra viagra canadabestialitypink porn stars viagra jokesclips of teen sexchicks suck horse cock online viagrasex with horsespainful analglory holes floridafree american bukkake cialis online buy viagrabig cock cum free cialisteen gay porn cialis side effects herbal viagra best price viagra purchase cialis cialis soft tabs cialis vs viagrafree fat girl webcamfree porn movie clipsoral penis suckingebony hardcore viagra pricepantyhose crossed legs cialis and levitralesbiennesblonde boobs buy viagra online