f93292e815
Auto-merge from cherry-pick concatenated both branch versions of Phase 9d
and 9e. This commit resolves the duplication:
- 9d now has 4 sub-items (9d.1–9d.4) combining both branches:
9d.1 ConesUtility (detailed: BFS, auto-placement, quantization)
+ Troyanov/Springborn refs (from literature analysis)
9d.2 Non-Euclidean cone extensions (RESEARCH)
+ Bobenko-Lutz 2025 + Crane 2018 (from literature analysis)
9d.3 StereographicUnwrapper + SphereUtility
9d.4 MobiusCenteringFunctional
- 9e keeps the detailed reviewer/meeting-prep version
+ adds mathematical references (Bobenko-Hoffmann-Springborn 2006)
- 9f (Polygon Laplacian, Alexa 2011/2020) retained as standalone section
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
479 lines
25 KiB
Markdown
479 lines
25 KiB
Markdown
# Development Roadmap
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> **Legend:** ✅ complete · 🔲 planned
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>
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> **Porting / research boundary:**
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> Phases 1–7 are direct ports of the Java original and its dissertation.
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> From Phase 8 onwards the work goes beyond the scope of the Java library.
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> Phase 8 (CGAL package) is infrastructure. Phase 9 is porting of remaining Java features.
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> Phase 10+ is independent research with no direct Java reference implementation.
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---
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## ◼ Porting complete — Phases 1–7
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```
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Phase 1 Clausen / Lobachevsky / ImLi₂ special functions ✅
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Phase 2 Hyper-ideal geometry (ζ, lᵢⱼ, αᵢⱼ, σᵢ, σᵢⱼ) ✅
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Phase 3 CGAL Surface_mesh infrastructure + all three functionals
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(Euclidean, Spherical, HyperIdeal)
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+ analytical Hessians for Euclidean + Spherical
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(HyperIdeal Hessian: symmetric FD — analytic deferred to 9b) ✅
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Phase 4 Newton solver (SimplicialLDLT + SparseQR fallback)
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+ Mesh I/O (OFF/OBJ/PLY) + example programs ✅ 68 tests
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Phase 5 Priority-BFS layout + CLI app + JSON/XML serialisation ✅ 95 tests
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Phase 6 Gauss–Bonnet check/enforce, tree-cotree cut graph (2g),
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exact hyperbolic trilateration, layout normalisation ✅ 121 tests
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Phase 7 MobiusMap, halfedge_uv, Möbius holonomy (SU(1,1)),
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period matrix τ∈ℍ + SL(2,ℤ) reduction,
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fundamental domain parallelogram + tiling ✅ 176 tests
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```
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---
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## ◼ Infrastructure — Phase 8: CGAL Package
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Goal: conformallab++ as a standalone CGAL package, submission-ready, fulfilling all
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CGAL package conventions with a traits-class design compatible with any CGAL-conforming
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mesh type.
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```
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8a Traits class & concepts
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→ include/CGAL/Conformal_map_traits.h
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Separates MeshType, KernelType, ScalarType from the algorithm.
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Enables use with any CGAL-compatible mesh, not just Surface_mesh.
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→ Concept checks via static_assert / CGAL_concept_check
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8b Public CGAL header hierarchy
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→ include/CGAL/Discrete_conformal_map.h (user-facing entry header)
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→ include/CGAL/Conformal_newton_solver.h
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→ include/CGAL/Conformal_layout.h
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→ include/CGAL/Conformal_cut_graph.h
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All existing include/conformallab/*.hpp remain as implementation details.
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8c CGAL-style documentation
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→ doc/Conformal_map/PackageDescription.txt
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→ doc/Conformal_map/fig/ (pipeline diagrams)
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→ Doxygen comments on all public concepts and functions
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→ User_manual.md + Reference_manual.md
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8d CGAL test format
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→ test/Conformal_map/CMakeLists.txt (CGAL-style CMake)
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Existing GTest tests remain; CGAL-format tests are added alongside.
