ConformalLabpp/doc/architecture/geometry-central-comparison.md

conformallab++ vs. geometry-central — Detailed Comparison

Purpose of this document.
conformallab++ and geometry-central (CMU, Keenan Crane's group) both implement discrete conformal equivalence of triangulated surfaces. They share the same mathematical core but diverge in algorithmic strategy, scope, and target audience. This document maps the overlap precisely, identifies what cannot and should not be adopted, and explains where a side-by-side study creates scientific added value.

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1 — Shared mathematical foundation

Both libraries implement the following chain:

Input mesh M  →  edge lengths ℓᵢⱼ  →  solve for u ∈ ℝᵛ
  such that  ℓ̃ᵢⱼ = e^{(uᵢ+uⱼ)/2} · ℓᵢⱼ  satisfies  Σα_v(u) = Θᵥ  ∀v

The variational framework that makes this a well-posed optimisation problem goes back to Bobenko & Springborn (2004). The hyperbolic (HyperIdeal) geometry is from Springborn (2020). The geometry-central implementation (Gillespie, Springborn & Crane, SIGGRAPH 2021) is an explicit extension of Springborn 2020 to intrinsic triangulations.

Key consequence: the mathematical problem is identical. Any difference in output is either a normalization convention or a bug in one of the two libraries — making cross-validation directly meaningful.

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2 — Algorithmic comparison

Dimension	conformallab++	geometry-central (Gillespie 2021)
Solver	Newton–Raphson, analytical Hessian	Newton or Yamabe gradient flow (user choice)
Convergence	Quadratic (8–20 iterations on typical meshes)	Newton: quadratic; Yamabe: linear (~hundreds of steps)
Triangulation	Fixed throughout — operates on original Surface_mesh	Ptolemaic flips applied before/during solve to reach intrinsic Delaunay
Hessian	SimplicialLDLT + SparseQR fallback; analytical for Euclidean/Spherical, FD for HyperIdeal	Assembled on the current (possibly flipped) triangulation
Mesh backend	CGAL Surface_mesh<Point_3>	geometry-central ManifoldSurfaceMesh
Geometry modes	Euclidean ✓ · Spherical ✓ · HyperIdeal ✓	Euclidean ✓ · Hyperbolic ✓ · Spherical ✗
Open meshes	✓ (boundary DOFs pinned)	✓

Why Newton on a fixed triangulation works well

The Euclidean and HyperIdeal energies are strictly convex after gauge-fixing. Newton therefore converges from u=0 in 8–20 iterations for any reasonable mesh. The Hessian is the cotangent Laplacian (Euclidean) or its hyperbolic analog — well-conditioned on Delaunay meshes, but can degrade on strongly non-Delaunay inputs.

What Ptolemaic flips add

A Ptolemaic flip replaces diagonal AC with BD in a quadrilateral under the constraint that the Ptolemy relation

AC · BD = AB · CD + AD · BC

holds. This is conformal-class-preserving — the new λ₀ values represent the same discrete conformal structure. The gain: the flipped triangulation is intrinsic Delaunay, which bounds the off-diagonal Hessian entries and prevents ill-conditioning on pathological inputs.

This is the only genuine algorithmic advantage geometry-central has for the shared sub-problem. It costs nothing mathematically and buys robustness on bad meshes.

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3 — Feature matrix: what exists where

Feature	conformallab++	geometry-central	Notes
Discrete conformal equivalence (Euclidean)	✓	✓	Shared core
Discrete conformal equivalence (Hyperbolic/HyperIdeal)	✓ (Springborn 2020 formulation)	✓ (Gillespie 2021 extension)	Mathematically equivalent
Discrete conformal equivalence (Spherical)	✓	✗	Unique to conformallab++
Analytical Hessian (Euclidean & Spherical)	✓	✓
Analytical Hessian (HyperIdeal)	FD (Phase 9b: analytical planned)	✓	gc has analytical version
Gauss–Bonnet check & enforce	✓	implicit in solver
Tree-cotree cut graph (2g seam edges)	✓	✗	Required for period matrix
Priority-BFS layout in ℝ²/S²/Poincaré disk	✓	partial (conformal param only)
Möbius holonomy SU(1,1)	✓	✗	Unique to conformallab++
Period matrix τ ∈ ℍ + SL(2,ℤ) reduction	✓	✗	Unique to conformallab++
Fundamental domain + tiling	✓	✗	Unique to conformallab++
Intrinsic Delaunay triangulation	✗	✓	gc has via SignpostIntrinsicTriangulation
Ptolemaic flips	✗	✓	gc's robustness mechanism
Heat method (geodesic distances)	✗	✓	Auxiliary tool in gc
YAML/declarative pipeline	✓ (Phase 8e spec)	✗
CGAL-package submission	✓ (Phase 8 target)	✗
JSON/XML serialisation	✓	✗
CLI app	✓	✗
Inversive distance functional (Luo 2004)	✗ (Phase 9a)	✗	Neither has it yet
Siegel period matrix Ω (genus g≥2)	✗ (Phase 10b)	✗

