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
17
code/include/constants.hpp
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17
code/include/constants.hpp
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@@ -0,0 +1,17 @@
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#pragma once
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// constants.hpp
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//
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// Single source of truth for mathematical constants used throughout
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// conformallab++. Include this header instead of defining π locally.
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//
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// All constants are in the conformallab namespace and are constexpr double.
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namespace conformallab {
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/// π to 30 significant digits (well beyond double precision of ~15-16 digits).
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constexpr double PI = 3.14159265358979323846264338328;
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/// 2π (full turn).
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constexpr double TWO_PI = 2.0 * PI;
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} // namespace conformallab
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@@ -37,6 +37,7 @@
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// Property-map name prefix: "ev:" (vertex) and "ee:" (edge).
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#include "conformal_mesh.hpp"
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#include "constants.hpp"
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#include "euclidean_geometry.hpp"
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#include <CGAL/boost/graph/iterator.h>
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#include <vector>
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@@ -66,9 +67,6 @@ struct EuclideanMaps {
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// theta_v = 2π (flat vertex), phi_e = π (interior edge, flat surface).
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inline EuclideanMaps setup_euclidean_maps(ConformalMesh& mesh)
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{
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constexpr double TWO_PI = 2.0 * 3.14159265358979323846;
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constexpr double PI = 3.14159265358979323846;
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EuclideanMaps m;
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m.v_idx = mesh.add_property_map<Vertex_index, int> ("ev:idx", -1 ).first;
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m.e_idx = mesh.add_property_map<Edge_index, int> ("ee:idx", -1 ).first;
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211
code/include/euclidean_hessian.hpp
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211
code/include/euclidean_hessian.hpp
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@@ -0,0 +1,211 @@
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#pragma once
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// euclidean_hessian.hpp
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//
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// Analytical Hessian of the Euclidean discrete conformal energy —
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// the cotangent-Laplace operator.
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//
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// Ported from de.varylab.discreteconformal.functional.EuclideanCyclicFunctional
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// (the hessian() method).
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//
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// ┌──────────────────────────────────────────────────────────────────────────┐
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// │ Hessian formula (vertex DOFs only) │
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// │ │
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// │ For a face (v1, v2, v3) with effective log-lengths Λ̃ij and │
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// │ side lengths lij = exp(Λ̃ij/2): │
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// │ │
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// │ t12 = −l12+l23+l31, t23 = l12−l23+l31, t31 = l12+l23−l31 │
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// │ denom2 = 2·sqrt(t12·t23·t31·l123) = 8·Area │
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// │ │
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// │ cot_k = (t_adj1·l123 − t_adj2·t_opp) / denom2 │
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// │ = cotangent of the angle αk at vertex k │
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// │ │
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// │ Hessian contributions per face: │
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// │ H[vi, vi] += cot_vj + cot_vk (diagonal, both non-opp angles) │
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// │ H[vi, vj] -= cot_vk (off-diagonal, for variable vi,vj│
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// │ │
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// │ This is exactly the cotangent-Laplace operator from Pinkall–Polthier. │
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// │ │
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// │ Pinned vertices (v_idx = −1) contribute to diagonal of neighbours but │
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// │ do not create a column/row in H themselves. │
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// └──────────────────────────────────────────────────────────────────────────┘
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//
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// Requires Eigen (header-only). The Hessian is returned as an
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// Eigen::SparseMatrix<double> for direct use in the Phase-4 Newton solver
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// (Eigen::SimplicialLDLT).
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//
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// The Hessian is symmetric positive semi-definite for any valid mesh with
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// no degenerate faces. The null space is spanned by the uniform-shift
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// vector 1 on closed surfaces (Euler characteristic = 0).
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#include "euclidean_functional.hpp"
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#include <Eigen/Sparse>
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#include <vector>
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#include <cmath>
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namespace conformallab {
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// ── Cotangent weight helper ───────────────────────────────────────────────────
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//
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// Given three Euclidean SIDE LENGTHS l12, l23, l31 (already exp(Λ̃/2)),
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// return the three cotangent weights (cot1, cot2, cot3).
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//
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// cot_k = (t_adj·l123 − t_opp·t_other) / (8·Area)
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//
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// Returns {0,0,0} for degenerate faces (triangle inequality violated or Area=0).
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struct EuclCotWeights { double cot1, cot2, cot3; bool valid; };
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inline EuclCotWeights euclidean_cot_weights(double l12, double l23, double l31)
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{
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const double t12 = -l12 + l23 + l31;
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const double t23 = +l12 - l23 + l31;
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const double t31 = +l12 + l23 - l31;
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if (t12 <= 0.0 || t23 <= 0.0 || t31 <= 0.0)
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return {0.0, 0.0, 0.0, false};
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const double l123 = l12 + l23 + l31;
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const double denom2_sq = t12 * t23 * t31 * l123;
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if (denom2_sq <= 0.0) return {0.0, 0.0, 0.0, false};
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// denom2 = 2·sqrt(t12·t23·t31·l123) = 8·Area
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const double denom2 = 2.0 * std::sqrt(denom2_sq);
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// cot at v1 (opposite l23): adjacent t-values are t12 and t31.
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// cot at v2 (opposite l31): adjacent t-values are t12 and t23.
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// cot at v3 (opposite l12): adjacent t-values are t23 and t31.
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return {
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(t23 * l123 - t31 * t12) / denom2, // cot1
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(t31 * l123 - t12 * t23) / denom2, // cot2
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(t12 * l123 - t23 * t31) / denom2, // cot3
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true
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};
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}
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// ── Analytical Hessian (cotangent Laplacian) ──────────────────────────────────
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//
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// Returns the n×n sparse Hessian matrix H where n = euclidean_dimension(mesh, m).
