# 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](https://depositonce.tu-berlin.de/items/8e2988b2-d991-45b5-aad5-9fb7988f3b2f) · CC BY-SA 4.0 |

Java reference implementation: [github.com/varylab/conformallab](https://github.com/varylab/conformallab)

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## References by module

| Reference | Used in |
|---|---|
| **Springborn** — *Ideal Hyperbolic Polyhedra and Discrete Uniformization*, Discrete & Computational Geometry (2020) | `hyper_ideal_geometry.hpp` — ζ₁₃/ζ₁₄/ζ₁₅ functions; `hyper_ideal_functional.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) | Bowers-Stephenson identity I_ij = (ℓ²−r_i²−r_j²)/(2 r_i r_j) used to initialise inversive distance from input geometry (Phase 9a.2) |
| **Glickenstein** — *Discrete conformal variations and scalar curvature on piecewise flat manifolds*, J. Differential Geometry 87 (2011) | Analytic Hessian of the inversive-distance functional (eq. 4.6) and cross-correspondence I_ij = cos θ_e between vertex-based (9a.2) and face-based (9a.1) circle packings |
| **Bobenko, Pinkall, Springborn** — *Discrete conformal maps and ideal hyperbolic polyhedra*, Geometry & Topology 14 (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) | Volume differential `2 dV = Σ_e aₑ dαₑ + Σ_v bᵥ dβᵥ` — foundation for the analytic HyperIdeal Hessian via Schläfli identity (Phase 9b-analytic, **new research beyond Java**) |
| **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 |
| **Crane, Soliman, Ben-Chen, Schröder** — *Optimal Cone Singularities for Conformal Flattening*, ACM SIGGRAPH (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. |
| **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. |
| **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. |
| **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. |
| **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. |
| **Bowers, Bowers, Lutz** — *Rigidity of circle polyhedra and hyperideal polyhedra: the tangency case* (2026). arXiv: [2601.22903](https://arxiv.org/abs/2601.22903) | **Phase 9b-analytic + Phase 10c'** (KoebePolyhedron): theoretical uniqueness/rigidity for hyperideal polyhedra in 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](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. |
| **Alexa** — *Discrete Laplacians on General Polygonal Meshes*, ACM TOG 39(6) (2020). DOI: [10.1145/3414685.3417840](https://doi.org/10.1145/3414685.3417840) | **Phase 9f** (extended journal version): error bounds, generalised polygon cotangent weights, convergence analysis. |

---

## 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](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. |
| **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.

---

## 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** — *Conformal Structures and Period Matrices of Polyhedral Surfaces* (2009) | Phase 10b: explicit algorithm for computing the discrete Siegel period matrix Ωᵢⱼ on a polyhedral surface from cotangent-weighted integration. |
| **Rivin, Springborn** — *The Schläfli formula in Einstein manifolds with boundary*, Electron. Res. Announc. AMS 5 (1999) | 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. |
| **Springborn** — *A discrete version of Liouville's theorem on conformal maps* (2019). 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. |
| **Springborn, Veselov** — *Quasiconformal distortion of projective transformations and discrete conformal maps*, Int. Math. Res. Not. (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](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. |
| **Sawhney, Crane** — *Boundary First Flattening*, ACM TOG 36(1) (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. |