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_euclidean_hessian.cpp
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213
code/tests/cgal/test_euclidean_hessian.cpp
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// test_euclidean_hessian.cpp
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//
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// Phase 3f — Euclidean cotangent-Laplace Hessian.
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//
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// The Hessian of the Euclidean discrete conformal energy is the well-known
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// cotangent-Laplace operator (Pinkall–Polthier 1993, Springborn 2008).
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//
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// Tests:
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// 1. Cotangent weights are analytically correct for simple triangles.
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// 2. Hessian is symmetric.
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// 3. Hessian has the null-space property H·1 = 0 (uniform-shift mode).
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// 4. Hessian is positive semi-definite (all eigenvalues ≥ 0).
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// 5. Finite-difference check H[i,j] ≈ (G_i(x+ε·eⱼ)−G_i(x−ε·eⱼ))/(2ε).
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//
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// All tests use meshes and maps built with Phase-3d infrastructure.
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#include "conformal_mesh.hpp"
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#include "mesh_builder.hpp"
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#include "euclidean_hessian.hpp"
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#include <gtest/gtest.h>
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#include <Eigen/Dense> // for dense conversion and eigenvalue solver
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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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// Cotangent weight: equilateral triangle → all cots = 1/√3 = cot(60°)
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// ════════════════════════════════════════════════════════════════════════════
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TEST(EuclideanHessian, CotWeights_EquilateralTriangle)
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{
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// Equilateral triangle with l = 1 (all log-lengths = 0).
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auto cw = euclidean_cot_weights(1.0, 1.0, 1.0);
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ASSERT_TRUE(cw.valid);
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const double expected = 1.0 / std::sqrt(3.0); // cot(60°)
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EXPECT_NEAR(cw.cot1, expected, 1e-12);
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EXPECT_NEAR(cw.cot2, expected, 1e-12);
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EXPECT_NEAR(cw.cot3, expected, 1e-12);
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}
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// ════════════════════════════════════════════════════════════════════════════
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// Cotangent weight: right-isosceles triangle (legs 1, hypotenuse √2)
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//
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// v1=(0,0): right angle → cot(90°) = 0
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// v2=(1,0), v3=(0,1): 45° angles → cot(45°) = 1
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// ════════════════════════════════════════════════════════════════════════════
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TEST(EuclideanHessian, CotWeights_RightIsoscelesTriangle)
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{
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// l12=1, l23=√2, l31=1
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auto cw = euclidean_cot_weights(1.0, std::sqrt(2.0), 1.0);
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ASSERT_TRUE(cw.valid);
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EXPECT_NEAR(cw.cot1, 0.0, 1e-12); // right angle at v1
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EXPECT_NEAR(cw.cot2, 1.0, 1e-12); // 45° at v2
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EXPECT_NEAR(cw.cot3, 1.0, 1e-12); // 45° at v3
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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(EuclideanHessian, HessianIsSymmetric)
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{
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auto mesh = make_quad_strip();
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auto maps = setup_euclidean_maps(mesh);
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compute_euclidean_lambda0_from_mesh(mesh, maps);
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int n = assign_euclidean_vertex_dof_indices(mesh, maps);
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std::vector<double> x(static_cast<std::size_t>(n), -0.1);
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auto H = euclidean_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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<< "Hessian must be symmetric";
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}
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// ════════════════════════════════════════════════════════════════════════════
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// Null-space property: H·1 = 0 for a closed surface (regular tetrahedron)
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//
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// The cotangent Laplacian on a closed mesh has the constant vector in its
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// null space (each row sums to zero).
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// ════════════════════════════════════════════════════════════════════════════
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TEST(EuclideanHessian, NullSpaceIsConstantVector_ClosedMesh)
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{
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auto mesh = make_tetrahedron();
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auto maps = setup_euclidean_maps(mesh);
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compute_euclidean_lambda0_from_mesh(mesh, maps);
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int n = assign_euclidean_vertex_dof_indices(mesh, maps);
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std::vector<double> x(static_cast<std::size_t>(n), 0.0);
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auto H = euclidean_hessian(mesh, x, maps);
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// 1-vector
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Eigen::VectorXd ones = Eigen::VectorXd::Ones(n);
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Eigen::VectorXd Hones = H * ones;
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EXPECT_NEAR(Hones.norm(), 0.0, 1e-10)
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<< "H·1 must be zero on a closed mesh (cotangent Laplacian null-space)";
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}
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// ════════════════════════════════════════════════════════════════════════════
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// Hessian is positive semi-definite: all eigenvalues ≥ 0
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//
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// Checked on a small mesh (regular tetrahedron, 4 vertices) using dense
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// self-adjoint eigenvalue decomposition (only feasible for small n).
