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Replaces the O(n·F) full-FD Hessian with an O(F·36) block-local variant
that exploits the per-face locality of the hyper-ideal functional. Both
variants are kept (full-FD as correctness reference, block-FD as default)
and proven to match to FD rounding tolerance on all test configurations.
Java parity note
────────────────
HyperIdealFunctional.java line 295-298 declares:
public boolean hasHessian() { return false; }
i.e. the upstream Java functional has NO Hessian implementation, analytic
or numerical. Both Hessian variants in this file are conformallab++
extensions beyond the Java port. Analytic Hessian via Schläfli-type
differentiation through (b_i, a_e) → l_ij → ζ_13/ζ_14/ζ_15 → α_ij/β_i
is deferred to a future PR.
Implementation
──────────────
* code/include/hyper_ideal_functional.hpp
- New pure-math helper face_angles_from_local_dofs() takes 6 input DOFs
(b1, b2, b3, a12, a23, a31) + variability flags and returns the 6
output angles (β1, β2, β3, α12, α23, α31).
- Used by block-FD Hessian as the inner loop; identical semantics to
the existing compute_face_angles().
* code/include/hyper_ideal_hessian.hpp
- hyper_ideal_hessian_block_fd() — new, default production path
- hyper_ideal_hessian_block_fd_sym() — symmetrised variant
- hyper_ideal_hessian() — full-FD baseline, kept for cross-validation
- hyper_ideal_hessian_sym() — symmetrised baseline
- Header docblock documents speed-up curve: ~33× at cathead.obj scale,
~1166× at brezel.obj scale.
Tests (7 new in test_hyper_ideal_hessian.cpp)
─────────────────────────────────────────────
* PureHelperMatchesMeshHelper — refactor sanity
* BlockFD_MatchesFullFD_ClosedTetrahedron
* BlockFD_MatchesFullFD_Open3FaceMesh (boundary edge path)
* BlockFD_MatchesFullFD_PinnedDOFs (partial-DOF path)
* BlockFD_IsPSD (Springborn 2020 convexity)
* BlockFD_SparsityMatchesFaceAdjacency (structural correctness)
* BlockFD_FasterThanFullFD (performance assertion: ≥ 3×)
Measured speed-up on the 200-face tet strip (603 DOFs):
full-FD: 226 591 µs
block-FD: 2 347 µs
ratio: 96.5×
The assertion uses ≥ 3× to leave wide CI-hardware tolerance.
Test count
──────────
CGAL suite: 184 → 191 (+7). Zero skips.
Why not full analytic now
─────────────────────────
Full analytic Hessian via the chain rule
(b_i, a_e) → l_ij → ζ_{13,14,15} → α_ij / β_i
requires Schläfli-type differentiation with multiple cases for the
ideal / hyper-ideal vertex mix. It would add another ~6× over
block-FD but at significantly higher implementation and verification
cost. Block-FD already removes the practical bottleneck for meshes
up to ~10k faces; analytic optimisation can land later when justified
by a concrete profiling result.
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
415 lines
18 KiB
C++
415 lines
18 KiB
C++
#pragma once
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// hyper_ideal_functional.hpp
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//
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// Energy and gradient of the hyper-ideal discrete conformal map 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.HyperIdealFunctional.
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//
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// ┌─────────────────────────────────────────────────────────────────────────┐
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// │ E(x) = Σ_faces U(f) − Σ_edges θ_e · a_e − Σ_vertices Θ_v · b_v │
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// │ │
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// │ ∂E/∂b_v = Σ_{faces adj. v} β_v(face) − Θ_v │
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// │ ∂E/∂a_e = α_e(face⁺) + α_e(face⁻) − θ_e │
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// └─────────────────────────────────────────────────────────────────────────┘
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//
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// DOF vector layout (matches getDimension() ordering):
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// x[v_idx[v]] = b_v for each variable vertex (log scale factor)
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// x[e_idx[e]] = a_e for each variable edge (intersection angle)
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// -1 in v_idx / e_idx means "pinned" (ideal / fixed at 0).
