ConformalLabpp/code/tests/cgal/test_euclidean_hessian.cpp

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

// test_euclidean_hessian.cpp
//
// Phase 3f — Euclidean cotangent-Laplace Hessian.
//
// The Hessian of the Euclidean discrete conformal energy is the well-known
// cotangent-Laplace operator (Pinkall–Polthier 1993, Springborn 2008).
//
// Tests:
//   1. Cotangent weights are analytically correct for simple triangles.
//   2. Hessian is symmetric.
//   3. Hessian has the null-space property H·1 = 0 (uniform-shift mode).
//   4. Hessian is positive semi-definite (all eigenvalues ≥ 0).
//   5. Finite-difference check H[i,j] ≈ (G_i(x+ε·eⱼ)−G_i(x−ε·eⱼ))/(2ε).
//
// All tests use meshes and maps built with Phase-3d infrastructure.

#include "conformal_mesh.hpp"
#include "mesh_builder.hpp"
#include "euclidean_hessian.hpp"
#include <gtest/gtest.h>
#include <Eigen/Dense>   // for dense conversion and eigenvalue solver
#include <cmath>
#include <vector>

using namespace conformallab;

// ════════════════════════════════════════════════════════════════════════════
// Cotangent weight: equilateral triangle → all cots = 1/√3 = cot(60°)
// ════════════════════════════════════════════════════════════════════════════

TEST(EuclideanHessian, CotWeights_EquilateralTriangle)
{
    // Equilateral triangle with l = 1 (all log-lengths = 0).
    auto cw = euclidean_cot_weights(1.0, 1.0, 1.0);

    ASSERT_TRUE(cw.valid);
    const double expected = 1.0 / std::sqrt(3.0);   // cot(60°)
    EXPECT_NEAR(cw.cot1, expected, 1e-12);
    EXPECT_NEAR(cw.cot2, expected, 1e-12);
    EXPECT_NEAR(cw.cot3, expected, 1e-12);
}

// ════════════════════════════════════════════════════════════════════════════
// Cotangent weight: right-isosceles triangle (legs 1, hypotenuse √2)
//
//   v1=(0,0): right angle → cot(90°) = 0
//   v2=(1,0), v3=(0,1): 45° angles → cot(45°) = 1
// ════════════════════════════════════════════════════════════════════════════

TEST(EuclideanHessian, CotWeights_RightIsoscelesTriangle)
{
    // l12=1, l23=√2, l31=1
    auto cw = euclidean_cot_weights(1.0, std::sqrt(2.0), 1.0);

    ASSERT_TRUE(cw.valid);
    EXPECT_NEAR(cw.cot1, 0.0, 1e-12);   // right angle at v1
    EXPECT_NEAR(cw.cot2, 1.0, 1e-12);   // 45° at v2
    EXPECT_NEAR(cw.cot3, 1.0, 1e-12);   // 45° at v3
}

// ════════════════════════════════════════════════════════════════════════════
// Hessian is symmetric: H[i,j] == H[j,i]
// ════════════════════════════════════════════════════════════════════════════

TEST(EuclideanHessian, HessianIsSymmetric)
{
    auto mesh = make_quad_strip();
    auto maps = setup_euclidean_maps(mesh);
    compute_euclidean_lambda0_from_mesh(mesh, maps);
    int n = assign_euclidean_vertex_dof_indices(mesh, maps);

    std::vector<double> x(static_cast<std::size_t>(n), -0.1);
    auto H = euclidean_hessian(mesh, x, maps);

    Eigen::MatrixXd Hd = Eigen::MatrixXd(H);
    EXPECT_NEAR((Hd - Hd.transpose()).norm(), 0.0, 1e-12)
        << "Hessian must be symmetric";
}

// ════════════════════════════════════════════════════════════════════════════
// Null-space property: H·1 = 0 for a closed surface (regular tetrahedron)
//
// The cotangent Laplacian on a closed mesh has the constant vector in its
// null space (each row sums to zero).
// ════════════════════════════════════════════════════════════════════════════

TEST(EuclideanHessian, NullSpaceIsConstantVector_ClosedMesh)
{
    auto mesh = make_tetrahedron();
    auto maps = setup_euclidean_maps(mesh);
    compute_euclidean_lambda0_from_mesh(mesh, maps);
    int n = assign_euclidean_vertex_dof_indices(mesh, maps);

    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
    auto H = euclidean_hessian(mesh, x, maps);

    // 1-vector
    Eigen::VectorXd ones = Eigen::VectorXd::Ones(n);
    Eigen::VectorXd Hones = H * ones;

