Möbius transformation

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.