8c353bb884
Phase 3d — EuclideanCyclicFunctional:
• euclidean_geometry.hpp: t-value / atan2 corner-angle formula with
centering trick (μ = (Λ̃₁₂+Λ̃₂₃+Λ̃₃₁)/6) for numerical stability
• euclidean_functional.hpp: EuclideanMaps bundle, gradient (G_v = Θ_v − Σα_v,
G_e = α_opp⁺ + α_opp⁻ − φ_e), 10-point GL path-integral energy,
gradient_check_euclidean — identical halfedge convention to SphericalFunctional
• test_euclidean_functional.cpp: 11 tests (1 skip) covering angle formula,
right-isosceles triangle, angle sum = π, degenerate detection, gradient
checks on triangle/quad-strip/tetrahedron/fan-5/mixed-pinned, NaN check
Phase 3e — Spherical gauge-fix:
• spherical_gauge_shift(): Newton + backtracking line search to find t*
where ΣG_v(x + t·1) = 0 (maximises E along the global scale direction);
bisection used when sign change is detectable, Newton+backtrack otherwise
• apply_spherical_gauge(): in-place wrapper
• 3 new tests: GaugeFix_SpherTetVertexZerosSumGv, GaugeFix_ApplyInPlace,
GaugeFix_AlreadyAtGaugeReturnsTNearZero
Total: 45 cgal tests pass, 3 skipped (@Ignore Hessian stubs, one per functional)
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
476 lines
20 KiB
C++
476 lines
20 KiB
C++
#pragma once
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// spherical_functional.hpp
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//
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// Energy and gradient of the spherical discrete conformal functional
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// evaluated on a ConformalMesh (CGAL::Surface_mesh).
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//
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// Ported from de.varylab.discreteconformal.functional.SphericalFunctional.
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//
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// ┌──────────────────────────────────────────────────────────────────────────┐
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// │ DOFs │
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// │ x[v_idx[v]] = u_v – conformal factor at vertex v │
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// │ x[e_idx[e]] = λ_e – edge log-length variable (optional) │
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// │ -1 means "pinned" (u_v = 0 / λ_e = λ°_e fixed) │
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// │ │
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// │ Effective log-length: Λ_ij = λ°_ij + u_i + u_j │
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// │ Spherical arc length: l_ij = 2·asin(min(exp(Λ_ij/2), 1)) │
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// │ │
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// │ Gradient: │
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// │ ∂E/∂u_v = Θ_v – Σ_{faces adj. v} α_v(face) │
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// │ ∂E/∂λ_e = α_opp(face⁺) + α_opp(face⁻) – π │
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// │ │
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// │ Energy: │
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// │ Computed as the Schläfli path integral │
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// │ E(x) = ∫₀¹ ⟨G(tx), x⟩ dt │
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// │ using 10-point Gauss-Legendre quadrature. This is mathematically │
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// │ exact for any conservative (curl-free) gradient G and is numerically │
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// │ accurate to ~10⁻¹⁰ for smooth angle functions. The gradient check │
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// │ therefore tests curl-freeness of G, which is the key integrability │
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// │ condition for the spherical discrete conformal functional. │
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// └──────────────────────────────────────────────────────────────────────────┘
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#include "conformal_mesh.hpp"
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#include "spherical_geometry.hpp"
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#include <CGAL/boost/graph/iterator.h>
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#include <vector>
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#include <cmath>
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#include <cstdint>
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namespace conformallab {
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// ── Property-map type aliases ─────────────────────────────────────────────────
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using SpherVMapD = ConformalMesh::Property_map<Vertex_index, double>;
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using SpherVMapI = ConformalMesh::Property_map<Vertex_index, int>;
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using SpherEMapD = ConformalMesh::Property_map<Edge_index, double>;
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using SpherEMapI = ConformalMesh::Property_map<Edge_index, int>;
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// ── Persistent map bundle ─────────────────────────────────────────────────────
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struct SphericalMaps {
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SpherVMapI v_idx; // DOF index per vertex (-1 = pinned / u_v = 0)
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SpherEMapI e_idx; // DOF index per edge (-1 = no edge DOF)
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SpherVMapD theta_v; // target cone angle Θ_v (default 2π)
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SpherEMapD theta_e; // target edge angle θ_e (default π)
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SpherEMapD lambda0; // base log-length λ°_e (default 0.0)
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};
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// Defaults: theta_v = 2π, theta_e = π, lambda0 = 0.
