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External reviewer pass over the literature references. Verified entries against arXiv/DOI/publisher and corrected misattributions that had propagated across the docs. Corrected citations (consistent across all docs): - Bowers-Bowers-Lutz 2026: title was the 2017 paper's -> "Rigidity of Koebe Polyhedra and Inversive Distance Circle Packings" - Liouville theorem: "Springborn 2019" -> Pinkall & Springborn, Geom. Dedicata 214 (2021) - Bobenko-Pinkall-Springborn: "G&T 14 (2010)" -> G&T 19(4) (2015), 2155-2215 - Optimal Cone Singularities: "Crane, Soliman, Ben-Chen, Schroeder" -> Soliman, Slepcev, Crane, ACM TOG 37(4) - Schlaefli formula: "Rivin, Springborn 1999" -> Rivin, Schlenker - Quasiconformal distortion: "Springborn, Veselov" -> Born, Buecking, Springborn (arXiv:1505.01341) - Period matrices: "Bobenko, Buecking 2009" (was Bobenko-Mercat-Schmies' title) -> Bobenko-Mercat-Schmies 2011 + genuine Bobenko-Buecking 2021 - Fabricated entry: "Alexa 2020, DOI 10.1145/3414685.3417840" pointed to an unrelated paper (Pixelor) -> Bunge, Herholz, Kazhdan, Botsch 2020 - Stripe Patterns: "Bonneel et al. 2015" -> Knoeppel, Crane, Pinkall, Schroeder 2015 Equation-number corrections (verified against the PDFs): - Glickenstein 2011 "eq. 4.6" -> "§5.2" (no such equation label exists) - Springborn 2020 "eq. 4.6" -> "§4 variational gradient" - inversive-distance attribution softened to classical inversive distance Other: - DBFEnergy bibliography (separate repo) and convergence half-sentence in novelty-statement.md §3.3 (Bobenko-Buecking 2021) - Status legend (implemented vs planned) at top of references.md - New Phase 12 (decorated DCE & geometric transition, Chain A, near-term) and Phase 13 (canonical tessellations & polyhedral realisation, Chain B capstone) in phases.md + research-track.md; 10c scope-boundary note clarifying infrastructure vs Lutz-specific algorithms Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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87 lines
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# References
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## Primary source
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This library implements the algorithms from:
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| **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](https://depositonce.tu-berlin.de/items/8e2988b2-d991-45b5-aad5-9fb7988f3b2f) · CC BY-SA 4.0 |
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Java reference implementation: [github.com/varylab/conformallab](https://github.com/varylab/conformallab)
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---
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## References by module
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> **Status-Konvention.** Die „Used in"-Spalte nennt das Modul *oder* die Phase.
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> Ein Verweis auf eine **ausgelieferte** Phase (Code existiert, getestet) ist mit
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> ✅ markiert; ein Verweis auf eine **geplante/Forschungs**-Phase mit 🔜. Nur die
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> ✅-Quellen sind Grundlage des aktuellen Codes; 🔜-Quellen belegen Roadmap-Ziele
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> (vgl. auch Abschnitt „Phase 10 references (future research)" unten und
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> `novelty-statement.md` §6 „What conformallab++ is not").
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>
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> | Marker | Bedeutung | Phasen |
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> |---|---|---|
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> | ✅ | implementiert & getestet | 9a.1, 9a.2, 9b-analytic, Cut-Graph, Hessians |
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> | 🔜 | geplant / Forschung | 9d.2, 9f, 10a, 10b, 10c |
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| Reference | Used in |
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| ✅ **Springborn** — *Ideal Hyperbolic Polyhedra and Discrete Uniformization*, Discrete & Computational Geometry **64** (2020), pp. 63–108. DOI: [10.1007/s00454-019-00132-8](https://doi.org/10.1007/s00454-019-00132-8) | `hyper_ideal_geometry.hpp` — ζ₁₃/ζ₁₄/ζ₁₅ functions; `hyper_ideal_functional.hpp` |
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| **Pinkall, Polthier** — *Computing Discrete Minimal Surfaces and Their Conjugates*, Experimental Mathematics (1993) | `euclidean_hessian.hpp` — cotangent Laplacian |
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| **Bobenko, Springborn** — *Variational Principles for Circle Patterns and Koebe's Theorem*, Transactions AMS (2004) | Variational angle-sum framework underlying all three functionals |
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| **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) |
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| **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. |
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| **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. |
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| **Bobenko, Pinkall, Springborn** — *Discrete conformal maps and ideal hyperbolic polyhedra*, Geometry & Topology **19**(4) (2015), pp. 2155–2215. arXiv: [1005.2698](https://arxiv.org/abs/1005.2698) | Face-based circle-packing functional (`CPEuclideanFunctional.java` → `cp_euclidean_functional.hpp`, Phase 9a.1) |
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| **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. |
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| **Erickson, Whittlesey** — *Greedy Optimal Homotopy and Homology Generators*, SODA (2005) | `cut_graph.hpp` — tree-cotree algorithm |
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| **Bobenko, Springborn** — *A Discrete Laplace–Beltrami Operator for Simplicial Surfaces*, Discrete & Computational Geometry (2007) | Background for cotangent weights |
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| **Desbrun, Kanso, Tong** — *Discrete Differential Forms for Computational Modeling*, SIGGRAPH Course Notes (2006) | Discrete exterior calculus background for Phase 10a |
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| **Soliman, Slepčev, Crane** — *Optimal Cone Singularities for Conformal Flattening*, ACM Transactions on Graphics **37**(4), Article 105 (2018). DOI: [10.1145/3197517.3201367](https://doi.org/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. |
