MÖBIUS TRANSFORMATIONS

f(z) = (az+b)/(cz+d) — circles map to circles on the Riemann sphere
Möbius transformations are the automorphisms of the Riemann sphere ℂ∪{∞}.

Every such map sends circles (and lines) to circles (and lines).

Fixed points:

Type:

tr² = (a+d)²
elliptic: tr²∈[0,4)
parabolic: tr²=4
hyperbolic: tr²>4
loxodromic: tr²∈ℂ\ℝ