Möbius transformation

Uncommon card · No. 186399

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f ( z ) = a z + b c z + d {\displaystyle f(z)={\frac {az+b}{cz+d}}} 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 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. These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The...

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