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>
326 lines
14 KiB
C++
326 lines
14 KiB
C++
#pragma once
|
||
// euclidean_functional.hpp
|
||
//
|
||
// Energy and gradient of the Euclidean discrete conformal functional
|
||
// (EuclideanCyclicFunctional) evaluated on a ConformalMesh.
|
||
//
|
||
// Ported from de.varylab.discreteconformal.functional.EuclideanCyclicFunctional.
|
||
//
|
||
// ┌──────────────────────────────────────────────────────────────────────────┐
|
||
// │ DOFs │
|
||
// │ x[v_idx[v]] = u_v – conformal factor at vertex v │
|
||
// │ x[e_idx[e]] = λ_e – edge log-length variable (optional) │
|
||
// │ -1 means "pinned" (u_v = 0 / λ_e = 0, only λ° contributes) │
|
||
// │ │
|
||
// │ Effective log-length (always additive, unlike SphericalFunctional): │
|
||
// │ Λ̃_ij = λ°_ij + u_i + u_j + (x[e_idx[e]] if variable, else 0) │
|
||
// │ │
|
||
// │ Side length: l_ij = exp(Λ̃_ij / 2) │
|
||
// │ │
|
||
// │ Gradient: │
|
||
// │ ∂E/∂u_v = Θ_v − Σ_{faces adj. v} α_v(face) │
|
||
// │ ∂E/∂λ_e = α_opp(face⁺) + α_opp(face⁻) − φ_e │
|
||
// │ │
|
||
// │ Energy: │
|
||
// │ Computed as the path integral E(x) = ∫₀¹ ⟨G(tx), x⟩ dt │
|
||
// │ using 10-point Gauss-Legendre quadrature (same as SphericalFunctional)│
|
||
// │ This is E(0)=0 by construction and exact for conservative G. │
|
||
// └──────────────────────────────────────────────────────────────────────────┘
|
||
//
|
||
// Halfedge convention (identical to SphericalFunctional):
|
||
// h0 = mesh.halfedge(f), h1 = next(h0), h2 = next(h1)
|
||
// v1 = source(h0), v2 = source(h1), v3 = source(h2)
|
||
// h_alpha[h0] = α3 (angle at v3, opposite edge h0 = v1v2)
|
||
// h_alpha[h1] = α1 (angle at v1, opposite edge h1 = v2v3)
|
||
// h_alpha[h2] = α2 (angle at v2, opposite edge h2 = v3v1)
|
||
//
|
||
// Property-map name prefix: "ev:" (vertex) and "ee:" (edge).
|
||
|
||
#include "conformal_mesh.hpp"
|
||
#include "euclidean_geometry.hpp"
|
||
#include <CGAL/boost/graph/iterator.h>
|
||
#include <vector>
|
||
#include <cmath>
|
||
#include <cstdint>
|
||
|
||
namespace conformallab {
|
||
|
||
// ── Property-map type aliases ─────────────────────────────────────────────────
|
||
|
||
using EuclVMapD = ConformalMesh::Property_map<Vertex_index, double>;
|
||
using EuclVMapI = ConformalMesh::Property_map<Vertex_index, int>;
|
||
using EuclEMapD = ConformalMesh::Property_map<Edge_index, double>;
|
||
using EuclEMapI = ConformalMesh::Property_map<Edge_index, int>;
|
||
|
||
// ── Persistent map bundle ─────────────────────────────────────────────────────
|
||
|
||
struct EuclideanMaps {
|
||
EuclVMapI v_idx; ///< DOF index per vertex (-1 = pinned / u_v = 0)
|
||
EuclEMapI e_idx; ///< DOF index per edge (-1 = no edge DOF)
|
||
EuclVMapD theta_v; ///< target cone angle Θ_v (default 2π)
|
||
EuclEMapD phi_e; ///< target edge turn angle φ_e (default π)
|
||
EuclEMapD lambda0; ///< base log-length λ°_e (default 0.0)
|
||
};
|
||
|
||
// Create and attach property maps with sensible defaults.
