Möbius transformations (also called linear fractional transformations) are maps from the complex plane to itself of the form

\[

f(z) = \frac{a z + b}{c z + d}

\]

where \(a, b, c, d \in \mathbf{C}\) and \(a d – b c \neq 0\). This definition extends to functions on the Riemann sphere, where the geometry of such transformations becomes more apparent. The following short video shows off a remarkable correspondence between symmetries of the sphere and Möbius transformations.

# Möbius Transformations

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