ConformalLabpp/doc/math/novelty-statement.md

Scientific Novelty Statement

Purpose. This document explicitly states what conformallab++ contributes that no other open-source C++ library provides, and for which research problems it is the right tool. It is intended as a reference for paper introductions, grant applications, and collaborator onboarding.

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1 — The one-sentence statement

conformallab++ is the only open-source C++ library that implements discrete conformal equivalence in all three geometric settings (Euclidean, Spherical, Hyper-ideal), with a complete downstream Teichmüller pipeline — period matrix τ ∈ ℍ, Möbius holonomy, and fundamental domain construction — in a single cohesive codebase targeting the CGAL ecosystem.

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2 — Unique features (no equivalent elsewhere in C++)

2.1 — Three geometry modes in one library

Mode	Space	Energy	Application
Euclidean	ℝ²	Σ log(ℓᵢⱼ/ℓ̃ᵢⱼ)²	Flat torus uniformization, texture atlasing
Spherical	S²	NSD variant	Constant positive curvature, Koebe's theorem
HyperIdeal	H² (Poincaré disk)	Springborn 2020 ζ-functions	Hyperbolic surfaces, genus g ≥ 2

No other open-source C++ library implements all three. geometry-central (CMU) has Euclidean and partial HyperIdeal but lacks the Spherical mode entirely.

2.2 — Period matrix τ with SL(2,ℤ) reduction

For a closed genus-1 surface, conformallab++ computes the complex modulus τ = ω_b/ω_a ∈ ℍ from the holonomy of the uniformizing flat metric, then reduces τ to the standard fundamental domain

F = { τ ∈ ℍ : |τ| ≥ 1,  |Re(τ)| ≤ 1/2 }

via the SL(2,ℤ) action. This identifies the conformal class of the surface in Teichmüller space T₁ ≅ ℍ/SL(2,ℤ).

No other open-source C++ library computes τ. The Java ConformalLab does, but requires the JVM and is not integrated with any modern mesh processing framework.

2.3 — Möbius holonomy in SU(1,1)

The holonomy representation ρ: π₁(Σ) → SU(1,1) is computed for closed surfaces of any genus. For the torus this gives the lattice generators ω_a, ω_b ∈ ℂ. For hyperbolic surfaces this gives deck transformations as Möbius maps acting on the Poincaré disk.

2.4 — Tree-cotree cut graph (Erickson–Whittlesey)

For a closed surface of genus g, the cut graph produces exactly 2g seam edges that cut the surface to a disk. This is required for layout, holonomy computation, and fundamental domain construction. The cut graph is not present in any other C++ conformal geometry library.

2.5 — Fundamental domain and tiling

From the holonomy generators, conformallab++ constructs the fundamental domain parallelogram and its lattice tiling for genus-1 surfaces. This is the discrete analog of the classical construction of a torus as ℂ/Λ.

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3 — What makes this a research tool, not just an implementation

3.1 — Variational framework, not heuristic

The energy functionals are derived from first principles (Bobenko–Springborn 2004). The Newton solver guarantees quadratic convergence to the global optimum for Euclidean and HyperIdeal modes (strict convexity). The solution is mathematically unique (up to Möbius normalisation) — not an approximation.

3.2 — Analytic Hessians

For Euclidean and Spherical modes, the Hessian is computed analytically from the cotangent Laplacian and its spherical analog. This gives exact derivatives, not finite-difference approximations, which is required for reproducible research.

3.3 — Discrete-to-smooth correspondence

The discrete period matrix τ_discrete is a computable invariant of the triangulated surface. Its convergence to the smooth Riemannian τ_smooth under mesh refinement is an open research question in general that this library is designed to investigate — though it has already been proven for the special class of ramified coverings of the Riemann sphere by Bobenko–Bücking (2021).

3.4 — Full test coverage of analytic invariants

176 CGAL tests verify mathematically provable properties:

• Gauss–Bonnet: Σ(2π−Θᵥ) = 2π·χ(M) to machine precision
• τ ∈ fundamental domain: three inequalities
• Holonomy closure: [T_a, T_b] = Id (abelian for genus 1)
• Gradient consistency: FD check at ε = 1e-5 for all three functionals

These are not regression tests — they verify mathematical correctness independently of the input mesh.

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4 — Target audience

Audience	Primary use
Discrete differential geometers	Computing τ, holonomy, uniformization for theoretical examples
Computational mathematicians	Benchmarking discrete-to-smooth convergence of τ
CGAL developers	Extending the CGAL parameterization package (Phase 8)
Computer graphics researchers	Conformal texture atlasing with exact angle preservation
Algebraic geometers	Numerical experiments on moduli spaces of tori

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5 — Relationship to the Java original

conformallab++ is a port of Stefan Sechelmann's Java ConformalLab (TU Berlin, ~850 commits, v1.0.0 2018, LGPL). The port:

• Replaces the custom Java halfedge structure (CoHDS) with CGAL::Surface_mesh
• Replaces JUnit tests with GTest + CGAL test format (176 tests)
• Adds Doxygen API documentation, CMake build, and CLI
• Is MIT licensed (the Java original is LGPL)
• Targets submission to the CGAL library as package Discrete_conformal_map

The mathematics is identical to the Java original. The C++ implementation is independently validated by the test suite and by agreement with Java outputs on shared test meshes (cathead, brezel, torus family).

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6 — What conformallab++ is not

• Not a mesh processing library. It operates on existing triangulated surfaces. Remeshing, smoothing, and simplification are outside its scope.
• Not a real-time renderer. The Newton solver is accurate but not optimised for interactive frame rates (though it converges in < 1 second for typical meshes).
• Not a distortion-minimisation tool. It computes the unique conformally equivalent metric, not a least-distortion UV map. Use libigl for the latter.
• Not complete for genus g ≥ 2. The Siegel period matrix Ω and full uniformization for higher genus are Phase 10 research targets, not yet implemented.