Möbius group

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form

of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) to the plane. 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 transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2,C). Together with its subgroups, it has numerous applications in mathematics and physics.

Möbius transformations are named in honor of August Ferdinand Möbius; they are also variously named homographies, homographic transformations, linear fractional transformations, bilinear transformations, or fractional linear transformations.

Möbius transformations are defined on the extended complex plane ${\displaystyle {\widehat {\mathbf {C} }}=\mathbf {C} \cup \{\infty \}}$ (i.e., the complex plane augmented by the point at infinity).

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