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. |

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.