Mobius Transformation

A Mobius Transformation (aka fractional linear functions) is this function:

f(z) := (a z+b)/(c z+d)

They differ from the inverse function 1/z [see Complex Inversion] by adding translation, rotation and scaling.

Because (for c ≠ 0)

(az+b)/(cz+d) = a/c * (cz + bc/a)/(cz + d) = a/c*(cz + d )/(cz + d) + (b - ad/c)/(cz + d)

so that

(az+b)/(cz+d) = a/c + (bc - ad)/c^2 * 1/(z + d/c).

This implies that arbitrary Moebius transformations are not more complicated than f(z) = 1/z. In particular: all Moebius transformations map lines and circles to lines and circles, or just circles to circles when stereographically projected to the sphere.

moebius gauss 001
moebius riemann 001
moebius riemann 001
map: z ⟶ f(z) = (1+a)*z/(z + a)
domain: polar grid 1/4 < |z| < 4
morph: 0 < a < 1
range: Riemann Sphere, anaglyph.
moebius rotate 001
moebius rotate 001

All these moebius transformations map 0 to -1 and infinity to +1. We can view the changing grid lines as “moebius rotation” around -1 and +1. The name is justified because these maps are true rotations when viewed on the Riemann sphere. The inverse function f(z) = 1/z is, on the Riemann sphere, 180 degree rotation around -1, +1. The changing grid lines therefore represent a homotopy from the identity to the inverse function and back to the identity.

moebius riemann

The previous moebius transformations, viewed on the Riemann sphere, are rotations around the x-axis.


loxodromic gauss 001
Loxodromic Moebius transformations with fixed points zero and infinity. A spiral grid is mapped to itsself.
loxodromic 1 1 001
Loxodromic Moebius transformation with fixed points -1, +1. Apply z ⟶ (1 - z)/ (1 + z) to the previous images.