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>
This commit is contained in:
Tarik Moussa
2026-05-25 23:59:47 +02:00
parent 979f30c80a
commit e8a118fa5f
2 changed files with 20 additions and 0 deletions

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@@ -68,3 +68,7 @@ builds on this paper and augments it with Ptolemaic flips.
| **Bobenko, Mercat, Schmies***Period Matrices of Polyhedral Surfaces*, in: Computational Approach to Riemann Surfaces (2011) | Discrete period matrices on polyhedral surfaces | | **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. | | **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. | | **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. |

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@@ -348,6 +348,12 @@ Phase 10 Global uniformization for genus g ≥ 2
10a Discrete holomorphic and harmonic 1-forms 10a Discrete holomorphic and harmonic 1-forms
→ Integrate basis 1-forms ωᵢ along b-cycles of the cut graph. → Integrate basis 1-forms ωᵢ along b-cycles of the cut graph.
Mathematical reference: Bobenko-Springborn 2004 §6 + Mercat 2001. Mathematical reference: Bobenko-Springborn 2004 §6 + Mercat 2001.
Knöppel, Crane, Pinkall, Schröder 2015 "Stripe
Patterns on Surfaces" (ACM SIGGRAPH 2015) —
application of discrete holomorphic 1-forms to
direction field design; provides an independent
C++ reference implementation (geometry-central)
for cross-validating the Phase 10a computation.
Java sources (partial, port-with-research): Java sources (partial, port-with-research):
CanonicalBasisUtility.java 337 lines (homology basis) CanonicalBasisUtility.java 337 lines (homology basis)
HomologyUtility.java 122 lines HomologyUtility.java 122 lines
@@ -367,6 +373,10 @@ Phase 10 Global uniformization for genus g ≥ 2
Maps and Convex Polyhedral Cusps" — uniformization Maps and Convex Polyhedral Cusps" — uniformization
theorem connecting cusps ↔ hyperideal vertices theorem connecting cusps ↔ hyperideal vertices
(bridges Phase 2/3 HyperIdeal geometry to 10b). (bridges Phase 2/3 HyperIdeal geometry to 10b).
Springborn 2019 "A discrete version of Liouville's
theorem on conformal maps" (arXiv:1911.00966) —
proves uniqueness/rigidity of the discrete conformal
structure; justifies that Ω is a conformal invariant.
Java partial reference: DiscreteRiemannUtility.java (186 lines). Java partial reference: DiscreteRiemannUtility.java (186 lines).
Requires: 10a. Requires: 10a.
Effort: ~1 week net after 10a. Effort: ~1 week net after 10a.
@@ -398,6 +408,12 @@ Phase 10 Global uniformization for genus g ≥ 2
Bobenko, Lutz 2024 IMRN (arXiv:2305.10988) — Bobenko, Lutz 2024 IMRN (arXiv:2305.10988) —
discrete uniformization theorem for decorated discrete uniformization theorem for decorated
piecewise Euclidean surfaces. piecewise Euclidean surfaces.
Springborn, Veselov 2015 "Quasiconformal distortion
of projective transformations and discrete conformal
maps" (Int. Math. Res. Not.) — error estimates for
the discrete-to-smooth conformal approximation;
quantifies how well H²/Γ approximates the smooth
hyperbolic metric.
Java reference: NONE — Java has the polygon + period matrix Java reference: NONE — Java has the polygon + period matrix
pieces but does not assemble them into pieces but does not assemble them into
a Fuchsian-group representation. a Fuchsian-group representation.