    EXPECT_NEAR(Hones.norm(), 0.0, 1e-10)
        << "H·1 must be zero on a closed mesh (cotangent Laplacian null-space)";
}

// ════════════════════════════════════════════════════════════════════════════
// Hessian is positive semi-definite: all eigenvalues ≥ 0
//
// Checked on a small mesh (regular tetrahedron, 4 vertices) using dense
// self-adjoint eigenvalue decomposition (only feasible for small n).
// ════════════════════════════════════════════════════════════════════════════

TEST(EuclideanHessian, HessianIsPositiveSemiDefinite)
{
    auto mesh = make_tetrahedron();
    auto maps = setup_euclidean_maps(mesh);
    compute_euclidean_lambda0_from_mesh(mesh, maps);
    int n = assign_euclidean_vertex_dof_indices(mesh, maps);

    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
    auto H = euclidean_hessian(mesh, x, maps);

    Eigen::MatrixXd Hd = Eigen::MatrixXd(H);
    Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> es(Hd);
    double min_ev = es.eigenvalues().minCoeff();

    EXPECT_GE(min_ev, -1e-10)
        << "All eigenvalues of the cotangent Laplacian must be ≥ 0; "
           "smallest = " << min_ev;
}

// ════════════════════════════════════════════════════════════════════════════
// Finite-difference Hessian check: single right-isosceles triangle
//
// H[i,j] ≈ (G_i(x+ε·eⱼ) − G_i(x−ε·eⱼ)) / (2ε)
// ════════════════════════════════════════════════════════════════════════════

TEST(EuclideanHessian, FDCheck_Triangle)
{
    auto mesh = make_triangle();
    auto maps = setup_euclidean_maps(mesh);
    compute_euclidean_lambda0_from_mesh(mesh, maps);
    int n = assign_euclidean_vertex_dof_indices(mesh, maps);

    std::vector<double> x(static_cast<std::size_t>(n), -0.1);

    EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
        << "FD Hessian check failed on right-isosceles triangle";
}

// ════════════════════════════════════════════════════════════════════════════
// Finite-difference Hessian check: quad strip (2 triangles, 1 interior edge)
// ════════════════════════════════════════════════════════════════════════════

TEST(EuclideanHessian, FDCheck_QuadStrip)
{
    auto mesh = make_quad_strip();
    auto maps = setup_euclidean_maps(mesh);
    compute_euclidean_lambda0_from_mesh(mesh, maps);
    int n = assign_euclidean_vertex_dof_indices(mesh, maps);

    std::vector<double> x(static_cast<std::size_t>(n), -0.1);

    EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
        << "FD Hessian check failed on quad strip";
}

// ════════════════════════════════════════════════════════════════════════════
// Finite-difference Hessian check: regular tetrahedron (closed, 4 faces)
// ════════════════════════════════════════════════════════════════════════════

TEST(EuclideanHessian, FDCheck_Tetrahedron)
{
    auto mesh = make_tetrahedron();
    auto maps = setup_euclidean_maps(mesh);
    compute_euclidean_lambda0_from_mesh(mesh, maps);
    int n = assign_euclidean_vertex_dof_indices(mesh, maps);

    std::vector<double> x(static_cast<std::size_t>(n), -0.15);

    EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
        << "FD Hessian check failed on regular tetrahedron";
}

// ════════════════════════════════════════════════════════════════════════════
// Finite-difference Hessian check: with mixed pinned/variable vertices
//
// One vertex pinned: the corresponding row/column must be absent from H
// while the diagonal of neighbouring variable vertices still gets the full
// cotangent contribution.
// ════════════════════════════════════════════════════════════════════════════

TEST(EuclideanHessian, FDCheck_MixedPinnedVertices)
{
    auto mesh = make_quad_strip();
    auto maps = setup_euclidean_maps(mesh);
    compute_euclidean_lambda0_from_mesh(mesh, maps);

    auto vit = mesh.vertices().begin();
    Vertex_index v0 = *vit++;
    Vertex_index v1 = *vit++;
    Vertex_index v2 = *vit++;
    Vertex_index v3 = *vit;

    maps.v_idx[v0] = -1;   // pinned
    maps.v_idx[v1] = 0;
    maps.v_idx[v2] = 1;
    maps.v_idx[v3] = 2;

    std::vector<double> x = {-0.1, -0.2, -0.15};

    EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
        << "FD Hessian check failed for mixed pinned/variable vertices";
}