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8e Declarative YAML pipeline
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→ Lightweight YAML format for reproducible experiments
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(specification in doc/api/cgal-package.md)
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→ Validator: checks require/provide tokens before execution
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→ CLI integration: conformallab_core --pipeline experiment.yml
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```
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---
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## ◼ Phase 9 — Mixed: remaining Java port + first research extensions
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> **Audit 2026-05-21:** Phase 9 was originally framed as "remaining
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> porting", but a closer look at the local Java repository revealed:
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> several Phase-9 items are **not** in Java at all (`InversiveDistanceFunctional`
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> does not exist; `HyperIdealFunctional.java:295-298` declares `hasHessian()=false`).
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> The plan below now distinguishes Java-port items from research items.
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> Full research catalogue: [`research-track.md`](research-track.md).
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```
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9a — Circle-packing functionals (split 2026-05-19)
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─────────────────────────────────────────────────────
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9a.1 CPEuclideanFunctional (Java port)
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→ cp_euclidean_functional.hpp
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Java source: CPEuclideanFunctional.java (260 lines)
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Mathematical reference: Bobenko-Pinkall-Springborn 2010
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Status: 🟡 PR #8 open, 10 tests passing.
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9a.2 Inversive-distance functional (RESEARCH, not a port)
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→ inversive_distance_functional.hpp
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Java source: NONE. Empirically verified.
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Mathematical reference: Luo 2004 + Bowers-Stephenson 2004 + Glickenstein 2011
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Status: 🟡 PR #8 open, 11 tests passing.
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Cross-validation: G_id(0) = G_eu(0) at 1e-10 (Glickenstein §5).
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9b — HyperIdeal Hessian (RESEARCH — Java has no Hessian at all)
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─────────────────────────────────────────────────────────────────
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9b Block-FD HyperIdeal Hessian
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→ Replace full FD in hyper_ideal_hessian.hpp
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Java source: NONE (HyperIdealFunctional.java:295-298 declares
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hasHessian()==false; Java has NO Hessian).
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Algorithm: per-face 6×6 block, scatter to global sparse matrix.
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Status: 🟡 PR #9 open, 7 tests passing, ~96× speed-up measured.
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9b-analytic Full analytic HyperIdeal Hessian via Schläfli identity
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→ planned, see research-track.md
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Mathematical source: Springborn 2020 §4 + Schläfli 1858/60
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+ Rivin, Springborn 1999 "The Schläfli formula in
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Einstein manifolds with boundary" (ERA-AMS 5)
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+ Cho-Kim 1999 + Glickenstein 2011 §4
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Algorithm: explicit chain rule through (bᵢ,aₑ) → ℓᵢⱼ → ζ₁₃/ζ₁₄/ζ₁₅ → αᵢⱼ/βᵢ
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Includes: short LaTeX correctness note in doc/math/.
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Effort: 10–14 days net. Trigger: profiling on V > 5000.
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9c — Genus g > 1 fundamental domain (Java port + research extensions)
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──────────────────────────────────────────────────────────────────────
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9c 4g-polygon boundary walk (genus g > 1)
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→ Extend compute_fundamental_domain() beyond genus 1
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Java sources: FundamentalPolygonUtility.java (698 lines)
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+ CanonicalFormUtility.java (532 lines)
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+ CuttingUtility / SurgeryUtility (~800 lines)
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Mathematical source: Poincaré 1882 + Sechelmann 2016 §5
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Research component: bridging to conformallab++ cut_graph.hpp
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+ holonomy infrastructure.
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Effort: ~2 weeks for fundamental polygon, +2 weeks for surgery
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layer, +1 week integration.
