In geometry, a part of mathematics, a Möbius transformation of the plane is a rational function of the form f ( from ) = a from + b c from + d {\displaystyle f(z)={\frac {az+b}{cz+d}}}
of a complex variable z; Here, the coefficients a, b, c, d are complex numbers that match ad-bc ≠ 0. Möbius transformations are named after August Ferdinand Möbius, but they are also called homographic transforms, linear fractional transformations or broken linear transformations. < / p> Overview
Möbius transformations are defined on the extended complex plane (i.e., the complex plane plus the point at infinity): C ^ = C & # x222A; { & # x221E; } . {\displaystyle {\widehat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}.}
This extensive complex can be considered as an atmosphere, the Riemann atmosphere, or as the complex projective line. Each Möbius transformation is a bijective angular image of the Riemann atmosphere on its own. Indeed, any such image is necessarily a Möbius transformation.
The collection of all Möbius transformations is a group under composition called the Möbius group. It is the automotive group of the Riemann atmosphere (when considered a Riemann surface) and is sometimes referred to as or ( C ^ ) {\displaystyle {\mbox{Aut}}({\widehat {\mathbb {C} }})\,} .
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