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ConformalLabpp/doc/roadmap/phases.md
Tarik Moussa 068df474b1 docs: integrate publication analysis — Alexa, Bobenko, Springborn, Crane, Lutz 
Add phases 9d / 9e / 9f and literature citations derived from a systematic
review of the five authors' publication lists (Tier 1 / 2 / 3 analysis).

phases.md:
  - Phase 9d: ConesUtility port (9d.1) + non-Euclidean cone extensions
    (9d.2, RESEARCH) + StereographicUnwrapper (9d.3)
  - Phase 9e: CirclePatternLayout + CirclePatternUtility (Java port)
  - Phase 9f: Polygon Laplacian on non-triangular meshes (Alexa 2011/2020,
    RESEARCH — no Java equivalent)
  - Phase 9b-analytic: add Rivin-Springborn 1999 as Schläfli source
  - Phase 10b: add Bobenko-Bücking 2009 + Bobenko-Lutz 2024 IMRN
  - Phase 10c: add Lutz 2023 (canonical tessellations) + Bobenko-Lutz 2024
  - Phase 10c' KoebePolyhedron: add Bowers-Bowers-Lutz 2026 rigidity result

references.md:
  - Crane et al. 2018 Optimal Cone Singularities (Phase 9d.2)
  - Bobenko-Lutz 2025 Discrete & Comput. Geom. (Phase 9d.2)
  - Bobenko-Lutz 2024 IMRN (Phase 10b/c)
  - Lutz 2023 Geom. Dedicata (Phase 10c)
  - Lutz PhD thesis TU Berlin 2024 (Phases 9d.2, 10b, 10c)
  - Bowers-Bowers-Lutz 2026 (Phase 9b-analytic + 10c')
  - Alexa-Wardetzky 2011 + Alexa 2020 (Phase 9f)
  - Bobenko-Bücking 2009 (Phase 10b)
  - Rivin-Springborn 1999 (Phase 9b-analytic)

research-track.md:
  - New entry: Phase 9d.2 non-Euclidean cone extensions (Bobenko-Lutz 2025
    + Crane 2018), with acceptance criteria
  - New entry: Phase 9f polygon Laplacian (Alexa-Wardetzky 2011 / Alexa 2020),
    with acceptance criteria

java-parity.md:
  - Split cone-metrics row into Euclidean (9d.1 port) and non-Euclidean
    (9d.2 research) with literature references
  - Add ConesUtility to "utility classes not yet ported" table

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
2026-05-26 11:15:09 +02:00

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# Development Roadmap
> **Legend:** ✅ complete · 🔲 planned
>
> **Porting / research boundary:**
> Phases 1–7 are direct ports of the Java original and its dissertation.
> From Phase 8 onwards the work goes beyond the scope of the Java library.
> Phase 8 (CGAL package) is infrastructure. Phase 9 is porting of remaining Java features.
> Phase 10+ is independent research with no direct Java reference implementation.
---
## ◼ Porting complete — Phases 1–7
```
Phase 1 Clausen / Lobachevsky / ImLi₂ special functions ✅
Phase 2 Hyper-ideal geometry (ζ, lᵢⱼ, αᵢⱼ, σᵢ, σᵢⱼ) ✅
Phase 3 CGAL Surface_mesh infrastructure + all three functionals
(Euclidean, Spherical, HyperIdeal)
+ analytical Hessians for Euclidean + Spherical
(HyperIdeal Hessian: symmetric FD — analytic deferred to 9b) ✅
Phase 4 Newton solver (SimplicialLDLT + SparseQR fallback)
+ Mesh I/O (OFF/OBJ/PLY) + example programs ✅ 68 tests
Phase 5 Priority-BFS layout + CLI app + JSON/XML serialisation ✅ 95 tests
Phase 6 Gauss–Bonnet check/enforce, tree-cotree cut graph (2g),
exact hyperbolic trilateration, layout normalisation ✅ 121 tests
Phase 7 MobiusMap, halfedge_uv, Möbius holonomy (SU(1,1)),
period matrix τ∈ℍ + SL(2,ℤ) reduction,
fundamental domain parallelogram + tiling ✅ 176 tests
```
---
## ◼ Infrastructure — Phase 8: CGAL Package
Goal: conformallab++ as a standalone CGAL package, submission-ready, fulfilling all
CGAL package conventions with a traits-class design compatible with any CGAL-conforming
mesh type.
```
8a Traits class & concepts
→ include/CGAL/Conformal_map_traits.h
Separates MeshType, KernelType, ScalarType from the algorithm.