|
||
// theta_v = 2π (flat vertex), phi_e = π (interior edge, flat surface).
|
||
inline EuclideanMaps setup_euclidean_maps(ConformalMesh& mesh)
|
||
{
|
||
constexpr double TWO_PI = 2.0 * 3.14159265358979323846;
|
||
constexpr double PI = 3.14159265358979323846;
|
||
|
||
EuclideanMaps m;
|
||
m.v_idx = mesh.add_property_map<Vertex_index, int> ("ev:idx", -1 ).first;
|
||
m.e_idx = mesh.add_property_map<Edge_index, int> ("ee:idx", -1 ).first;
|
||
m.theta_v= mesh.add_property_map<Vertex_index, double>("ev:theta", TWO_PI ).first;
|
||
m.phi_e = mesh.add_property_map<Edge_index, double>("ee:phi", PI ).first;
|
||
m.lambda0= mesh.add_property_map<Edge_index, double>("ee:lam0", 0.0 ).first;
|
||
return m;
|
||
}
|
||
|
||
// Assign DOF indices 0..n-1 for all vertices only (no edge DOFs).
|
||
inline int assign_euclidean_vertex_dof_indices(ConformalMesh& mesh, EuclideanMaps& m)
|
||
{
|
||
int idx = 0;
|
||
for (auto v : mesh.vertices()) m.v_idx[v] = idx++;
|
||
return idx;
|
||
}
|
||
|
||
// Assign DOF indices for all vertices AND all edges.
|
||
inline int assign_euclidean_all_dof_indices(ConformalMesh& mesh, EuclideanMaps& m)
|
||
{
|
||
int idx = 0;
|
||
for (auto v : mesh.vertices()) m.v_idx[v] = idx++;
|
||
for (auto e : mesh.edges()) m.e_idx[e] = idx++;
|
||
return idx;
|
||
}
|
||
|
||
// Count variable DOFs (vertices + edges).
|
||
inline int euclidean_dimension(const ConformalMesh& mesh, const EuclideanMaps& m)
|
||
{
|
||
int dim = 0;
|
||
for (auto v : mesh.vertices()) if (m.v_idx[v] >= 0) ++dim;
|
||
for (auto e : mesh.edges()) if (m.e_idx[e] >= 0) ++dim;
|
||
return dim;
|
||
}
|
||
|
||
// Set lambda0 from mesh vertex positions (Euclidean):
|
||
// λ°_e = 2·log(|p_i − p_j|) (natural log of Euclidean edge length squared)
|
||
//
|
||
// This gives exp(Λ̃_ij / 2) = l_ij at x=0.
|
||
inline void compute_euclidean_lambda0_from_mesh(ConformalMesh& mesh, EuclideanMaps& m)
|
||
{
|
||
for (auto e : mesh.edges()) {
|
||
auto h = mesh.halfedge(e);
|
||
auto p1 = mesh.point(mesh.source(h));
|
||
auto p2 = mesh.point(mesh.target(h));
|
||
double dx = p1.x() - p2.x();
|
||
double dy = p1.y() - p2.y();
|
||
double dz = p1.z() - p2.z();
|
||
double len = std::sqrt(dx*dx + dy*dy + dz*dz);
|
||
if (len > 1e-15)
|
||
m.lambda0[e] = 2.0 * std::log(len);
|
||
else
|
||
m.lambda0[e] = -30.0; // degenerate edge
|
||
}
|
||
}
|
||
|
||
// ── Internal helpers ──────────────────────────────────────────────────────────
|
||
|
||
static inline double eucl_dof_val(int idx, const std::vector<double>& x)
|
||
{
|
||
return idx >= 0 ? x[static_cast<std::size_t>(idx)] : 0.0;
|
||
}
|
||
|
||
static inline std::size_t eucl_hidx(Halfedge_index h)
|
||
{
|
||
return static_cast<std::size_t>(static_cast<std::uint32_t>(h));
|
||
}
|
||
|
||
// ── Gradient ──────────────────────────────────────────────────────────────────
|
||
//
|
||
// G_v = Θ_v − Σ_{faces adj. v} α_v(face)
|
||
// G_e = α_opp(face⁺) + α_opp(face⁻) − φ_e
|
||
//
|
||
// Corner-angle storage (h_alpha):
|
||
// h_alpha[h] = corner angle OPPOSITE to the edge of halfedge h in its face.
