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910 lines
46 KiB
C++
910 lines
46 KiB
C++
#pragma once
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// Copyright (c) 2024-2026 Tarik Moussa.
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// SPDX-License-Identifier: MIT
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// layout.hpp
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//
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// Phase 5/6/7 — Layout / embedding: DOF vector → vertex coordinates in the
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// target geometry via BFS-trilateration.
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//
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// ┌──────────────────────────────────────────────────────────────────────────┐
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// │ Algorithm (all three geometries) │
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// │ │
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// │ 1. For every edge e compute the updated length l_e from the DOF x. │
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// │ 2. Choose root face = largest 3-D area face (quality heuristic). │
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// │ 3. Priority BFS over the dual graph: faces are processed in order of │
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// │ shortest distance (BFS depth) from the root face, minimising error │
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// │ accumulation. Equal depth: arbitrary tie-break. │
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// │ 4. For each face: trilaterate the one unplaced vertex from the two │
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// │ already-placed vertices on the shared edge. │
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// │ 5. If a CutGraph is supplied, cut edges are treated as boundary; │
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// │ holonomy is recorded for each cut edge. │
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// │ 6. After the first connected component, unvisited faces start new │
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// │ BFS sweeps (multi-component support). │
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// │ │
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// │ For open meshes: globally consistent placement. │
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// │ For closed meshes without CutGraph: first (shallowest) visit wins, │
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// │ has_seam = true. │
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// │ For closed meshes with CutGraph: each vertex placed exactly once; │
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// │ holonomy = translation (Euclidean) or Möbius map (hyperbolic). │
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// └──────────────────────────────────────────────────────────────────────────┘
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//
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// Trilateration
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// Euclidean — exact (analytic formula in ℝ²)
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// Spherical — exact (spherical law of cosines on S²)
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// Hyperbolic — exact (hyperbolic law of cosines + Möbius maps,
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// Poincaré disk model)
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//
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// Normalisation
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// normalise_euclidean(layout) — centroid → origin, major axis → x-axis (PCA)
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// normalise_hyperbolic(layout) — face-area-weighted iterative Möbius centering
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// normalise_spherical(layout) — rotate centroid to north pole (Rodrigues)
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//
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// Holonomy (genus-g surfaces with CutGraph)
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// HolonomyData.translations[i] — translation ω_i (Euclidean / spherical)
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// HolonomyData.mobius_maps[i] — Möbius map T_i (hyperbolic)
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// For a flat torus: ω_1, ω_2 are the lattice generators; τ = ω_2/ω_1 ∈ ℍ.
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//
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// Texture atlas
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// Layout2D.halfedge_uv[h.idx()] — UV of source(h) as seen from face(h).
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// At seam edges the two opposite halfedges carry different UV values,
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// enabling proper per-halfedge UV for GPU texture atlasing.
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//
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// Möbius map
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// MobiusMap::from_three(z1→w1, z2→w2, z3→w3) — fit a Möbius transformation
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// to three point correspondences (via 3×3 complex linear system).
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#include "conformal_mesh.hpp"
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#include "euclidean_functional.hpp"
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#include "spherical_functional.hpp"
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#include "hyper_ideal_functional.hpp"
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#include "cut_graph.hpp"
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#include <Eigen/Dense>
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#include <vector>
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#include <queue>
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#include <complex>
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#include <cmath>
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#include <algorithm>
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#include <limits>
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#include <fstream>
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#include <string>
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namespace conformallab {
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// ── Möbius map ────────────────────────────────────────────────────────────────
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//
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// T(z) = (a·z + b) / (c·z + d)
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//
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// For hyperbolic holonomy the map is an orientation-preserving isometry of
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// the Poincaré disk (SU(1,1) element).
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// ─────────────────────────────────────────────────────────────────────────────
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/// Möbius transformation `T(z) = (a·z + b) / (c·z + d)` of the Riemann
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/// sphere; restricted to SU(1,1) for hyperbolic holonomy on the
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/// Poincaré disk.
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struct MobiusMap {
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/// Complex scalar used for all entries.
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using C = std::complex<double>;
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C a{1.0, 0.0}; ///< Top-left coefficient.
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C b{0.0, 0.0}; ///< Top-right coefficient.
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C c{0.0, 0.0}; ///< Bottom-left coefficient.
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C d{1.0, 0.0}; ///< Bottom-right coefficient.
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/// Apply the transformation to a complex point.
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C apply(C z) const { return (a * z + b) / (c * z + d); }
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/// Apply the transformation to a 2-D real point (interpreted as `x + iy`).
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Eigen::Vector2d apply(const Eigen::Vector2d& p) const {
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C w = apply(C(p.x(), p.y()));
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return Eigen::Vector2d(w.real(), w.imag());
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}
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/// Identity map.
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static MobiusMap identity() { return {C(1), C(0), C(0), C(1)}; }
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/// Inverse map.
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MobiusMap inverse() const { return {d, -b, -c, a}; }
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/// Composition `(*this) ∘ T`, i.e. apply `T` first then `*this`.
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MobiusMap compose(const MobiusMap& T) const {
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return { a*T.a + b*T.c, a*T.b + b*T.d,
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c*T.a + d*T.c, c*T.b + d*T.d };
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}
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/// `true` iff the map is the identity up to tolerance `tol`.
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bool is_identity(double tol = 1e-9) const {
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if (std::abs(d) < 1e-14) return false;
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C a_ = a/d, b_ = b/d, c_ = c/d;
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return std::abs(a_ - C(1)) < tol && std::abs(b_) < tol && std::abs(c_) < tol;
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}
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/// Fit T to three point correspondences T(z_i) = w_i.
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/// Sets d=1 and solves the 3×3 complex linear system.
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/// Returns identity when the system is (near-)singular.
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static MobiusMap from_three(C z1, C w1, C z2, C w2, C z3, C w3) {
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Eigen::Matrix3cd M;
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M << z1, C(1), -w1*z1,
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z2, C(1), -w2*z2,
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z3, C(1), -w3*z3;
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Eigen::Vector3cd rhs; rhs << w1, w2, w3;
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Eigen::ColPivHouseholderQR<Eigen::Matrix3cd> qr(M);
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if (qr.rank() < 3) return identity();
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Eigen::Vector3cd sol = qr.solve(rhs);
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return { sol[0], sol[1], sol[2], C(1.0) };
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}
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};
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// ── Result types ──────────────────────────────────────────────────────────────
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/// Result of a 2-D layout (`euclidean_layout`, `hyper_ideal_layout`):
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/// per-vertex UV coordinates plus a per-half-edge UV atlas for seamed
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/// textures.