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//
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// Usage
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// ─────
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// auto mesh = make_tetrahedron();
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// auto maps = setup_hyper_ideal_maps(mesh);
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// int n = assign_all_dof_indices(mesh, maps); // all variable
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// std::vector<double> x(n, 1.0);
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// auto res = evaluate_hyper_ideal(mesh, x, maps);
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// // res.energy, res.gradient
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#include "conformal_mesh.hpp"
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#include "hyper_ideal_geometry.hpp"
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#include "hyper_ideal_utility.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 VMapD = ConformalMesh::Property_map<Vertex_index, double>;
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using VMapI = ConformalMesh::Property_map<Vertex_index, int>;
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using EMapD = ConformalMesh::Property_map<Edge_index, double>;
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using EMapI = ConformalMesh::Property_map<Edge_index, int>;
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// ── Persistent map bundle ─────────────────────────────────────────────────────
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struct HyperIdealMaps {
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VMapI v_idx; // DOF index per vertex (-1 = pinned / ideal point)
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EMapI e_idx; // DOF index per edge (-1 = fixed)
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VMapD theta_v; // target cone angle Θ_v (parameter, not variable)
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EMapD theta_e; // target intersection angle θ_e
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};
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// Add all needed persistent property maps and return handles.
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// Defaults: theta_v = 2π (regular cone), theta_e = π (orthogonal circles).
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inline HyperIdealMaps setup_hyper_ideal_maps(ConformalMesh& mesh)
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{
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HyperIdealMaps m;
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m.v_idx = mesh.add_property_map<Vertex_index, int> ("v:idx", -1 ).first;
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m.e_idx = mesh.add_property_map<Edge_index, int> ("e:idx", -1 ).first;
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m.theta_v = mesh.add_property_map<Vertex_index, double>("v:theta", 2.0*PI ).first;
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m.theta_e = mesh.add_property_map<Edge_index, double>("e:theta", PI ).first;
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return m;
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}
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// Count variable DOFs: #variable_vertices + #variable_edges.
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inline int hyper_ideal_dimension(const ConformalMesh& mesh, const HyperIdealMaps& 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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// Assign DOF indices 0..n-1: vertices first, then edges.
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// All vertices and edges become variable. Returns total DOF count.
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inline int assign_all_dof_indices(ConformalMesh& mesh, HyperIdealMaps& 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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// ── Evaluation result ─────────────────────────────────────────────────────────
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struct HyperIdealResult {
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double energy = 0.0;
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std::vector<double> gradient; // empty when gradient was not requested
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};
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// ── Internal helpers ──────────────────────────────────────────────────────────
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// Get the DOF value from x, or 0.0 if pinned.
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static inline double 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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// Convert a CGAL halfedge index to a plain std::size_t (for vector indexing).
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static inline std::size_t 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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// ── Pure-math face-angle kernel ──────────────────────────────────────────────
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//
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// Computes the six per-face angle outputs (β₁, β₂, β₃, α₁₂, α₂₃, α₃₁) from
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// the six local DOF inputs (b₁, b₂, b₃, a₁₂, a₂₃, a₃₁) and the variability
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// flags (vᵢb). This is the pure functional core of `compute_face_angles`
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// — no mesh, no property maps, no global x vector.
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//
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// Why exposed as a free function (Phase 9b):
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// ─────────────────────────────────────────
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// The block-FD Hessian (`hyper_ideal_hessian_block_fd`) perturbs only the
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// 6 DOFs adjacent to a single face at a time, recomputes the 6 angle
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// outputs of that face, and uses the local 6×6 Jacobian to scatter into
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// the global Hessian. Working through a pure 6→6 function (instead of
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// perturbing the full x and re-running the gradient over all faces)
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// reduces the cost of the Hessian from O(F·n) to O(F·36).
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//
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// The clamping logic (negative b → 0.01, negative a → 0) mirrors
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// HyperIdealFunctional.java's defensive behaviour (lines 122-127 of the
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// Java original); this keeps the FD perturbation regime well-defined.
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struct FaceAngleOutputs {
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double beta1, beta2, beta3; ///< interior angles at v₁,v₂,v₃
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double alpha12, alpha23, alpha31; ///< dihedral angles at e₁₂,e₂₃,e₃₁
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};
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inline FaceAngleOutputs face_angles_from_local_dofs(
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double b1, double b2, double b3,
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double a12, double a23, double a31,
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bool v1b, bool v2b, bool v3b)
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{
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// Same defensive clamps as compute_face_angles.
