ConformalLabpp/code/include/gauss_bonnet.hpp

#pragma once
// Copyright (c) 2024-2026 Tarik Moussa.
// SPDX-License-Identifier: MIT

// gauss_bonnet.hpp
//
// Phase 6 — Gauss–Bonnet consistency check for prescribed target angles.
//
// Before calling newton_*() with custom target angles, verify that
// the angle defect sum matches the topology.
//
// ┌─────────────────────────────────────────────────────────────────────────┐
// │  Geometry         Identity to satisfy                                   │
// │  ─────────────────────────────────────────────────────────────────────  │
// │  Euclidean/flat   Σ_v (2π − Θ_v)  =  2π · χ(M)   (exact equality)    │
// │  Spherical        Σ_v (2π − Θ_v)  >  0            (sufficient, χ > 0) │
// │                                                                         │
// │  HyperIdeal — NOT SUPPORTED by this header.                            │
// │  The correct hyperbolic Gauss–Bonnet identity is                        │
// │    Σ_v (2π − Θ_v) − Area(M) = 2π · χ(M)                               │
// │  which differs from the Euclidean identity by the Area(M) > 0 term.    │
// │  Computing Area(M) from the HyperIdeal DOFs is non-trivial.            │
// │  gauss_bonnet_sum(mesh, HyperIdealMaps) and                             │
// │  enforce_gauss_bonnet(mesh, HyperIdealMaps) are therefore DELETED.     │
// │  Do NOT call check_gauss_bonnet before newton_hyper_ideal —            │
// │  it is not needed; the HyperIdeal energy is strictly convex so Newton  │
// │  converges without a pre-check.                                         │
// └─────────────────────────────────────────────────────────────────────────┘
//
// If the Euclidean/Spherical check fails, no conformal factor can realise
// the target angles and Newton will silently fail to converge.
//
// PRECONDITION — closed meshes only.  Every function here sums (2π − Θ_v)
// over ALL vertices.  On a mesh with boundary the boundary vertices carry a
// (π − Θ_v) term instead, so the identity Σ(2π−Θ_v) = 2π·χ does NOT hold and
// `check_gauss_bonnet` will (correctly) throw.  For open meshes pin the
// boundary directly and skip the Gauss–Bonnet check (see the CLI's
// flattening path).
//
// API:
//   int    euler_characteristic(mesh)
//   int    genus(mesh)
//   double gauss_bonnet_sum(mesh, EuclideanMaps/SphericalMaps) — Σ(2π − Θ_v)
//   double gauss_bonnet_rhs(mesh)            — 2π · χ(M)
//   double gauss_bonnet_deficit(mesh, maps)  — lhs − rhs  (0 = satisfied)
//   void   check_gauss_bonnet(mesh, maps [, tol])   — throws if violated
//   void   enforce_gauss_bonnet(mesh, maps)  — shifts θ_v by uniform Δ
//   (HyperIdealMaps overloads are deleted — see box above)

#include "conformal_mesh.hpp"
#include "euclidean_functional.hpp"
#include "spherical_functional.hpp"
#include "hyper_ideal_functional.hpp"
#include "constants.hpp"

#include <stdexcept>
#include <sstream>
#include <cmath>
#include <string>

namespace conformallab {

// ── Topology helpers ──────────────────────────────────────────────────────────

/// Euler characteristic χ = V − E + F.
/// For closed orientable surfaces: χ = 2 − 2g.
inline int euler_characteristic(const ConformalMesh& mesh)
{
    return static_cast<int>(mesh.number_of_vertices())
         - static_cast<int>(mesh.number_of_edges())
         + static_cast<int>(mesh.number_of_faces());
}

/// Genus of a closed orientable surface: g = (2 − χ) / 2.
/// Returns 0 for open meshes (boundary present) — callers should check.
inline int genus(const ConformalMesh& mesh)
{
    int chi = euler_characteristic(mesh);
    return (2 - chi) / 2;
}

// ── Left-hand side  Σ(2π − Θ_v) ─────────────────────────────────────────────

/// Sum `Σ_v (2π − Θ_v)` for a raw vertex → angle property map.
inline double gauss_bonnet_sum(
    const ConformalMesh&                                      mesh,
    const ConformalMesh::Property_map<Vertex_index, double>&  theta)
{
    double s = 0.0;
    for (auto v : mesh.vertices())
        s += TWO_PI - theta[v];
    return s;
}

/// `gauss_bonnet_sum` for the Euclidean-functional property bundle.
inline double gauss_bonnet_sum(const ConformalMesh& m, const EuclideanMaps&  mp)
    { return gauss_bonnet_sum(m, mp.theta_v); }
/// `gauss_bonnet_sum` for the Spherical-functional property bundle.
inline double gauss_bonnet_sum(const ConformalMesh& m, const SphericalMaps&  mp)
    { return gauss_bonnet_sum(m, mp.theta_v); }