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struct Layout2D {
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/// uv[v.idx()] — primary 2-D position (first / shallowest-BFS-depth visit).
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std::vector<Eigen::Vector2d> uv;
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/// halfedge_uv[h.idx()] — UV of source(h) as seen from face(h).
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///
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/// For interior (non-seam) halfedges: equals uv[source(h).idx()].
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/// For seam halfedges: carries the UV from the virtual unfolding across
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/// the cut (i.e. the trilaterated position that was NOT used as the
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/// primary uv). This gives each face its own copy of a seam vertex,
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/// enabling a proper GPU texture atlas without vertex duplication.
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///
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/// Size = mesh.number_of_halfedges(). Border halfedges = (0,0).
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std::vector<Eigen::Vector2d> halfedge_uv;
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bool success = false; ///< `true` iff the BFS placed every vertex.
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bool has_seam = false; ///< `true` when a vertex was reached via two paths.
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};
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/// Result of a 3-D layout (`spherical_layout`): per-vertex positions on S².
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struct Layout3D {
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std::vector<Eigen::Vector3d> pos; ///< Per-vertex spherical positions.
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bool success = false; ///< `true` iff the BFS placed every vertex.
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bool has_seam = false; ///< `true` when a vertex was reached via two paths.
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};
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/// Per-cut-edge holonomy.
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///
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/// Euclidean: translations[i] = ω_i (translation vector for cut_edge_indices[i]).
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/// For a flat torus ω_1, ω_2 are the lattice generators; τ = ω_2/ω_1 ∈ ℍ.
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///
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/// Hyperbolic: mobius_maps[i] = T_i (Möbius isometry of the Poincaré disk).
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/// T_i(p_actual) = p_virtual — maps the placed vertex position to the
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/// trilaterated virtual position obtained by continuing the unfolding across
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/// the cut.
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struct HolonomyData {
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std::vector<Eigen::Vector2d> translations; ///< Euclidean / spherical translation per cut edge.
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std::vector<MobiusMap> mobius_maps; ///< Hyperbolic Möbius isometry per cut edge (Phase 7).
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std::vector<std::size_t> cut_edge_indices; ///< Index (in the cut-graph edge list) of each holonomy entry.
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};
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// ── Internal helpers ──────────────────────────────────────────────────────────
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namespace detail {
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// ─────────────────────────────────────────────────────────────────────────────
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// Priority-BFS queue entry
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// Faces closer to the root (lower depth) are processed first, reducing the
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// accumulation of trilateration errors for later-placed vertices.
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// ─────────────────────────────────────────────────────────────────────────────
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struct BFSEntry {
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int depth; ///< BFS depth = max(depth[v_src], depth[v_tgt]) + 1
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Halfedge_index h; ///< halfedge pointing INTO the face to be placed
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/// min-heap: smaller depth = higher priority
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bool operator>(const BFSEntry& o) const { return depth > o.depth; }
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};
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using BFSQueue = std::priority_queue<BFSEntry,
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std::vector<BFSEntry>,
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std::greater<BFSEntry>>;
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// ─────────────────────────────────────────────────────────────────────────────
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// Euclidean trilateration — LEFT (CCW) side of p_a → p_b
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// ─────────────────────────────────────────────────────────────────────────────
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inline Eigen::Vector2d trilaterate_2d(
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const Eigen::Vector2d& pa, const Eigen::Vector2d& pb,
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double da, double db)
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{
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Eigen::Vector2d ab = pb - pa;
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double d = ab.norm();
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if (d < 1e-14) return pa;
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Eigen::Vector2d e1 = ab / d;
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Eigen::Vector2d e2(-e1.y(), e1.x());
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double t = (da*da - db*db + d*d) / (2.0 * d);
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double s2 = da*da - t*t;
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double s = (s2 > 0.0) ? std::sqrt(s2) : 0.0;
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return pa + t*e1 + s*e2;
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}
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// ─────────────────────────────────────────────────────────────────────────────
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// Spherical trilateration (S²) — LEFT side of geodesic arc pa → pb
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// ─────────────────────────────────────────────────────────────────────────────
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inline Eigen::Vector3d trilaterate_sph(
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const Eigen::Vector3d& pa, const Eigen::Vector3d& pb,
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double da, double db)
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{
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double c = pa.dot(pb);
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double denom = 1.0 - c*c;
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if (denom < 1e-14) return pa;
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double cda = std::cos(da), cdb = std::cos(db);
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double alpha = (cda - c*cdb) / denom;
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double beta = (cdb - c*cda) / denom;
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Eigen::Vector3d cross = pa.cross(pb);
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double cn2 = cross.squaredNorm();
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double gamma2 = 1.0 - alpha*alpha - beta*beta - 2.0*alpha*beta*c;
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double gamma = (gamma2 > 0.0 && cn2 > 1e-28) ? std::sqrt(gamma2 / cn2) : 0.0;
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Eigen::Vector3d p = alpha*pa + beta*pb + gamma*cross;
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double n = p.norm();
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return (n > 1e-14) ? (p / n) : pa;
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}
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// ─────────────────────────────────────────────────────────────────────────────
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// Hyperbolic trilateration — exact via Möbius + hyperbolic law of cosines.
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// LEFT (CCW) side of pa → pb in the Poincaré disk.
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// ─────────────────────────────────────────────────────────────────────────────
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inline Eigen::Vector2d trilaterate_hyp(
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const Eigen::Vector2d& pa, const Eigen::Vector2d& pb,
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double D, double da, double db)
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{
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using C = std::complex<double>;
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if (D < 1e-12 || da < 1e-12) return pa;
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if (std::sinh(da) * std::sinh(D) < 1e-14) return pa;
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C a(pa.x(), pa.y()), b(pb.x(), pb.y());
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auto fwd = [](C z, C c_) { return (z-c_)/(C(1)-std::conj(c_)*z); };
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auto inv = [](C w, C c_) { return (w+c_)/(C(1)+std::conj(c_)*w); };
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double theta_b = std::arg(fwd(b, a));
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double cos_alpha = (std::cosh(da)*std::cosh(D) - std::cosh(db))
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/ (std::sinh(da)*std::sinh(D));
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cos_alpha = std::max(-1.0, std::min(1.0, cos_alpha));
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C pc_origin = std::tanh(da*0.5) * std::exp(C(0.0, theta_b + std::acos(cos_alpha)));
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C pc_c = inv(pc_origin, a);
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return Eigen::Vector2d(pc_c.real(), pc_c.imag());
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}
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// ─────────────────────────────────────────────────────────────────────────────
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// 3-D face area (cross product / 2)
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// ─────────────────────────────────────────────────────────────────────────────
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inline double face_3d_area(const ConformalMesh& mesh, Face_index f)
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{
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Halfedge_index h = mesh.halfedge(f);
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auto pA = mesh.point(mesh.source(h));
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auto pB = mesh.point(mesh.target(h));
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auto pC = mesh.point(mesh.target(mesh.next(h)));
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Eigen::Vector3d ab(pB.x()-pA.x(), pB.y()-pA.y(), pB.z()-pA.z());
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Eigen::Vector3d ac(pC.x()-pA.x(), pC.y()-pA.y(), pC.z()-pA.z());
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return 0.5 * ab.cross(ac).norm();
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}
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// ─────────────────────────────────────────────────────────────────────────────
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// Best root face: largest 3-D area, with 1.5× bonus for interior faces.
