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.

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.