ConformalLabpp/doc/math/references.md

References

Primary source

This library implements the algorithms from:

Sechelmann — Variational Methods for Discrete Surface Parameterization: Applications and Implementation, Doctoral thesis, TU Berlin 2016	The complete mathematical foundation: discrete conformal equivalence, variational angle-sum functionals, Newton solver, uniformization, period matrices, holonomy. DOI: 10.14279/depositonce-5415 · CC BY-SA 4.0

Java reference implementation: github.com/varylab/conformallab

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References by module

Status-Konvention. Die „Used in"-Spalte nennt das Modul oder die Phase. Ein Verweis auf eine ausgelieferte Phase (Code existiert, getestet) ist mit ✅ markiert; ein Verweis auf eine geplante/Forschungs-Phase mit 🔜. Nur die ✅-Quellen sind Grundlage des aktuellen Codes; 🔜-Quellen belegen Roadmap-Ziele (vgl. auch Abschnitt „Phase 10 references (future research)" unten und novelty-statement.md §6 „What conformallab++ is not").

Marker	Bedeutung	Phasen
✅	implementiert & getestet	9a.1, 9a.2, 9b-analytic, Cut-Graph, Hessians
🔜	geplant / Forschung	9d.2, 9f, 10a, 10b, 10c

Reference	Used in
✅ Springborn — Ideal Hyperbolic Polyhedra and Discrete Uniformization, Discrete & Computational Geometry 64 (2020), pp. 63–108. DOI: 10.1007/s00454-019-00132-8	hyper_ideal_geometry.hpp — ζ₁₃/ζ₁₄/ζ₁₅ functions; hyper_ideal_functional.hpp
✅ Kolpakov, Mednykh — A Formula for the Volume of a Hyperbolic Tetrahedron, arXiv: math/0603097 (2006)	Tetrahedron volume with one ideal vertex: calculateTetrahedronVolumeWithIdealVertexAtGamma in hyper_ideal_utility.hpp (Phase 9b analytic Hessian)
✅ Meyerhoff, Ushijima — A Note on the Dirichlet Domain, in: The Epstein Birthday Schrift (2006)	Tetrahedron volume with three ideal vertices: calculateTetrahedronVolumeFullyIdeal in hyper_ideal_utility.hpp
Pinkall, Polthier — Computing Discrete Minimal Surfaces and Their Conjugates, Experimental Mathematics (1993)	euclidean_hessian.hpp — cotangent Laplacian
Bobenko, Springborn — Variational Principles for Circle Patterns and Koebe's Theorem, Transactions AMS (2004)	Variational angle-sum framework underlying all three functionals
Luo — Combinatorial Yamabe Flow on Surfaces, Communications in Contemporary Mathematics (2004)	Inversive-distance functional — new research in Phase 9a.2 (no Java original; implemented from this paper + Glickenstein 2011 + Bowers-Stephenson 2004)
Bowers, Stephenson — Uniformizing dessins and Belyĭ maps via circle packing, Memoirs of the AMS 170(805) (2004)	Introduces inversive-distance circle packings (used in Phase 9a.2). Hinweis: die zur Initialisierung benutzte Formel I_ij = (ℓ²−r_i²−r_j²)/(2 r_i r_j) ist die klassische inversive Distanz (vgl. Glickenstein §5.2: ℓ²=r_i²+r_j²+2r_ir_jη), nicht eine eigene „Bowers-Stephenson-Identität" — B–S liefern die Packungstheorie, nicht diese Formel.
Glickenstein — Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds, J. Differential Geometry 87(2) (2011), pp. 201–238	Analytic Hessian of the inversive-distance functional. ⚠️ Korrektur: die Arbeit nummeriert Gleichungen nicht im Format „(4.6)" — der Verweis ist durch die §5.2-Parametrisierung ℓ²_ij = r²_i + r²_j + 2 r_i r_j η_ij zu ersetzen. Cross-correspondence: η_ij ist die inversive Distanz und entspricht dem Kosinus des Supplements des Schnittwinkels (Schnitt bei arccos(−η_ij)) — also I_ij = cos θ_e nur bis aufs Vorzeichen/Supplement, nicht wörtlich.
✅ Bobenko, Pinkall, Springborn — Discrete conformal maps and ideal hyperbolic polyhedra, Geometry & Topology 19(4) (2015), pp. 2155–2215. arXiv: 1005.2698 (first posted 2010)	Face-based circle-packing functional (CPEuclideanFunctional.java → cp_euclidean_functional.hpp, Phase 9a.1)