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// ─────────────────────────────────────────────────────────────────────────────
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inline Face_index best_root_face(const ConformalMesh& mesh)
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{
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Face_index best;
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double best_score = -1.0;
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for (auto f : mesh.faces()) {
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double area = face_3d_area(mesh, f);
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bool interior = true;
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for (auto h : CGAL::halfedges_around_face(mesh.halfedge(f), mesh))
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if (mesh.is_border(mesh.opposite(h))) { interior = false; break; }
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double score = area * (interior ? 1.5 : 1.0);
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if (score > best_score) { best_score = score; best = f; }
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}
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return best;
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}
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// ─────────────────────────────────────────────────────────────────────────────
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// Euclidean bounding box of placed vertices (for multi-component offset)
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// ─────────────────────────────────────────────────────────────────────────────
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inline void bbox_2d(const std::vector<Eigen::Vector2d>& uv,
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const std::vector<bool>& placed,
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double& xmax)
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{
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xmax = 0.0;
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for (std::size_t i = 0; i < uv.size(); ++i)
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if (placed[i]) xmax = std::max(xmax, uv[i].x());
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}
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// ─────────────────────────────────────────────────────────────────────────────
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// Möbius centering — unweighted (fallback / Phase 6 compat.)
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// ─────────────────────────────────────────────────────────────────────────────
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inline void center_poincare_disk(std::vector<Eigen::Vector2d>& uv)
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{
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if (uv.empty()) return;
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using C = std::complex<double>;
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C c(0);
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for (auto& p : uv) c += C(p.x(), p.y());
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c /= static_cast<double>(uv.size());
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double r = std::abs(c);
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if (r > 0.999) c *= 0.999/r;
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if (r < 1e-12) return;
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for (auto& p : uv) {
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C z(p.x(), p.y());
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C w = (z-c)/(C(1)-std::conj(c)*z);
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p = {w.real(), w.imag()};
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}
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}
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// ─────────────────────────────────────────────────────────────────────────────
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// Face-area-weighted iterative Möbius centering (Fréchet mean, Phase 7).
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// weights[i] = Voronoi area of vertex i.
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// ─────────────────────────────────────────────────────────────────────────────
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inline void center_poincare_disk_weighted(
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std::vector<Eigen::Vector2d>& uv, const std::vector<double>& weights)
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{
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if (uv.empty()) return;
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using C = std::complex<double>;
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double total_w = 0.0; for (double w : weights) total_w += w;
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if (total_w < 1e-14) { center_poincare_disk(uv); return; }
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C c(0);
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for (int iter = 0; iter < 30; ++iter) {
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C mt(0);
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for (std::size_t i = 0; i < uv.size(); ++i) {
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C z(uv[i].x(), uv[i].y());
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mt += weights[i] * (z-c)/(C(1)-std::conj(c)*z);
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}
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mt /= total_w;
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C c_new = (mt+c)/(C(1)+std::conj(c)*mt);
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if (std::abs(c_new-c) < 1e-12) { c = c_new; break; }
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c = c_new;
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}
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double r = std::abs(c);
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if (r > 0.999) c *= 0.999/r;
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if (r < 1e-12) return;
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for (auto& p : uv) {
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C z(p.x(), p.y());
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C w = (z-c)/(C(1)-std::conj(c)*z);
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p = {w.real(), w.imag()};
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}
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}
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} // namespace detail
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/// Compute per-vertex area weights (sum of 1/3 of each adjacent triangle area).
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/// Used by area-weighted layout normalisation routines.
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inline std::vector<double> compute_vertex_area_weights(const ConformalMesh& mesh)
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{
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std::vector<double> w(mesh.number_of_vertices(), 0.0);
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for (auto f : mesh.faces()) {
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double area = detail::face_3d_area(mesh, f);
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for (auto h : CGAL::halfedges_around_face(mesh.halfedge(f), mesh))
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w[static_cast<std::size_t>(mesh.target(h).idx())] += area / 3.0;
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}
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return w;
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}
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// ── Layout normalisation ──────────────────────────────────────────────────────
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/// Euclidean canonical normalisation: translate centroid to the origin
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/// and rotate the principal axis of the UV cloud onto the x-axis.
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inline void normalise_euclidean(Layout2D& layout)
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{
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if (!layout.success || layout.uv.empty()) return;
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const std::size_t n = layout.uv.size();
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Eigen::Vector2d mean = Eigen::Vector2d::Zero();
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for (auto& p : layout.uv) mean += p;
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mean /= static_cast<double>(n);
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for (auto& p : layout.uv) p -= mean;
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for (auto& p : layout.halfedge_uv) p -= mean;
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Eigen::Matrix2d cov = Eigen::Matrix2d::Zero();
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for (auto& p : layout.uv) cov += p * p.transpose();
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cov /= static_cast<double>(n);
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Eigen::SelfAdjointEigenSolver<Eigen::Matrix2d> eig(cov);
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Eigen::Vector2d major = eig.eigenvectors().col(1);
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double angle = -std::atan2(major.y(), major.x());
|
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Eigen::Matrix2d R;
|
||
R << std::cos(angle), -std::sin(angle),
|
||
std::sin(angle), std::cos(angle);
|
||
for (auto& p : layout.uv) p = R * p;
|
||
for (auto& p : layout.halfedge_uv) p = R * p;
|
||
}
|
||
|
||
/// Hyperbolic — face-area-weighted iterative Möbius centering.
