Möbius Transformation Calculator

Generically 2 (solve cz²+(d−a)z−b=0). The identity has all points fixed. Parabolic (like z+1) has exactly 1. The character of fixed points classifies: elliptic (2 on circle), hyperbolic (2 on line), loxodromic (2 general), parabolic (1).

They decompose into: translation (z+b), rotation/scaling (az), inversion (1/z). Each preserves circles (treating lines as circles through ∞). Composition preserves the property.

(z₁,z₂;z₃,z₄) = (z₁−z₃)(z₂−z₄)/((z₁−z₄)(z₂−z₃)). Möbius transformations preserve it. The unique T mapping z₁→0, z₂→1, z₃→∞ is T(z) = (z−z₁)(z₂−z₃)/((z−z₃)(z₂−z₁)).