Mobius Transformation Calculator
Calculate Mobius (linear fractional) transformations and analyze their properties.
Transformation f(z) = (az + b) / (cz + d)
Input Point z
Matrix Representation
[a b] [1.00+0.00i 0.00+0.00i]
[c d] = [0.00+0.00i 1.00+0.00i]
f(z) = w
1.0000 + 1.0000i
|w| = 1.4142, arg(w) = 45.00 deg
TType
Parabolic
det|det|
1.0000
Fixed Points
No fixed points found
Pole
At infinity
Properties
- Conformal (angle-preserving)
- Maps circles to circles (or lines)
- Preserves cross-ratio
- Forms a group under composition
About Mobius Transformations
Definition
A Mobius transformation is a complex function of the form f(z) = (az+b)/(cz+d) where ad-bc != 0. They are the only conformal automorphisms of the Riemann sphere.
Applications
- Conformal mapping in fluid dynamics
- Hyperbolic geometry
- Special relativity (Lorentz group)
- Computer graphics and visualization