A Geometrical Representation and Visualization of Möbius Transformation Groups

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Kento Nakamura and Kazushi Ahara: A Geometrical Representation and Visualization of Möbius Transformation Groups. In: Bridges 2017, Pages 159–166.



In this paper, we propose a system which enables us to create computer graphics arts originating from Kleinian groups easily and intuitively. Using this system, we can realize various types of Möbius transformations by arranging some circles and lines on the plane. The system uses the efficient rendering algorithm called IIS, introduced in the previous paper by the authors. In the 3D space, making a change circles and lines into spheres and planes, we extend the system to the 3D graphics of Kleinian groups.

Extended Abstract


 author      = {Kento Nakamura and Kazushi Ahara},
 title       = {A Geometrical Representation and Visualization of M\"{o}bius Transformation Groups},
 pages       = {159--166},
 booktitle   = {Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture},
 year        = {2017},
 editor      = {David Swart, Carlo H. S\'equin, and Krist\'of Fenyvesi},
 isbn        = {978-1-938664-22-9},
 issn        = {1099-6702},
 publisher   = {Tessellations Publishing},
 address     = {Phoenix, Arizona},
 note        = {Available online at \url{http://archive.bridgesmathart.org/2017/bridges2017-159.pdf}}

Used References

[1] David Mumford, Caroline Series, David Wright. Indra’s Pearls. Cambridge University Press, 2002.

[2] Kento Nakamura and Kazushi Ahara. A New Algorithm for Rendering Kissing Schottky Groups. In Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture, Pages 367-370. Tessellations Publishing, 2016. http://archive.bridgesmathart.org/2016/bridges2016-367.html

[3] Iwaniec, Tadeusz and Martin, Gaven. The Liouville theorem. Analysis and topology, Pages 339-361. World Scientific Publishing, 1998.


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