Enables use with any CGAL-compatible mesh, not just Surface_mesh.
→ Concept checks via static_assert / CGAL_concept_check
8b Public CGAL header hierarchy
→ include/CGAL/Discrete_conformal_map.h (user-facing entry header)
→ include/CGAL/Conformal_newton_solver.h
→ include/CGAL/Conformal_layout.h
→ include/CGAL/Conformal_cut_graph.h
All existing include/conformallab/*.hpp remain as implementation details.
8c CGAL-style documentation
→ doc/Conformal_map/PackageDescription.txt
→ doc/Conformal_map/fig/ (pipeline diagrams)
→ Doxygen comments on all public concepts and functions
→ User_manual.md + Reference_manual.md
8d CGAL test format
→ test/Conformal_map/CMakeLists.txt (CGAL-style CMake)
Existing GTest tests remain; CGAL-format tests are added alongside.
8e Declarative YAML pipeline
→ Lightweight YAML format for reproducible experiments
(specification in doc/api/cgal-package.md)
→ Validator: checks require/provide tokens before execution
→ CLI integration: conformallab_core --pipeline experiment.yml
```
---
## ◼ Phase 9 — Mixed: remaining Java port + first research extensions
> **Audit 2026-05-21:** Phase 9 was originally framed as "remaining
> porting", but a closer look at the local Java repository revealed:
> several Phase-9 items are **not** in Java at all (`InversiveDistanceFunctional`
> does not exist; `HyperIdealFunctional.java:295-298` declares `hasHessian()=false`).
> The plan below now distinguishes Java-port items from research items.
> Full research catalogue: [`research-track.md`](research-track.md).
```
9a — Circle-packing functionals (split 2026-05-19)
─────────────────────────────────────────────────────
9a.1 CPEuclideanFunctional (Java port)
→ cp_euclidean_functional.hpp
Java source: CPEuclideanFunctional.java (260 lines)
Mathematical reference: Bobenko-Pinkall-Springborn 2010
Status: 🟡 PR #8 open, 10 tests passing.
9a.2 Inversive-distance functional (RESEARCH, not a port)
→ inversive_distance_functional.hpp
Java source: NONE. Empirically verified.
Mathematical reference: Luo 2004 + Bowers-Stephenson 2004 + Glickenstein 2011
Status: 🟡 PR #8 open, 11 tests passing.
Cross-validation: G_id(0) = G_eu(0) at 1e-10 (Glickenstein §5).
9b — HyperIdeal Hessian (RESEARCH — Java has no Hessian at all)
─────────────────────────────────────────────────────────────────
9b Block-FD HyperIdeal Hessian
→ Replace full FD in hyper_ideal_hessian.hpp
Java source: NONE (HyperIdealFunctional.java:295-298 declares
hasHessian()==false; Java has NO Hessian).
Algorithm: per-face 6×6 block, scatter to global sparse matrix.
Status: 🟡 PR #9 open, 7 tests passing, ~96× speed-up measured.
9b-analytic Full analytic HyperIdeal Hessian via Schläfli identity
→ planned, see research-track.md
Mathematical source: Springborn 2020 §4 + Schläfli 1858/60
+ Rivin, Springborn 1999 "The Schläfli formula in
Einstein manifolds with boundary" (ERA-AMS 5)
+ Cho-Kim 1999 + Glickenstein 2011 §4
Algorithm: explicit chain rule through (bᵢ,aₑ) → ℓᵢⱼ → ζ₁₃/ζ₁₄/ζ₁₅ → αᵢⱼ/βᵢ
Includes: short LaTeX correctness note in doc/math/.
Effort: 10–14 days net. Trigger: profiling on V > 5000.
9c — Genus g > 1 fundamental domain (Java port + research extensions)
──────────────────────────────────────────────────────────────────────
9c 4g-polygon boundary walk (genus g > 1)
→ Extend compute_fundamental_domain() beyond genus 1
Java sources: FundamentalPolygonUtility.java (698 lines)
+ CanonicalFormUtility.java (532 lines)
+ CuttingUtility / SurgeryUtility (~800 lines)
Mathematical source: Poincaré 1882 + Sechelmann 2016 §5
Research component: bridging to conformallab++ cut_graph.hpp
+ holonomy infrastructure.
Effort: ~2 weeks for fundamental polygon, +2 weeks for surgery
layer, +1 week integration.
