Moebius transformations

A Möbius transformation is a mathematical function that maps points from one complex plane to another, typically represented as f(z) = (az + b)/(cz + d), where a, b, c, and d are complex numbers. This transformation preserves angles and the shapes of small figures, meaning it maintains the local structure of figures, though it may distort their size and position. Commonly used in complex analysis and geometry, Möbius transformations can turn circles into other circles and handle mappings of infinity, serving as powerful tools to analyze complex relationships and symmetries.