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/include/spherical_hessian.hpp
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code/include/spherical_hessian.hpp
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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
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// (the hessian() method, vertex DOFs).
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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 spherical face (v1, v2, v3) with vertex angles α1, α2, α3: │
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// │ │
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// │ Spherical cotangent weight for edge (vi, vj) with opposite vk: │
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// │ β_k = (π − αi − αj + αk) / 2 │
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// │ w_k = cot(β_k) = 1/tan(β_k) │
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// │ │
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// │ Euclidean limit: α1+α2+α3 → π, β_k → αk, w_k → cot(αk). ✓ │
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// │ │
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// │ Hessian contributions per face: │
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// │ H[vi, vi] += w_ij + w_ik (diagonal: weights of incident edges) │
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// │ H[vi, vj] -= w_ij (off-diagonal: weight of edge ij) │
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// │ │
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// │ where w_ij is the weight of the edge between vi and vj (opposite vk): │
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// │ w_ij = cot(β_k) with β_k = (π − αi − αj + αk) / 2. │
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// └──────────────────────────────────────────────────────────────────────────┘
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//
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// Requires Eigen (header-only). Returns Eigen::SparseMatrix<double>.
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#include "spherical_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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// ── Spherical cotangent weight helper ────────────────────────────────────────
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//
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// Given the three face angles α1, α2, α3 of a spherical triangle, return the
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// three edge cotangent weights w1 (edge opp v1), w2 (edge opp v2), w3 (edge opp v3).
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//
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// w_k = cot(β_k) where β_k = (π − α_adj1 − α_adj2 + α_opp) / 2
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// = (π − αi − αj + αk) / 2 for edge (vi,vj), opposite vk
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//
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// Mapping in our CGAL halfedge convention:
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// h0 = halfedge(f): edge v1-v2 → opposite v3 → w = cot(β3), β3=(π-α1-α2+α3)/2
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// h1: edge v2-v3 → opposite v1 → w = cot(β1), β1=(π-α2-α3+α1)/2
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// h2: edge v3-v1 → opposite v2 → w = cot(β2), β2=(π-α3-α1+α2)/2
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//
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// Returns valid=false if any β_k is out of range (degenerate face).
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struct SpherCotWeights { double w12, w23, w31; bool valid; };
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inline SpherCotWeights spherical_cot_weights(double alpha1, double alpha2, double alpha3)
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{
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// β for each edge:
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// β3 = (π - α1 - α2 + α3)/2 — weight for edge v1-v2 (opposite v3)
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// β1 = (π - α2 - α3 + α1)/2 — weight for edge v2-v3 (opposite v1)
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// β2 = (π - α3 - α1 + α2)/2 — weight for edge v3-v1 (opposite v2)
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const double beta3 = (PI - alpha1 - alpha2 + alpha3) * 0.5;
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const double beta1 = (PI - alpha2 - alpha3 + alpha1) * 0.5;
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const double beta2 = (PI - alpha3 - alpha1 + alpha2) * 0.5;
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// Each β_k must be in (0, π/2] for the weight to be positive and well-defined.
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// For degenerate or very flat triangles some β may be ≤ 0 or ≥ π/2.
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if (beta1 <= 0.0 || beta2 <= 0.0 || beta3 <= 0.0) return {0.0, 0.0, 0.0, false};
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const double tb1 = std::tan(beta1);
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const double tb2 = std::tan(beta2);
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const double tb3 = std::tan(beta3);
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if (std::abs(tb1) < 1e-15 || std::abs(tb2) < 1e-15 || std::abs(tb3) < 1e-15)
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return {0.0, 0.0, 0.0, false};
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// w_ij = cot(β_k) where β_k is for the edge opposite vk.
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// w12 is for edge v1-v2 (opposite v3): cot(β3)
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// w23 is for edge v2-v3 (opposite v1): cot(β1)
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// w31 is for edge v3-v1 (opposite v2): cot(β2)
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return {1.0 / tb3, 1.0 / tb1, 1.0 / tb2, true};
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}
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// ── Analytical Hessian ────────────────────────────────────────────────────────
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//
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// Returns the n×n sparse Hessian matrix H where n = spherical_dimension(mesh, m).
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// x – current DOF vector.
