Möbius Transformation

A Mobius 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. Mobius transformations are the projective transformations of the complex projective line, and together with its subgroups they form the Mobius group, which has numerous applications across mathematics and physics; they are named in honor of the mathematician August Ferdinand Mobius and are examples of homographies, linear fractional transformations, bilinear transformations and, in relativity theory, spin transformations. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/