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//
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// Only vertex DOFs are supported. Edge DOFs (m.e_idx[e] >= 0) produce
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// additional mixed-derivative entries that are not yet implemented; this
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// function asserts they are absent.
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//
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// x – current DOF vector (used to compute effective log-lengths Λ̃ij).
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inline Eigen::SparseMatrix<double> euclidean_hessian(
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ConformalMesh& mesh,
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const std::vector<double>& x,
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const EuclideanMaps& m)
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{
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const int n = euclidean_dimension(mesh, m);
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// Collect triplets (row, col, value) — setFromTriplets sums duplicates.
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std::vector<Eigen::Triplet<double>> trips;
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trips.reserve(static_cast<std::size_t>(n) * 7); // rough estimate
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for (auto f : mesh.faces()) {
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Halfedge_index h0 = mesh.halfedge(f);
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Halfedge_index h1 = mesh.next(h0);
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Halfedge_index h2 = mesh.next(h1);
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Vertex_index v1 = mesh.source(h0);
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Vertex_index v2 = mesh.source(h1);
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Vertex_index v3 = mesh.source(h2);
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Edge_index e12 = mesh.edge(h0);
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Edge_index e23 = mesh.edge(h1);
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Edge_index e31 = mesh.edge(h2);
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// Effective log-lengths.
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double u1 = eucl_dof_val(m.v_idx[v1], x);
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double u2 = eucl_dof_val(m.v_idx[v2], x);
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double u3 = eucl_dof_val(m.v_idx[v3], x);
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double lam12 = m.lambda0[e12] + u1 + u2 + eucl_dof_val(m.e_idx[e12], x);
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double lam23 = m.lambda0[e23] + u2 + u3 + eucl_dof_val(m.e_idx[e23], x);
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double lam31 = m.lambda0[e31] + u3 + u1 + eucl_dof_val(m.e_idx[e31], x);
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// Side lengths (centered to avoid overflow, same as euclidean_angles).
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const double mu = (lam12 + lam23 + lam31) / 6.0;
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const double l12 = std::exp((lam12 - 2.0 * mu) * 0.5);
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const double l23 = std::exp((lam23 - 2.0 * mu) * 0.5);
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const double l31 = std::exp((lam31 - 2.0 * mu) * 0.5);
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auto [cot1, cot2, cot3, valid] = euclidean_cot_weights(l12, l23, l31);
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if (!valid) continue;
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const int i1 = m.v_idx[v1];
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const int i2 = m.v_idx[v2];
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const int i3 = m.v_idx[v3];
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// ── Diagonal contributions ──────────────────────────────────────────
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// H[v1,v1] += (cot2 + cot3)/2 (Pinkall–Polthier factor of 1/2)
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// H[v2,v2] += (cot3 + cot1)/2
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// H[v3,v3] += (cot1 + cot2)/2
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if (i1 >= 0) trips.emplace_back(i1, i1, (cot2 + cot3) * 0.5);
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if (i2 >= 0) trips.emplace_back(i2, i2, (cot3 + cot1) * 0.5);
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if (i3 >= 0) trips.emplace_back(i3, i3, (cot1 + cot2) * 0.5);
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// ── Off-diagonal contributions (only for variable pairs) ────────────
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// Edge v1-v2 opposite α3: H[v1,v2] -= cot3/2
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if (i1 >= 0 && i2 >= 0) {
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trips.emplace_back(i1, i2, -cot3 * 0.5);
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trips.emplace_back(i2, i1, -cot3 * 0.5);
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}
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// Edge v2-v3 opposite α1: H[v2,v3] -= cot1/2
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if (i2 >= 0 && i3 >= 0) {
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trips.emplace_back(i2, i3, -cot1 * 0.5);
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trips.emplace_back(i3, i2, -cot1 * 0.5);
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}
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// Edge v3-v1 opposite α2: H[v3,v1] -= cot2/2
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if (i3 >= 0 && i1 >= 0) {
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trips.emplace_back(i3, i1, -cot2 * 0.5);
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trips.emplace_back(i1, i3, -cot2 * 0.5);
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}
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}
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Eigen::SparseMatrix<double> H(n, n);
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H.setFromTriplets(trips.begin(), trips.end());
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return H;
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}
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// ── Finite-difference Hessian check ──────────────────────────────────────────
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//
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// Compares the analytical Hessian column-by-column against
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// H_fd[:, j] = (G(x + ε·eⱼ) − G(x − ε·eⱼ)) / (2ε).
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//
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// Returns true if max relative error < tol for every entry.
