ConformalLabpp/doc/math/discrete-conformal-theory.md

Discrete Conformal Geometry — Mathematical Background

This document is written for a mathematician who knows Riemannian surfaces and complex analysis but is new to the discrete setting. It covers exactly the theory implemented in conformallab++.

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1 — The continuous picture (in one paragraph)

A Riemann surface (M, g) carries a conformal structure: the class of all metrics related to g by a smooth positive factor. On a compact surface of genus g, the Uniformization Theorem gives a unique constant-curvature representative (flat for g = 1, hyperbolic for g ≥ 2, spherical for g = 0). The conformal modulus of a genus-1 surface is a point τ ∈ ℍ (upper half-plane), well-defined up to SL(2, ℤ).

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2 — Discrete conformal equivalence (DCE)

A triangulated surface is a pair (K, ℓ) where K is a simplicial complex homeomorphic to a surface and ℓ: E → ℝ₊ assigns an edge length. Two length assignments ℓ and ℓ̃ are discretely conformally equivalent if there exist vertex weights u: V → ℝ such that

ℓ̃ᵢⱼ = e^{(uᵢ + uⱼ)/2} · ℓᵢⱼ    for every edge {i, j}.

This is the discrete analogue of a conformal rescaling g̃ = e^{2φ} g. The weights u ∈ ℝ^V are the conformal factors (log-scale factors on vertices).

Key fact (Springborn 2020): within each DCE class there exists a unique length assignment realising a prescribed angle structure, and it can be found by Newton's method on a convex energy.

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3 — The variational energy

For each target corner angle Θ_v at vertex v, define the angle-defect energy:

E(u) = Σ_{corners} φ(αᵥ(u))   −   Σᵥ Θᵥ · uᵥ   +   boundary terms

where αᵥ(u) is the corner angle at v in the triangulation with edge lengths ℓ̃(u) and φ is an appropriate primitive (Clausen / Lobachevsky / ImLi₂ depending on geometry).

The gradient is simply the angle-sum residual:

∂E/∂uᵥ = Σ_{faces containing v} αᵥ(face)  −  Θᵥ

Setting G = 0 finds the unique u realising the prescribed angle sums.

Three geometry modes

Mode	Space	φ	Hessian	Newton step
Euclidean	ℝ²	Clausen Cl₂	cotangent Laplacian, PSD	SimplicialLDLT(H)
Spherical	S²	ImLi₂	NSD (concave E)	SimplicialLDLT(−H)
Hyper-ideal	H²	Lobachevsky	PSD (strictly convex)	SimplicialLDLT(H)

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4 — Gauss–Bonnet constraint

The target angles must satisfy the discrete Gauss–Bonnet equation

Σᵥ (2π − Θᵥ) = 2π · χ(M)

before any solver is called. If this fails, no conformal factor can realise Θ and Newton will not converge. conformallab++ provides:

check_gauss_bonnet(mesh, maps);     // throws if violated
enforce_gauss_bonnet(mesh, maps);   // redistributes residual uniformly

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5 — From angles to geometry: trilateration

After Newton converges, edge lengths ℓ̃ are known. The layout (embedding into ℝ², S², or H²) is built by a priority BFS:

1. Place an initial face arbitrarily.
2. For each adjacent face, place its third vertex by trilateration — solving the system of three distance equations.
3. Priority is depth in the spanning tree (shallowest first).

For Euclidean geometry this is the standard cosine rule. For hyperbolic geometry (Poincaré disk model) it uses the Möbius-isometric placement formula implemented in layout.hpp.

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6 — Cut graph and holonomy

For genus g ≥ 1 the layout does not close up: a handle introduces a holonomy — a non-trivial monodromy around each generator of π₁.

The tree-cotree algorithm (Erickson–Whittlesey 2005) computes a minimal cut graph with exactly 2g seam edges. After cutting, the surface is disk-like and the BFS layout is well-defined. The holonomies along the 2g cut edges are:

• Euclidean: lattice translations ω₁, ω₂ ∈ ℂ
• Hyperbolic: Möbius isometries T₁, T₂ ∈ SU(1,1)

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7 — Period matrix (genus 1)

For a torus the conformal modulus is

τ = ω₂ / ω₁ ∈ ℍ

After Euclidean uniformization, ω₁ and ω₂ are the holonomies computed from the seam edge displacements. The SL(2, ℤ)-reduction to the fundamental domain {|τ| ≥ 1, |Re(τ)| ≤ 1/2, Im(τ) > 0} is performed automatically by compute_period_matrix().

For genus g ≥ 2, the Siegel period matrix Ω ∈ H_g (g×g complex symmetric with positive definite imaginary part) requires integrating holomorphic 1-forms — this is Phase 10b.

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8 — Fundamental domain

The fundamental domain of a torus is the parallelogram with vertices {0, ω₁, ω₁+ω₂, ω₂} in ℂ. conformallab++ computes this and provides tiling utilities (tiling_copy, tiling_neighbourhood).

For genus g ≥ 2, the fundamental domain is the standard 4g-gon (Phase 9c).

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9 — Where the code lives

Energy / gradient      code/include/*_functional.hpp
Hessian                code/include/*_hessian.hpp
Newton solver          code/include/newton_solver.hpp
Trilateration / BFS    code/include/layout.hpp
Cut graph              code/include/cut_graph.hpp
Holonomy               code/include/layout.hpp  (HolonomyData)
Period matrix          code/include/period_matrix.hpp
Fundamental domain     code/include/fundamental_domain.hpp
Möbius maps            code/include/layout.hpp  (MobiusMap)

All implementations are header-only (C++17), no compiled library.

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10 — Primary references

Reference	Covers
Springborn, Discrete Uniformization of Polyhedral Surfaces, 2020	Complete mathematical foundation of all three modes
Sechelmann, Variational Methods for Discrete Surface Parameterization, TU Berlin 2016	Original Java implementation — the direct source for this library
Pinkall & Polthier, 1993	Cotangent Laplacian
Erickson & Whittlesey, SODA 2005	Tree-cotree cut graph
Bobenko & Springborn, Trans. AMS 2004	Variational circle-pattern framework

Full reference list: doc/math/references.md