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// lambda0 = 0 means exp(λ°/2)=1, i.e., l=π — degenerate unless u_i<0.
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// For real meshes, set lambda0 from mesh geometry via
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// compute_lambda0_from_mesh() below.
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inline SphericalMaps setup_spherical_maps(ConformalMesh& mesh)
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{
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SphericalMaps m;
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m.v_idx = mesh.add_property_map<Vertex_index, int> ("sv:idx", -1 ).first;
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m.e_idx = mesh.add_property_map<Edge_index, int> ("se:idx", -1 ).first;
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m.theta_v= mesh.add_property_map<Vertex_index, double>("sv:theta", 2.0*PI_SPHER).first;
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m.theta_e= mesh.add_property_map<Edge_index, double>("se:theta", PI_SPHER ).first;
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m.lambda0= mesh.add_property_map<Edge_index, double>("se:lam0", 0.0 ).first;
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return m;
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}
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// Assign DOF indices 0..n-1 for all vertices (only vertex DOFs).
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inline int assign_vertex_dof_indices(ConformalMesh& mesh, SphericalMaps& m)
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{
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int idx = 0;
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for (auto v : mesh.vertices()) m.v_idx[v] = idx++;
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return idx;
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}
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// Assign DOF indices for all vertices AND edges.
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inline int assign_all_spherical_dof_indices(ConformalMesh& mesh, SphericalMaps& m)
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{
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int idx = 0;
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for (auto v : mesh.vertices()) m.v_idx[v] = idx++;
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for (auto e : mesh.edges()) m.e_idx[e] = idx++;
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return idx;
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}
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// Count variable DOFs.
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inline int spherical_dimension(const ConformalMesh& mesh, const SphericalMaps& m)
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{
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int dim = 0;
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for (auto v : mesh.vertices()) if (m.v_idx[v] >= 0) ++dim;
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for (auto e : mesh.edges()) if (m.e_idx[e] >= 0) ++dim;
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return dim;
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}
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// Set lambda0 from mesh vertex positions (unit-sphere assumed):
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// λ°_e = 2·log(sin(l_e / 2)) where l_e = arccos(p_i · p_j).
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// Requires vertices to lie on the unit sphere.
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inline void compute_lambda0_from_mesh(ConformalMesh& mesh, SphericalMaps& m)
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{
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for (auto e : mesh.edges()) {
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auto h = mesh.halfedge(e);
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auto p1 = mesh.point(mesh.source(h));
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auto p2 = mesh.point(mesh.target(h));
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// Dot product (works for unit-sphere vertices).
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double dot = p1.x()*p2.x() + p1.y()*p2.y() + p1.z()*p2.z();
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dot = std::max(-1.0, std::min(1.0, dot));
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double l_e = std::acos(dot); // spherical arc length
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double half_sin = std::sin(l_e * 0.5); // = exp(λ°/2)
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if (half_sin > 1e-15)
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m.lambda0[e] = 2.0 * std::log(half_sin);
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else
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m.lambda0[e] = -30.0; // very short edge: essentially 0
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}
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}
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// ── Evaluation result ─────────────────────────────────────────────────────────
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struct SphericalResult {
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double energy = 0.0;
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std::vector<double> gradient;
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};
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// ── Internal helpers ──────────────────────────────────────────────────────────
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static inline double spher_dof_val(int idx, const std::vector<double>& x)
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{
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return idx >= 0 ? x[static_cast<std::size_t>(idx)] : 0.0;
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}
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static inline std::size_t spher_hidx(Halfedge_index h)
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{
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return static_cast<std::size_t>(static_cast<std::uint32_t>(h));
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}
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// ── Gradient only (no energy) ─────────────────────────────────────────────────
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// Compute gradient G(x).
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// G_v = Θ_v − Σ_faces α_v(face)
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// G_e = α_opp(face+) + α_opp(face−) − θ_e
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//
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// The corner angle α_v is stored on halfedges using the convention:
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// h_alpha[h] = corner angle at source(prev(h)) = corner angle at the vertex
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// ACROSS FROM the edge of halfedge h in its face.