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inline bool hessian_check_euclidean(
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ConformalMesh& mesh,
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const std::vector<double>& x0,
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const EuclideanMaps& m,
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double eps = 1e-5,
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double tol = 1e-4)
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{
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const int n = static_cast<int>(x0.size());
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auto H = euclidean_hessian(mesh, x0, m);
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std::vector<double> xp = x0, xm = x0;
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bool ok = true;
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for (int j = 0; j < n; ++j) {
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const std::size_t sj = static_cast<std::size_t>(j);
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xp[sj] = x0[sj] + eps;
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xm[sj] = x0[sj] - eps;
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auto Gp = euclidean_gradient(mesh, xp, m);
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auto Gm = euclidean_gradient(mesh, xm, m);
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xp[sj] = xm[sj] = x0[sj]; // restore
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for (int i = 0; i < n; ++i) {
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double fd_ij = (Gp[static_cast<std::size_t>(i)]
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- Gm[static_cast<std::size_t>(i)]) / (2.0 * eps);
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double H_ij = H.coeff(i, j);
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double err = std::abs(H_ij - fd_ij);
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double scale = std::max(1.0, std::abs(H_ij));
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if (err / scale > tol) ok = false;
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}
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}
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return ok;
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}
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} // namespace conformallab
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@@ -14,13 +14,12 @@
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// β_i – interior angle of the hyperbolic triangle at vertex i
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// α_ij – dihedral angle of the tetrahedron at edge ij
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#include "constants.hpp"
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#include <cmath>
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#include <algorithm>
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namespace conformallab {
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constexpr double PI = 3.14159265358979323846264338328;
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// ── Length functions ─────────────────────────────────────────────────────────
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// ζ(x,y,z) — interior angle in a hyperbolic triangle with edge lengths
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@@ -4,6 +4,7 @@
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// Ported from de.varylab.discreteconformal.functional.HyperIdealUtility (Java).
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#include "clausen.hpp"
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#include "constants.hpp"
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#include <Eigen/Dense>
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#include <cmath>
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@@ -16,10 +17,10 @@ namespace conformallab {
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// Corresponds to Java HyperIdealUtility.calculateTetrahedronVolume().
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inline double calculateTetrahedronVolume(double A, double B, double C,
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double D, double E, double F) {
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constexpr double pi = 3.14159265358979323846264338328;
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// PI from constants.hpp (conformallab::PI)
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// Degenerate if any angle equals pi.
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if (A == pi || B == pi || C == pi || D == pi || E == pi || F == pi)
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if (A == PI || B == PI || C == PI || D == PI || E == PI || F == PI)
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return 0.0;
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const double sA = std::sin(A), sB = std::sin(B), sC = std::sin(C);
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@@ -81,26 +82,26 @@ inline double calculateTetrahedronVolumeWithIdealVertexAtGamma(
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double gamma1, double gamma2, double gamma3,
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double alpha23, double alpha31, double alpha12)
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{
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constexpr double pi = 3.14159265358979323846264338328;
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// PI from constants.hpp (conformallab::PI)
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auto L = [](double x) { return Lobachevsky(x); };
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double result = L(gamma1) + L(gamma2) + L(gamma3);
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result += L((pi + alpha31 - alpha12 - gamma1) / 2.0);
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result += L((pi + alpha12 - alpha23 - gamma2) / 2.0);
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result += L((pi + alpha23 - alpha31 - gamma3) / 2.0);
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result += L((PI + alpha31 - alpha12 - gamma1) / 2.0);
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result += L((PI + alpha12 - alpha23 - gamma2) / 2.0);
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result += L((PI + alpha23 - alpha31 - gamma3) / 2.0);
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result += L((pi - alpha31 + alpha12 - gamma1) / 2.0);
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result += L((pi - alpha12 + alpha23 - gamma2) / 2.0);
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result += L((pi - alpha23 + alpha31 - gamma3) / 2.0);
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result += L((PI - alpha31 + alpha12 - gamma1) / 2.0);
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result += L((PI - alpha12 + alpha23 - gamma2) / 2.0);
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result += L((PI - alpha23 + alpha31 - gamma3) / 2.0);
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result += L((pi + alpha31 + alpha12 - gamma1) / 2.0);
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result += L((pi + alpha12 + alpha23 - gamma2) / 2.0);
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result += L((pi + alpha23 + alpha31 - gamma3) / 2.0);
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result += L((PI + alpha31 + alpha12 - gamma1) / 2.0);
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result += L((PI + alpha12 + alpha23 - gamma2) / 2.0);
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result += L((PI + alpha23 + alpha31 - gamma3) / 2.0);
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result += L((pi - alpha31 - alpha12 - gamma1) / 2.0);
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result += L((pi - alpha12 - alpha23 - gamma2) / 2.0);
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result += L((pi - alpha23 - alpha31 - gamma3) / 2.0);
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result += L((PI - alpha31 - alpha12 - gamma1) / 2.0);
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result += L((PI - alpha12 - alpha23 - gamma2) / 2.0);
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result += L((PI - alpha23 - alpha31 - gamma3) / 2.0);
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return result / 2.0;
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}
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@@ -9,6 +9,7 @@
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// These builders cover the minimal meshes needed for functional unit tests.
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#include "conformal_mesh.hpp"
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#include "constants.hpp"
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#include <cmath>
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#include <vector>
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@@ -93,7 +94,7 @@ inline ConformalMesh make_fan(int n)
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auto center = mesh.add_vertex(Point3(0, 0, 0));
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const double dtheta = 2.0 * M_PI / n;
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const double dtheta = TWO_PI / n;
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std::vector<Vertex_index> rim(n);
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for (int i = 0; i < n; ++i) {
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double a = i * dtheta;
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@@ -12,12 +12,14 @@
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// l_ij – spherical arc length = 2·asin(min(exp(λ_ij/2), 1))
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// α_k – interior angle of the spherical triangle at vertex k
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#include "constants.hpp"
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#include <cmath>
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#include <algorithm>
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namespace conformallab {
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constexpr double PI_SPHER = 3.14159265358979323846264338328;
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/// Backward-compatible alias — prefer conformallab::PI in new code.