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// ════════════════════════════════════════════════════════════════════════════
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TEST(EuclideanHessian, HessianIsPositiveSemiDefinite)
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{
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auto mesh = make_tetrahedron();
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auto maps = setup_euclidean_maps(mesh);
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compute_euclidean_lambda0_from_mesh(mesh, maps);
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int n = assign_euclidean_vertex_dof_indices(mesh, maps);
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std::vector<double> x(static_cast<std::size_t>(n), 0.0);
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auto H = euclidean_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 min_ev = es.eigenvalues().minCoeff();
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EXPECT_GE(min_ev, -1e-10)
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<< "All eigenvalues of the cotangent Laplacian must be ≥ 0; "
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"smallest = " << min_ev;
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}
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// ════════════════════════════════════════════════════════════════════════════
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// Finite-difference Hessian check: single right-isosceles triangle
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//
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// H[i,j] ≈ (G_i(x+ε·eⱼ) − G_i(x−ε·eⱼ)) / (2ε)
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// ════════════════════════════════════════════════════════════════════════════
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TEST(EuclideanHessian, FDCheck_Triangle)
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{
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auto mesh = make_triangle();
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auto maps = setup_euclidean_maps(mesh);
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compute_euclidean_lambda0_from_mesh(mesh, maps);
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int n = assign_euclidean_vertex_dof_indices(mesh, maps);
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std::vector<double> x(static_cast<std::size_t>(n), -0.1);
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EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
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<< "FD Hessian check failed on right-isosceles triangle";
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}
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// ════════════════════════════════════════════════════════════════════════════
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// Finite-difference Hessian check: quad strip (2 triangles, 1 interior edge)
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// ════════════════════════════════════════════════════════════════════════════
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TEST(EuclideanHessian, FDCheck_QuadStrip)
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{
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auto mesh = make_quad_strip();
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auto maps = setup_euclidean_maps(mesh);
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compute_euclidean_lambda0_from_mesh(mesh, maps);
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int n = assign_euclidean_vertex_dof_indices(mesh, maps);
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std::vector<double> x(static_cast<std::size_t>(n), -0.1);
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EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
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<< "FD Hessian check failed on quad strip";
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}
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// ════════════════════════════════════════════════════════════════════════════
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// Finite-difference Hessian check: regular tetrahedron (closed, 4 faces)
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// ════════════════════════════════════════════════════════════════════════════
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TEST(EuclideanHessian, FDCheck_Tetrahedron)
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{
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auto mesh = make_tetrahedron();
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auto maps = setup_euclidean_maps(mesh);
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compute_euclidean_lambda0_from_mesh(mesh, maps);
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int n = assign_euclidean_vertex_dof_indices(mesh, maps);
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std::vector<double> x(static_cast<std::size_t>(n), -0.15);
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EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
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<< "FD Hessian check failed on regular tetrahedron";
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}
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// ════════════════════════════════════════════════════════════════════════════
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// Finite-difference Hessian check: with mixed pinned/variable vertices
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//
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// One vertex pinned: the corresponding row/column must be absent from H
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// while the diagonal of neighbouring variable vertices still gets the full
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// cotangent contribution.
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// ════════════════════════════════════════════════════════════════════════════
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TEST(EuclideanHessian, FDCheck_MixedPinnedVertices)
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{
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auto mesh = make_quad_strip();
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auto maps = setup_euclidean_maps(mesh);
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compute_euclidean_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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Vertex_index v3 = *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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maps.v_idx[v3] = 2;
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std::vector<double> x = {-0.1, -0.2, -0.15};
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EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
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<< "FD Hessian check failed for mixed pinned/variable vertices";
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}
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