e28aee7051
Ziel: einem interessierten Mathematiker ermöglichen, die bisherige Arbeit
unabhängig zu validieren und eigene Forschung beizutragen.
Neu:
doc/math/discrete-conformal-theory.md
Kompakte mathematische Einführung (DCE, Variationsprinzip, drei
Geometriemodi, Holonomie, Periodenmatrix) für Riemann-Flächen-Kenner.
doc/math/validation.md
Analytisch bekannte Sollwerte + wie man sie mit dem Code prüft:
Gauss–Bonnet (χ), τ ∈ Fundamentaldomäne (3 Invarianten), Symmetrie-
Argumente für τ=i (4-fach) und τ=e^{iπ/3} (6-fach), Newton-Konvergenz,
Gradienten-Check (FD), Holonomie-Kommutator. Reviewer-Checkliste.
CONTRIBUTING.md (Root)
Gitea/GitHub-Standard: CONTRIBUTING.md im Root-Verzeichnis als
Kurzreferenz mit Links zu doc/contributing.md und den Math-Docs.
code/data/off/torus_4x4.off — 16 Vertices, 32 Flächen, Genus 1
code/data/off/torus_8x8.off — 64 Vertices, 128 Flächen, Genus 1
code/data/off/torus_hex_6x6.off — 36 Vertices, 72 Flächen, 6-fach Sym.
Aktualisiert:
README.md — 158 → 170 Tests, zwei neue Math-Links in Tabelle
doc/api/tests.md — 28 Suiten, 170 Tests, 1 Skip (korrigiert)
doc/contributing.md — Testzähler 158+2 → 170+1
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
176 lines
6.0 KiB
Markdown
176 lines
6.0 KiB
Markdown
# Discrete Conformal Geometry — Mathematical Background
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This document is written for a mathematician who knows Riemannian surfaces and
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complex analysis but is new to the *discrete* setting. It covers exactly the
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theory implemented in conformallab++.
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---
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## 1 — The continuous picture (in one paragraph)
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A **Riemann surface** (M, g) carries a conformal structure: the class of all
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metrics related to g by a smooth positive factor. On a compact surface of genus
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g, the Uniformization Theorem gives a unique constant-curvature representative
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(flat for g = 1, hyperbolic for g ≥ 2, spherical for g = 0). The conformal
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modulus of a genus-1 surface is a point τ ∈ ℍ (upper half-plane), well-defined
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up to SL(2, ℤ).
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---
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## 2 — Discrete conformal equivalence (DCE)
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A **triangulated surface** is a pair (K, ℓ) where K is a simplicial complex
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homeomorphic to a surface and ℓ: E → ℝ₊ assigns an edge length. Two
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length assignments ℓ and ℓ̃ are **discretely conformally equivalent** if there
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exist vertex weights u: V → ℝ such that
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```
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ℓ̃ᵢⱼ = e^{(uᵢ + uⱼ)/2} · ℓᵢⱼ for every edge {i, j}.
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```
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This is the discrete analogue of a conformal rescaling g̃ = e^{2φ} g.
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The weights u ∈ ℝ^V are the *conformal factors* (log-scale factors on vertices).
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**Key fact** (Springborn 2020): within each DCE class there exists a unique
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length assignment realising a prescribed angle structure, and it can be found
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by Newton's method on a convex energy.
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---
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## 3 — The variational energy
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For each target corner angle **Θ_v** at vertex v, define the **angle-defect energy**:
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```
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E(u) = Σ_{corners} φ(αᵥ(u)) − Σᵥ Θᵥ · uᵥ + boundary terms
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```
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where αᵥ(u) is the corner angle at v in the triangulation with edge lengths
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ℓ̃(u) and φ is an appropriate primitive (Clausen / Lobachevsky / ImLi₂ depending
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on geometry).
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The **gradient** is simply the angle-sum residual:
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```
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∂E/∂uᵥ = Σ_{faces containing v} αᵥ(face) − Θᵥ
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```
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Setting G = 0 finds the unique u realising the prescribed angle sums.