|
||
inline void normalise_hyperbolic(Layout2D& layout, const ConformalMesh& mesh)
|
||
{
|
||
if (!layout.success || layout.uv.empty()) return;
|
||
auto weights = compute_vertex_area_weights(mesh);
|
||
detail::center_poincare_disk_weighted(layout.uv, weights);
|
||
// halfedge_uv follows the same Möbius map
|
||
detail::center_poincare_disk(layout.halfedge_uv);
|
||
}
|
||
/// Hyperbolic canonical normalisation, mesh-free fallback: uniform
|
||
/// (unweighted) iterative Möbius centring of the Poincaré disk.
|
||
inline void normalise_hyperbolic(Layout2D& layout) // fallback without mesh
|
||
{
|
||
if (!layout.success || layout.uv.empty()) return;
|
||
detail::center_poincare_disk(layout.uv);
|
||
detail::center_poincare_disk(layout.halfedge_uv);
|
||
}
|
||
|
||
/// Spherical canonical normalisation: rotate the layout so that the
|
||
/// per-vertex centroid (projected back to S²) coincides with the north
|
||
/// pole (Rodrigues rotation).
|
||
inline void normalise_spherical(Layout3D& layout)
|
||
{
|
||
if (!layout.success || layout.pos.empty()) return;
|
||
Eigen::Vector3d mean = Eigen::Vector3d::Zero();
|
||
for (auto& p : layout.pos) mean += p;
|
||
double len = mean.norm(); if (len < 1e-12) return;
|
||
mean /= len;
|
||
Eigen::Vector3d north(0, 0, 1);
|
||
Eigen::Vector3d axis = mean.cross(north);
|
||
double sin_a = axis.norm(), cos_a = mean.dot(north);
|
||
if (sin_a < 1e-12) return;
|
||
axis /= sin_a;
|
||
Eigen::Matrix3d K;
|
||
K << 0, -axis.z(), axis.y(),
|
||
axis.z(), 0, -axis.x(),
|
||
-axis.y(), axis.x(), 0;
|
||
Eigen::Matrix3d Rot = Eigen::Matrix3d::Identity() + sin_a*K + (1-cos_a)*K*K;
|
||
for (auto& p : layout.pos) p = Rot * p;
|
||
}
|
||
|
||
// ─────────────────────────────────────────────────────────────────────────────
|
||
// BFS placement helpers shared across all three layout functions.
|
||
//
|
||
// set_face_huv: populate halfedge_uv for the three halfedges of face f,
|
||
// given that the face was entered via halfedge h_enter.
|
||
// h_enter: source=v_src, target=v_tgt → huv = uv[v_src]
|
||
// next(h_enter): source=v_tgt, target=v_new → huv = uv[v_tgt]
|
||
// prev(h_enter): source=v_new, target=v_src → huv = p_new (trilaterated)
|
||
// ─────────────────────────────────────────────────────────────────────────────
|
||
namespace detail {
|
||
|
||
inline void set_face_huv_2d(
|
||
std::vector<Eigen::Vector2d>& huv,
|
||
const ConformalMesh& mesh,
|
||
Halfedge_index h,
|
||
const std::vector<Eigen::Vector2d>& uv,
|
||
const Eigen::Vector2d& p_new)
|
||
{
|
||
auto s = [](std::size_t x){ return x; };
|
||
huv[s(static_cast<std::size_t>(h.idx()))] = uv[static_cast<std::size_t>(mesh.source(h).idx())];
|
||
huv[s(static_cast<std::size_t>(mesh.next(h).idx()))] = uv[static_cast<std::size_t>(mesh.target(h).idx())];
|
||
huv[s(static_cast<std::size_t>(mesh.prev(h).idx()))] = p_new;
|
||
}
|
||
|
||
inline void set_root_huv_2d(
|
||
std::vector<Eigen::Vector2d>& huv,
|
||
const ConformalMesh& mesh,
|
||
Face_index f,
|
||
const std::vector<Eigen::Vector2d>& uv)
|
||
{
|
||
for (auto hf : CGAL::halfedges_around_face(mesh.halfedge(f), mesh))
|
||
huv[static_cast<std::size_t>(hf.idx())] = uv[static_cast<std::size_t>(mesh.source(hf).idx())];
|
||
}
|
||
|
||
} // namespace detail
|
||
|
||
// ── Euclidean layout ──────────────────────────────────────────────────────────
|
||
|
||
/// Embed the mesh in ℝ² using the Euclidean metric encoded in x.
|
||
///
|
||
/// Runs priority-BFS trilateration: places vertices in order of BFS depth
|
||
/// from the root face (largest 3D area), so errors accumulate last.
|
||
/// For closed genus-g surfaces a CutGraph must be supplied — otherwise
|
||
/// the layout will have a seam discontinuity (Layout2D::has_seam = true).
|
||
///
|
||
/// \param mesh Input surface mesh. Must have lambda0 and v_idx set in maps.
|
||
/// \param x DOF vector returned by newton_euclidean().
|
||
/// \param maps EuclideanMaps (lambda0, v_idx, e_idx).
|
||
/// \param cut Optional cut graph (compute_cut_graph()). Pass nullptr for
|
||
/// open meshes or if seams are acceptable.
|
||
/// \param holonomy If non-null and cut != nullptr, receives the lattice
|
||
/// translations ω_i ∈ ℂ per cut edge.
|
||
/// Pass to compute_period_matrix() for the conformal modulus τ.
|
||
/// \param normalise If true, calls normalise_euclidean() on the result:
|
||
/// centroid → origin, major axis → x-axis (PCA).
|
||
/// \return Layout2D with .uv[v] (per-vertex UV) and
|
||
/// .halfedge_uv[h] (per-halfedge UV for texture atlasing).
|
||
///
|
||
/// \note halfedge_uv differs from uv at seam edges: the two sides of a cut
|
||
/// carry different UV coordinates for proper GPU texture atlasing.
