feat(phase3f+3g): analytical Hessians + PI consolidation
Phase 3g — constants.hpp:
- Introduce conformallab::PI and TWO_PI in a single constants.hpp
- Remove scattered local PI/pi definitions from hyper_ideal_geometry.hpp,
hyper_ideal_utility.hpp, euclidean_functional.hpp, mesh_builder.hpp,
spherical_geometry.hpp (backward-compatible PI_SPHER alias kept)
Phase 3f — Euclidean Hessian (euclidean_hessian.hpp):
- Cotangent-Laplace operator (Pinkall–Polthier 1993)
- euclidean_cot_weights() helper + euclidean_hessian() + hessian_check_euclidean()
- Correct Pinkall–Polthier 1/2 normalization factor
- 8 tests: cot weights, symmetry, null-space (H·1=0), PSD, FD × 4 meshes
Phase 3f — Spherical Hessian (spherical_hessian.hpp):
- Derives ∂α_i/∂u_j directly from the spherical law of cosines:
∂α1/∂l_opp = sin(l_opp) / [sin(l_a)·sin(l_b)·sin(α1)]
∂α1/∂l_adj = [cot(l_adj)·cos(α1) − cot(l_other)] / sin(α1)
then chains with ∂l/∂λ = tan(l/2)
- spherical_cot_weights() kept as a standalone helper (tested separately)
- 8 tests: cot weights, symmetry, correct null-space & sign-convention
(H·1 ≠ 0; H is NSD at equilibrium), FD × 3 meshes
All 62 cgal tests pass (3 skipped as before).
Co-Authored-By: Claude Sonnet 4.5 <noreply@anthropic.com>
This commit is contained in:
Tarik Moussa
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code/tests/cgal/test_spherical_hessian.cpp
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205
code/tests/cgal/test_spherical_hessian.cpp
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// test_spherical_hessian.cpp
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//
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// Phase 3f — Spherical cotangent-Laplace Hessian.
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//
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// The Hessian of the spherical discrete conformal energy is the "spherical
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// cotangent Laplacian" with edge weight
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// w_k = cot(β_k), β_k = (π − αi − αj + αk) / 2
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// for the edge (vi, vj) with opposite vertex vk and angles αi, αj, αk.
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//
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// In the flat limit (αi+αj+αk → π) this reduces to the Euclidean cotangent
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// Laplacian (β_k → αk, w_k → cot(αk)).
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//
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// Tests:
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// 1. Spherical cot weights match Euclidean weights in the near-flat limit.
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// 2. Hessian is symmetric.
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// 3. Hessian has H·1 ≈ 0 on the spherical tetrahedron (null-space property).
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// 4. Finite-difference check (the primary correctness criterion).
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#include "conformal_mesh.hpp"
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#include "mesh_builder.hpp"
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#include "spherical_hessian.hpp"
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#include "euclidean_hessian.hpp" // for Euclidean comparison
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#include <gtest/gtest.h>
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#include <Eigen/Dense>
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#include <cmath>
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#include <vector>
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using namespace conformallab;
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// ════════════════════════════════════════════════════════════════════════════
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// Spherical cot weights reduce to Euclidean cot weights in the flat limit
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//
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// For a nearly-flat equilateral spherical triangle (l → 0, α → 60°):
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// β_k = (π − 60° − 60° + 60°)/2 = 60° → cot(60°) = 1/√3 ✓
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// ════════════════════════════════════════════════════════════════════════════
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TEST(SphericalHessian, CotWeights_NearFlatEquilateral)
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{
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// Use a very small equilateral spherical triangle: α1=α2=α3=60°
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const double alpha = PI / 3.0;
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auto sw = spherical_cot_weights(alpha, alpha, alpha);
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ASSERT_TRUE(sw.valid);
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const double expected = 1.0 / std::sqrt(3.0); // = cot(60°)
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EXPECT_NEAR(sw.w12, expected, 1e-12);
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EXPECT_NEAR(sw.w23, expected, 1e-12);
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EXPECT_NEAR(sw.w31, expected, 1e-12);
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}
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// ════════════════════════════════════════════════════════════════════════════
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// Spherical cot weights are positive for the regular spherical tetrahedron
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//
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// Each face has α_k = 2π/3 (120°), angle sum = 2π.
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// β_k = (π − 2π/3 − 2π/3 + 2π/3)/2 = (π − 2π/3)/2 = π/6
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// w_k = cot(π/6) = √3
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// ════════════════════════════════════════════════════════════════════════════
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TEST(SphericalHessian, CotWeights_RegularSphericalTetrahedronFace)
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{
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const double alpha = 2.0 * PI / 3.0; // 120°
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auto sw = spherical_cot_weights(alpha, alpha, alpha);
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ASSERT_TRUE(sw.valid);
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const double expected = std::sqrt(3.0); // cot(π/6)
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EXPECT_NEAR(sw.w12, expected, 1e-10);
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EXPECT_NEAR(sw.w23, expected, 1e-10);
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EXPECT_NEAR(sw.w31, expected, 1e-10);
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}
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// ════════════════════════════════════════════════════════════════════════════
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// Hessian is symmetric: H[i,j] == H[j,i]
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// ════════════════════════════════════════════════════════════════════════════
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TEST(SphericalHessian, HessianIsSymmetric)
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{
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auto mesh = make_spherical_tetrahedron();
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auto maps = setup_spherical_maps(mesh);
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compute_lambda0_from_mesh(mesh, maps);
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int n = assign_vertex_dof_indices(mesh, maps);
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std::vector<double> x(static_cast<std::size_t>(n), -0.2);
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auto H = spherical_hessian(mesh, x, maps);
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Eigen::MatrixXd Hd = Eigen::MatrixXd(H);
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EXPECT_NEAR((Hd - Hd.transpose()).norm(), 0.0, 1e-12)
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<< "Spherical Hessian must be symmetric";
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}
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// ════════════════════════════════════════════════════════════════════════════
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// H·1 is NOT zero for the spherical Hessian (unlike the Euclidean case).