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```
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9d — Cone metrics + sphere utilities (Java port + research extension)
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────────────────────────────────────────────────────────────────────
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```
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9d.1 ConesUtility (Java port: unwrapper/ConesUtility.java)
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→ cone_singularities.hpp
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Fills the "⚠️ data structure only" gap in java-parity.md:
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- Detect interior cone vertices (angle deficit ≠ 0)
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- BFS path from cone to mesh boundary → cut edge set
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- Auto-placement: conjugate gradient on Θ-gradient magnitude
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- Quantization: snap cone angles to π/2, π/3, π/6 for
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quad / triangle / hexagonal atlas targets
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Java reference: unwrapper/ConesUtility.java
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Mathematical reference: Troyanov 1991 + Springborn 2020 §3
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9d.2 Non-Euclidean cone extensions (RESEARCH, not in Java)
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→ extend ConesUtility to HyperIdeal + Spherical modes
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Java source: NONE — Java ConesUtility is Euclidean-only.
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Mathematical reference:
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Bobenko, Lutz 2025 "Decorated Discrete Conformal Equivalence in
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Non-Euclidean Geometries" (Discrete & Comput. Geom. 2025,
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arXiv:2310.17529) §3 — decorated DCE framework unifying cone
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singularities and cusps in hyperbolic + spherical geometry.
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Crane, Soliman, Ben-Chen, Schröder 2018 "Optimal Cone Singularities
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for Conformal Flattening" (ACM SIGGRAPH 2018) — L¹-optimal
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automatic cone placement; directly applicable to 9d.2 algorithm.
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Status: 🔲 planned
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9d.3 StereographicUnwrapper + SphereUtility (Java port)
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→ stereographic_layout.hpp
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Stereographic projection S²→ℂ∪{∞} + Möbius centering for
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genus-0 surfaces. Converts spherical DCE output to a flat 2-D atlas.
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Java reference: unwrapper/StereographicUnwrapper.java (266 lines)
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9d.4 MobiusCenteringFunctional (Java port, optional upgrade)
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→ integrate into layout.hpp normalise_hyperbolic()
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Variational Möbius centering via Lorentz geometry:
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E = Σ log(-⟨x,p⟩/√(-⟨x,x⟩))
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Supplies gradient + Hessian — replaces iterative Fréchet mean.
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Java reference: functional/MobiusCenteringFunctional.java
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```
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9e — Circle pattern layout (Java port — complement to Phase 9a.1)
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────────────────────────────────────────────────────────────────────
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```
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9e CirclePatternLayout + CirclePatternUtility (Java port)
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→ circle_pattern_layout.hpp
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Phase 9a.1 ported the CPEuclidean energy + solver; this phase
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adds the embedding step (ρ values → actual vertex positions in ℝ²).
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- CirclePatternUtility: compute per-face radii ρ via NTR solver
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- CirclePatternLayout: embed from ρ values (intersection-angle model)
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- CPEuclideanRotation: rotation-invariant CP functional variant
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Java references: unwrapper/circlepattern/CirclePattern{Layout,Utility}.java
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unwrapper/circlepattern/CPEuclideanRotation.java
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Mathematical reference: Bobenko-Springborn 2004 variational principle
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+ Bobenko-Hoffmann-Springborn 2006 "Minimal
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surfaces from circle patterns" (Discrete &
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Comput. Geom. 35, 2006).
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```
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9f — Polygon Laplacian (RESEARCH — no Java equivalent)
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──────────────────────────────────────────────────────
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```
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9f Discrete Laplacian on general polygonal meshes
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→ polygon_laplacian.hpp
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Java source: NONE
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Mathematical reference:
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Alexa, Wardetzky 2011 "Discrete Laplacians on General Polygonal
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Meshes" (ACM SIGGRAPH 2011) — virtual-node construction,
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polygon cotangent weights extending the Pinkall-Polthier formula.
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Alexa 2020 "Discrete Laplacians on General Polygonal Meshes"
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(ACM TOG 39, 2020) — extended journal treatment, error bounds.
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Enables: DCE energy evaluation on quad-dominant / Voronoi /
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polygon meshes without forced triangulation.
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Replaces euclidean_hessian.hpp for non-triangular inputs.
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Status: 🔲 planned (pure research, no Java source)
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Effort: medium (~2 weeks core + tests; +1 week Newton integration).