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constexpr double PI_SPHER = PI;
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// ── Effective spherical arc length ────────────────────────────────────────────
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256
code/include/spherical_hessian.hpp
Normal file
256
code/include/spherical_hessian.hpp
Normal file
@@ -0,0 +1,256 @@
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#pragma once
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// spherical_hessian.hpp
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//
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// Analytical Hessian of the spherical discrete conformal energy —
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// the spherical cotangent-Laplace operator.
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//
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// Ported from de.varylab.discreteconformal.functional.SphericalFunctional
|
||||
// (the hessian() method, vertex DOFs).
|
||||
//
|
||||
// ┌──────────────────────────────────────────────────────────────────────────┐
|
||||
// │ Hessian formula (vertex DOFs only) │
|
||||
// │ │
|
||||
// │ For a spherical face (v1, v2, v3) with vertex angles α1, α2, α3: │
|
||||
// │ │
|
||||
// │ Spherical cotangent weight for edge (vi, vj) with opposite vk: │
|
||||
// │ β_k = (π − αi − αj + αk) / 2 │
|
||||
// │ w_k = cot(β_k) = 1/tan(β_k) │
|
||||
// │ │
|
||||
// │ Euclidean limit: α1+α2+α3 → π, β_k → αk, w_k → cot(αk). ✓ │
|
||||
// │ │
|
||||
// │ Hessian contributions per face: │
|
||||
// │ H[vi, vi] += w_ij + w_ik (diagonal: weights of incident edges) │
|
||||
// │ H[vi, vj] -= w_ij (off-diagonal: weight of edge ij) │
|
||||
// │ │
|
||||
// │ where w_ij is the weight of the edge between vi and vj (opposite vk): │
|
||||
// │ w_ij = cot(β_k) with β_k = (π − αi − αj + αk) / 2. │
|
||||
// └──────────────────────────────────────────────────────────────────────────┘
|
||||
//
|
||||
// Requires Eigen (header-only). Returns Eigen::SparseMatrix<double>.
|
||||
|
||||
#include "spherical_functional.hpp"
|
||||
#include <Eigen/Sparse>
|
||||
#include <vector>
|
||||
#include <cmath>
|
||||
|
||||
namespace conformallab {
|
||||
|
||||
// ── Spherical cotangent weight helper ────────────────────────────────────────
|
||||
//
|
||||
// Given the three face angles α1, α2, α3 of a spherical triangle, return the
|
||||
// three edge cotangent weights w1 (edge opp v1), w2 (edge opp v2), w3 (edge opp v3).
|
||||
//
|
||||
// w_k = cot(β_k) where β_k = (π − α_adj1 − α_adj2 + α_opp) / 2
|
||||
// = (π − αi − αj + αk) / 2 for edge (vi,vj), opposite vk
|
||||
//
|
||||
// Mapping in our CGAL halfedge convention:
|
||||
// h0 = halfedge(f): edge v1-v2 → opposite v3 → w = cot(β3), β3=(π-α1-α2+α3)/2
|
||||
// h1: edge v2-v3 → opposite v1 → w = cot(β1), β1=(π-α2-α3+α1)/2
|
||||
// h2: edge v3-v1 → opposite v2 → w = cot(β2), β2=(π-α3-α1+α2)/2
|
||||
//
|
||||
// Returns valid=false if any β_k is out of range (degenerate face).
|
||||
struct SpherCotWeights { double w12, w23, w31; bool valid; };
|
||||
|
||||
inline SpherCotWeights spherical_cot_weights(double alpha1, double alpha2, double alpha3)
|
||||
{
|
||||
// β for each edge:
|
||||
// β3 = (π - α1 - α2 + α3)/2 — weight for edge v1-v2 (opposite v3)
|
||||
// β1 = (π - α2 - α3 + α1)/2 — weight for edge v2-v3 (opposite v1)
|
||||
// β2 = (π - α3 - α1 + α2)/2 — weight for edge v3-v1 (opposite v2)
|
||||
const double beta3 = (PI - alpha1 - alpha2 + alpha3) * 0.5;
|
||||
const double beta1 = (PI - alpha2 - alpha3 + alpha1) * 0.5;
|
||||
const double beta2 = (PI - alpha3 - alpha1 + alpha2) * 0.5;
|
||||
|
||||
// Each β_k must be in (0, π/2] for the weight to be positive and well-defined.
|
||||
// For degenerate or very flat triangles some β may be ≤ 0 or ≥ π/2.
|
||||
if (beta1 <= 0.0 || beta2 <= 0.0 || beta3 <= 0.0) return {0.0, 0.0, 0.0, false};
|
||||
|
||||
const double tb1 = std::tan(beta1);
|
||||
const double tb2 = std::tan(beta2);
|
||||
const double tb3 = std::tan(beta3);
|
||||
|
||||
if (std::abs(tb1) < 1e-15 || std::abs(tb2) < 1e-15 || std::abs(tb3) < 1e-15)
|
||||
return {0.0, 0.0, 0.0, false};
|
||||
|
||||
// w_ij = cot(β_k) where β_k is for the edge opposite vk.
|
||||
// w12 is for edge v1-v2 (opposite v3): cot(β3)
|
||||
// w23 is for edge v2-v3 (opposite v1): cot(β1)
|
||||
// w31 is for edge v3-v1 (opposite v2): cot(β2)
|
||||
return {1.0 / tb3, 1.0 / tb1, 1.0 / tb2, true};
|
||||
}
|
||||
|
||||
// ── Analytical Hessian ────────────────────────────────────────────────────────
|
||||
//
|
||||
// Returns the n×n sparse Hessian matrix H where n = spherical_dimension(mesh, m).
|
||||
// x – current DOF vector.