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| **Bobenko, Lutz** — *Decorated Discrete Conformal Equivalence in Non-Euclidean Geometries*, Discrete & Computational Geometry (2025). arXiv: [2310.17529](https://arxiv.org/abs/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. |
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| **Bobenko, Lutz** — *Decorated Discrete Conformal Maps and Convex Polyhedral Cusps*, IMRN 2024(12), pp. 9505–9534. arXiv: [2305.10988](https://arxiv.org/abs/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. |
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| **Lutz** — *Canonical Tessellations of Decorated Hyperbolic Surfaces*, Geometriae Dedicata 217 (2023). arXiv: [2206.13461](https://arxiv.org/abs/2206.13461) | **Phase 10c**: canonical Delaunay tessellations in Penner coordinates; unifies the decorated framework with the fundamental domain construction for genus g ≥ 2. |
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| **Lutz** — *Decorated Discrete Conformal Equivalence, Canonical Tessellations, and Polyhedral Realization* (PhD thesis, TU Berlin, 2024). DOI: [10.14279/depositonce-20357](https://doi.org/10.14279/depositonce-20357) | Comprehensive single reference for Phases 9d.2, 10b, 10c — collects Bobenko-Lutz 2024/2025 and Lutz 2023 with complete proofs. |
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| **Bowers, Bowers, Lutz** — *Rigidity of Koebe Polyhedra and Inversive Distance Circle Packings* (2026). arXiv: [2601.22903](https://arxiv.org/abs/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. |
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| **Alexa, Wardetzky** — *Discrete Laplacians on General Polygonal Meshes*, ACM SIGGRAPH (2011). DOI: [10.1145/1964921.1964997](https://doi.org/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. |
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| **Bunge, Herholz, Kazhdan, Botsch** — *Polygon Laplacian Made Simple*, Computer Graphics Forum **39**(2) (2020), pp. 303–313. DOI: [10.1111/cgf.13931](https://doi.org/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](https://doi.org/10.1145/3386569.3392389).) |
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---
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## geometry-central cross-reference *(optional comparison track)*
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> These references relate to an alternative implementation of the same
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> mathematical problem. They are not prerequisites for conformallab++,
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> but are relevant for cross-validation and possible algorithmic adoptions
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> (→ GC-1/2/3 in the phase roadmap, → Section 9 in `validation.md`).
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| Reference | Relevance |
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| **Gillespie, Springborn, Crane** — *Discrete Conformal Equivalence of Polyhedral Surfaces*, ACM SIGGRAPH 2021. DOI: [10.1145/3450626.3459763](https://doi.org/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. |
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| **Sharp, Soliman, Crane** — *Navigating Intrinsic Triangulations*, ACM SIGGRAPH 2019 | Algorithmic basis for `SignpostIntrinsicTriangulation` in geometry-central — relevant for GC-2 (optional pre-conditioning). |
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**Note on Springborn 2020:**
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The paper *"Ideal Hyperbolic Polyhedra and Discrete Uniformization"*
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(Springborn, Discrete & Computational Geometry 2020) is **already implemented in
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conformallab++** — it is the direct reference for the HyperIdeal geometry mode
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(`hyper_ideal_geometry.hpp`). The geometry-central implementation (Gillespie 2021)
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builds on this paper and augments it with Ptolemaic flips.
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---
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## Phase 10 references (future research)
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| Reference | Relevant for |
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| **Farkas, Kra** — *Riemann Surfaces*, Springer GTM 71 | Siegel period matrix, Teichmüller theory |
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| **Siegel** — *Topics in Complex Function Theory, Vol. 2*, Wiley | Siegel upper half-space H_g, Sp(2g,ℤ) reduction |
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| **Bobenko, Mercat, Schmies** — *Period Matrices of Polyhedral Surfaces*, in: Computational Approach to Riemann Surfaces (2011) | Discrete period matrices on polyhedral surfaces |
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| **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](https://doi.org/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. |
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| **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. |
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| **Pinkall, Springborn** — *A discrete version of Liouville's theorem on conformal maps*, Geometriae Dedicata **214** (2021), pp. 389–398. arXiv: [1911.00966](https://arxiv.org/abs/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. |
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| **Born, Bücking, Springborn** — *Quasiconformal distortion of projective transformations and discrete conformal maps*, arXiv: [1505.01341](https://arxiv.org/abs/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. |
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| **Knöppel, Crane, Pinkall, Schröder** — *Stripe Patterns on Surfaces*, ACM SIGGRAPH (2015). DOI: [10.1145/2766890](https://doi.org/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. |
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| **Sawhney, Crane** — *Boundary First Flattening*, ACM TOG **37**(1), Article 5 (2017). DOI: [10.1145/3132705](https://doi.org/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. |
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