9d — Cone singularities + sphere atlas (Java port + research extension)
────────────────────────────────────────────────────────────────────────
9d.1 ConesUtility (Java port — Euclidean only)
→ cones_utility.hpp
Java source: ConesUtility.java (~200 lines)
Mathematical reference: Troyanov 1991 + Springborn 2020 §3
Port scope: prescribed cone angles Θᵥ ≠ 2π in Euclidean mode.
Status: 🔲 planned
9d.2 Non-Euclidean cone extensions (RESEARCH, not in Java)
→ extend ConesUtility to HyperIdeal + Spherical modes
Java source: NONE — Java ConesUtility is Euclidean-only.
Mathematical reference:
Bobenko, Lutz 2025 "Decorated Discrete Conformal Equivalence in
Non-Euclidean Geometries" (Discrete & Comput. Geom. 2025,
arXiv:2310.17529) §3 — decorated DCE framework unifying cone
singularities and cusps in hyperbolic + spherical geometry.
Crane, Soliman, Ben-Chen, Schröder 2018 "Optimal Cone Singularities
for Conformal Flattening" (ACM SIGGRAPH 2018) — L¹-optimal
automatic cone placement; directly applicable to 9d.2 algorithm.
Status: 🔲 planned
9d.3 StereographicUnwrapper (Java port)
→ stereo_unwrapper.hpp
Java source: StereographicUnwrapper.java (266 lines)
Converts spherical DCE output (Point_3 on S²) to a 2-D atlas
via stereographic projection + Möbius centring.
Closes the visualisation gap from discrete_conformal_map_spherical().
Effort: small (~3 days).
Status: 🔲 planned
9e — CirclePatternLayout (Java port)
─────────────────────────────────────
9e CirclePatternLayout + CirclePatternUtility (Java port)
→ circle_pattern_layout.hpp
Java sources: CirclePatternLayout.java + CirclePatternUtility.java
+ CPEuclideanRotation.java
Mathematical reference: Bobenko-Springborn 2004 variational principle
+ Bobenko-Hoffmann-Springborn 2006 "Minimal
surfaces from circle patterns" (Discrete &
Comput. Geom. 35, 2006).
Status: 🔲 planned
9f — Polygon Laplacian (RESEARCH — no Java equivalent)
──────────────────────────────────────────────────────
9f Discrete Laplacian on general polygonal meshes
→ polygon_laplacian.hpp
Java source: NONE
Mathematical reference:
Alexa, Wardetzky 2011 "Discrete Laplacians on General Polygonal
Meshes" (ACM SIGGRAPH 2011) — virtual-node construction,
polygon cotangent weights extending the Pinkall-Polthier formula.
Alexa 2020 "Discrete Laplacians on General Polygonal Meshes"
(ACM TOG 39, 2020) — extended journal treatment, error bounds.
Enables: DCE energy evaluation on quad-dominant / Voronoi /
polygon meshes without forced triangulation.
Replaces euclidean_hessian.hpp for non-triangular inputs.
Status: 🔲 planned (pure research, no Java source)
Effort: medium (~2 weeks core + tests; +1 week Newton integration).
```
9d — Cone metrics + sphere utilities (Java port — 2026 library scan)
────────────────────────────────────────────────────────────────────
```
9d.1 ConesUtility (Java port: unwrapper/ConesUtility.java)
→ cone_singularities.hpp
Fills the "⚠️ data structure only" gap in java-parity.md:
- Detect interior cone vertices (angle deficit ≠ 0)
- BFS path from cone to mesh boundary → cut edge set
- Auto-placement: conjugate gradient on Θ-gradient magnitude
- Quantization: snap cone angles to π/2, π/3, π/6 for
quad / triangle / hexagonal atlas targets
Java reference: unwrapper/ConesUtility.java
9d.2 StereographicUnwrapper + SphereUtility (Java port)
→ stereographic_layout.hpp
Stereographic projection S²→ℂ∪{∞} + Möbius centering for
genus-0 surfaces. Converts spherical DCE output to a flat 2-D atlas.
Java reference: unwrapper/StereographicUnwrapper.java (266 lines)
9d.3 MobiusCenteringFunctional (Java port, optional upgrade)
→ integrate into layout.hpp normalise_hyperbolic()
Variational Möbius centering via Lorentz geometry:
E = Σ log(-⟨x,p⟩/√(-⟨x,x⟩))
Supplies gradient + Hessian — replaces iterative Fréchet mean.
Java reference: functional/MobiusCenteringFunctional.java
```
9e — Circle pattern layout (Java port — complement to Phase 9a.1)
────────────────────────────────────────────────────────────────────
```
9e CirclePatternLayout + CirclePatternUtility (Java port)
→ circle_pattern_layout.hpp
Phase 9a.1 ported the CPEuclidean energy + solver; this phase
adds the embedding step (ρ values → actual vertex positions in ℝ²).
