f(z) = (az+b)/(cz+d) maps circles and lines to circles and lines
Domain (z-plane)
Image (w-plane)
Fixed points: —
det(ad−bc): 1
Colored grid in domain maps to circles/lines in image.
Möbius maps are the automorphisms of the Riemann sphere — conformal bijections preserving circles.
A Möbius transformation f(z) = (az+b)/(cz+d) with ad−bc≠0 is the most general
conformal automorphism of the Riemann sphere ℂ∪{∞}. They form the group PSL(2,ℂ).
Key property: circles and lines map to circles and lines (lines are "circles through ∞").
Every Möbius map is classified by its trace tr²/(ad−bc): elliptic (rotation-like),
hyperbolic (dilation-like), loxodromic (both), or parabolic (one fixed point).