|
||
inline Layout2D euclidean_layout(
|
||
ConformalMesh& mesh,
|
||
const std::vector<double>& x,
|
||
const EuclideanMaps& maps,
|
||
const CutGraph* cut = nullptr,
|
||
HolonomyData* holonomy = nullptr,
|
||
bool normalise = false)
|
||
{
|
||
const std::size_t nv = mesh.number_of_vertices();
|
||
const std::size_t nf = mesh.number_of_faces();
|
||
const std::size_t nh = mesh.number_of_halfedges();
|
||
Layout2D result;
|
||
result.uv.assign(nv, Eigen::Vector2d::Zero());
|
||
result.halfedge_uv.assign(nh, Eigen::Vector2d::Zero());
|
||
if (nf == 0) return result;
|
||
|
||
auto get_u = [&](Vertex_index v) {
|
||
int iv = maps.v_idx[v]; return (iv>=0) ? x[static_cast<std::size_t>(iv)] : 0.0; };
|
||
auto get_ue = [&](Edge_index e) {
|
||
int ie = maps.e_idx[e]; return (ie>=0) ? x[static_cast<std::size_t>(ie)] : 0.0; };
|
||
auto edge_len = [&](Halfedge_index h) {
|
||
Edge_index e = mesh.edge(h);
|
||
double lam = maps.lambda0[e] + get_u(mesh.source(h)) + get_u(mesh.target(h)) + get_ue(e);
|
||
return std::exp(lam * 0.5); };
|
||
|
||
std::vector<bool> vertex_placed(nv, false);
|
||
std::vector<bool> face_placed(nf, false);
|
||
std::vector<int> vertex_depth(nv, std::numeric_limits<int>::max());
|
||
|
||
auto place_component = [&](Face_index f_root, Eigen::Vector2d offset) {
|
||
Halfedge_index h0 = mesh.halfedge(f_root);
|
||
Halfedge_index h1 = mesh.next(h0), h2 = mesh.next(h1);
|
||
Vertex_index vA = mesh.source(h0), vB = mesh.source(h1), vC = mesh.source(h2);
|
||
double lAB = edge_len(h0), lBC = edge_len(h1), lCA = edge_len(h2);
|
||
result.uv[vA.idx()] = offset;
|
||
result.uv[vB.idx()] = offset + Eigen::Vector2d(lAB, 0.0);
|
||
result.uv[vC.idx()] = detail::trilaterate_2d(result.uv[vA.idx()], result.uv[vB.idx()], lCA, lBC);
|
||
vertex_placed[vA.idx()] = vertex_placed[vB.idx()] = vertex_placed[vC.idx()] = true;
|
||
vertex_depth[vA.idx()] = vertex_depth[vB.idx()] = vertex_depth[vC.idx()] = 0;
|
||
face_placed[f_root.idx()] = true;
|
||
detail::set_root_huv_2d(result.halfedge_uv, mesh, f_root, result.uv);
|
||
|
||
detail::BFSQueue pq;
|
||
auto enqueue = [&](Face_index f) {
|
||
for (auto hf : CGAL::halfedges_around_face(mesh.halfedge(f), mesh)) {
|
||
Halfedge_index ho = mesh.opposite(hf);
|
||
if (mesh.is_border(ho)) continue;
|
||
if (cut && cut->is_cut(mesh.edge(hf))) continue;
|
||
Face_index fadj = mesh.face(ho);
|
||
if (!face_placed[fadj.idx()]) {
|
||
int d = std::max(vertex_depth[mesh.source(ho).idx()],
|
||
vertex_depth[mesh.target(ho).idx()]) + 1;
|
||
pq.push({d, ho});
|
||
}
|
||
}
|
||
};
|
||
enqueue(f_root);
|
||
|
||
while (!pq.empty()) {
|
||
auto [depth, h] = pq.top(); pq.pop();
|
||
Face_index f = mesh.face(h);
|
||
if (face_placed[f.idx()]) continue;
|
||
|
||
Vertex_index v_src = mesh.source(h), v_tgt = mesh.target(h);
|
||
Vertex_index v_new = mesh.target(mesh.next(h));
|
||
Eigen::Vector2d p = detail::trilaterate_2d(
|
||
result.uv[v_src.idx()], result.uv[v_tgt.idx()],
|
||
edge_len(mesh.prev(h)), edge_len(mesh.next(h)));
|
||
|
||
if (!vertex_placed[v_new.idx()]) {
|
||
result.uv[v_new.idx()] = p;
|
||
vertex_placed[v_new.idx()] = true;
|
||
vertex_depth[v_new.idx()] = depth;
|
||
} else {
|
||
result.has_seam = true;
|
||
}
|
||
// halfedge_uv: p is the UV of v_new in THIS face (may differ from uv[v_new] at seam)
|
||
detail::set_face_huv_2d(result.halfedge_uv, mesh, h, result.uv, p);
|
||
face_placed[f.idx()] = true;
|
||
enqueue(f);
|
||
}
|
||
};
|
||
|
||
place_component(detail::best_root_face(mesh), Eigen::Vector2d::Zero());
|
||
for (auto f : mesh.faces()) {
|
||
if (face_placed[f.idx()]) continue;
|
||
double xmax; detail::bbox_2d(result.uv, vertex_placed, xmax);
|
||
place_component(f, Eigen::Vector2d(xmax * 1.1 + 1.0, 0.0));
|
||
}
|
||
result.success = true;
|
||
|
||
// ── Holonomy ──────────────────────────────────────────────────────────────
|
||
if (cut && holonomy) {
|
||
holonomy->cut_edge_indices = cut->cut_edge_indices;
|
||
holonomy->translations.clear();
|
||
holonomy->translations.reserve(cut->cut_edge_indices.size());
|
||
holonomy->mobius_maps.clear();
|
||
for (std::size_t ce_idx : cut->cut_edge_indices) {
|
||
Edge_index e = *std::next(mesh.edges().begin(), static_cast<std::ptrdiff_t>(ce_idx));
|
||
Halfedge_index h = mesh.halfedge(e);
|
||
Halfedge_index ho = mesh.opposite(h);
|
||
Halfedge_index hx = mesh.is_border(ho) ? h : ho;
|
||
if (mesh.is_border(hx)) { holonomy->translations.push_back(Eigen::Vector2d::Zero()); continue; }
|
||
Vertex_index vs = mesh.source(hx), vt = mesh.target(hx), vn = mesh.target(mesh.next(hx));
|
||
if (!vertex_placed[vn.idx()]) { holonomy->translations.push_back(Eigen::Vector2d::Zero()); continue; }
|
||
Eigen::Vector2d p_tri = detail::trilaterate_2d(
|
||
result.uv[vs.idx()], result.uv[vt.idx()],
|
||
edge_len(mesh.prev(hx)), edge_len(mesh.next(hx)));
|
||
// Store seam UV for the cut-crossing halfedges
|
||
detail::set_face_huv_2d(result.halfedge_uv, mesh, hx, result.uv, p_tri);
|
||
holonomy->translations.push_back(p_tri - result.uv[vn.idx()]);
|
||
}
|
||
}
|
||
|
||
if (normalise) normalise_euclidean(result);
|
||
return result;
|
||
}
|
||
|
||
// ── Spherical layout ──────────────────────────────────────────────────────────
|
||
|
||
/// Embed the mesh on the unit sphere S² using the spherical metric encoded in x.