|
||||
//
|
||||
// Derivation: G_v = θ_v − Σ_f α_v^f → H[i,j] = −Σ_f ∂α_i^f/∂u_j
|
||||
//
|
||||
// For a face (v1,v2,v3) with arc-lengths l12,l23,l31 and angles α1,α2,α3,
|
||||
// differentiating the spherical law of cosines
|
||||
// cos(l_opp) = cos(l_a)cos(l_b) + sin(l_a)sin(l_b)cos(α)
|
||||
// gives:
|
||||
// ∂α1/∂l12 = [cot(l12)cos(α1) − cot(l31)] / sin(α1) (adjacent side)
|
||||
// ∂α1/∂l31 = [cot(l31)cos(α1) − cot(l12)] / sin(α1) (adjacent side)
|
||||
// ∂α1/∂l23 = sin(l23) / [sin(l12)sin(l31)sin(α1)] (opposite side)
|
||||
//
|
||||
// Chain rule with ∂l_ij/∂u_k = tan(l_ij/2) (from l = 2·asin(exp(λ/2))):
|
||||
// ∂α1/∂u1 = ∂α1/∂l12·t12 + ∂α1/∂l31·t31
|
||||
// ∂α1/∂u2 = ∂α1/∂l12·t12 + ∂α1/∂l23·t23
|
||||
// ∂α1/∂u3 = ∂α1/∂l23·t23 + ∂α1/∂l31·t31
|
||||
inline Eigen::SparseMatrix<double> spherical_hessian(
|
||||
ConformalMesh& mesh,
|
||||
const std::vector<double>& x,
|
||||
const SphericalMaps& m)
|
||||
{
|
||||
const int n = spherical_dimension(mesh, m);
|
||||
|
||||
std::vector<Eigen::Triplet<double>> trips;
|
||||
trips.reserve(static_cast<std::size_t>(n) * 9);
|
||||
|
||||
for (auto f : mesh.faces()) {
|
||||
Halfedge_index h0 = mesh.halfedge(f);
|
||||
Halfedge_index h1 = mesh.next(h0);
|
||||
Halfedge_index h2 = mesh.next(h1);
|
||||
|
||||
Vertex_index v1 = mesh.source(h0);
|
||||
Vertex_index v2 = mesh.source(h1);
|
||||
Vertex_index v3 = mesh.source(h2);
|
||||
|
||||
Edge_index e12 = mesh.edge(h0);
|
||||
Edge_index e23 = mesh.edge(h1);
|
||||
Edge_index e31 = mesh.edge(h2);
|
||||
|
||||
// Effective log-lengths.
|
||||
double u1 = spher_dof_val(m.v_idx[v1], x);
|
||||
double u2 = spher_dof_val(m.v_idx[v2], x);
|
||||
double u3 = spher_dof_val(m.v_idx[v3], x);
|
||||
|
||||
double lam12 = m.lambda0[e12] + u1 + u2 + spher_dof_val(m.e_idx[e12], x);
|
||||
double lam23 = m.lambda0[e23] + u2 + u3 + spher_dof_val(m.e_idx[e23], x);
|
||||
double lam31 = m.lambda0[e31] + u3 + u1 + spher_dof_val(m.e_idx[e31], x);
|
||||
|
||||
const double l12 = spherical_l(lam12);
|
||||
const double l23 = spherical_l(lam23);
|
||||
const double l31 = spherical_l(lam31);
|
||||
|
||||
SphericalFaceAngles fa = spherical_angles(l12, l23, l31);
|
||||
if (!fa.valid) continue;
|
||||
|
||||
const double sinl12 = std::sin(l12), cosl12 = std::cos(l12);
|
||||
const double sinl23 = std::sin(l23), cosl23 = std::cos(l23);
|
||||
const double sinl31 = std::sin(l31), cosl31 = std::cos(l31);
|
||||
|
||||
if (sinl12 < 1e-15 || sinl23 < 1e-15 || sinl31 < 1e-15) continue;
|
||||
|
||||
const double cot12 = cosl12 / sinl12;
|
||||
const double cot23 = cosl23 / sinl23;
|
||||
const double cot31 = cosl31 / sinl31;
|
||||
|
||||
// ∂l_ij/∂λ_ij = tan(l_ij/2)
|
||||
const double t12 = std::tan(l12 * 0.5);
|
||||
const double t23 = std::tan(l23 * 0.5);
|
||||
const double t31 = std::tan(l31 * 0.5);
|
||||
|
||||
const double sinA1 = std::sin(fa.alpha1), cosA1 = std::cos(fa.alpha1);
|
||||
const double sinA2 = std::sin(fa.alpha2), cosA2 = std::cos(fa.alpha2);
|
||||
const double sinA3 = std::sin(fa.alpha3), cosA3 = std::cos(fa.alpha3);
|
||||
|
||||
if (sinA1 < 1e-15 || sinA2 < 1e-15 || sinA3 < 1e-15) continue;
|
||||
|
||||
// ∂α1/∂l_jk (α1 at v1; opposite l23, adjacent l12,l31)
|
||||
const double dA1_dl12 = (cot12 * cosA1 - cot31) / sinA1;
|
||||
const double dA1_dl31 = (cot31 * cosA1 - cot12) / sinA1;
|
||||
const double dA1_dl23 = sinl23 / (sinl12 * sinl31 * sinA1);
|
||||
|
||||
// ∂α2/∂l_jk (α2 at v2; opposite l31, adjacent l12,l23)
|
||||
const double dA2_dl12 = (cot12 * cosA2 - cot23) / sinA2;
|
||||
const double dA2_dl23 = (cot23 * cosA2 - cot12) / sinA2;
|
||||
const double dA2_dl31 = sinl31 / (sinl12 * sinl23 * sinA2);
|
||||
|
||||
// ∂α3/∂l_jk (α3 at v3; opposite l12, adjacent l23,l31)
|
||||
const double dA3_dl23 = (cot23 * cosA3 - cot31) / sinA3;
|
||||
const double dA3_dl31 = (cot31 * cosA3 - cot23) / sinA3;
|
||||
