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// Copyright (c) 2009 INRIA Sophia-Antipolis (France). // All rights reserved. // // This file is part of CGAL (www.cgal.org). // // $URL: https://github.com/CGAL/cgal/blob/v6.1/Mesh_3/include/CGAL/Mesh_3/Lloyd_move.h $ // $Id: include/CGAL/Mesh_3/Lloyd_move.h b26b07a1242 $ // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial // // // Author(s) : Stephane Tayeb // //****************************************************************************** // File Description : Lloyd move function //****************************************************************************** #ifndef CGAL_MESH_3_LLOYD_MOVE_H #define CGAL_MESH_3_LLOYD_MOVE_H #include #include #include #include #include #include #include #include #include #include #include #include #include namespace CGAL { namespace Mesh_3 { template > class Lloyd_move { typedef typename C3T3::Triangulation Tr; typedef typename Tr::Geom_traits GT; typedef typename Tr::Vertex_handle Vertex_handle; typedef typename Tr::Edge Edge; typedef typename Tr::Facet Facet; typedef typename Tr::Cell_handle Cell_handle; typedef typename Tr::Bare_point Bare_point; typedef typename Tr::Weighted_point Weighted_point; typedef typename std::vector Facet_vector; typedef typename std::vector Cell_vector; typedef typename GT::FT FT; typedef typename GT::Point_2 Point_2; typedef typename GT::Vector_3 Vector_3; typedef typename GT::Tetrahedron_3 Tetrahedron_3; typedef typename GT::Plane_3 Plane_3; typedef typename GT::Aff_transformation_3 Aff_transformation_3; public: typedef SizingField Sizing_field; /** * @brief Returns the move to apply on `v` according to Lloyd optimization * function. */ Vector_3 operator()(const Vertex_handle& v, const Cell_vector& incident_cells, const C3T3& c3t3, const Sizing_field& sizing_field = Sizing_field() ) const { #ifdef CGAL_MESH_3_OPTIMIZER_DEBUG_VERBOSE std::cout << "computing move of: " << &*v << " pos: " << c3t3.triangulation().point(v) << " dim: " << c3t3.in_dimension(v) << std::endl; #endif switch ( c3t3.in_dimension(v) ) { case 3: return lloyd_move_inside_domain(v,incident_cells,c3t3,sizing_field); break; case 2: return lloyd_move_on_boundary(v,c3t3,sizing_field); break; case 1: case 0: case -1: // Don't move edge or corner vertices // N.B.: dimension = -1 is possible if we added points on a far sphere // during initialization return CGAL::NULL_VECTOR; break; default: // Should not happen CGAL_assertion(false); return CGAL::NULL_VECTOR; break; } return CGAL::NULL_VECTOR; } #if defined(CGAL_MESH_3_OPTIMIZER_VERBOSE) \ || defined (CGAL_MESH_3_EXPORT_PERFORMANCE_DATA) static std::string name() { return std::string("Lloyd"); } #endif private: /** * Project_on_plane defines `operator()` to project a point object on the plane `plane`. */ struct Project_on_plane { Project_on_plane(const Plane_3& plane, const C3T3& c3t3) : plane_(plane), c3t3_(c3t3) { } Bare_point operator()(const Bare_point& p) const { return c3t3_.triangulation().geom_traits(). construct_projected_point_3_object()(plane_, p); } private: const Plane_3& plane_; const C3T3& c3t3_; }; /** * `To_2d` defines `operator()` to transform from `Bare_point` to `Point_2`. */ struct To_2d { To_2d(const Aff_transformation_3& to_2d) : to_2d_(to_2d) {} Point_2 operator()(const Bare_point& p) const { return Point_2(to_2d_.transform(p).x(), to_2d_.transform(p).y()); } private: const Aff_transformation_3& to_2d_; }; /** * `To_3d` defines `operator()` to transform from `Point_2` to `Bare_point`. */ struct To_3d { To_3d(const Aff_transformation_3& to_3d) : to_3d_(to_3d) {} Bare_point operator()(const Point_2& p) const { return to_3d_.transform((Bare_point(p.x(),p.y(),0))); } private: const Aff_transformation_3& to_3d_; }; /** * Returns the move for the inside vertex `v`. */ Vector_3 lloyd_move_inside_domain(const Vertex_handle& v, const Cell_vector& incident_cells, const C3T3& c3t3, const Sizing_field& sizing_field) const { // Move data Vector_3 move = CGAL::NULL_VECTOR; FT sum_masses(0); // ----------------------------------- // Tessellate Voronoi cell // ----------------------------------- // Stores vertex that have already been used typedef CGAL::Hash_handles_with_or_without_timestamps Hash_fct; typedef boost::unordered_set Vertex_container; typedef typename Vertex_container::iterator VC_it; Vertex_container treated_vertices; for (typename