|
||
///
|
||
/// Runs priority-BFS trilateration using the spherical law of cosines.
|
||
/// Typical use: genus-0 (sphere-like) surfaces after newton_spherical().
|
||
///
|
||
/// \param mesh Input genus-0 surface mesh.
|
||
/// \param x DOF vector returned by newton_spherical().
|
||
/// \param maps SphericalMaps.
|
||
/// \param cut Optional cut graph (rarely needed for genus-0).
|
||
/// \param holonomy If non-null, receives rotational holonomies (spherical).
|
||
/// \param normalise If true, calls normalise_spherical(): rotates the centroid
|
||
/// to the north pole (Rodrigues rotation formula).
|
||
/// \return Layout3D with .xyz[v] ∈ S² ⊂ ℝ³ for each vertex.
|
||
inline Layout3D spherical_layout(
|
||
ConformalMesh& mesh,
|
||
const std::vector<double>& x,
|
||
const SphericalMaps& maps,
|
||
const CutGraph* cut = nullptr,
|
||
HolonomyData* holonomy = nullptr,
|
||
bool normalise = false)
|
||
{
|
||
const std::size_t nv = mesh.number_of_vertices();
|
||
const std::size_t nf = mesh.number_of_faces();
|
||
Layout3D result;
|
||
result.pos.assign(nv, Eigen::Vector3d::Zero());
|
||
if (nf == 0) return result;
|
||
|
||
auto get_u = [&](Vertex_index v) {
|
||
int iv = maps.v_idx[v]; return (iv>=0) ? x[static_cast<std::size_t>(iv)] : 0.0; };
|
||
auto get_ue = [&](Edge_index e) {
|
||
int ie = maps.e_idx[e]; return (ie>=0) ? x[static_cast<std::size_t>(ie)] : 0.0; };
|
||
auto arc_len = [&](Halfedge_index h) {
|
||
Edge_index e = mesh.edge(h);
|
||
double lam = maps.lambda0[e] + get_u(mesh.source(h)) + get_u(mesh.target(h)) + get_ue(e);
|
||
return 2.0 * std::asin(std::min(std::exp(lam*0.5), 1.0)); };
|
||
|
||
std::vector<bool> vertex_placed(nv, false);
|
||
std::vector<bool> face_placed(nf, false);
|
||
std::vector<int> vertex_depth(nv, std::numeric_limits<int>::max());
|
||
|
||
auto place_component = [&](Face_index f_root) {
|
||
Halfedge_index h0 = mesh.halfedge(f_root);
|
||
Halfedge_index h1 = mesh.next(h0), h2 = mesh.next(h1);
|
||
Vertex_index vA = mesh.source(h0), vB = mesh.source(h1), vC = mesh.source(h2);
|
||
double lAB = arc_len(h0), lBC = arc_len(h1), lCA = arc_len(h2);
|
||
result.pos[vA.idx()] = Eigen::Vector3d(0, 0, 1);
|
||
result.pos[vB.idx()] = Eigen::Vector3d(std::sin(lAB), 0, std::cos(lAB));
|
||
result.pos[vC.idx()] = detail::trilaterate_sph(result.pos[vA.idx()], result.pos[vB.idx()], lCA, lBC);
|
||
vertex_placed[vA.idx()] = vertex_placed[vB.idx()] = vertex_placed[vC.idx()] = true;
|
||
vertex_depth[vA.idx()] = vertex_depth[vB.idx()] = vertex_depth[vC.idx()] = 0;
|
||
face_placed[f_root.idx()] = true;
|
||
|
||
detail::BFSQueue pq;
|
||
auto enqueue = [&](Face_index f) {
|
||
for (auto hf : CGAL::halfedges_around_face(mesh.halfedge(f), mesh)) {
|
||
Halfedge_index ho = mesh.opposite(hf);
|
||
if (mesh.is_border(ho)) continue;
|
||
if (cut && cut->is_cut(mesh.edge(hf))) continue;
|
||
Face_index fadj = mesh.face(ho);
|
||
if (!face_placed[fadj.idx()]) {
|
||
int d = std::max(vertex_depth[mesh.source(ho).idx()],
|
||
vertex_depth[mesh.target(ho).idx()]) + 1;
|
||
pq.push({d, ho});
|
||
}
|
||
}
|
||
};
|
||
enqueue(f_root);
|
||
|
||
while (!pq.empty()) {
|
||
auto [depth, h] = pq.top(); pq.pop();
|
||
Face_index f = mesh.face(h);
|
||
if (face_placed[f.idx()]) continue;
|
||
Vertex_index vs = mesh.source(h), vt = mesh.target(h), vn = mesh.target(mesh.next(h));
|
||
Eigen::Vector3d p = detail::trilaterate_sph(
|
||
result.pos[vs.idx()], result.pos[vt.idx()],
|
||
arc_len(mesh.prev(h)), arc_len(mesh.next(h)));
|
||
if (!vertex_placed[vn.idx()]) {
|
||
result.pos[vn.idx()] = p; vertex_placed[vn.idx()] = true; vertex_depth[vn.idx()] = depth;
|
||
} else result.has_seam = true;
|
||
face_placed[f.idx()] = true; enqueue(f);
|
||
}
|
||
};
|
||
|
||
place_component(detail::best_root_face(mesh));
|
||
for (auto f : mesh.faces()) if (!face_placed[f.idx()]) place_component(f);
|
||
result.success = true;
|
||
|
||
if (cut && holonomy) {
|
||
holonomy->cut_edge_indices = cut->cut_edge_indices;
|
||
holonomy->translations.clear();
|
||
holonomy->translations.reserve(cut->cut_edge_indices.size());
|
||
holonomy->mobius_maps.clear();
|
||
for (std::size_t ce_idx : cut->cut_edge_indices) {
|
||
Edge_index e = *std::next(mesh.edges().begin(), static_cast<std::ptrdiff_t>(ce_idx));
|
||
Halfedge_index hx = mesh.is_border(mesh.opposite(mesh.halfedge(e)))
|
||
? mesh.halfedge(e) : mesh.opposite(mesh.halfedge(e));
|
||
if (mesh.is_border(hx)) { holonomy->translations.push_back(Eigen::Vector2d::Zero()); continue; }
|
||
Vertex_index vs = mesh.source(hx), vt = mesh.target(hx), vn = mesh.target(mesh.next(hx));
|
||
if (!vertex_placed[vn.idx()]) { holonomy->translations.push_back(Eigen::Vector2d::Zero()); continue; }
|
||
Eigen::Vector3d p_tri = detail::trilaterate_sph(
|
||
result.pos[vs.idx()], result.pos[vt.idx()],
|
||
arc_len(mesh.prev(hx)), arc_len(mesh.next(hx)));
|
||
Eigen::Vector3d diff = p_tri - result.pos[vn.idx()];
|
||
// Note: spherical holonomy is geometrically a 3-D rotation, not a 2-D
|
||
// translation. The Vector2d here stores only the (x,y) component of the
|
||
// S²-position difference across the cut, which is an approximation.