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### Three geometry modes
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| Mode | Space | φ | Hessian | Newton step |
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|---|---|---|---|---|
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| Euclidean | ℝ² | Clausen Cl₂ | cotangent Laplacian, PSD | SimplicialLDLT(H) |
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| Spherical | S² | ImLi₂ | NSD (concave E) | SimplicialLDLT(−H) |
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| Hyper-ideal | H² | Lobachevsky | PSD (strictly convex) | SimplicialLDLT(H) |
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---
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## 4 — Gauss–Bonnet constraint
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The target angles **must** satisfy the discrete Gauss–Bonnet equation
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```
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Σᵥ (2π − Θᵥ) = 2π · χ(M)
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```
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before any solver is called. If this fails, no conformal factor can realise Θ
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and Newton will not converge. conformallab++ provides:
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```cpp
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check_gauss_bonnet(mesh, maps); // throws if violated
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enforce_gauss_bonnet(mesh, maps); // redistributes residual uniformly
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```
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---
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## 5 — From angles to geometry: trilateration
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After Newton converges, edge lengths ℓ̃ are known. The layout
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(embedding into ℝ², S², or H²) is built by a **priority BFS**:
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1. Place an initial face arbitrarily.
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2. For each adjacent face, place its third vertex by *trilateration* —
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solving the system of three distance equations.
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3. Priority is depth in the spanning tree (shallowest first).
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For Euclidean geometry this is the standard cosine rule.
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For hyperbolic geometry (Poincaré disk model) it uses the Möbius-isometric
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placement formula implemented in `layout.hpp`.
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---
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## 6 — Cut graph and holonomy
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For genus g ≥ 1 the layout does not close up: a handle introduces a
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**holonomy** — a non-trivial monodromy around each generator of π₁.
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The **tree-cotree algorithm** (Erickson–Whittlesey 2005) computes a minimal
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cut graph with exactly 2g seam edges. After cutting, the surface is disk-like
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and the BFS layout is well-defined. The holonomies along the 2g cut edges are:
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- **Euclidean:** lattice translations ω₁, ω₂ ∈ ℂ
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- **Hyperbolic:** Möbius isometries T₁, T₂ ∈ SU(1,1)
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---
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## 7 — Period matrix (genus 1)
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For a torus the **conformal modulus** is
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```
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τ = ω₂ / ω₁ ∈ ℍ
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```
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After Euclidean uniformization, ω₁ and ω₂ are the holonomies computed from
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the seam edge displacements. The SL(2, ℤ)-reduction to the fundamental
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domain {|τ| ≥ 1, |Re(τ)| ≤ 1/2, Im(τ) > 0} is performed automatically by
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`compute_period_matrix()`.
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For genus g ≥ 2, the Siegel period matrix Ω ∈ H_g (g×g complex symmetric with
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positive definite imaginary part) requires integrating holomorphic 1-forms — this
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is Phase 10b.
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---
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## 8 — Fundamental domain
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The fundamental domain of a torus is the parallelogram with vertices
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{0, ω₁, ω₁+ω₂, ω₂} in ℂ. conformallab++ computes this and provides tiling
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utilities (`tiling_copy`, `tiling_neighbourhood`).
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For genus g ≥ 2, the fundamental domain is the standard 4g-gon (Phase 9c).
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---
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## 9 — Where the code lives
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```
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Energy / gradient code/include/*_functional.hpp
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Hessian code/include/*_hessian.hpp
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Newton solver code/include/newton_solver.hpp
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Trilateration / BFS code/include/layout.hpp
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Cut graph code/include/cut_graph.hpp
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Holonomy code/include/layout.hpp (HolonomyData)
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Period matrix code/include/period_matrix.hpp
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Fundamental domain code/include/fundamental_domain.hpp
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Möbius maps code/include/layout.hpp (MobiusMap)
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```
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All implementations are header-only (C++17), no compiled library.
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---
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## 10 — Primary references
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| Reference | Covers |
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| Springborn, *Discrete Uniformization of Polyhedral Surfaces*, 2020 | Complete mathematical foundation of all three modes |
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| Sechelmann, *Variational Methods for Discrete Surface Parameterization*, TU Berlin 2016 | Original Java implementation — the direct source for this library |
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| Pinkall & Polthier, 1993 | Cotangent Laplacian |
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| Erickson & Whittlesey, SODA 2005 | Tree-cotree cut graph |
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| Bobenko & Springborn, Trans. AMS 2004 | Variational circle-pattern framework |
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Full reference list: [doc/math/references.md](references.md)
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