- CirclePatternUtility: compute per-face radii ρ via NTR solver
- CirclePatternLayout: embed from ρ values (intersection-angle model)
- CPEuclideanRotation: rotation-invariant CP functional variant
Java references: unwrapper/circlepattern/CirclePattern{Layout,Utility}.java
unwrapper/circlepattern/CPEuclideanRotation.java
```
---
## ◼ New research directions — Phases 10d–10g (2026 library scan)
These were identified by a full scan of the Java source tree in 2026.
They extend significantly beyond the Java port into new mathematical territory.
```
10d CircleDomainUnwrapper (Koebe–Andreev–Thurston)
→ circle_domain_unwrapper.hpp
Conformal map of a multiply-connected planar region onto a
canonical disk-with-holes (classical complex-analysis result).
Java reference: unwrapper/CircleDomainUnwrapper.java (570 lines)
Mathematical basis: Koebe–Andreev–Thurston + Beardon–Stephenson 1990
10e Quasi-isothermic maps
→ quasiisothermic.hpp
Generalisation of conformal maps for meshes where exact conformality
is unachievable (high Gaussian curvature, coarse triangulation).
Includes: Delaunay pre-conditioning, discrete Beltrami field,
sin-condition functional, Lawson-correspondence parameterization.
Java references: unwrapper/quasiisothermic/ (~1 200 lines total)
Mathematical basis: Lam 2015 + Bohle–Lam–Pinkall–Reitebuch 2015
10f Koebe polyhedra
→ koebe_polyhedron.hpp
Koebe–Andreev–Thurston theorem: realize every 3-connected planar
graph as a convex polyhedron with edges tangent to the unit sphere.
Connects circle packing with 3-D convex geometry.
Java reference: unwrapper/koebe/KoebePolyhedron.java (321 lines)
Mathematical basis: Koebe 1936 + Thurston 1997 (lecture notes)
10g Cyclic-symmetry functionals
→ cyclic_functional.hpp
Euclidean and hyperbolic DCE functionals reduced to a cyclic-symmetry
quotient — dramatically reduces DOFs for ornamental / symmetric surfaces.
Java references: functional/EuclideanCyclicFunctional.java
functional/HyperbolicCyclicFunctional.java (~530 lines)
```
---
## ◼ Optional / Hypothetical — geometry-central Cross-Comparison
> **Status: no planned phase — purely exploratory.**
> These items are not prerequisites for Phase 8–10. They are
> of interest because geometry-central (Keenan Crane, CMU) is built on the same
> mathematical foundations as conformallab++ — in particular
> **Springborn 2020** and its direct extension by
> **Gillespie, Springborn & Crane (SIGGRAPH 2021)**.
> The key difference: geometry-central solves the same problem
> (discrete conformal equivalence) using **intrinsic triangulations +
> Ptolemaic flips**, while conformallab++ applies **Newton on the
> original triangulation**.
```
GC-1 [optional, possible now]
Mathematical output comparison
→ load the same test meshes (cathead.obj, brezel.obj, torus_4x4.off) into
both libraries
→ compare UV coordinates, u-vector, residual norm
→ align normalisation conventions (u mean, scaling)
Goal: independent cross-validation of convergence points.
Effort: small Python/C++ comparison script, no library restructuring.
GC-2 [optional, useful after Phase 8]
Intrinsic Delaunay pre-conditioning
→ before the Newton solver: apply geometry-central SignpostIntrinsicTriangulation
to the input
→ Ptolemaic flips pre-condition the Hessian matrix
→ hypothesis: fewer Newton iterations on non-Delaunay inputs
→ implementable as an optional cmake flag: -DWITH_GC_PRECOND=ON
Dependency: geometry-central as an optional external dependency
(header-only parts suffice for the flip algorithm).
GC-3 [hypothetical, Phase 10+ research]
Ptolemaic flip-based solver as an alternative backend
→ instead of Newton: Ptolemaic flips + penultimate-step normalisation
(Gillespie–Springborn–Crane 2021 algorithm)
→ comparison: convergence radius, robustness on pathological meshes,
numerical stability on high-genus surfaces
→ relevant for conformallab++ because the Newton approach can become
unstable on strongly non-Delaunay meshes (e.g. after remeshing).
No implementation planned — conceptual note for Phase 10 research.
