IIsegoria

Interactive projective geometry

Circles, poles
& projective change.

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 visionFirst watch whole grids bend through a pole; then inspect the local third-order residue that no Möbius approximation can remove.

Möbius transformations

Fractional linear, globally geometric

T(z) = (az + b)/(cz + d),   ad − bc ≠ 0

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—one reason the geometry remains so rigid.

Invariant family

Circles include lines

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

Angles survive

T′(z) = (ad − bc)/(cz + d)² is nonzero wherever T is finite, so the map is conformal away from its pole.

Schwarzian

A third-order obstruction

S(f) = f‴/f′ − (3/2)(f″/f′)²

S(f) vanishes exactly when f is locally Möbius. It ignores postcomposition by a Möbius map.

Chain rule

Projective curvature composes

S(f ∘ g) = (S(f) ∘ g)(g′)² + S(g)

The squared derivative is the natural weight of a quadratic differential.

Useful examples

A compact Schwarzian gallery

  • S(zⁿ) = (1 − n²)/(2z²)
  • S(eᵃᶻ) = −a²/2
  • S(tan az) = 2a²
  • S(log z) = 1/(2z²)