|
||
// h_alpha[h0] = α3, h_alpha[h1] = α1, h_alpha[h2] = α2
|
||
// (same convention as SphericalFunctional)
|
||
inline std::vector<double> euclidean_gradient(
|
||
ConformalMesh& mesh,
|
||
const std::vector<double>& x,
|
||
const EuclideanMaps& m)
|
||
{
|
||
const int n = euclidean_dimension(mesh, m);
|
||
std::vector<double> G(static_cast<std::size_t>(n), 0.0);
|
||
|
||
// Per-halfedge corner-angle storage.
|
||
const std::size_t nh = mesh.number_of_halfedges();
|
||
std::vector<double> h_alpha(nh, 0.0);
|
||
|
||
// ── Pass 1: compute corner angles per face ────────────────────────────────
|
||
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 (always additive: u_i + u_j regardless of DOF status)
|
||
double u1 = eucl_dof_val(m.v_idx[v1], x);
|
||
double u2 = eucl_dof_val(m.v_idx[v2], x);
|
||
double u3 = eucl_dof_val(m.v_idx[v3], x);
|
||
|
||
double lam12 = m.lambda0[e12] + u1 + u2 + eucl_dof_val(m.e_idx[e12], x);
|
||
double lam23 = m.lambda0[e23] + u2 + u3 + eucl_dof_val(m.e_idx[e23], x);
|
||
double lam31 = m.lambda0[e31] + u3 + u1 + eucl_dof_val(m.e_idx[e31], x);
|
||
|
||
auto fa = euclidean_angles(lam12, lam23, lam31);
|
||
if (!fa.valid) continue; // degenerate triangle: contributes 0
|
||
|
||
// h_alpha[h] = corner angle OPPOSITE to h's edge:
|
||
// h0 (edge v1v2) → opposite corner at v3 → α3
|
||
// h1 (edge v2v3) → opposite corner at v1 → α1
|
||
// h2 (edge v3v1) → opposite corner at v2 → α2
|
||
h_alpha[eucl_hidx(h0)] = fa.alpha3;
|
||
h_alpha[eucl_hidx(h1)] = fa.alpha1;
|
||
h_alpha[eucl_hidx(h2)] = fa.alpha2;
|
||
}
|
||
|
||
// ── Pass 2: accumulate vertex gradient ───────────────────────────────────
|
||
// G_v = Θ_v − Σ h_alpha[prev(h)] for each incoming non-border h to v.
|
||
for (auto v : mesh.vertices()) {
|
||
int iv = m.v_idx[v];
|
||
if (iv < 0) continue;
|
||
double sum_alpha = 0.0;
|
||
for (auto h : CGAL::halfedges_around_target(v, mesh)) {
|
||
if (mesh.is_border(h)) continue;
|
||
sum_alpha += h_alpha[eucl_hidx(mesh.prev(h))];
|
||
}
|
||
G[static_cast<std::size_t>(iv)] = m.theta_v[v] - sum_alpha;
|
||
}
|
||
|
||
// ── Pass 3: accumulate edge gradient ─────────────────────────────────────
|
||
// G_e = α_opp(f⁺) + α_opp(f⁻) − φ_e
|
||
// α_opp of edge e in face f = h_alpha[halfedge h of e pointing INTO f].