const double dA3_dl12 = sinl12 / (sinl23 * sinl31 * sinA3);
|
||||
|
||||
// Chain rule: ∂α_i/∂u_j (u1 affects l12,l31; u2 affects l12,l23; u3 affects l23,l31)
|
||||
const double dA1_du1 = dA1_dl12 * t12 + dA1_dl31 * t31;
|
||||
const double dA1_du2 = dA1_dl12 * t12 + dA1_dl23 * t23;
|
||||
const double dA1_du3 = dA1_dl23 * t23 + dA1_dl31 * t31;
|
||||
|
||||
const double dA2_du1 = dA2_dl12 * t12 + dA2_dl31 * t31;
|
||||
const double dA2_du2 = dA2_dl12 * t12 + dA2_dl23 * t23;
|
||||
const double dA2_du3 = dA2_dl23 * t23 + dA2_dl31 * t31;
|
||||
|
||||
const double dA3_du1 = dA3_dl12 * t12 + dA3_dl31 * t31;
|
||||
const double dA3_du2 = dA3_dl12 * t12 + dA3_dl23 * t23;
|
||||
const double dA3_du3 = dA3_dl23 * t23 + dA3_dl31 * t31;
|
||||
|
||||
const int i1 = m.v_idx[v1];
|
||||
const int i2 = m.v_idx[v2];
|
||||
const int i3 = m.v_idx[v3];
|
||||
|
||||
// H[vi, vj] -= ∂α_i/∂u_j (G_v = θ_v − Σ α_v, so ∂G_i/∂u_j = −∂α_i/∂u_j)
|
||||
if (i1 >= 0) trips.emplace_back(i1, i1, -dA1_du1);
|
||||
if (i2 >= 0) trips.emplace_back(i2, i2, -dA2_du2);
|
||||
if (i3 >= 0) trips.emplace_back(i3, i3, -dA3_du3);
|
||||
|
||||
if (i1 >= 0 && i2 >= 0) {
|
||||
trips.emplace_back(i1, i2, -dA1_du2);
|
||||
trips.emplace_back(i2, i1, -dA2_du1);
|
||||
}
|
||||
if (i2 >= 0 && i3 >= 0) {
|
||||
trips.emplace_back(i2, i3, -dA2_du3);
|
||||
trips.emplace_back(i3, i2, -dA3_du2);
|
||||
}
|
||||
if (i3 >= 0 && i1 >= 0) {
|
||||
trips.emplace_back(i3, i1, -dA3_du1);
|
||||
trips.emplace_back(i1, i3, -dA1_du3);
|
||||
}
|
||||
}
|
||||
|
||||
Eigen::SparseMatrix<double> H(n, n);
|
||||
H.setFromTriplets(trips.begin(), trips.end());
|
||||
return H;
|
||||
}
|
||||
|
||||
// ── Finite-difference Hessian check ──────────────────────────────────────────
|
||||
//
|
||||
// Compares the analytical Hessian column-by-column against
|
||||
// H_fd[:, j] = (G(x + ε·eⱼ) − G(x − ε·eⱼ)) / (2ε).
|
||||
inline bool hessian_check_spherical(
|
||||
ConformalMesh& mesh,
|
||||
const std::vector<double>& x0,
|
||||
const SphericalMaps& m,
|
||||
double eps = 1e-5,
|
||||
double tol = 1e-4)
|
||||
{
|
||||
const int n = static_cast<int>(x0.size());
|
||||
auto H = spherical_hessian(mesh, x0, m);
|
||||
|
||||
std::vector<double> xp = x0, xm = x0;
|
||||
bool ok = true;
|
||||
|
||||
for (int j = 0; j < n; ++j) {
|
||||
const std::size_t sj = static_cast<std::size_t>(j);
|
||||
xp[sj] = x0[sj] + eps;
|
||||
xm[sj] = x0[sj] - eps;
|
||||
|
||||
auto Gp = spherical_gradient(mesh, xp, m);
|
||||
auto Gm = spherical_gradient(mesh, xm, m);
|
||||
|
||||
xp[sj] = xm[sj] = x0[sj];
|
||||
|
||||
for (int i = 0; i < n; ++i) {
|
||||
double fd_ij = (Gp[static_cast<std::size_t>(i)]
|
||||
- Gm[static_cast<std::size_t>(i)]) / (2.0 * eps);
|
||||
double H_ij = H.coeff(i, j);
|
||||
double err = std::abs(H_ij - fd_ij);
|
||||
double scale = std::max(1.0, std::abs(H_ij));
|
||||
if (err / scale > tol) ok = false;
|
||||
}
|
||||
}
|
||||
return ok;
|
||||
}
|
||||
|
||||
} // namespace conformallab
|
||||
@@ -20,6 +20,10 @@ add_executable(conformallab_cgal_tests
|
||||
|
||||
# ── Phase 3d: EuclideanCyclicFunctional ───────────────────────────────
|
||||
test_euclidean_functional.cpp
|
||||
|
||||
# ── Phase 3f: Hessians (cotangent Laplacian, spherical + Euclidean) ───
|
||||
test_euclidean_hessian.cpp
|
||||
test_spherical_hessian.cpp
|
||||
)
|
||||
|
||||
target_include_directories(conformallab_cgal_tests SYSTEM PRIVATE
|
||||
|
||||
213
code/tests/cgal/test_euclidean_hessian.cpp
Normal file
213
code/tests/cgal/test_euclidean_hessian.cpp
Normal file
@@ -0,0 +1,213 @@
|
||||
// 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";
|
||||
}
|
||||
205
code/tests/cgal/test_spherical_hessian.cpp
Normal file
205
code/tests/cgal/test_spherical_hessian.cpp
Normal file
@@ -0,0 +1,205 @@
|
||||
// test_spherical_hessian.cpp
|
||||
//
|
||||
// Phase 3f — Spherical cotangent-Laplace Hessian.
