Editing Möbius transformation (section)

{{Short description|Rational function of the form (az + b)/(cz + d)}}
{{distinguish|Möbius transform|Möbius function}}

In [[geometry]] and [[complex analysis]], a '''Möbius transformation''' of the [[complex plane]] is a [[rational function]] of the form
<math display="block">f(z) = \frac{a z + b}{c z + d}</math>
of one [[complex number|complex]] variable {{mvar|z}}; here the coefficients {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} are complex numbers satisfying {{math|''ad'' − ''bc'' ≠ 0}}.

Geometrically, a Möbius transformation can be obtained by first applying the inverse [[stereographic projection]] from the plane to the unit [[sphere]], moving and rotating the sphere to a new location and orientation in space, and then applying a stereographic projection to map from the sphere back to the plane.{{sfn|Arnold|Rogness|2008|loc=Theorem 1}} These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle.

The Möbius transformations are the [[projective transformation]]s of the [[complex projective line]]. They form a [[group (mathematics)|group]] called the '''Möbius group''', which is the [[projective linear group]] {{nowrap|PGL(2, '''C''')}}. Together with its [[subgroup]]s, it has numerous applications in mathematics and physics.

[[Möbius geometry|Möbius geometries]] and their transformations generalize this case to any number of dimensions over other fields.

Möbius transformations are named in honor of [[August Ferdinand Möbius]]; they are an example of [[homography|homographies]], [[linear fractional transformation]]s, bilinear transformations, and spin transformations (in relativity theory).<ref>
{{Cite book|last=Needham|first=Tristan|url=https://press.princeton.edu/books/hardcover/9780691203690/visual-differential-geometry-and-forms|title=Differential Geometry and Forms; A Mathematical Drama in Five Acts|publisher=Princeton University Press|year=2021|isbn=9780691203690 |at=p. 77, footnote 16}}</ref>