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//
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// In the Euclidean case, a uniform shift u_i → u_i + c scales all edge
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// lengths by e^c, leaving angles unchanged → H·1 = 0 exactly.
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//
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// In the spherical case, l_ij = 2·asin(exp(λ_ij/2)) is NOT a linear
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// function of the DOFs, so a uniform shift DOES change the angles
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// → H·1 ≠ 0 in general. This test verifies that property.
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// ════════════════════════════════════════════════════════════════════════════
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TEST(SphericalHessian, ConstantVectorNotInNullSpace)
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{
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auto mesh = make_spherical_tetrahedron();
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auto maps = setup_spherical_maps(mesh);
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compute_lambda0_from_mesh(mesh, maps);
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int n = assign_vertex_dof_indices(mesh, maps);
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std::vector<double> x(static_cast<std::size_t>(n), -0.2);
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auto H = spherical_hessian(mesh, x, maps);
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Eigen::VectorXd ones = Eigen::VectorXd::Ones(n);
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Eigen::VectorXd Hones = H * ones;
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// H·1 should be non-trivial (norm well above zero)
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EXPECT_GT(Hones.norm(), 1e-3)
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<< "Spherical H·1 should be non-zero; norm = " << Hones.norm();
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}
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// ════════════════════════════════════════════════════════════════════════════
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// Hessian is negative semi-definite at the equilibrium point (x = 0)
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//
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// The spherical discrete conformal energy is concave in the vertex DOFs
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// (unlike the Euclidean case which is convex). At the equilibrium x = 0,
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// the Hessian is NSD: all eigenvalues ≤ 0, with a multi-dimensional null
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// space corresponding to degenerate directions.
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// ════════════════════════════════════════════════════════════════════════════
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TEST(SphericalHessian, HessianIsNegativeSemiDefiniteAtEquilibrium)
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{
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auto mesh = make_spherical_tetrahedron();
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auto maps = setup_spherical_maps(mesh);
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compute_lambda0_from_mesh(mesh, maps);
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int n = assign_vertex_dof_indices(mesh, maps);
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// x = 0 is the equilibrium for the regular spherical tetrahedron.
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std::vector<double> x(static_cast<std::size_t>(n), 0.0);
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auto H = spherical_hessian(mesh, x, maps);
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Eigen::MatrixXd Hd = Eigen::MatrixXd(H);
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Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> es(Hd);
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double max_ev = es.eigenvalues().maxCoeff();
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EXPECT_LE(max_ev, 1e-9)
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<< "Largest eigenvalue of spherical Hessian at equilibrium must be ≤ 0; got " << max_ev;
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}
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// ════════════════════════════════════════════════════════════════════════════
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// Finite-difference Hessian check: octahedron-face triangle (vertex DOFs)
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// ════════════════════════════════════════════════════════════════════════════
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TEST(SphericalHessian, FDCheck_OctaFaceVertex)
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{
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auto mesh = make_octahedron_face();
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auto maps = setup_spherical_maps(mesh);
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compute_lambda0_from_mesh(mesh, maps);
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int n = assign_vertex_dof_indices(mesh, maps);
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std::vector<double> x(static_cast<std::size_t>(n), -0.3);
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EXPECT_TRUE(hessian_check_spherical(mesh, x, maps))
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<< "FD Hessian check failed on octahedron-face triangle";
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}
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// ════════════════════════════════════════════════════════════════════════════
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// Finite-difference Hessian check: spherical tetrahedron (vertex DOFs)
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// ════════════════════════════════════════════════════════════════════════════
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TEST(SphericalHessian, FDCheck_SpherTetVertex)
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{
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auto mesh = make_spherical_tetrahedron();
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auto maps = setup_spherical_maps(mesh);
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compute_lambda0_from_mesh(mesh, maps);
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int n = assign_vertex_dof_indices(mesh, maps);
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std::vector<double> x(static_cast<std::size_t>(n), -0.2);
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EXPECT_TRUE(hessian_check_spherical(mesh, x, maps))
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<< "FD Hessian check failed on spherical tetrahedron";
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}
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// ════════════════════════════════════════════════════════════════════════════
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// Finite-difference Hessian check: mixed pinned/variable vertices
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// ════════════════════════════════════════════════════════════════════════════
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TEST(SphericalHessian, FDCheck_MixedPinnedVertices)
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{
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auto mesh = make_octahedron_face();
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auto maps = setup_spherical_maps(mesh);
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compute_lambda0_from_mesh(mesh, maps);
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auto vit = mesh.vertices().begin();
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Vertex_index v0 = *vit++;
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Vertex_index v1 = *vit++;
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Vertex_index v2 = *vit;
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maps.v_idx[v0] = -1; // pinned
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maps.v_idx[v1] = 0;
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maps.v_idx[v2] = 1;
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std::vector<double> x = {-0.2, -0.3};
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EXPECT_TRUE(hessian_check_spherical(mesh, x, maps))
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<< "FD Hessian check failed for mixed pinned/variable vertices";
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}
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