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// This convention makes both the vertex and edge gradient accumulators natural.
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inline std::vector<double> spherical_gradient(
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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<double> G(static_cast<std::size_t>(n), 0.0);
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// Temporary per-halfedge corner-angle storage.
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// h_alpha[h] = corner angle at the vertex opposite to the edge of h.
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const std::size_t nh = mesh.number_of_halfedges();
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std::vector<double> h_alpha(nh, 0.0);
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// ── Pass 1: compute corner angles per face ────────────────────────────────
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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-length Λ_ij = λ°_ij + u_i + u_j
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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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double l12 = spherical_l(lam12);
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double l23 = spherical_l(lam23);
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double l31 = spherical_l(lam31);
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SphericalFaceAngles fa = spherical_angles(l12, l23, l31);
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if (!fa.valid) continue; // degenerate face: contributes 0
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// Store convention: h_alpha[h] = corner angle at source(prev(h))
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// h0 (e12): opposite vertex is v3 → source(prev(h0)) = source(h2) = v3 → α3
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// h1 (e23): opposite vertex is v1 → source(prev(h1)) = source(h0) = v1 → α1
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// h2 (e31): opposite vertex is v2 → source(prev(h2)) = source(h1) = v2 → α2
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h_alpha[spher_hidx(h0)] = fa.alpha3;
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h_alpha[spher_hidx(h1)] = fa.alpha1;
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h_alpha[spher_hidx(h2)] = fa.alpha2;
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}
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// ── Pass 2: accumulate gradient ───────────────────────────────────────────
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// Vertex: G_v = Θ_v − Σ h_alpha[prev(h)] for each incoming non-border h to v.
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for (auto v : mesh.vertices()) {
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int iv = m.v_idx[v];
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if (iv < 0) continue;
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double sum_alpha = 0.0;
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for (auto h : CGAL::halfedges_around_target(v, mesh)) {
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if (mesh.is_border(h)) continue;
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sum_alpha += h_alpha[spher_hidx(mesh.prev(h))];
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}
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G[static_cast<std::size_t>(iv)] = m.theta_v[v] - sum_alpha;
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}
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// Edge: G_e for λ_e additive (Λ_ij = λ°_ij + u_i + u_j + λ_e).
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//
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// From the Schläfli identity applied to the spherical face,
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// the contribution of edge DOF λ_e from face f is:
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// a_f = (2·α_opp − S_f) / 2 where S_f = Σ angles in face f.
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//
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// Summing over both adjacent faces:
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// G_e = a_f+ + a_f−
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// = α_opp⁺ + α_opp⁻ − (S_f⁺ + S_f⁻) / 2 − θ_e
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//
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// For flat (Euclidean) triangles S_f = π, recovering the familiar
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// α_opp⁺ + α_opp⁻ − π formula. For spherical triangles S_f > π.
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for (auto e : mesh.edges()) {
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int ie = m.e_idx[e];
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if (ie < 0) continue;
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auto h = mesh.halfedge(e);
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auto ho = mesh.opposite(h);
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double sum = 0.0;
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if (!mesh.is_border(h)) {
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double alpha_opp = h_alpha[spher_hidx(h)];
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double S_f = alpha_opp
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+ h_alpha[spher_hidx(mesh.next(h))]
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+ h_alpha[spher_hidx(mesh.prev(h))];
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sum += (2.0 * alpha_opp - S_f) * 0.5;
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}
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if (!mesh.is_border(ho)) {
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double alpha_opp = h_alpha[spher_hidx(ho)];
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double S_f = alpha_opp
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+ h_alpha[spher_hidx(mesh.next(ho))]
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+ h_alpha[spher_hidx(mesh.prev(ho))];
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sum += (2.0 * alpha_opp - S_f) * 0.5;
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}
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G[static_cast<std::size_t>(ie)] = sum - m.theta_e[e];
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}
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return G;
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}
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// ── Energy via Gauss-Legendre path integral ───────────────────────────────────
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//
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// E(x) = ∫₀¹ ⟨G(tx), x⟩ dt
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//
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// This is the correct potential for any conservative G = ∇E.
