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

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	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 circle polyhedra and hyperideal polyhedra: the tangency case (2026). arXiv: 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	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	Phase 9f (extended journal version): error bounds, generalised polygon cotangent weights, convergence analysis.

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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 — 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	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	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	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.