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```
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---
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## ◼ New research directions — Phases 10d–10g (2026 library scan)
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These were identified by a full scan of the Java source tree in 2026.
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They extend significantly beyond the Java port into new mathematical territory.
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```
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10d CircleDomainUnwrapper (Koebe–Andreev–Thurston)
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→ circle_domain_unwrapper.hpp
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Conformal map of a multiply-connected planar region onto a
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canonical disk-with-holes (classical complex-analysis result).
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Java reference: unwrapper/CircleDomainUnwrapper.java (570 lines)
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Mathematical basis: Koebe–Andreev–Thurston + Beardon–Stephenson 1990
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10e Quasi-isothermic maps
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→ quasiisothermic.hpp
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Generalisation of conformal maps for meshes where exact conformality
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is unachievable (high Gaussian curvature, coarse triangulation).
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Includes: Delaunay pre-conditioning, discrete Beltrami field,
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sin-condition functional, Lawson-correspondence parameterization.
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Java references: unwrapper/quasiisothermic/ (~1 200 lines total)
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Mathematical basis: Lam 2015 + Bohle–Lam–Pinkall–Reitebuch 2015
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10f Koebe polyhedra
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→ koebe_polyhedron.hpp
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Koebe–Andreev–Thurston theorem: realize every 3-connected planar
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graph as a convex polyhedron with edges tangent to the unit sphere.
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Connects circle packing with 3-D convex geometry.
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Java reference: unwrapper/koebe/KoebePolyhedron.java (321 lines)
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Mathematical basis: Koebe 1936 + Thurston 1997 (lecture notes)
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10g Cyclic-symmetry functionals
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→ cyclic_functional.hpp
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Euclidean and hyperbolic DCE functionals reduced to a cyclic-symmetry
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quotient — dramatically reduces DOFs for ornamental / symmetric surfaces.
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Java references: functional/EuclideanCyclicFunctional.java
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functional/HyperbolicCyclicFunctional.java (~530 lines)
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```
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---
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## ◼ Optional / Hypothetical — geometry-central Cross-Comparison
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> **Status: no planned phase — purely exploratory.**
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> These items are not prerequisites for Phase 8–10. They are
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> of interest because geometry-central (Keenan Crane, CMU) is built on the same
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> mathematical foundations as conformallab++ — in particular
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> **Springborn 2020** and its direct extension by
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> **Gillespie, Springborn & Crane (SIGGRAPH 2021)**.
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> The key difference: geometry-central solves the same problem
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> (discrete conformal equivalence) using **intrinsic triangulations +
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> Ptolemaic flips**, while conformallab++ applies **Newton on the
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> original triangulation**.
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```
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GC-1 [optional, possible now]
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Mathematical output comparison
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→ load the same test meshes (cathead.obj, brezel.obj, torus_4x4.off) into
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both libraries
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→ compare UV coordinates, u-vector, residual norm
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→ align normalisation conventions (u mean, scaling)
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Goal: independent cross-validation of convergence points.
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Effort: small Python/C++ comparison script, no library restructuring.
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GC-2 [optional, useful after Phase 8]
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Intrinsic Delaunay pre-conditioning
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→ before the Newton solver: apply geometry-central SignpostIntrinsicTriangulation
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to the input
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→ Ptolemaic flips pre-condition the Hessian matrix
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→ hypothesis: fewer Newton iterations on non-Delaunay inputs
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→ implementable as an optional cmake flag: -DWITH_GC_PRECOND=ON
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Dependency: geometry-central as an optional external dependency
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(header-only parts suffice for the flip algorithm).
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GC-3 [hypothetical, Phase 10+ research]
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Ptolemaic flip-based solver as an alternative backend
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→ instead of Newton: Ptolemaic flips + penultimate-step normalisation
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(Gillespie–Springborn–Crane 2021 algorithm)
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→ comparison: convergence radius, robustness on pathological meshes,
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numerical stability on high-genus surfaces
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→ relevant for conformallab++ because the Newton approach can become
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unstable on strongly non-Delaunay meshes (e.g. after remeshing).