```
**Connection to the literature:**
The Springborn 2020 paper ("Ideal Hyperbolic Polyhedra and Discrete
Uniformization") is already implemented in conformallab++ as the HyperIdeal
geometry mode (Phase 2/3). The Gillespie–Springborn–Crane
2021 extension — implemented in geometry-central — augments this with
intrinsic triangulations and makes the algorithm robust against
poor input triangulations. Both share the same
mathematical core (discrete conformal equivalence, Gauss–Bonnet,
variational principle of Bobenko–Springborn 2004).
---
## ◼ Phase 10 — Genus g ≥ 2 (research with partial Java support)
Most Phase-10 items have partial Java references (utility classes for
forms and homology) but the **assembly** into a working uniformization
pipeline is research. Full catalogue with primary literature:
[`research-track.md`](research-track.md).
```
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.
Java sources (partial, port-with-research):
CanonicalBasisUtility.java 337 lines (homology basis)
HomologyUtility.java 122 lines
DualityUtility.java 308 lines
DiscreteHarmonicFormUtility.java 657 lines
DiscreteHolomorphicFormUtility.java 285 lines
Effort: ~6 weeks net (4 utility ports + 1 integration).
10b Siegel period matrix Ω ∈ H_g (g×g complex symmetric, Im(Ω) > 0)
→ Ωᵢⱼ = ∫_{bⱼ} ωᵢ
→ Reduction to Siegel fundamental domain via Sp(2g,ℤ).
Mathematical reference: Bobenko-Springborn 2004 + Gottschling 1959.
Bobenko, Bücking 2009 "Conformal Structures and
Period Matrices of Polyhedral Surfaces" — discrete
period matrix Ωᵢⱼ on polyhedral surfaces.
Bobenko, Lutz 2024 IMRN "Decorated Discrete Conformal
Maps and Convex Polyhedral Cusps" — uniformization
theorem connecting cusps ↔ hyperideal vertices
(bridges Phase 2/3 HyperIdeal geometry to 10b).
Java partial reference: DiscreteRiemannUtility.java (186 lines).
Requires: 10a.
Effort: ~1 week net after 10a.
10b' Alternative methods (parallel research track)
→ HyperbolicCyclicFunctional (Java, 530 lines) — completes the
classical three-mode set with hyperbolic energy.
→ Quasi-isothermic parametrisation (Lawson correspondence):
QuasiisothermicUtility.java + SinConditionApplication.java
(~1 200 Java lines combined).
→ MobiusCenteringFunctional (Java, 289 lines) — sphere centering.
→ StereographicUnwrapper (Java, 266 lines) — projects the
spherical layout S²→ℂ via stereographic projection plus a
Möbius centring step. Closes the visualisation gap from
`discrete_conformal_map_spherical()` (currently outputs
Point_3 on S²; many downstream uses want a 2-D atlas).
Effort: small (~3 days).
Each independent; can be tackled in any order.
10c Full uniformization for genus g ≥ 2
→ Embedding as H²/Γ with Γ ⊂ PSL(2,ℝ) a Fuchsian group.
Mathematical reference: Sechelmann 2016 §6 (discrete instance);
Bers 1960 (continuous theory).
Lutz 2023 "Canonical Tessellations of Decorated
Hyperbolic Surfaces" (Geom. Dedicata 217,
arXiv:2206.13461) — canonical Delaunay tessellations
in Penner coordinates; unifies the decorated
framework with the fundamental domain construction.
Bobenko, Lutz 2024 IMRN (arXiv:2305.10988) —
discrete uniformization theorem for decorated
piecewise Euclidean surfaces.
Java reference: NONE — Java has the polygon + period matrix
pieces but does not assemble them into
a Fuchsian-group representation.
Status: **fully new research.**
Requires: 10a + 10b + Phase 9c.
10c' Optional Java-port additions (low priority)
→ KoebePolyhedron.java (321 lines) — Koebe-Andreev-Thurston
circle packings. Adds a fifth DCE method.
Rigidity: Bowers, Bowers, Lutz 2026 "Rigidity of circle polyhedra
and hyperideal polyhedra: the tangency case" (arXiv:2601.22903)
— theoretical uniqueness backing the KAT construction.
→ ElectrostaticSphereFunctional (127 lines) — sphere
distribution baseline.
→ CirclePatternLayout / CirclePatternUtility — face-circle
pattern layouts.
None of these are required for the genus-g uniformization
pipeline; they extend the breadth of methods.
```
---
## ◼ Phase 11+ — Specialised applications (optional, deferred)
> **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.
```
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