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4 — What should be adopted — and what should not

Adopt: Ptolemaic pre-conditioning (GC-2, after Phase 8)

What: a single preprocessing pass that Delaunay-izes the input triangulation via Ptolemaic flips, updates λ₀ accordingly, then hands off to the existing Newton pipeline unchanged.

Why it fits:

• Preserves the conformal class — mathematically sound
• Drop-in before compute_euclidean_lambda0_from_mesh(), no interface change
• Does not touch cut graph, holonomy, period matrix
• ~200 lines of code, one new test suite

Interface sketch:

// include/preprocessing_delaunay.hpp  (Phase GC-2)
void ptolemy_delaunay(ConformalMesh& mesh, EuclideanMaps& maps);
// Flips edges until all faces satisfy the Delaunay condition.
// Updates maps.lambda0[e] via the Ptolemy relation after each flip.
// Prerequisite: compute_euclidean_lambda0_from_mesh() already called.
// Postcondition: mesh is an intrinsic Delaunay triangulation of the same surface.

Adopt partially: analytical HyperIdeal Hessian

geometry-central has a closed-form Hessian for the hyperbolic energy. conformallab++ currently uses finite differences (Phase 9b plans analytical). The geometry-central implementation can serve as a reference for Phase 9b — not a code copy, but a mathematical cross-check.

Do not adopt: full intrinsic triangulations as architecture

SignpostIntrinsicTriangulation is geometry-central's core data structure. It tracks vertex positions as (face, barycentric coordinates) rather than 3D points. Replacing CGAL::Surface_mesh with this would require:

1. Rewriting the cut graph algorithm (which works on halfedges of the original mesh and must survive across flips — non-trivial bookkeeping)
2. Tracking seam edges through flip events for holonomy computation
3. Abandoning the CGAL package target (Phase 8) — CGAL's mesh concepts are extrinsic
4. Losing the 3D layout output (Poincaré disk, sphere) which clients depend on

Verdict: the architecture incompatibility is fundamental, not incidental. The period matrix pipeline requires a stable topological cut that does not survive arbitrary flip sequences. This is not a solvable engineering problem within the current project scope — it would be a different project.

Do not adopt: Yamabe flow

Newton converges in 8–20 iterations; Yamabe flow needs hundreds. The only reason to use Yamabe flow is when the Hessian is indefinite (which happens in the Spherical case — and conformallab++ already handles this with the correct sign flip in the energy). There is no mesh type where Yamabe flow beats Newton on metrics that conformallab++ targets.

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5 — Where cross-comparison creates scientific added value

5.1 — Independent cross-validation of the shared core

The discrete conformal equivalence problem for Euclidean and HyperIdeal geometry is implemented independently in two codebases, by different groups, with different algorithms. Agreement on:

• the u-vector (after normalization)
• UV coordinates (up to Möbius transformation)
• the residual ‖G(u*)‖ at convergence

would constitute mutual validation without ground truth. This is the same methodology used in numerical PDE literature to validate independent solvers.

Concrete protocol:

For each test mesh (cathead.obj, brezel.obj, torus_4x4.off, torus_hex_6x6.off):
  1. Load into both libraries with identical vertex ordering
  2. Run conformallab++ Newton solver  →  u_clab, UV_clab
  3. Run geometry-central solver        →  u_gc,   UV_gc
  4. Normalize both (subtract mean, divide by scale)
  5. Report  max|u_clab[v] - u_gc[v]|  and  mean conformal distortion difference

Expected: agreement to ≤ 1e-8 on well-conditioned meshes. Discrepancy would indicate a bug or a normalization mismatch worth investigating.

5.2 — Convergence study: Newton with vs. without Ptolemaic pre-conditioning

Hypothesis: on non-Delaunay meshes (e.g. a torus refined by subdivision without re-meshing), Ptolemaic Delaunay pre-conditioning reduces Newton iteration count.