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if (v1b && v2b && a12 < 0.0) a12 = 0.0;
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if (v2b && v3b && a23 < 0.0) a23 = 0.0;
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if (v3b && v1b && a31 < 0.0) a31 = 0.0;
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if (v1b && b1 < 0.0) b1 = 0.01;
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if (v2b && b2 < 0.0) b2 = 0.01;
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if (v3b && b3 < 0.0) b3 = 0.01;
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double l12 = lij(b1, b2, a12, v1b, v2b);
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double l23 = lij(b2, b3, a23, v2b, v3b);
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double l31 = lij(b3, b1, a31, v3b, v1b);
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if (l12 < 1E-12 && l23 < 1E-12 && l31 < 1E-12)
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l12 = l23 = l31 = 1E-12;
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FaceAngleOutputs o;
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if (l12 > l23 + l31) {
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o.beta1 = 0.0; o.beta2 = 0.0; o.beta3 = PI;
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o.alpha12 = PI; o.alpha23 = 0.0; o.alpha31 = 0.0;
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} else if (l23 > l12 + l31) {
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o.beta1 = PI; o.beta2 = 0.0; o.beta3 = 0.0;
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o.alpha12 = 0.0; o.alpha23 = PI; o.alpha31 = 0.0;
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} else if (l31 > l12 + l23) {
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o.beta1 = 0.0; o.beta2 = PI; o.beta3 = 0.0;
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o.alpha12 = 0.0; o.alpha23 = 0.0; o.alpha31 = PI;
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} else {
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o.beta1 = zeta(l12, l31, l23);
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o.beta2 = zeta(l23, l12, l31);
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o.beta3 = zeta(l31, l23, l12);
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o.alpha12 = alpha_ij(a12, a23, a31, b1, b2, b3,
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o.beta1, o.beta2, o.beta3, v1b, v2b, v3b);
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o.alpha23 = alpha_ij(a23, a31, a12, b2, b3, b1,
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o.beta2, o.beta3, o.beta1, v2b, v3b, v1b);
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o.alpha31 = alpha_ij(a31, a12, a23, b3, b1, b2,
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o.beta3, o.beta1, o.beta2, v3b, v1b, v2b);
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}
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return o;
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}
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// ── Per-face angle kernel ─────────────────────────────────────────────────────
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struct FaceAngles {
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double alpha12, alpha23, alpha31; // dihedral angles at each edge
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double beta1, beta2, beta3; // interior angles at each vertex
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double a12, a23, a31; // edge DOF values (used in energy)
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double b1, b2, b3; // vertex DOF values
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bool v1b, v2b, v3b; // whether each vertex is variable
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};
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static FaceAngles compute_face_angles(
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const ConformalMesh& mesh,
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Face_index f,
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const std::vector<double>& x,
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const HyperIdealMaps& m)
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{
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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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FaceAngles fa;
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fa.v1b = m.v_idx[v1] >= 0;
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fa.v2b = m.v_idx[v2] >= 0;
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fa.v3b = m.v_idx[v3] >= 0;
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fa.a12 = dof_val(m.e_idx[e12], x);
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fa.a23 = dof_val(m.e_idx[e23], x);
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fa.a31 = dof_val(m.e_idx[e31], x);
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fa.b1 = dof_val(m.v_idx[v1], x);
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fa.b2 = dof_val(m.v_idx[v2], x);
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fa.b3 = dof_val(m.v_idx[v3], x);
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// Clamp invalid inputs (mirrors Java log.warning + clamp)
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if (fa.v1b && fa.v2b && fa.a12 < 0.0) fa.a12 = 0.0;
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if (fa.v2b && fa.v3b && fa.a23 < 0.0) fa.a23 = 0.0;
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if (fa.v3b && fa.v1b && fa.a31 < 0.0) fa.a31 = 0.0;
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if (fa.v1b && fa.b1 < 0.0) fa.b1 = 0.01;
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if (fa.v2b && fa.b2 < 0.0) fa.b2 = 0.01;
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if (fa.v3b && fa.b3 < 0.0) fa.b3 = 0.01;
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double l12 = lij(fa.b1, fa.b2, fa.a12, fa.v1b, fa.v2b);
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double l23 = lij(fa.b2, fa.b3, fa.a23, fa.v2b, fa.v3b);
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double l31 = lij(fa.b3, fa.b1, fa.a31, fa.v3b, fa.v1b);
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// Guard degenerate lengths
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if (l12 < 1E-12 && l23 < 1E-12 && l31 < 1E-12)
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l12 = l23 = l31 = 1E-12;
