Möbius Transformation

A Möbius transformation is a rational function of one complex variable, of the form f of z equal to the quantity a z plus b, divided by c z plus d, where the coefficients a, b, c and d are complex numbers satisfying a d minus b c not equal to zero. Geometrically, such a transformation can be obtained by applying the inverse stereographic projection to move a point from the plane onto a sphere, rotating and repositioning that sphere, and then projecting back down onto the plane, and these transformations preserve angles while mapping every line to a line or a circle and every circle to a line or a circle. Möbius transformations are the projective transformations of the complex projective line, and together with its subgroups they form the Möbius group, which has numerous applications across mathematics and physics; they are named in honor of the mathematician August Ferdinand Möbius and are examples of homographies, linear fractional transformations, bilinear transformations and, in relativity theory, spin transformations.