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// Uses 10-point Gauss-Legendre quadrature; error ≈ O(h²⁰) for smooth G.
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//
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// 10-point GL nodes and weights on [0, 1] (transformed from [-1, 1]):
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// t_k = (1 + s_k) / 2, w_k = w_GL_k / 2
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inline double spherical_energy(
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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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// 10-point Gauss-Legendre nodes and weights on [-1, 1].
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static const double gl_s[10] = {
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-0.9739065285171717, -0.8650633666889845,
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-0.6794095682990244, -0.4333953941292472,
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-0.1488743389816312, 0.1488743389816312,
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0.4333953941292472, 0.6794095682990244,
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0.8650633666889845, 0.9739065285171717
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};
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static const double gl_w[10] = {
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0.0666713443086881, 0.1494513491505806,
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0.2190863625159820, 0.2692667193099963,
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0.2955242247147529, 0.2955242247147529,
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0.2692667193099963, 0.2190863625159820,
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0.1494513491505806, 0.0666713443086881
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};
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const std::size_t n = x.size();
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double E = 0.0;
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for (int k = 0; k < 10; ++k) {
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double t = (1.0 + gl_s[k]) * 0.5; // node on [0, 1]
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double wt = gl_w[k] * 0.5; // weight on [0, 1]
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// Evaluate G(t·x)
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std::vector<double> tx(n);
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for (std::size_t i = 0; i < n; ++i) tx[i] = t * x[i];
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auto G = spherical_gradient(mesh, tx, m);
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// Accumulate ⟨G(tx), x⟩ · wt
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double dot = 0.0;
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for (std::size_t i = 0; i < n; ++i) dot += G[i] * x[i];
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E += wt * dot;
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}
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return E;
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}
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// ── Full evaluation (energy + gradient) ──────────────────────────────────────
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inline SphericalResult evaluate_spherical(
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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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bool need_energy = true,
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bool need_gradient = true)
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{
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SphericalResult res;
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if (need_gradient)
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res.gradient = spherical_gradient(mesh, x, m);
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if (need_energy)
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res.energy = spherical_energy(mesh, x, m);
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return res;
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}
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// ── Finite-difference gradient check ─────────────────────────────────────────
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//
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// Tests |G[i] − fd[i]| / max(1, |G[i]|) < tol for all DOFs.
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// Same defaults as the hyper-ideal gradient check (Java FunctionalTest).
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inline bool gradient_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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auto G = spherical_gradient(mesh, x0, m);
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const int n = static_cast<int>(G.size());
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std::vector<double> xp = x0, xm = x0;
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bool ok = true;
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for (int i = 0; i < n; ++i) {
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std::size_t si = static_cast<std::size_t>(i);
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xp[si] = x0[si] + eps;
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xm[si] = x0[si] - eps;
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double Ep = spherical_energy(mesh, xp, m);
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double Em = spherical_energy(mesh, xm, m);
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xp[si] = xm[si] = x0[si]; // restore
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double fd = (Ep - Em) / (2.0 * eps);
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double err = std::abs(G[si] - fd);
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double scale = std::max(1.0, std::abs(G[si]));
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if (err / scale > tol) ok = false;
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}
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return ok;
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}
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// ── Gauge-fix for closed spherical surfaces ───────────────────────────────────
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//
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// On a closed (boundaryless) spherical surface the energy E(u + t·1) has a
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// unique maximum w.r.t. t ∈ ℝ (the "global scale" gauge mode). Without
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// fixing this gauge the functional is unbounded below and no solver converges.
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//
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// This function returns the scalar shift t* such that
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// Σ_v G_v(x + t*·1_v) = 0
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// i.e., the derivative of E(u+t·1) w.r.t. t is zero at t = t*.
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//
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// Apply the shift by adding t* to every vertex DOF in x.
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//
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// Implementation: bisection on f(t) = Σ_v G_v(x + t·1_v).
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// f is strictly monotone decreasing (second derivative < 0) for a convex
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// functional, so bisection converges in O(log₂(2·bracket/tol)) iterations.