// gauss_bonnet_sum for HyperIdealMaps is intentionally DELETED.
// The correct hyperbolic Gauss–Bonnet identity is
//   Σ(2π−Θ_v) − Area(M) = 2π·χ(M)
// not the Euclidean form Σ(2π−Θ_v) = 2π·χ(M).  Providing this overload
// would silently skip the Area term, making check_gauss_bonnet always
// fail for valid hyperbolic targets (e.g. a genus-2 mesh with Θ_v=2π
// gives Σ(2π−Θ_v)=0 but 2π·χ=−4π → deficit=4π ≠ 0 every time).
// Use newton_hyper_ideal directly — no pre-check is needed because the
// HyperIdeal energy is strictly convex (Springborn 2020 Theorem 1.3).
inline double gauss_bonnet_sum(const ConformalMesh&, const HyperIdealMaps&) = delete;

// ── Right-hand side  2π · χ(M) ───────────────────────────────────────────────

/// Right-hand side of Gauss-Bonnet: `2π · χ(M)`.
inline double gauss_bonnet_rhs(const ConformalMesh& mesh)
{
    return TWO_PI * static_cast<double>(euler_characteristic(mesh));
}

// ── Deficit: lhs − rhs  (0 = Gauss–Bonnet satisfied) ─────────────────────────

/// Gauss-Bonnet deficit `lhs − rhs`; zero iff the identity is satisfied.
template <typename Maps>
inline double gauss_bonnet_deficit(const ConformalMesh& mesh, const Maps& maps)
{
    return gauss_bonnet_sum(mesh, maps) - gauss_bonnet_rhs(mesh);
}

/// Throws `std::runtime_error` if `|lhs − 2π·χ| > tol`.
/// Overload accepting a precomputed `lhs`.
inline void check_gauss_bonnet(const ConformalMesh& mesh,
                                double lhs,
                                double tol = 1e-8)
{
    double rhs = gauss_bonnet_rhs(mesh);
    double def = lhs - rhs;
    if (std::abs(def) > tol) {
        std::ostringstream msg;
        msg << "Gauss–Bonnet violated:\n"
            << "  Σ(2π−Θ_v) = " << lhs
            << "  expected 2π·χ = " << rhs
            << "  (χ = " << euler_characteristic(mesh)
            << ", genus = " << genus(mesh) << ")\n"
            << "  deficit = " << def;
        throw std::runtime_error(msg.str());
    }
}

/// Throws `std::runtime_error` if Gauss-Bonnet is violated by more than `tol`.
template <typename Maps>
inline void check_gauss_bonnet(const ConformalMesh& mesh,
                                const Maps& maps,
                                double tol = 1e-8)
{
    check_gauss_bonnet(mesh, gauss_bonnet_sum(mesh, maps), tol);
}

// ── enforce_gauss_bonnet — adjust θ_v by uniform Δ ───────────────────────────
//
// Adds δ = (lhs − rhs) / V to every θ_v so that Gauss–Bonnet holds exactly.
// After this call, check_gauss_bonnet() will not throw (up to floating-point).
// Modifies ALL vertices' θ_v (no v_idx filtering) — the shift is a property
// of the target angles, independent of which vertices are free DOFs.

/// Distribute the Gauss-Bonnet deficit uniformly across all `Θ_v`:
/// add `δ = (lhs − rhs) / V` to every entry so that the identity holds
/// exactly afterwards.  Overload for a raw property map.
inline void enforce_gauss_bonnet(
    ConformalMesh&                                           mesh,
    ConformalMesh::Property_map<Vertex_index, double>&       theta)
{
    double lhs   = gauss_bonnet_sum(mesh, theta);
    double rhs   = gauss_bonnet_rhs(mesh);
    // Adding δ to every θ_v decreases the sum Σ(2π−θ_v) by V·δ.
    // We need lhs − V·δ = rhs, so δ = (lhs − rhs) / V.
    double delta = (lhs - rhs) / static_cast<double>(mesh.number_of_vertices());
    for (auto v : mesh.vertices())
        theta[v] += delta;
}

/// Distribute the Gauss-Bonnet deficit uniformly across `maps.theta_v`.
/// Supported for EuclideanMaps and SphericalMaps only.
/// HyperIdealMaps overload is deleted — see header comment for why.
template <typename Maps>
inline void enforce_gauss_bonnet(ConformalMesh& mesh, Maps& maps)
{
    enforce_gauss_bonnet(mesh, maps.theta_v);
}

// enforce_gauss_bonnet for HyperIdealMaps is intentionally DELETED.
// The Euclidean identity Σ(2π−Θ_v)=2π·χ is not the correct pre-condition
// for HyperIdeal.  Calling this function would silently shift Θ_v to
// satisfy the wrong identity, producing incorrect target angles.
inline void enforce_gauss_bonnet(ConformalMesh&, HyperIdealMaps&) = delete;

} // namespace conformallab