A bijective conformal map of the extended complex plane or Riemann sphere. A Möbius transformation has the form f(z)=(az + b)/(cz + d), where a,b,c,d are complex numbers such that ad ≠ bc. Note that f(z) is understood to be ∞ if cz + d = 0 and f(∞) = a/c if c ≠ 0 and f(∞) = ∞ if c = 0. The Möbius transformations form a group under composition; if the extended complex plane is identified with the projective complex line (see projective space), then this group is PGL(2,ℂ).