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//
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// Parameters:
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// bracket – initial search interval [−bracket, +bracket] (default 50)
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// tol – absolute tolerance on t* (default 1e-8)
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//
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// Returns 0.0 if the zero cannot be bracketed (already at gauge maximum,
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// or open surface — no shift needed).
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inline double spherical_gauge_shift(
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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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double bracket = 50.0,
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double tol = 1e-8)
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{
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// Helper: sum of all vertex gradient components at x + t·1_v.
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auto sum_Gv = [&](double t) -> double {
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// Build a shifted copy of x (only vertex DOFs shifted).
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std::vector<double> xt = x;
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for (auto v : mesh.vertices()) {
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int iv = m.v_idx[v];
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if (iv >= 0)
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xt[static_cast<std::size_t>(iv)] += t;
|
||
}
|
||
auto G = spherical_gradient(mesh, xt, m);
|
||
double sum = 0.0;
|
||
for (auto v : mesh.vertices()) {
|
||
int iv = m.v_idx[v];
|
||
if (iv >= 0)
|
||
sum += G[static_cast<std::size_t>(iv)];
|
||
}
|
||
return sum;
|
||
};
|
||
|
||
// ── Try bisection first (works when there is a sign change in [−bracket, +bracket]) ──
|
||
double a = -bracket, b = bracket;
|
||
double fa = sum_Gv(a), fb = sum_Gv(b);
|
||
|
||
if (fa * fb <= 0.0) {
|
||
for (int iter = 0; iter < 120; ++iter) {
|
||
double c = 0.5 * (a + b);
|
||
double fc = sum_Gv(c);
|
||
if (std::abs(fc) < tol || (b - a) < tol) return c;
|
||
if (fa * fc < 0.0) { b = c; fb = fc; }
|
||
else { a = c; fa = fc; }
|
||
}
|
||
return 0.5 * (a + b);
|
||
}
|
||
|
||
// ── No sign change (zero may lie at a domain boundary). ───────────────────
|
||
// Use damped Newton's method with backtracking line search.
|
||
// f'(t) estimated by forward finite difference.
|
||
// When the Newton step overshoots the valid domain (ΣG_v jumps back up
|
||
// because faces become degenerate), backtracking halves the step until
|
||
// |f| strictly decreases.
|
||
const double fd_eps = 1e-5;
|
||
double t = 0.0;
|
||
double ft = sum_Gv(t);
|
||
|
||
for (int iter = 0; iter < 120; ++iter) {
|
||
if (std::abs(ft) < tol) return t;
|
||
|
||
double ftp = sum_Gv(t + fd_eps);
|
||
double dft = (ftp - ft) / fd_eps;
|
||
if (std::abs(dft) < 1e-14) return t; // gradient flat — give up
|
||
|
||
double dt_raw = -ft / dft;
|
||
|
||
// Backtracking line search: halve dt until |f| decreases.
|
||
double alpha = 1.0;
|
||
bool improved = false;
|
||
for (int back = 0; back < 40; ++back) {
|
||
double t_try = t + alpha * dt_raw;
|
||
t_try = std::max(-bracket, std::min(bracket, t_try));
|
||
double ft_try = sum_Gv(t_try);
|
||
if (std::abs(ft_try) < std::abs(ft)) {
|
||
t = t_try;
|
||
ft = ft_try;
|
||
improved = true;
|
||
break;
|
||
}
|
||
alpha *= 0.5;
|
||
}
|
||
if (!improved) return t; // cannot reduce further
|
||
}
|
||
return t;
|
||
}
|
||
|
||
// Apply the gauge shift in-place: x_v ← x_v + t* for all variable vertices.
|
||
inline void apply_spherical_gauge(
|
||
ConformalMesh& mesh,
|
||
std::vector<double>& x,
|
||
const SphericalMaps& m,
|
||
double bracket = 50.0,
|
||
double tol = 1e-8)
|
||
{
|
||
double t = spherical_gauge_shift(mesh, x, m, bracket, tol);
|
||
for (auto v : mesh.vertices()) {
|
||
int iv = m.v_idx[v];
|
||
if (iv >= 0)
|
||
x[static_cast<std::size_t>(iv)] += t;
|
||
}
|
||
}
|
||
|
||
} // namespace conformallab
|