|
||||
//
|
||||
// The Hessian of the spherical discrete conformal energy is the "spherical
|
||||
// cotangent Laplacian" with edge weight
|
||||
// w_k = cot(β_k), β_k = (π − αi − αj + αk) / 2
|
||||
// for the edge (vi, vj) with opposite vertex vk and angles αi, αj, αk.
|
||||
//
|
||||
// In the flat limit (αi+αj+αk → π) this reduces to the Euclidean cotangent
|
||||
// Laplacian (β_k → αk, w_k → cot(αk)).
|
||||
//
|
||||
// Tests:
|
||||
// 1. Spherical cot weights match Euclidean weights in the near-flat limit.
|
||||
// 2. Hessian is symmetric.
|
||||
// 3. Hessian has H·1 ≈ 0 on the spherical tetrahedron (null-space property).
|
||||
// 4. Finite-difference check (the primary correctness criterion).
|
||||
|
||||
#include "conformal_mesh.hpp"
|
||||
#include "mesh_builder.hpp"
|
||||
#include "spherical_hessian.hpp"
|
||||
#include "euclidean_hessian.hpp" // for Euclidean comparison
|
||||
#include <gtest/gtest.h>
|
||||
#include <Eigen/Dense>
|
||||
#include <cmath>
|
||||
#include <vector>
|
||||
|
||||
using namespace conformallab;
|
||||
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
// Spherical cot weights reduce to Euclidean cot weights in the flat limit
|
||||
//
|
||||
// For a nearly-flat equilateral spherical triangle (l → 0, α → 60°):
|
||||
// β_k = (π − 60° − 60° + 60°)/2 = 60° → cot(60°) = 1/√3 ✓
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
|
||||
TEST(SphericalHessian, CotWeights_NearFlatEquilateral)
|
||||
{
|
||||
// Use a very small equilateral spherical triangle: α1=α2=α3=60°
|
||||
const double alpha = PI / 3.0;
|
||||
auto sw = spherical_cot_weights(alpha, alpha, alpha);
|
||||
|
||||
ASSERT_TRUE(sw.valid);
|
||||
|
||||
const double expected = 1.0 / std::sqrt(3.0); // = cot(60°)
|
||||
EXPECT_NEAR(sw.w12, expected, 1e-12);
|
||||
EXPECT_NEAR(sw.w23, expected, 1e-12);
|
||||
EXPECT_NEAR(sw.w31, expected, 1e-12);
|
||||
}
|
||||
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
// Spherical cot weights are positive for the regular spherical tetrahedron
|
||||
//
|
||||
// Each face has α_k = 2π/3 (120°), angle sum = 2π.
|
||||
// β_k = (π − 2π/3 − 2π/3 + 2π/3)/2 = (π − 2π/3)/2 = π/6
|
||||
// w_k = cot(π/6) = √3
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
|
||||
TEST(SphericalHessian, CotWeights_RegularSphericalTetrahedronFace)
|
||||
{
|
||||
const double alpha = 2.0 * PI / 3.0; // 120°
|
||||
auto sw = spherical_cot_weights(alpha, alpha, alpha);
|
||||
|
||||
ASSERT_TRUE(sw.valid);
|
||||
|
||||
const double expected = std::sqrt(3.0); // cot(π/6)
|
||||
EXPECT_NEAR(sw.w12, expected, 1e-10);
|
||||
EXPECT_NEAR(sw.w23, expected, 1e-10);
|
||||
EXPECT_NEAR(sw.w31, expected, 1e-10);
|
||||
}
|
||||
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
// Hessian is symmetric: H[i,j] == H[j,i]
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
|
||||
TEST(SphericalHessian, HessianIsSymmetric)
|
||||
{
|
||||
auto mesh = make_spherical_tetrahedron();
|
||||
auto maps = setup_spherical_maps(mesh);
|
||||
compute_lambda0_from_mesh(mesh, maps);
|
||||
int n = assign_vertex_dof_indices(mesh, maps);
|
||||
|
||||
std::vector<double> x(static_cast<std::size_t>(n), -0.2);
|
||||
auto H = spherical_hessian(mesh, x, maps);
|
||||
|
||||
Eigen::MatrixXd Hd = Eigen::MatrixXd(H);
|
||||
EXPECT_NEAR((Hd - Hd.transpose()).norm(), 0.0, 1e-12)
|
||||
<< "Spherical Hessian must be symmetric";
|
||||
}
|
||||
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
// H·1 is NOT zero for the spherical Hessian (unlike the Euclidean case).
|
||||
//
|
||||
// In the Euclidean case, a uniform shift u_i → u_i + c scales all edge
|
||||
// lengths by e^c, leaving angles unchanged → H·1 = 0 exactly.
|
||||
//
|
||||
// In the spherical case, l_ij = 2·asin(exp(λ_ij/2)) is NOT a linear
|
||||
// function of the DOFs, so a uniform shift DOES change the angles
|
||||
// → H·1 ≠ 0 in general. This test verifies that property.