|
||
// For accurate spherical holonomy (rotation axis + angle) use the full
|
||
// 3-D positions in result.pos[] directly. Phase 10+ will replace this
|
||
// with a proper SO(3) representation.
|
||
holonomy->translations.push_back(Eigen::Vector2d(diff.x(), diff.y()));
|
||
}
|
||
}
|
||
|
||
if (normalise) normalise_spherical(result);
|
||
return result;
|
||
}
|
||
|
||
// ── HyperIdeal layout (Poincaré disk) — exact trilateration ──────────────────
|
||
|
||
/// Embed the mesh in the Poincaré disk (H²) using the hyperbolic metric encoded in x.
|
||
///
|
||
/// Runs priority-BFS trilateration using exact Möbius-isometric placement:
|
||
/// each new vertex is located by solving the hyperbolic law of cosines and
|
||
/// applying a Möbius map to position it in the disk.
|
||
/// For closed genus-g surfaces (g ≥ 1) a CutGraph is required.
|
||
///
|
||
/// \param mesh Input genus-g surface mesh (g ≥ 1).
|
||
/// \param x DOF vector returned by newton_hyper_ideal()
|
||
/// (vertex b_v and edge a_e variables).
|
||
/// \param maps HyperIdealMaps (lambda0, v_idx, e_idx).
|
||
/// \param cut CutGraph from compute_cut_graph(). Required for closed surfaces.
|
||
/// \param holonomy If non-null, receives the Möbius maps T_i ∈ SU(1,1) per cut edge.
|
||
/// Pass to compute_period_matrix() for holonomy analysis.
|
||
/// \param normalise If true, calls normalise_hyperbolic(): iterative face-area-weighted
|
||
/// Möbius centring (Fréchet mean, 30 iterations) → disk origin.
|
||
/// \return Layout2D with .uv[v] ∈ Poincaré disk (|uv| < 1) for each vertex,
|
||
/// and .halfedge_uv[h] for seam-aware texture atlasing.
|
||
///
|
||
/// \note All vertex positions satisfy |uv[v]| < 1 (inside the Poincaré disk)
|
||
/// if the metric is hyperbolic. Points on or outside the boundary indicate
|
||
/// a non-hyperbolic metric (Gauss–Bonnet violation).
|
||
inline Layout2D hyper_ideal_layout(
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||
ConformalMesh& mesh,
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||
const std::vector<double>& x,
|
||
const HyperIdealMaps& maps,
|
||
const CutGraph* cut = nullptr,
|
||
HolonomyData* holonomy = nullptr,
|
||
bool normalise = false)
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||
{
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||
const std::size_t nv = mesh.number_of_vertices();
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||
const std::size_t nf = mesh.number_of_faces();
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||
const std::size_t nh = mesh.number_of_halfedges();
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||
Layout2D result;
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||
result.uv.assign(nv, Eigen::Vector2d::Zero());
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||
result.halfedge_uv.assign(nh, Eigen::Vector2d::Zero());
|
||
if (nf == 0) return result;
|
||
|
||
auto get_b = [&](Vertex_index v) {
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||
int iv = maps.v_idx[v]; return (iv>=0) ? x[static_cast<std::size_t>(iv)] : 0.0; };
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auto get_a = [&](Edge_index e) {
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||
int ie = maps.e_idx[e]; return (ie>=0) ? x[static_cast<std::size_t>(ie)] : 0.0; };
|
||
auto hyp_dist = [&](Halfedge_index h) {
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||
double bi = get_b(mesh.source(h)), bj = get_b(mesh.target(h)), a = get_a(mesh.edge(h));
|
||
return std::acosh(std::max(1.0, std::cosh(bi+a*0.5)*std::cosh(bj+a*0.5)-std::sinh(bi)*std::sinh(bj))); };
|
||
|
||
std::vector<bool> vertex_placed(nv, false);
|
||
std::vector<bool> face_placed(nf, false);
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||
std::vector<int> vertex_depth(nv, std::numeric_limits<int>::max());
|
||
|
||
auto place_component = [&](Face_index f_root) {
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||
Halfedge_index h0 = mesh.halfedge(f_root);
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||
Halfedge_index h1 = mesh.next(h0), h2 = mesh.next(h1);
|
||
Vertex_index vA = mesh.source(h0), vB = mesh.source(h1), vC = mesh.source(h2);
|
||
double dAB = hyp_dist(h0), dBC = hyp_dist(h1), dCA = hyp_dist(h2);
|
||
result.uv[vA.idx()] = Eigen::Vector2d::Zero();
|
||
result.uv[vB.idx()] = Eigen::Vector2d(std::tanh(dAB*0.5), 0.0);
|
||
result.uv[vC.idx()] = detail::trilaterate_hyp(result.uv[vA.idx()], result.uv[vB.idx()], dAB, dCA, dBC);
|
||
vertex_placed[vA.idx()] = vertex_placed[vB.idx()] = vertex_placed[vC.idx()] = true;
|
||
vertex_depth[vA.idx()] = vertex_depth[vB.idx()] = vertex_depth[vC.idx()] = 0;
|
||
face_placed[f_root.idx()] = true;
|
||
detail::set_root_huv_2d(result.halfedge_uv, mesh, f_root, result.uv);
|
||
|
||
detail::BFSQueue pq;
|
||
auto enqueue = [&](Face_index f) {
|
||
