# Public Headers (`code/include/`)

All algorithms are header-only. Include the headers you need directly —
there is no compiled library to link against (only GTest for tests and
CGAL/Eigen for the CGAL-dependent headers).

## Core mesh type

| Header | Description |
|---|---|
| `conformal_mesh.hpp` | `ConformalMesh` = `CGAL::Surface_mesh<Point3>`. Property-map naming convention. Index type aliases. `GeometryType` enum. |
| `constants.hpp` | `conformallab::PI`, `TWO_PI` |
| `mesh_builder.hpp` | `make_triangle()` / `make_tetrahedron()` / `make_quad_strip()` / `make_fan()` / `make_open_cylinder()` — test mesh factories |
| `mesh_io.hpp` | `load_mesh()` / `save_mesh()` via `CGAL::IO` (OFF / OBJ / PLY) |
| `mesh_utils.hpp` | `cgal_to_eigen()` — convert `ConformalMesh` vertex positions to `Eigen::MatrixXd` |

## Special functions

| Header | Description |
|---|---|
| `clausen.hpp` | `clausen_cl2(θ)` (Clausen Cl₂), `lobachevsky(θ)` (Л), `im_li2(θ)` (ImLi₂ = imaginary part of dilogarithm) |

## HyperIdeal geometry (H²)

| Header | Description |
|---|---|
| `hyper_ideal_geometry.hpp` | `ζ₁₃`, `ζ₁₄`, `ζ₁₅` (Springborn 2020), `l_from_zeta()`, `alpha_ij()`, `beta_i()`, `sigma_i()`, `sigma_ij()` |
| `hyper_ideal_utility.hpp` | Tetrahedron volumes (Meyerhoff formula, Kolpakov–Mednykh) |
| `hyper_ideal_visualization_utility.hpp` | Poincaré disk projection, circumcircle helpers, Lorentz boost |
| `hyper_ideal_functional.hpp` | `HyperIdealMaps`, `setup_hyper_ideal_maps()`, `compute_hyper_ideal_lambda0_from_mesh()`, `assign_all_dof_indices()`, `hyper_ideal_gradient()`, `hyper_ideal_energy()` |
| `hyper_ideal_hessian.hpp` | `hyper_ideal_hessian()` — symmetric FD Hessian (Phase 9b: analytic) |

## Spherical geometry (S²)

| Header | Description |
|---|---|
| `spherical_geometry.hpp` | Spherical arc-length, half-angle formula, spherical law of cosines |
| `spherical_functional.hpp` | `SphericalMaps`, `setup_spherical_maps()`, `compute_spherical_lambda0_from_mesh()`, `spherical_gradient()`, `spherical_energy()` |
| `spherical_hessian.hpp` | `spherical_hessian()` — analytic Hessian via ∂α/∂u from law of cosines |

## Euclidean geometry (ℝ²)

| Header | Description |
|---|---|
| `euclidean_geometry.hpp` | Euclidean corner angle (t-value / atan2), edge lengths from DOF vector |
| `euclidean_functional.hpp` | `EuclideanMaps`, `setup_euclidean_maps()`, `compute_euclidean_lambda0_from_mesh()`, `euclidean_gradient()`, `euclidean_energy()` |
| `euclidean_hessian.hpp` | `euclidean_hessian()` — cotangent Laplacian (Pinkall–Polthier 1993) |

## Solver

| Header | Description |
|---|---|
| `newton_solver.hpp` | `newton_euclidean()`, `newton_spherical()`, `newton_hyper_ideal()`, `solve_linear_system()` (SimplicialLDLT + SparseQR fallback), `NewtonResult` struct |

## Preprocessing

| Header | Description |
|---|---|
| `gauss_bonnet.hpp` | `euler_characteristic()`, `genus()`, `gauss_bonnet_sum()`, `gauss_bonnet_rhs()`, `gauss_bonnet_deficit()`, `check_gauss_bonnet()`, `enforce_gauss_bonnet()` |

## Layout and holonomy

| Header | Description |
|---|---|
| `cut_graph.hpp` | `CutGraph` struct, `compute_cut_graph()` — tree-cotree algorithm (Erickson–Whittlesey 2005), produces 2g seam edges |
| `layout.hpp` | `euclidean_layout()`, `spherical_layout()`, `hyper_ideal_layout()`, `normalise_{euclidean,hyperbolic,spherical}()`. Structs: `Layout2D`, `Layout3D`, `HolonomyData`. `MobiusMap` (T(z)=(az+b)/(cz+d), `from_three`, `compose`, `inverse`, `apply`). Priority-BFS, `halfedge_uv`. |

## Post-processing

| Header | Description |
|---|---|
| `period_matrix.hpp` | `PeriodData`, `compute_period_matrix()`, `reduce_to_fundamental_domain()`, `is_in_fundamental_domain()` — period ratio τ = ω₂/ω₁ ∈ ℍ, SL(2,ℤ) reduction |
| `fundamental_domain.hpp` | `FundamentalDomain`, `compute_fundamental_domain()` (genus 1: CCW parallelogram; genus > 1: empty, TODO Phase 9c), `tiling_copy()`, `tiling_neighbourhood()` |
| `serialization.hpp` | `save_result_json()`, `load_result_json()`, `save_result_xml()`, `load_result_xml()`, `save_layout_off()` |