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//
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// Derivation: G_v = θ_v − Σ_f α_v^f → H[i,j] = −Σ_f ∂α_i^f/∂u_j
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//
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// For a face (v1,v2,v3) with arc-lengths l12,l23,l31 and angles α1,α2,α3,
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// differentiating the spherical law of cosines
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// cos(l_opp) = cos(l_a)cos(l_b) + sin(l_a)sin(l_b)cos(α)
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// gives:
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// ∂α1/∂l12 = [cot(l12)cos(α1) − cot(l31)] / sin(α1) (adjacent side)
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// ∂α1/∂l31 = [cot(l31)cos(α1) − cot(l12)] / sin(α1) (adjacent side)
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// ∂α1/∂l23 = sin(l23) / [sin(l12)sin(l31)sin(α1)] (opposite side)
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//
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// Chain rule with ∂l_ij/∂u_k = tan(l_ij/2) (from l = 2·asin(exp(λ/2))):
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// ∂α1/∂u1 = ∂α1/∂l12·t12 + ∂α1/∂l31·t31
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// ∂α1/∂u2 = ∂α1/∂l12·t12 + ∂α1/∂l23·t23
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// ∂α1/∂u3 = ∂α1/∂l23·t23 + ∂α1/∂l31·t31
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inline Eigen::SparseMatrix<double> spherical_hessian(
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ConformalMesh& mesh,
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const std::vector<double>& x,
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const SphericalMaps& m)
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{
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const int n = spherical_dimension(mesh, m);
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std::vector<Eigen::Triplet<double>> trips;
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trips.reserve(static_cast<std::size_t>(n) * 9);
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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 = spher_dof_val(m.v_idx[v1], x);
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double u2 = spher_dof_val(m.v_idx[v2], x);
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double u3 = spher_dof_val(m.v_idx[v3], x);
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double lam12 = m.lambda0[e12] + u1 + u2 + spher_dof_val(m.e_idx[e12], x);
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double lam23 = m.lambda0[e23] + u2 + u3 + spher_dof_val(m.e_idx[e23], x);
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double lam31 = m.lambda0[e31] + u3 + u1 + spher_dof_val(m.e_idx[e31], x);
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const double l12 = spherical_l(lam12);
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const double l23 = spherical_l(lam23);
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const double l31 = spherical_l(lam31);
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SphericalFaceAngles fa = spherical_angles(l12, l23, l31);
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if (!fa.valid) continue;
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const double sinl12 = std::sin(l12), cosl12 = std::cos(l12);
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const double sinl23 = std::sin(l23), cosl23 = std::cos(l23);
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const double sinl31 = std::sin(l31), cosl31 = std::cos(l31);
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if (sinl12 < 1e-15 || sinl23 < 1e-15 || sinl31 < 1e-15) continue;
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const double cot12 = cosl12 / sinl12;
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const double cot23 = cosl23 / sinl23;
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const double cot31 = cosl31 / sinl31;
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// ∂l_ij/∂λ_ij = tan(l_ij/2)
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const double t12 = std::tan(l12 * 0.5);
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const double t23 = std::tan(l23 * 0.5);
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const double t31 = std::tan(l31 * 0.5);
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const double sinA1 = std::sin(fa.alpha1), cosA1 = std::cos(fa.alpha1);
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const double sinA2 = std::sin(fa.alpha2), cosA2 = std::cos(fa.alpha2);
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const double sinA3 = std::sin(fa.alpha3), cosA3 = std::cos(fa.alpha3);
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if (sinA1 < 1e-15 || sinA2 < 1e-15 || sinA3 < 1e-15) continue;
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// ∂α1/∂l_jk (α1 at v1; opposite l23, adjacent l12,l31)
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const double dA1_dl12 = (cot12 * cosA1 - cot31) / sinA1;
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const double dA1_dl31 = (cot31 * cosA1 - cot12) / sinA1;
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const double dA1_dl23 = sinl23 / (sinl12 * sinl31 * sinA1);
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// ∂α2/∂l_jk (α2 at v2; opposite l31, adjacent l12,l23)
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const double dA2_dl12 = (cot12 * cosA2 - cot23) / sinA2;
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const double dA2_dl23 = (cot23 * cosA2 - cot12) / sinA2;
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const double dA2_dl31 = sinl31 / (sinl12 * sinl23 * sinA2);
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// ∂α3/∂l_jk (α3 at v3; opposite l12, adjacent l23,l31)
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const double dA3_dl23 = (cot23 * cosA3 - cot31) / sinA3;
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const double dA3_dl31 = (cot31 * cosA3 - cot23) / sinA3;
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const double dA3_dl12 = sinl12 / (sinl23 * sinl31 * sinA3);
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// Chain rule: ∂α_i/∂u_j (u1 affects l12,l31; u2 affects l12,l23; u3 affects l23,l31)
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const double dA1_du1 = dA1_dl12 * t12 + dA1_dl31 * t31;
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const double dA1_du2 = dA1_dl12 * t12 + dA1_dl23 * t23;
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const double dA1_du3 = dA1_dl23 * t23 + dA1_dl31 * t31;
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const double dA2_du1 = dA2_dl12 * t12 + dA2_dl31 * t31;
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const double dA2_du2 = dA2_dl12 * t12 + dA2_dl23 * t23;
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const double dA2_du3 = dA2_dl23 * t23 + dA2_dl31 * t31;
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const double dA3_du1 = dA3_dl12 * t12 + dA3_dl31 * t31;
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const double dA3_du2 = dA3_dl12 * t12 + dA3_dl23 * t23;
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const double dA3_du3 = dA3_dl23 * t23 + dA3_dl31 * t31;
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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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// H[vi, vj] -= ∂α_i/∂u_j (G_v = θ_v − Σ α_v, so ∂G_i/∂u_j = −∂α_i/∂u_j)
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if (i1 >= 0) trips.emplace_back(i1, i1, -dA1_du1);
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if (i2 >= 0) trips.emplace_back(i2, i2, -dA2_du2);
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if (i3 >= 0) trips.emplace_back(i3, i3, -dA3_du3);
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if (i1 >= 0 && i2 >= 0) {
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trips.emplace_back(i1, i2, -dA1_du2);
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trips.emplace_back(i2, i1, -dA2_du1);
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}
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if (i2 >= 0 && i3 >= 0) {
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trips.emplace_back(i2, i3, -dA2_du3);
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trips.emplace_back(i3, i2, -dA3_du2);
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}
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if (i3 >= 0 && i1 >= 0) {
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trips.emplace_back(i3, i1, -dA3_du1);
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trips.emplace_back(i1, i3, -dA1_du3);
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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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inline bool hessian_check_spherical(
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ConformalMesh& mesh,
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const std::vector<double>& x0,
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const SphericalMaps& 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 = spherical_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 = spherical_gradient(mesh, xp, m);
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auto Gm = spherical_gradient(mesh, xm, m);
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xp[sj] = xm[sj] = x0[sj];
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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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