Interactive projective geometry
Möbius transformations are the fractional-linear symmetries of the Riemann sphere. The Schwarzian is the first derivative that notices when a holomorphic map is not one of them.
Two scales of visionI take it in this order: first watch whole grids bend through a pole, then inspect the local third-order residue that no Möbius approximation can remove.
Möbius transformations
The exceptional point z = −d/c maps to ∞, while ∞ maps to a/c when c ≠ 0. Matrices represent these maps only up to a common nonzero scale.
Composition is matrix multiplication, and inversion is matrix inversion, which is one reason I find the geometry stays so rigid.
Invariant family
On the Riemann sphere, an ordinary line is a circle through ∞. Möbius maps therefore send every circle or line to another circle or line.
Local geometry
T′(z) = (ad − bc)/(cz + d)² is nonzero wherever T is finite, so the map is conformal away from its pole.
Schwarzian
S(f) vanishes exactly when f is locally Möbius. It ignores postcomposition by a Möbius map.
Chain rule
The squared derivative is the natural weight of a quadratic differential.
The examples I keep to hand