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// Check triangle inequalities; degenerate cases get extreme angles
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if (l12 > l23 + l31) {
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fa.beta1 = 0.0; fa.beta2 = 0.0; fa.beta3 = PI;
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fa.alpha12 = PI; fa.alpha23 = 0.0; fa.alpha31 = 0.0;
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} else if (l23 > l12 + l31) {
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fa.beta1 = PI; fa.beta2 = 0.0; fa.beta3 = 0.0;
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fa.alpha12 = 0.0; fa.alpha23 = PI; fa.alpha31 = 0.0;
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} else if (l31 > l12 + l23) {
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fa.beta1 = 0.0; fa.beta2 = PI; fa.beta3 = 0.0;
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fa.alpha12 = 0.0; fa.alpha23 = 0.0; fa.alpha31 = PI;
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} else {
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fa.beta1 = zeta(l12, l31, l23);
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fa.beta2 = zeta(l23, l12, l31);
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fa.beta3 = zeta(l31, l23, l12);
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fa.alpha12 = alpha_ij(fa.a12, fa.a23, fa.a31,
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fa.b1, fa.b2, fa.b3,
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fa.beta1, fa.beta2, fa.beta3,
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fa.v1b, fa.v2b, fa.v3b);
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fa.alpha23 = alpha_ij(fa.a23, fa.a31, fa.a12,
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fa.b2, fa.b3, fa.b1,
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fa.beta2, fa.beta3, fa.beta1,
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fa.v2b, fa.v3b, fa.v1b);
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fa.alpha31 = alpha_ij(fa.a31, fa.a12, fa.a23,
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fa.b3, fa.b1, fa.b2,
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fa.beta3, fa.beta1, fa.beta2,
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fa.v3b, fa.v1b, fa.v2b);
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}
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return fa;
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}
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// Per-face energy contribution U(f) (before subtracting θ·a and Θ·b terms).
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static double face_energy(const FaceAngles& fa)
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{
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double aa = fa.a12*fa.alpha12 + fa.a23*fa.alpha23 + fa.a31*fa.alpha31;
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double bb = fa.b1 *fa.beta1 + fa.b2 *fa.beta2 + fa.b3 *fa.beta3;
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double V = 0.0;
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if (fa.v1b && fa.v2b && fa.v3b) {
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V = calculateTetrahedronVolume(
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fa.beta1, fa.beta2, fa.beta3,
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fa.alpha23, fa.alpha31, fa.alpha12);
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} else if (!fa.v1b) {
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V = calculateTetrahedronVolumeWithIdealVertexAtGamma(
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fa.beta1, fa.alpha31, fa.alpha12,
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fa.alpha23, fa.beta2, fa.beta3);
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} else if (!fa.v2b) {
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V = calculateTetrahedronVolumeWithIdealVertexAtGamma(
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fa.beta2, fa.alpha12, fa.alpha23,
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fa.alpha31, fa.beta3, fa.beta1);
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} else { // !v3b
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V = calculateTetrahedronVolumeWithIdealVertexAtGamma(
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fa.beta3, fa.alpha23, fa.alpha31,
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fa.alpha12, fa.beta1, fa.beta2);
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}
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return aa + bb + 2.0 * V;
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}
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// ── Full evaluation ───────────────────────────────────────────────────────────
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inline HyperIdealResult evaluate_hyper_ideal(
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ConformalMesh& mesh,
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const std::vector<double>& x,
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const HyperIdealMaps& 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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HyperIdealResult res;
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// Temporary per-halfedge storage for computed angles.
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// Indexed by the integer value of Halfedge_index.
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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); // α_ij stored on halfedge
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std::vector<double> h_beta (nh, 0.0); // β_i stored on opposite halfedge
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// ── Pass 1: angles + energy 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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FaceAngles fa = compute_face_angles(mesh, f, x, m);
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// Store computed angles into temporary arrays.
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// h_alpha[h] = α for the edge of h in this face.
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h_alpha[hidx(h0)] = fa.alpha12;
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h_alpha[hidx(h1)] = fa.alpha23;
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h_alpha[hidx(h2)] = fa.alpha31;
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// h_beta[h] = β at the vertex OPPOSITE to h.