for (auto hf : CGAL::halfedges_around_face(mesh.halfedge(f), mesh)) {
|
||
Halfedge_index ho = mesh.opposite(hf);
|
||
if (mesh.is_border(ho)) continue;
|
||
if (cut && cut->is_cut(mesh.edge(hf))) continue;
|
||
Face_index fadj = mesh.face(ho);
|
||
if (!face_placed[fadj.idx()]) {
|
||
int d = std::max(vertex_depth[mesh.source(ho).idx()],
|
||
vertex_depth[mesh.target(ho).idx()]) + 1;
|
||
pq.push({d, ho});
|
||
}
|
||
}
|
||
};
|
||
enqueue(f_root);
|
||
|
||
while (!pq.empty()) {
|
||
auto [depth, h] = pq.top(); pq.pop();
|
||
Face_index f = mesh.face(h);
|
||
if (face_placed[f.idx()]) continue;
|
||
Vertex_index vs = mesh.source(h), vt = mesh.target(h), vn = mesh.target(mesh.next(h));
|
||
double D = hyp_dist(h), da = hyp_dist(mesh.prev(h)), db = hyp_dist(mesh.next(h));
|
||
Eigen::Vector2d p = detail::trilaterate_hyp(result.uv[vs.idx()], result.uv[vt.idx()], D, da, db);
|
||
if (!vertex_placed[vn.idx()]) {
|
||
result.uv[vn.idx()] = p; vertex_placed[vn.idx()] = true; vertex_depth[vn.idx()] = depth;
|
||
} else result.has_seam = true;
|
||
detail::set_face_huv_2d(result.halfedge_uv, mesh, h, result.uv, p);
|
||
face_placed[f.idx()] = true; enqueue(f);
|
||
}
|
||
};
|
||
|
||
place_component(detail::best_root_face(mesh));
|
||
for (auto f : mesh.faces()) if (!face_placed[f.idx()]) place_component(f);
|
||
result.success = true;
|
||
|
||
// ── Möbius-map holonomy ───────────────────────────────────────────────────
|
||
if (cut && holonomy) {
|
||
holonomy->cut_edge_indices = cut->cut_edge_indices;
|
||
holonomy->translations.clear();
|
||
holonomy->mobius_maps.clear();
|
||
holonomy->mobius_maps.reserve(cut->cut_edge_indices.size());
|
||
for (std::size_t ce_idx : cut->cut_edge_indices) {
|
||
Edge_index e = *std::next(mesh.edges().begin(), static_cast<std::ptrdiff_t>(ce_idx));
|
||
Halfedge_index hcut = mesh.halfedge(e), ho = mesh.opposite(hcut);
|
||
Halfedge_index hx = mesh.is_border(ho) ? hcut : ho;
|
||
if (mesh.is_border(hx)) { holonomy->mobius_maps.push_back(MobiusMap::identity()); continue; }
|
||
Vertex_index vs = mesh.source(hx), vt = mesh.target(hx), vn = mesh.target(mesh.next(hx));
|
||
if (!vertex_placed[vn.idx()]) { holonomy->mobius_maps.push_back(MobiusMap::identity()); continue; }
|
||
double D = hyp_dist(hx), da = hyp_dist(mesh.prev(hx)), db = hyp_dist(mesh.next(hx));
|
||
Eigen::Vector2d p_tri = detail::trilaterate_hyp(result.uv[vs.idx()], result.uv[vt.idx()], D, da, db);
|
||
detail::set_face_huv_2d(result.halfedge_uv, mesh, hx, result.uv, p_tri);
|
||
using C = std::complex<double>;
|
||
// Möbius deck transformation T across cut edge (vs,vt):
|
||
// T is the unique Möbius isometry of the Poincaré disk that:
|
||
// - fixes vs and vt (z1=w1, z2=w2: the cut-edge endpoints are
|
||
// identified across the seam, so T maps each to itself)
|
||
// - maps vn (placed side) → p_tri (virtual side)
|
||
// This uniquely determines the hyperbolic translation/rotation
|
||
// along the geodesic through vs and vt.
|
||
holonomy->mobius_maps.push_back(MobiusMap::from_three(
|
||
C(result.uv[vs.idx()].x(), result.uv[vs.idx()].y()),
|
||
C(result.uv[vs.idx()].x(), result.uv[vs.idx()].y()),
|
||
C(result.uv[vt.idx()].x(), result.uv[vt.idx()].y()),
|
||
C(result.uv[vt.idx()].x(), result.uv[vt.idx()].y()),
|
||
C(result.uv[vn.idx()].x(), result.uv[vn.idx()].y()),
|
||
C(p_tri.x(), p_tri.y())));
|
||
}
|
||
}
|
||
|
||
if (normalise) normalise_hyperbolic(result, mesh);
|
||
return result;
|
||
}
|
||
|
||
// ── Convenience: save layout as OFF ──────────────────────────────────────────
|
||
|
||
/// Write a 2-D layout to disk in OFF format with z = 0. Convenience
|
||
/// helper for quickly inspecting the UV result in any OFF viewer.
|
||
inline void save_layout_off(
|
||
const std::string& path, ConformalMesh& mesh, const Layout2D& layout)
|
||
{
|
||
std::ofstream ofs(path);
|
||
ofs << "OFF\n" << mesh.number_of_vertices() << " " << mesh.number_of_faces() << " 0\n";
|
||
for (auto v : mesh.vertices()) { auto& p = layout.uv[v.idx()]; ofs << p.x() << " " << p.y() << " 0\n"; }
|
||
for (auto f : mesh.faces()) {
|
||
ofs << "3";
|
||
for (auto h : CGAL::halfedges_around_face(mesh.halfedge(f), mesh)) ofs << " " << mesh.target(h).idx();
|
||
ofs << "\n";
|
||
}
|
||
}
|
||
|
||
/// Write a 3-D (spherical) layout to disk in OFF format.
|
||
inline void save_layout_off(
|
||
const std::string& path, ConformalMesh& mesh, const Layout3D& layout)
|
||
{
|
||
std::ofstream ofs(path);
|
||
ofs << "OFF\n" << mesh.number_of_vertices() << " " << mesh.number_of_faces() << " 0\n";
|
||
for (auto v : mesh.vertices()) { auto& p = layout.pos[v.idx()]; ofs << p.x() << " " << p.y() << " " << p.z() << "\n"; }
|
||
for (auto f : mesh.faces()) {
|
||
ofs << "3";
|
||
for (auto h : CGAL::halfedges_around_face(mesh.halfedge(f), mesh)) ofs << " " << mesh.target(h).idx();
|
||
ofs << "\n";
|
||
}
|
||
}
|
||
|
||
} // namespace conformallab
|