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No implementation planned — conceptual note for Phase 10 research.
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```
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**Connection to the literature:**
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The Springborn 2020 paper ("Ideal Hyperbolic Polyhedra and Discrete
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Uniformization") is already implemented in conformallab++ as the HyperIdeal
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geometry mode (Phase 2/3). The Gillespie–Springborn–Crane
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2021 extension — implemented in geometry-central — augments this with
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intrinsic triangulations and makes the algorithm robust against
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poor input triangulations. Both share the same
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mathematical core (discrete conformal equivalence, Gauss–Bonnet,
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variational principle of Bobenko–Springborn 2004).
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---
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## ◼ Phase 10 — Genus g ≥ 2 (research with partial Java support)
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Most Phase-10 items have partial Java references (utility classes for
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forms and homology) but the **assembly** into a working uniformization
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pipeline is research. Full catalogue with primary literature:
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[`research-track.md`](research-track.md).
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```
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Phase 10 Global uniformization for genus g ≥ 2
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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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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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CanonicalBasisUtility.java 337 lines (homology basis)
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HomologyUtility.java 122 lines
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DualityUtility.java 308 lines
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DiscreteHarmonicFormUtility.java 657 lines
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DiscreteHolomorphicFormUtility.java 285 lines
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Effort: ~6 weeks net (4 utility ports + 1 integration).
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10b Siegel period matrix Ω ∈ H_g (g×g complex symmetric, Im(Ω) > 0)
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→ Ωᵢⱼ = ∫_{bⱼ} ωᵢ
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→ Reduction to Siegel fundamental domain via Sp(2g,ℤ).
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Mathematical reference: Bobenko-Springborn 2004 + Gottschling 1959.
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Bobenko, Bücking 2009 "Conformal Structures and
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Period Matrices of Polyhedral Surfaces" — discrete
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period matrix Ωᵢⱼ on polyhedral surfaces.
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Bobenko, Lutz 2024 IMRN "Decorated Discrete Conformal
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Maps and Convex Polyhedral Cusps" — uniformization
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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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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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Requires: 10a.
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Effort: ~1 week net after 10a.
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10b' Alternative methods (parallel research track)
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→ HyperbolicCyclicFunctional (Java, 530 lines) — completes the
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classical three-mode set with hyperbolic energy.
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→ Quasi-isothermic parametrisation (Lawson correspondence):
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QuasiisothermicUtility.java + SinConditionApplication.java
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(~1 200 Java lines combined).
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→ MobiusCenteringFunctional (Java, 289 lines) — sphere centering.
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→ StereographicUnwrapper (Java, 266 lines) — projects the
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spherical layout S²→ℂ via stereographic projection plus a
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Möbius centring step. Closes the visualisation gap from
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`discrete_conformal_map_spherical()` (currently outputs
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Point_3 on S²; many downstream uses want a 2-D atlas).
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Effort: small (~3 days).
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Each independent; can be tackled in any order.
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10c Full uniformization for genus g ≥ 2
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→ Embedding as H²/Γ with Γ ⊂ PSL(2,ℝ) a Fuchsian group.
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Mathematical reference: Sechelmann 2016 §6 (discrete instance);
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Bers 1960 (continuous theory).
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Lutz 2023 "Canonical Tessellations of Decorated
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Hyperbolic Surfaces" (Geom. Dedicata 217,
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arXiv:2206.13461) — canonical Delaunay tessellations
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in Penner coordinates; unifies the decorated
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framework with the fundamental domain construction.
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Bobenko, Lutz 2024 IMRN (arXiv:2305.10988) —
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discrete uniformization theorem for decorated
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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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pieces but does not assemble them into
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a Fuchsian-group representation.
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Status: **fully new research.**
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Requires: 10a + 10b + Phase 9c.
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10c' Optional Java-port additions (low priority)
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→ KoebePolyhedron.java (321 lines) — Koebe-Andreev-Thurston
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circle packings. Adds a fifth DCE method.
