Möbius Transformations

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Overwiev

A Möbius transformation is a bijective conformal map of the extended complex plane (i.e. the complex plane augmented by the point at infinity):



The set of all Möbius transformations forms a group under composition called the Möbius group.

The Möbius group is the automorphism group of the Riemann sphere, sometimes denoted



Certain subgroups of the Möbius group form the automorphism groups of the other simply-connected Riemann surfaces (the complex plane and the hyperbolic plane). As such, Möbius transformations play an important role in the theory of Riemann surfaces. The fundamental group of every Riemann surface is a discrete subgroup of the Möbius group (see Fuchsian group and Kleinian group). Möbius transformations are also closely related to isometries of hyperbolic 3-manifolds.

A particularly important subgroup of the Möbius group is the modular group; it is central to the theory of many fractals, modular forms, elliptic curves and Pellian equations.

In physics, the identity component of the Lorentz group acts on the celestial sphere the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativistic velocities will see the pattern of constellations as seen near the Earth continuously transform according to infinitesimal Möbius transformations. This observation is often taken as the starting point of twistor theory.

Definition

The general form of a Möbius transformation is given by



where a, b, c, d are any complex numbers satisfying ad − bc ≠ 0. (If ad = bc the rational function defined above is a constant.) This definition can be extended to the whole Riemann sphere (the complex plane plus the point at infinity).



Reference:Wikipedia