Möbius Transformations

In 1827 August Möbius introduced fractional linear transformations in *Der Barycentrische Calcul*. Fifty-five years later Poincaré realized those same maps are the isometries of the hyperbolic plane. In 2017, Maximilian Nickel and Douwe Kiela (Facebook AI Research, NeurIPS) published Poincaré embeddings: WordNet's 82,000-noun hierarchy fits into a 5-dimensional Poincaré disk with MAP 0.86, beating a 200-dimensional Euclidean embedding at 0.81. A 19th-century geometric construction became state of the art in 21st-century graph machine learning.

• **Poincaré Embeddings (Nickel-Kiela, NeurIPS 2017):** WordNet's 82,000-node hierarchy in a 5D Poincaré disk reaches MAP 0.86 versus Euclidean's 0.81 at 200D
• **Computer vision (homography):** the cross-ratio of four collinear points is the only projective invariant, used directly in OpenCV's `findHomography`
• **Circle packing (Thurston, 1985):** Stephenson's CirclePack algorithm uses Möbius maps to build conformal flattenings of 3D meshes
• **SL(2,R) monodromy:** Heun and Painlevé ODEs are described by Möbius actions of the fundamental group, central to integrable systems and string theory

Предварительные знания

• Conformal Mappings

The Möbius Group

Möbius transformations form a group under composition, isomorphic to PSL(2,C). Every conformal automorphism of the Riemann sphere is a Möbius transformation, making the group fundamental to complex analysis and geometry.

**Möbius transformation:** f(z) = (az+b)/(cz+d), ad−bc ≠ 0 **Group structure:** Möb ≅ PSL(2,C) = SL(2,C)/{±I} - Composition ↔ matrix multiplication: [a,b;c,d] ∘ [e,f;g,h] - Identity: f(z) = z ↔ I = [1,0;0,1] - Inverse: f⁻¹(z) = (dz−b)/(−cz+a) **Subgroups:** - PSL(2,R): automorphisms of the upper half-plane - PSU(1,1): automorphisms of the unit disk

Fixed Points and Classification

Every non-identity Möbius transformation has exactly 1 or 2 fixed points on the Riemann sphere. The number and nature of fixed points classifies all Möbius transformations into three geometric types.

**Fixed points:** solve f(z) = z, i.e., (az+b)/(cz+d) = z → cz² + (d−a)z − b = 0 **Classification by trace τ = a+d (in SL(2,C)):** - **Elliptic:** τ ∈ (−2,2) real → two conjugate fixed points - Rotates around fixed points; |τ|<2 - Example: z ↦ e^{iθ}z (pure rotation) - **Hyperbolic:** τ ∈ R, |τ|>2 → two real fixed points - Stretches between fixed points - **Parabolic:** τ = ±2 → one fixed point (double root) - Translation-like: conjugate to z ↦ z+1

The Cross-Ratio

The cross-ratio is the unique Möbius-invariant. It encodes the relative positions of four points on the Riemann sphere and is preserved under every Möbius transformation.

**Cross-ratio:** (z₁,z₂;z₃,z₄) = (z₁−z₃)(z₂−z₄) / [(z₁−z₄)(z₂−z₃)] **Möbius invariance:** for any Möbius f: (f(z₁),f(z₂);f(z₃),f(z₄)) = (z₁,z₂;z₃,z₄) **Key uses:** - Three points → one (determine the map) - Real cross-ratio ↔ four points on a circle or line - Hyperbolic distance in the Poincaré model: d(z₁,z₂) = |log(z₁,z₂;p,q)| where p,q are ideal boundary points

The Poincaré Disk Model

The Poincaré disk model of hyperbolic geometry lives inside the unit disk. Its isometries are exactly the Möbius transformations preserving the disk: the connection between complex analysis and non-Euclidean geometry.

**Poincaré disk:** H² = {|z| < 1} with metric ds = 2|dz|/(1−|z|²) **Isometries:** Möbius transformations of the form: f(z) = e^{iθ} · (z−z₀)/(1−z̄₀z) **Geodesics:** arcs of circles perpendicular to the unit circle ∂D **Hyperbolic distance:** d(z₁,z₂) = 2·arctanh|(z₁−z₂)/(1−z̄₁z₂)| **Group:** PSU(1,1) = automorphisms of D

Key Ideas

• **Möbius group:** Möb ≅ PSL(2,C); composition = matrix multiplication
• **Classification:** elliptic/hyperbolic/parabolic by trace τ; 2 / 2 / 1 fixed points
• **Cross-ratio:** Möbius-invariant; real ↔ four concyclic points
• **Poincaré disk:** isometries = Möbius transforms preserving |z|<1; geodesics ⊥ ∂D

Connected Topics

Möbius transformations connect complex analysis to geometry and physics:

• Conformal Mappings — Möbius transforms are the simplest and most important conformal maps
• Analytic Continuation — Möbius transforms on the Riemann sphere appear in analytic continuation over complex paths
• Complex Analysis in Physics — PSL(2,C) is the Lorentz group; Möbius transforms act on the relativistic celestial sphere

Вопросы для размышления

• Why does PSL(2,C) rather than GL(2,C) appear as the Möbius group?
• How does the cross-ratio prove that Möbius transformations map generalized circles to generalized circles?
• Why is the Poincaré disk model of hyperbolic geometry relevant to machine learning embedding problems?

Связанные уроки

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