Schläfli — On the multiple integral ∫dx dy …, Quarterly Journal of Pure and Applied Mathematics (1858/60)	Klassische Schläfli-Differentialformel (dV = −½ Σ_e ℓ_e dθ_e). ⚠️ Hinweis: die in Phase 9b-analytic benutzte Randterm-Form 2 dV = Σ_e aₑ dαₑ + Σ_v bᵥ dβᵥ steht nicht bei Schläfli 1858, sondern ist die verallgemeinerte Fassung für Mannigfaltigkeiten mit Rand → korrekter Beleg: Rivin–Schlenker 1999 (Phase-10-Liste). Schläfli 1858 nur als historischer Ursprung zitieren.
Erickson, Whittlesey — Greedy Optimal Homotopy and Homology Generators, SODA (2005)	cut_graph.hpp — tree-cotree algorithm
Bobenko, Springborn — A Discrete Laplace–Beltrami Operator for Simplicial Surfaces, Discrete & Computational Geometry (2007)	Background for cotangent weights
Desbrun, Kanso, Tong — Discrete Differential Forms for Computational Modeling, SIGGRAPH Course Notes (2006)	Discrete exterior calculus background for Phase 10a
Soliman, Slepčev, Crane — Optimal Cone Singularities for Conformal Flattening, ACM Transactions on Graphics 37(4), Article 105 (2018). DOI: 10.1145/3197517.3201367	L¹-optimal automatic cone placement — Phase 9d.2 (non-Euclidean cone extensions). Provides the optimisation algorithm for choosing cone positions automatically; complements Bobenko-Lutz 2025 on non-Euclidean settings.
Bobenko, Lutz — Decorated Discrete Conformal Equivalence in Non-Euclidean Geometries, Discrete & Computational Geometry (2025). arXiv: 2310.17529	Phase 9d.2: extends DCE to hyperbolic + spherical geometry with Penner-coordinate decorations; unifies cone singularities and hyperideal cusps in one algebraic framework.
Bobenko, Lutz — Decorated Discrete Conformal Maps and Convex Polyhedral Cusps, IMRN 2024(12), pp. 9505–9534. arXiv: 2305.10988	Phase 10b/10c: discrete uniformization theorem for decorated piecewise Euclidean surfaces; connects Phase 2/3 hyperideal vertices (cusps at ∞) to the period matrix and fundamental domain.
Lutz — Canonical Tessellations of Decorated Hyperbolic Surfaces, Geometriae Dedicata 217 (2023). arXiv: 2206.13461	Phase 10c: canonical Delaunay tessellations in Penner coordinates; unifies the decorated framework with the fundamental domain construction for genus g ≥ 2.
Lutz — Decorated Discrete Conformal Equivalence, Canonical Tessellations, and Polyhedral Realization (PhD thesis, TU Berlin, 2024). DOI: 10.14279/depositonce-20357	Comprehensive single reference for Phases 9d.2, 10b, 10c — collects Bobenko-Lutz 2024/2025 and Lutz 2023 with complete proofs.
Bowers, Bowers, Lutz — Rigidity of Koebe Polyhedra and Inversive Distance Circle Packings (2026). arXiv: 2601.22903	Phase 9b-analytic + Phase 10c' (KoebePolyhedron): theoretical uniqueness/rigidity for Koebe polyhedra and inversive-distance circle packings (incl. the tangency case); supports correctness of the analytic Hessian and the KAT construction.
Alexa, Wardetzky — Discrete Laplacians on General Polygonal Meshes, ACM SIGGRAPH (2011). DOI: 10.1145/1964921.1964997	Phase 9f: virtual-node polygon Laplacian extending Pinkall-Polthier to non-triangular meshes. Enables DCE on quad/Voronoi meshes without triangulation.
Bunge, Herholz, Kazhdan, Botsch — Polygon Laplacian Made Simple, Computer Graphics Forum 39(2) (2020), pp. 303–313. DOI: 10.1111/cgf.13931	Phase 9f: virtual-vertex polygon Laplacian — fügt pro Polygon einen virtuellen Knoten ein (impliziter Triangle-Fan), erweitert die cotangent-Diskretisierung auf nicht-konvexe/nicht-planare Polygone. (Alternative DEC-Variante: de Goes, Butts, Desbrun, Discrete Differential Operators on Polygonal Meshes, ACM TOG 39(4) (2020), DOI 10.1145/3386569.3392389.)