|
||
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);
|
||
double sum = -m.phi_e[e];
|
||
if (!mesh.is_border(h)) sum += h_alpha[eucl_hidx(h)];
|
||
if (!mesh.is_border(ho)) sum += h_alpha[eucl_hidx(ho)];
|
||
G[static_cast<std::size_t>(ie)] = sum;
|
||
}
|
||
|
||
return G;
|
||
}
|
||
|
||
// ── Energy via Gauss-Legendre path integral ───────────────────────────────────
|
||
//
|
||
// E(x) = ∫₀¹ ⟨G(tx), x⟩ dt (10-point GL quadrature, same as SphericalFunctional)
|
||
inline double euclidean_energy(
|
||
ConformalMesh& mesh,
|
||
const std::vector<double>& x,
|
||
const EuclideanMaps& m)
|
||
{
|
||
static const double gl_s[10] = {
|
||
-0.9739065285171717, -0.8650633666889845,
|
||
-0.6794095682990244, -0.4333953941292472,
|
||
-0.1488743389816312, 0.1488743389816312,
|
||
0.4333953941292472, 0.6794095682990244,
|
||
0.8650633666889845, 0.9739065285171717
|
||
};
|
||
static const double gl_w[10] = {
|
||
0.0666713443086881, 0.1494513491505806,
|
||
0.2190863625159820, 0.2692667193099963,
|
||
0.2955242247147529, 0.2955242247147529,
|
||
0.2692667193099963, 0.2190863625159820,
|
||
0.1494513491505806, 0.0666713443086881
|
||
};
|
||
|
||
const std::size_t n = x.size();
|
||
double E = 0.0;
|
||
|
||
for (int k = 0; k < 10; ++k) {
|
||
double t = (1.0 + gl_s[k]) * 0.5;
|
||
double wt = gl_w[k] * 0.5;
|
||
|
||
std::vector<double> tx(n);
|
||
for (std::size_t i = 0; i < n; ++i) tx[i] = t * x[i];
|
||
|
||
auto G = euclidean_gradient(mesh, tx, m);
|
||
|
||
double dot = 0.0;
|
||
for (std::size_t i = 0; i < n; ++i) dot += G[i] * x[i];
|
||
E += wt * dot;
|
||
}
|
||
return E;
|
||
}
|
||
|
||
// ── Full evaluation (energy + gradient) ──────────────────────────────────────
|
||
|
||
struct EuclideanResult {
|
||
double energy = 0.0;
|
||
std::vector<double> gradient;
|
||
};
|
||
|
||
inline EuclideanResult evaluate_euclidean(
|
||
ConformalMesh& mesh,
|
||
const std::vector<double>& x,
|
||
const EuclideanMaps& m,
|
||
bool need_energy = true,
|
||
bool need_gradient = true)
|
||
{
|
||
EuclideanResult res;
|
||
if (need_gradient)
|
||
res.gradient = euclidean_gradient(mesh, x, m);
|
||
if (need_energy)
|
||
res.energy = euclidean_energy(mesh, x, m);
|
||
return res;
|
||
}
|
||
|
||
// ── Finite-difference gradient check ─────────────────────────────────────────
|
||
//
|
||
// Tests |G[i] − (E(x+εeᵢ) − E(x−εeᵢ))/(2ε)| / max(1,|G[i]|) < tol
|
||
// for all variable DOFs.
|
||
inline bool gradient_check_euclidean(
|
||
ConformalMesh& mesh,
|
||
const std::vector<double>& x0,
|
||
const EuclideanMaps& m,
|
||
double eps = 1e-5,
|
||
double tol = 1e-4)
|
||
{
|
||
auto G = euclidean_gradient(mesh, x0, m);
|
||
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 = euclidean_energy(mesh, xp, m);
|
||
double Em = euclidean_energy(mesh, xm, m);
|
||
|
||
xp[si] = xm[si] = x0[si]; // restore
|
||
|
||
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
|