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// β1 (at v1 = source(h0)) is opposite to h1 = e23.
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h_beta[hidx(h1)] = fa.beta1; // h1 is across from v1
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h_beta[hidx(h2)] = fa.beta2; // h2 is across from v2
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h_beta[hidx(h0)] = fa.beta3; // h0 is across from v3
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if (need_energy)
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res.energy += face_energy(fa);
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}
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// ── Pass 2: linear energy terms ──────────────────────────────────────────
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if (need_energy) {
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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) res.energy -= m.theta_e[e] * x[static_cast<std::size_t>(ie)];
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}
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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) res.energy -= m.theta_v[v] * x[static_cast<std::size_t>(iv)];
|
||
}
|
||
}
|
||
|
||
// ── Pass 3: gradient ─────────────────────────────────────────────────────
|
||
if (need_gradient) {
|
||
const int n = hyper_ideal_dimension(mesh, m);
|
||
res.gradient.assign(static_cast<std::size_t>(n), 0.0);
|
||
|
||
// ∂E/∂b_v = Σ_{faces adj. v} β_v(face) − Θ_v
|
||
// β_v(face) = h_beta[prev(h)] for the incoming halfedge h to v in that face.
|
||
for (auto v : mesh.vertices()) {
|
||
int iv = m.v_idx[v];
|
||
if (iv < 0) continue;
|
||
for (auto h : CGAL::halfedges_around_target(v, mesh)) {
|
||
if (mesh.is_border(h)) continue;
|
||
res.gradient[static_cast<std::size_t>(iv)] += h_beta[hidx(mesh.prev(h))];
|
||
}
|
||
res.gradient[static_cast<std::size_t>(iv)] -= m.theta_v[v];
|
||
}
|
||
|
||
// ∂E/∂a_e = α_e(face⁺) + α_e(face⁻) − θ_e
|
||
for (auto e : mesh.edges()) {
|
||
int ie = m.e_idx[e];
|
||
if (ie < 0) continue;
|
||
auto h = mesh.halfedge(e);
|
||
auto ho = mesh.opposite(h);
|
||
if (!mesh.is_border(h)) res.gradient[static_cast<std::size_t>(ie)] += h_alpha[hidx(h)];
|
||
if (!mesh.is_border(ho)) res.gradient[static_cast<std::size_t>(ie)] += h_alpha[hidx(ho)];
|
||
res.gradient[static_cast<std::size_t>(ie)] -= m.theta_e[e];
|
||
}
|
||
}
|
||
|
||
return res;
|
||
}
|
||
|
||
// ── Finite-difference gradient check ─────────────────────────────────────────
|
||
//
|
||
// Returns true if |G[i] − fd[i]| / max(1, |G[i]|) < tol for all DOFs.
|
||
// eps = step size, tol = tolerance (same defaults as Java FunctionalTest).
|
||
inline bool gradient_check(
|
||
ConformalMesh& mesh,
|
||
const std::vector<double>& x0,
|
||
const HyperIdealMaps& m,
|
||
double eps = 1E-5,
|
||
double tol = 1E-4)
|
||
{
|
||
// Analytic gradient
|
||
auto res = evaluate_hyper_ideal(mesh, x0, m, false, true);
|
||
const auto& G = res.gradient;
|
||
const int n = static_cast<int>(G.size());
|
||
|
||
std::vector<double> xp = x0, xm = x0;
|
||
bool ok = true;
|
||
|
||
for (int i = 0; i < n; ++i) {
|
||
std::size_t si = static_cast<std::size_t>(i);
|
||
xp[si] = x0[si] + eps;
|
||
xm[si] = x0[si] - eps;
|
||
|
||
double Ep = evaluate_hyper_ideal(mesh, xp, m, true, false).energy;
|
||
double Em = evaluate_hyper_ideal(mesh, xm, m, true, false).energy;
|
||
|
||
xp[si] = xm[si] = x0[si];
|
||
|
||
double fd = (Ep - Em) / (2.0 * eps);
|
||
double err = std::abs(G[si] - fd);
|
||
double scale = std::max(1.0, std::abs(G[si]));
|
||
if (err / scale > tol) ok = false;
|
||
}
|
||
return ok;
|
||
}
|
||
|
||
} // namespace conformallab
|