# 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 |

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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](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 |