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Rigidity: Bowers, Bowers, Lutz 2026 "Rigidity of circle polyhedra
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and hyperideal polyhedra: the tangency case" (arXiv:2601.22903)
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— theoretical uniqueness backing the KAT construction.
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→ ElectrostaticSphereFunctional (127 lines) — sphere
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distribution baseline.
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→ CirclePatternLayout / CirclePatternUtility — face-circle
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pattern layouts.
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None of these are required for the genus-g uniformization
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pipeline; they extend the breadth of methods.
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```
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---
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## ◼ Phase 11+ — Specialised applications (optional, deferred)
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> **Status:** out-of-scope for v1.0 but recorded here so that future
|
||
> contributors don't re-discover them. Both items live in the Java
|
||
> repo as plugin sub-packages and would benefit from porting *only*
|
||
> after Phase 10 is complete (they require the period-matrix and
|
||
> fundamental-domain infrastructure to be in place first).
|
||
|
||
```
|
||
11a Schottky uniformisation
|
||
Java plugin: plugin/schottky/* (~12 Java files, ~3 000 LoC)
|
||
Mathematical basis: Schottky group — discrete subgroup
|
||
Γ ⊂ PSL(2,ℂ) generated by hyperbolic loxodromic
|
||
elements, fundamental domain a sphere with
|
||
2g disjoint discs removed.
|
||
Use case: "handlebody" uniformisation, complement of
|
||
Phase 10c's Fuchsian-group representation
|
||
(Schottky represents Riemann surfaces as
|
||
quotients of domains in S² rather than of H²).
|
||
Requires: Phase 10b (period matrix) + working
|
||
Möbius-group machinery from Phase 7.
|
||
Effort: very large (4–6 weeks) — significant Java
|
||
code, complex-analytic algorithms,
|
||
substantial test design.
|
||
|
||
11b Riemann maps (planar conformal mapping)
|
||
Java plugin: plugin/riemannmap/* (~6 Java files, ~1 500 LoC)
|
||
Mathematical basis: Riemann mapping theorem — every simply
|
||
connected proper subdomain of ℂ is conformally
|
||
equivalent to the unit disc. Discrete version
|
||
via circle packing or Schwarz-Christoffel-like
|
||
formulae.
|
||
Use case: Texture mapping of bounded planar regions;
|
||
classical conformal mapping for engineering
|
||
applications (electrostatics, fluid flow).
|
||
Requires: Phase 10b' QuasiisothermicUtility or the
|
||
CP-Euclidean machinery from Phase 9a.1
|
||
(depending on the chosen discrete-Riemann
|
||
algorithm).
|
||
Effort: large (3–4 weeks) — smaller than Schottky
|
||
but still substantial. Heavy on
|
||
visualisation; consider porting only the
|
||
algorithmic core.
|
||
|
||
11c Multiply-connected planar conformal maps (CircleDomainUnwrapper)
|
||
Java source: unwrapper/CircleDomainUnwrapper.java (570 LoC)
|
||
Mathematical basis: Riemann mapping theorem for multiply-
|
||
connected domains — every n-connected planar
|
||
region is conformally equivalent to a disk
|
||
with (n−1) round holes (Koebe's "general
|
||
uniformization theorem", 1909).
|
||
Use case: Classical complex-analysis problems —
|
||
conformal mapping of an annulus, a torus
|
||
slit on a plane, fluid flow around obstacles,
|
||
electrostatics with multiple conductors.
|
||
**A use-case class conformallab++ does not
|
||
currently cover.**
|
||
Requires: Phase 10b' QuasiisothermicUtility or the
|
||
CP-Euclidean machinery from Phase 9a.1.
|
||
Effort: large (~2 weeks).
|
||
|
||
All three items are tracked here so the project memory is preserved;
|
||
none of them are roadmap commitments. See `research-track.md` for the
|
||
formal research-versus-port classification before starting any.
|
||
```
|