Measurable quantities:

• Newton iterations to ‖G‖ < 1e-8
• Hessian condition number κ(H) at u = 0
• Wall-clock time

This comparison requires only GC-2 to be implemented in conformallab++ and would answer the question: how bad does a mesh have to be before Ptolemaic pre-conditioning pays off?

Publication potential: a short note or conference contribution comparing the two approaches on a systematic mesh quality benchmark would be self-contained and novel — neither library has published this comparison.

5.3 — Period matrix and holonomy as differentiating contribution

geometry-central deliberately stops at the conformal parameterization. The period matrix τ and Möbius holonomy computation in conformallab++ extend the pipeline into Teichmüller theory territory that geometry-central does not address.

This is the strongest scientific differentiator: conformallab++ can compute

τ = ω_b / ω_a  ∈ ℍ,  SL(2,ℤ)-reduced

for any closed genus-1 surface, and (in Phase 10) the Siegel matrix Ω for genus g≥2. No other open-source C++ library does this. Cross-comparison with geometry-central makes this gap explicit and positions conformallab++ as the more complete tool for Teichmüller-theoretic applications.

5.4 — Spherical geometry as unique contribution

The Spherical geometry mode (angle sums → 4π on a sphere, NSD Hessian with sign flip) has no counterpart in geometry-central. A mathematician interested in conformal maps on surfaces of positive curvature (constant curvature +1) has no alternative in open-source C++.

5.5 — Validation of Springborn 2020 in two independent implementations

Springborn 2020 ("Ideal Hyperbolic Polyhedra and Discrete Uniformization") is the shared theoretical reference for both the HyperIdeal geometry mode in conformallab++ (Phase 2/3) and the hyperbolic component of Gillespie 2021 in geometry-central. Cross-checking the ζ-function values (ζ₁₃, ζ₁₄, ζ₁₅) and the resulting angle sums on the same meshes would validate both implementations of the paper — a service to the discrete geometry community.

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6 — Demarcation: where the comparison ends

Topic	conformallab++	geometry-central	Comparable?
u-vector at convergence	✓	✓	✓ after normalization
UV parameterization	✓	✓	✓ up to Möbius
Convergence speed (iterations)	✓	✓ (Newton mode)	✓ direct
HyperIdeal angle sums	✓	✓	✓
Spherical angle sums	✓	✗	✗
Period matrix τ	✓	✗	✗
Holonomy T_a, T_b	✓	✗	✗
Fundamental domain	✓	✗	✗
Intrinsic Delaunay quality	not tracked	✓	—
Mesh topology handling	CGAL halfedge	gc manifold mesh	not comparable
Scalability (large meshes)	not benchmarked yet	benchmarked in paper	comparable if same mesh

The comparison is meaningful and complete for the shared conformal core. It ends where conformallab++ continues into Teichmüller theory (holonomy, τ, fundamental domain) — that region has no counterpart in geometry-central and must be validated by analytic invariants alone (→ doc/math/validation.md).

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7 — Practical roadmap for the comparison

Step	When	What	Effort
GC-1a	Now	Manual UV comparison on cathead.obj — run both, diff u-vectors	1 day
GC-1b	Now	Add normalization utility to conformallab++ (normalize_u_vector())	2h
GC-1c	After Phase 8	Automated comparison script (Python or small C++ binary)	2 days
GC-2	After Phase 8	ptolemy_delaunay() preprocessing pass	1 week
GC-bench	After GC-2	Convergence study: Newton ± Ptolemaic pre-conditioning on 10 meshes	1 week
GC-paper	Phase 10	Short note on the comparison — period matrix as differentiator	—

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8 — References

Reference	Role in this comparison
Bobenko, Springborn — Variational Principles for Circle Patterns, Trans. AMS (2004)	Shared variational foundation for all three geometry modes
Springborn — Ideal Hyperbolic Polyhedra and Discrete Uniformization, DCG (2020)	Mathematical basis for HyperIdeal in conformallab++ AND for Gillespie 2021
Gillespie, Springborn, Crane — Discrete Conformal Equivalence of Polyhedral Surfaces, SIGGRAPH (2021)	geometry-central implementation; introduces Ptolemaic flips
Sharp, Soliman, Crane — Navigating Intrinsic Triangulations, SIGGRAPH (2019)	geometry-central SignpostIntrinsicTriangulation — basis for GC-2
Sechelmann — doctoral thesis, TU Berlin (2016)	conformallab++ primary source; covers period matrix, holonomy, all three modes