docs: add 4 remaining Tier-2 papers (Springborn 2019, Springborn-Veselov 2015, Crane 2017 BFF, Stripe Patterns 2015)
phases.md: - 10b: Springborn 2019 discrete Liouville theorem (uniqueness of Ω) - 10c: Springborn-Veselov 2015 quasiconformal distortion (error bounds) - 10a: Knöppel-Crane-Pinkall-Schröder 2015 Stripe Patterns (cross-validation ref) references.md (Phase 10 section, 4 new rows): - Springborn 2019 arXiv:1911.00966 → Phase 10b uniqueness - Springborn-Veselov 2015 Int. Math. Res. Not. → Phase 10c error analysis - Knöppel-Crane-Pinkall-Schröder 2015 SIGGRAPH → Phase 10a cross-validation - Sawhney-Crane 2017 BFF ACM TOG → complementary method to Phase 9d Completes the literature integration started in the previous commit: all Tier-2 papers from the Alexa/Bobenko/Springborn/Crane/Lutz analysis are now documented in the roadmap. Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
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@@ -68,3 +68,7 @@ builds on this paper and augments it with Ptolemaic flips.
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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, 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** — *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. |
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| **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. |
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| **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. |
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| **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. |
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| **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. |
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| **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. |
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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 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. |
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@@ -348,6 +348,12 @@ Phase 10 Global uniformization for genus g ≥ 2
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10a Discrete holomorphic and harmonic 1-forms
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10a Discrete holomorphic and harmonic 1-forms
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→ Integrate basis 1-forms ωᵢ along b-cycles of the cut graph.
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→ Integrate basis 1-forms ωᵢ along b-cycles of the cut graph.
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Mathematical reference: Bobenko-Springborn 2004 §6 + Mercat 2001.
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Mathematical reference: Bobenko-Springborn 2004 §6 + Mercat 2001.
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Knöppel, Crane, Pinkall, Schröder 2015 "Stripe
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Patterns on Surfaces" (ACM SIGGRAPH 2015) —
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application of discrete holomorphic 1-forms to
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direction field design; provides an independent
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C++ reference implementation (geometry-central)
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for cross-validating the Phase 10a computation.
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Java sources (partial, port-with-research):
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Java sources (partial, port-with-research):
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CanonicalBasisUtility.java 337 lines (homology basis)
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CanonicalBasisUtility.java 337 lines (homology basis)
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HomologyUtility.java 122 lines
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HomologyUtility.java 122 lines
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@@ -367,6 +373,10 @@ Phase 10 Global uniformization for genus g ≥ 2
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Maps and Convex Polyhedral Cusps" — uniformization
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Maps and Convex Polyhedral Cusps" — uniformization
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theorem connecting cusps ↔ hyperideal vertices
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theorem connecting cusps ↔ hyperideal vertices
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(bridges Phase 2/3 HyperIdeal geometry to 10b).
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(bridges Phase 2/3 HyperIdeal geometry to 10b).
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Springborn 2019 "A discrete version of Liouville's
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theorem on conformal maps" (arXiv:1911.00966) —
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proves uniqueness/rigidity of the discrete conformal
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structure; justifies that Ω is a conformal invariant.
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Java partial reference: DiscreteRiemannUtility.java (186 lines).
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Java partial reference: DiscreteRiemannUtility.java (186 lines).
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Requires: 10a.
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Requires: 10a.
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Effort: ~1 week net after 10a.
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Effort: ~1 week net after 10a.
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@@ -398,6 +408,12 @@ Phase 10 Global uniformization for genus g ≥ 2
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Bobenko, Lutz 2024 IMRN (arXiv:2305.10988) —
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Bobenko, Lutz 2024 IMRN (arXiv:2305.10988) —
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discrete uniformization theorem for decorated
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discrete uniformization theorem for decorated
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piecewise Euclidean surfaces.
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piecewise Euclidean surfaces.
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Springborn, Veselov 2015 "Quasiconformal distortion
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of projective transformations and discrete conformal
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maps" (Int. Math. Res. Not.) — error estimates for
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the discrete-to-smooth conformal approximation;
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quantifies how well H²/Γ approximates the smooth
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hyperbolic metric.
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Java reference: NONE — Java has the polygon + period matrix
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Java reference: NONE — Java has the polygon + period matrix
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pieces but does not assemble them into
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pieces but does not assemble them into
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a Fuchsian-group representation.
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a Fuchsian-group representation.
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