Möbius transformation

Möbius transformation

English

Alternative forms

• Mobius transformation
• Moebius transformation

Etymology

Named for German mathematician and theoretical astronomer August Ferdinand Möbius (1790–1868).

Noun

Möbius transformation (plural Möbius transformations)

1. (geometry, complex analysis) A transformation of the extended complex plane that is a rational function of the form f(z) = (az + b) / (cz + d), where a, b, c, d are complex numbers such that ad − bc ≠ 0; an automorphism of the complex projective line.
   • 2008, J. Vernon Armitage, John Parker, Jørgensen's inequality for Non-Archimedean Metric Spaces, Mikhail Kapranov, Sergii Kolyada, Yu. I. Manin, Pieter Moree, Leonid Potyagailo (editors), Geometry and Dynamics of Groups and Spaces: In Memory of Alexander Reznikov, page 97,

     Jørgensen's inequality gives a necessary condition for a non-elementary group of Möbius transformations to be discrete.

   • 2012, Martin Delacourt, Petr Kůrka, Finite State Transducers of Modular Möbius Number Systems, Branislav Rovan, Vladimiro Sassone, Peter Widmayer (editors), Mathematical Foundations of Computer Science 2012, 37th International Symposium, MFCS 2012, Proceedings, Springer, LNCS 7464, page 323,

     Modular Möbius number systems consist of Möbius transformations with integer coefficients and unit determinant.

   • 2013, Angel Cano, Juan Pablo Navarrete, Seade Kuri José Antonio, Complex Kleinian Groups, page 1,

     Classical Kleinian groups are discrete subgroups of Möbius transformations which act on the Riemann sphere with a nonempty region of discontinuity.

Hypernyms

• automorphism
• conformal mapping
• linear fractional transformation
  • homography, projective transformation

Holonyms

• Möbius group

Translations

transformation of the complex plane

Dutch: Möbius-transformatie f French: transformation de Möbius f German: Möbiustransformation f Italian: trasformazione di Möbius f Portuguese: transformação de Möbius f Spanish: transformación de Möbius f