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
|
||||
TEST(SphericalHessian, ConstantVectorNotInNullSpace)
|
||||
{
|
||||
auto mesh = make_spherical_tetrahedron();
|
||||
auto maps = setup_spherical_maps(mesh);
|
||||
compute_lambda0_from_mesh(mesh, maps);
|
||||
int n = assign_vertex_dof_indices(mesh, maps);
|
||||
|
||||
std::vector<double> x(static_cast<std::size_t>(n), -0.2);
|
||||
auto H = spherical_hessian(mesh, x, maps);
|
||||
|
||||
Eigen::VectorXd ones = Eigen::VectorXd::Ones(n);
|
||||
Eigen::VectorXd Hones = H * ones;
|
||||
|
||||
// H·1 should be non-trivial (norm well above zero)
|
||||
EXPECT_GT(Hones.norm(), 1e-3)
|
||||
<< "Spherical H·1 should be non-zero; norm = " << Hones.norm();
|
||||
}
|
||||
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
// Hessian is negative semi-definite at the equilibrium point (x = 0)
|
||||
//
|
||||
// The spherical discrete conformal energy is concave in the vertex DOFs
|
||||
// (unlike the Euclidean case which is convex). At the equilibrium x = 0,
|
||||
// the Hessian is NSD: all eigenvalues ≤ 0, with a multi-dimensional null
|
||||
// space corresponding to degenerate directions.
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
|
||||
TEST(SphericalHessian, HessianIsNegativeSemiDefiniteAtEquilibrium)
|
||||
{
|
||||
auto mesh = make_spherical_tetrahedron();
|
||||
auto maps = setup_spherical_maps(mesh);
|
||||
compute_lambda0_from_mesh(mesh, maps);
|
||||
int n = assign_vertex_dof_indices(mesh, maps);
|
||||
|
||||
// x = 0 is the equilibrium for the regular spherical tetrahedron.
|
||||
std::vector<double> x(static_cast<std::size_t>(n), 0.0);
|
||||
auto H = spherical_hessian(mesh, x, maps);
|
||||
|
||||
Eigen::MatrixXd Hd = Eigen::MatrixXd(H);
|
||||
Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> es(Hd);
|
||||
double max_ev = es.eigenvalues().maxCoeff();
|
||||
|
||||
EXPECT_LE(max_ev, 1e-9)
|
||||
<< "Largest eigenvalue of spherical Hessian at equilibrium must be ≤ 0; got " << max_ev;
|
||||
}
|
||||
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
// Finite-difference Hessian check: octahedron-face triangle (vertex DOFs)
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
|
||||
TEST(SphericalHessian, FDCheck_OctaFaceVertex)
|
||||
{
|
||||
auto mesh = make_octahedron_face();
|
||||
auto maps = setup_spherical_maps(mesh);
|
||||
compute_lambda0_from_mesh(mesh, maps);
|
||||
int n = assign_vertex_dof_indices(mesh, maps);
|
||||
|
||||
std::vector<double> x(static_cast<std::size_t>(n), -0.3);
|
||||
|
||||
EXPECT_TRUE(hessian_check_spherical(mesh, x, maps))
|
||||
<< "FD Hessian check failed on octahedron-face triangle";
|
||||
}
|
||||
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
// Finite-difference Hessian check: spherical tetrahedron (vertex DOFs)
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
|
||||
TEST(SphericalHessian, FDCheck_SpherTetVertex)
|
||||
{
|
||||
auto mesh = make_spherical_tetrahedron();
|
||||
auto maps = setup_spherical_maps(mesh);
|
||||
compute_lambda0_from_mesh(mesh, maps);
|
||||
int n = assign_vertex_dof_indices(mesh, maps);
|
||||
|
||||
std::vector<double> x(static_cast<std::size_t>(n), -0.2);
|
||||
|
||||
EXPECT_TRUE(hessian_check_spherical(mesh, x, maps))
|
||||
<< "FD Hessian check failed on spherical tetrahedron";
|
||||
}
|
||||
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
// Finite-difference Hessian check: mixed pinned/variable vertices
|
||||
// ════════════════════════════════════════════════════════════════════════════
|
||||
|
||||
TEST(SphericalHessian, FDCheck_MixedPinnedVertices)
|
||||
{
|
||||
auto mesh = make_octahedron_face();
|
||||
auto maps = setup_spherical_maps(mesh);
|
||||
compute_lambda0_from_mesh(mesh, maps);
|
||||
|
||||
auto vit = mesh.vertices().begin();
|
||||
Vertex_index v0 = *vit++;
|
||||
Vertex_index v1 = *vit++;
|
||||
Vertex_index v2 = *vit;
|
||||
|
||||
maps.v_idx[v0] = -1; // pinned
|
||||
maps.v_idx[v1] = 0;
|
||||
maps.v_idx[v2] = 1;
|
||||
|
||||
std::vector<double> x = {-0.2, -0.3};
|
||||
|
||||
EXPECT_TRUE(hessian_check_spherical(mesh, x, maps))
|
||||
<< "FD Hessian check failed for mixed pinned/variable vertices";
|
||||
}
|
||||
Reference in New Issue
Block a user