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geometry-central cross-reference (optional comparison track)

These references relate to an alternative implementation of the same mathematical problem. They are not prerequisites for conformallab++, but are relevant for cross-validation and possible algorithmic adoptions (→ GC-1/2/3 in the phase roadmap, → Section 9 in validation.md).

Reference	Relevance
Gillespie, Springborn, Crane — Discrete Conformal Equivalence of Polyhedral Surfaces, ACM SIGGRAPH 2021. DOI: 10.1145/3450626.3459763	Implemented in geometry-central. Extends Springborn 2020 with intrinsic triangulations and Ptolemaic flips. Solves the same DCE problem as conformallab++, but with a different algorithm.
Sharp, Soliman, Crane — Navigating Intrinsic Triangulations, ACM SIGGRAPH 2019	Algorithmic basis for SignpostIntrinsicTriangulation in geometry-central — relevant for GC-2 (optional pre-conditioning).

Note on Springborn 2020:
The paper "Ideal Hyperbolic Polyhedra and Discrete Uniformization" (Springborn, Discrete & Computational Geometry 2020) is already implemented in conformallab++ — it is the direct reference for the HyperIdeal geometry mode (hyper_ideal_geometry.hpp). The geometry-central implementation (Gillespie 2021) builds on this paper and augments it with Ptolemaic flips.

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Phase 10 references (future research)

Reference	Relevant for
Farkas, Kra — Riemann Surfaces, Springer GTM 71	Siegel period matrix, Teichmüller theory
Siegel — Topics in Complex Function Theory, Vol. 2, Wiley	Siegel upper half-space H_g, Sp(2g,ℤ) reduction
Bobenko, Mercat, Schmies — Period Matrices of Polyhedral Surfaces, in: Computational Approach to Riemann Surfaces (2011)	Discrete period matrices on polyhedral surfaces
Bobenko, Bücking — Convergence of discrete period matrices and discrete holomorphic integrals for ramified coverings of the Riemann sphere, Math. Phys. Anal. Geom. 24, Art. 23 (2021). DOI: 10.1007/s11040-021-09394-2	Phase 10b: discrete Siegel period matrix Ωᵢⱼ from cotangent-weighted integration plus the convergence result Ω_discrete → Ω_smooth under refinement (für ramified coverings) — belegt die Diskret-zu-glatt-Aussage in `novelty-statement.md §3.3.
Rivin, Schlenker — The Schläfli formula in Einstein manifolds with boundary, Electron. Res. Announc. AMS 5 (1999), pp. 18–23	Phase 9b-analytic: modern form of the Schläfli identity 2 dV = Σ aₑ dαₑ for manifolds with boundary — the bilinear form used to derive the analytic HyperIdeal Hessian.
Pinkall, Springborn — A discrete version of Liouville's theorem on conformal maps, Geometriae Dedicata 214 (2021), pp. 389–398. arXiv: 1911.00966	Phase 10b uniqueness: proves that the discrete conformal structure (and hence Ω) is a conformal invariant — the discrete Liouville theorem. Justifies that conformallab++ outputs a canonical representative.
Born, Bücking, Springborn — Quasiconformal distortion of projective transformations and discrete conformal maps, arXiv: 1505.01341 (2015)	Phase 10c error analysis: quantifies how well the discrete H²/Γ embedding approximates the smooth hyperbolic metric; error bounds for the Fuchsian group representation.
Knöppel, Crane, Pinkall, Schröder — Stripe Patterns on Surfaces, ACM SIGGRAPH (2015). DOI: 10.1145/2766890	Phase 10a cross-validation: applies discrete holomorphic 1-forms to direction field design; geometry-central provides an independent C++ implementation to cross-check the Phase 10a DiscreteHolomorphicFormUtility port.
Sawhney, Crane — Boundary First Flattening, ACM TOG 37(1), Article 5 (2017). DOI: 10.1145/3132705	Complementary method to Phase 9d: boundary-prescribed conformal flattening — user specifies boundary shape, interior conforms freely. Contrast: conformallab++ prescribes cone angles in the interior; BFF prescribes the boundary. Alternative approach for applications needing controlled boundary.