Möbius transformation (Möbius function)

Description

In geometry and complex analysis, a Möbius transformation of the plane is a rational function of one complex variable. 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 projective transformations of the complex projective line and they are also variously named homographies, homographic transformations, linear fractional transformations, bilinear transformations, or fractional linear transformations. The set of all Möbius transformations forms a group under composition. This group can be given the structure of a complex manifold in such a way that composition and inversion are holomorphic maps. The coefficients of the Möbius function are complex numbers satisfying: ad − bc ≠ 0.