From e8a118fa5ffddfdb48560c97c26235957848f646 Mon Sep 17 00:00:00 2001 From: Tarik Moussa Date: Mon, 25 May 2026 23:59:47 +0200 Subject: [PATCH] docs: add 4 remaining Tier-2 papers (Springborn 2019, Springborn-Veselov 2015, Crane 2017 BFF, Stripe Patterns 2015) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 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 --- doc/math/references.md | 4 ++++ doc/roadmap/phases.md | 16 ++++++++++++++++ 2 files changed, 20 insertions(+) diff --git a/doc/math/references.md b/doc/math/references.md index 07ef752..3de48d2 100644 --- a/doc/math/references.md +++ b/doc/math/references.md @@ -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, 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. | diff --git a/doc/roadmap/phases.md b/doc/roadmap/phases.md index ee6ab16..2db01de 100644 --- a/doc/roadmap/phases.md +++ b/doc/roadmap/phases.md @@ -348,6 +348,12 @@ Phase 10 Global uniformization for genus g ≥ 2 10a Discrete holomorphic and harmonic 1-forms → Integrate basis 1-forms ωᵢ along b-cycles of the cut graph. 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): CanonicalBasisUtility.java 337 lines (homology basis) HomologyUtility.java 122 lines @@ -367,6 +373,10 @@ Phase 10 Global uniformization for genus g ≥ 2 Maps and Convex Polyhedral Cusps" — uniformization theorem connecting cusps ↔ hyperideal vertices (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). Requires: 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) — discrete uniformization theorem for decorated 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 pieces but does not assemble them into a Fuchsian-group representation.