Möbius transformation
= Möbius transformation
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= Möbius map
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A Möbius transformation of the Riemann sphere has the form
$$
z\longmapsto\frac{az+b}{cz+d},
\qquad ad-bc\ne0.
$$
The group is generated by translations, nonzero complex scalings, and inversion $z\mapsto1/z$.