Cell_vector::const_iterator cit = incident_cells.begin(); cit != incident_cells.end(); ++cit) { const int& k = (*cit)->index(v); // For each vertex of the cell for ( int i=1 ; i<4 ; ++i ) { const Vertex_handle& v1 = (*cit)->vertex((k+i)&3); std::pair is_insert_successful = treated_vertices.insert(v1); if ( ! is_insert_successful.second ) // vertex has already been treated continue; // Vertex has not been treated: turn around edge(v,v1) turn_around_edge(v, Edge(*cit,k,(k+i)&3), c3t3, move, sum_masses, sizing_field); } } CGAL_assertion(sum_masses != 0.0); return move / sum_masses; } /** * Returns the move for the on-boundary vertex `v`. */ Vector_3 lloyd_move_on_boundary(const Vertex_handle& v, const C3T3& c3t3, const Sizing_field& sizing_field) const { CGAL_precondition(c3t3.in_dimension(v) == 2); // get all surface delaunay ball point std::vector points = extract_lloyd_boundary_points(v,c3t3); switch(points.size()) { case 0: // could happen if there is an isolated surface point into mesh case 1: // don't do anything, as the point is already on the surface { return CGAL::NULL_VECTOR; break; } case 2: // segment centroid { const Bare_point& a = points.front(); const Bare_point& b = points.back(); return centroid_segment_move(v, a, b, c3t3, sizing_field); break; } case 3: // triangle centroid { const Bare_point& a = points.at(0); const Bare_point& b = points.at(1); const Bare_point& c = points.at(2); return centroid_triangle_move(v, a, b, c, c3t3, sizing_field); break; } default: // >= 4 points, centroid + projection return centroid_general_move(v, points.begin(), points.end(), c3t3, sizing_field); break; } return CGAL::NULL_VECTOR; } /** * Returns a vector containing the surface delaunay ball centers of the surface * facets that are incident to vertex `v`. */ std::vector extract_lloyd_boundary_points(const Vertex_handle& v, const C3T3& c3t3) const { const Tr& tr = c3t3.triangulation(); typename GT::Construct_point_3 cp = tr.geom_traits().construct_point_3_object(); Facet_vector incident_facets; incident_facets.reserve(64); tr.finite_incident_facets(v, std::back_inserter(incident_facets)); std::vector points; points.reserve(64); const Weighted_point& position = tr.point(v); // Get c3t3's facets incident to v, and add their surface delaunay ball // center to output for ( typename Facet_vector::iterator fit = incident_facets.begin() ; fit != incident_facets.end() ; ++fit ) { if ( ! c3t3.is_in_complex(*fit) ) continue; const Cell_handle cell = fit->first; const int i = fit->second; // In the case of a periodic triangulation, the incident facets of a point // do not necessarily have the same offsets. Worse, the surface centers // might not have the same offset as their facet. Thus, no solution except // calling a function 'get_closest_point(p, q)' that simply returns q // for a non-periodic triangulation, and checks all possible offsets for // periodic triangulations points.push_back(tr.get_closest_point(cp(position), cell->get_facet_surface_center(i))); } return points; } /** * Returns the move from `v` to the centroid of the segment `[a,b]`. */ Vector_3 centroid_segment_move(const Vertex_handle& v, const Bare_point& a, const Bare_point& b, const C3T3& c3t3, const Sizing_field& sizing_field) const { typename GT::Construct_point_3 cp = c3t3.triangulation().geom_traits().construct_point_3_object(); typename GT::Construct_vector_3 vector = c3t3.triangulation().geom_traits().construct_vector_3_object(); const Weighted_point position = c3t3.triangulation().point(v); const Bare_point& p = cp(position); FT da = density_1d(a, v, sizing_field); FT db = density_1d(b, v, sizing_field); CGAL_assertion( !is_zero(da + db) ); return ( (vector(p,a)*da + vector(p,b)*db) / (da + db) ); } /** * Returns the move from `v` to the centroid of triangle `[a,b,c]`. */ Vector_3 centroid_triangle_move(const Vertex_handle& v, const Bare_point& a, const Bare_point& b, const Bare_point& c, const C3T3& c3t3, const Sizing_field& sizing_field) const { typename GT::Construct_point_3 cp = c3t3.triangulation().geom_traits().construct_point_3_object(); typename GT::Construct_vector_3 vector = c3t3.triangulation().geom_traits().construct_vector_3_object(); const Weighted_point& position = c3t3.triangulation().point(v); const Bare_point& p = cp(position); FT da = density_2d(a,v,sizing_field); FT db = density_2d(b,v,sizing_field); FT dc = density_2d(c,v,sizing_field); CGAL_assertion( !is_zero(da+db+dc) ); return ( (da*vector(p,a) + db*vector(p,b) + dc*vector(p,c)) / (da+db+dc) ); } /** * Compute the approximate centroid of the intersection between the 3D voronoi * cell and the boundary. The input is the set of intersection points between * Voronoi edges and the boundary. */ template Vector_3 centroid_general_move(const Vertex_handle& v, ForwardIterator first, ForwardIterator last, const C3T3& c3t3, const Sizing_field& sizing_field) const { CGAL_assertion(std::distance(first,last) > 3); // Fit plane using point-based PCA: compute least square fitting plane Plane_3 plane; Bare_point point; CGAL::linear_least_squares_fitting_3(first, last, plane, point, Dimension_tag<0>(), c3t3.triangulation().geom_traits(), Default_diagonalize_traits()); // Project all points to the plane std::transform(first, last, first, Project_on_plane(plane, c3t3)); CGAL_assertion(std::distance(first, last) > 3); // Get 2D-3D transformations Aff_transformation_3 to_3d = compute_to_3d_transform(plane, *first, c3t3); Aff_transformation_3 to_2d = to_3d.inverse(); // Transform to 2D std::vector points_2d; points_2d.reserve(std::distance(first,last)); std::transform(first, last, std::back_inserter(points_2d), To_2d(to_2d)); // Compute 2D convex hull CGAL_assertion(points_2d.size() > 3); std::vector ch_2d; // AF: We have to debug CGAL::convex_hull_2 = ch_akl_toussaint // as it triggers filter failures unnecessarily CGAL::ch_graham_andrew(points_2d.begin(),points_2d.end(), std::back_inserter(ch_2d)); // Lift back convex hull to 3D std::vector polygon_3d; polygon_3d.reserve(ch_2d.size()); std::transform(ch_2d.begin(), ch_2d.end(), std::back_inserter(polygon_3d), To_3d(to_3d)); // Compute centroid using quadrature sizing return centroid_3d_polygon_move(v, polygon_3d.begin(), polygon_3d.end(), c3t3, sizing_field); } /** * Returns the move from `v` to the centroid of polygon `[first,last]`. * The polygon has to be convex. */ template Vector_3 centroid_3d_polygon_move(const Vertex_handle& v, ForwardIterator first, ForwardIterator last, const C3T3& c3t3, const Sizing_field& sizing_field) const { CGAL_precondition(std::distance(first,last) >= 3); typename GT::Compute_area_3 area = c3t3.triangulation().geom_traits().compute_area_3_object(); typename GT::Construct_centroid_3 centroid = c3t3.triangulation().geom_traits().construct_centroid_3_object(); typename GT::Construct_point_3 cp = c3t3.triangulation().geom_traits().construct_point_3_object(); typename GT::Construct_vector_3 vector = c3t3.triangulation().geom_traits().construct_vector_3_object(); // Vertex current position const Weighted_point& vertex_weighted_position = c3t3.triangulation().point(v); const Bare_point& vertex_position = cp(vertex_weighted_position); // Use as reference point to triangulate const Bare_point& a = *first++; const Bare_point* b = &(*first++); // Treat first point (optimize density_2d call) const Bare_point& c = *first++; const Bare_point& triangle_centroid = centroid(a,*b,c); FT density = density_2d(triangle_centroid, v, sizing_field); FT sum_masses = density * area(a,*b,c); Vector_3 move = sum_masses * vector(vertex_position, triangle_centroid); b = &c; // Next points while ( first != last ) { const Bare_point& c = *first++; const Bare_point& triangle_centroid = centroid(a,*b,c); FT density = density_2d(triangle_centroid, v, sizing_field); FT mass = density * area(a,*b,c); move = move + mass * vector(vertex_position, triangle_centroid); sum_masses += mass; // Go one step forward b = &c; } CGAL_assertion(sum_masses != 0); return move / sum_masses; } /** * Return the transformation from `reference_point` to `plane`. */ Aff_transformation_3 compute_to_3d_transform(const Plane_3& plane, const Bare_point& reference_point, const C3T3& c3t3) const { typename GT::Construct_base_vector_3 base = c3t3.triangulation().geom_traits().construct_base_vector_3_object(); typename GT::Construct_orthogonal_vector_3 orthogonal_vector = c3t3.triangulation().geom_traits().construct_orthogonal_vector_3_object(); Vector_3 u = base(plane, 1); u = u / CGAL::sqrt(u*u); Vector_3 v = base(plane, 2); v = v / CGAL::sqrt(v*v); Vector_3 w = orthogonal_vector(plane); w = w / CGAL::sqrt(w*w); return Aff_transformation_3(u.x(),v.x(),w.x(),reference_point.x(), u.y(),v.y(),w.y(),reference_point.y(), u.z(),v.z(),w.z(),reference_point.z()); } /** * Return density_1d */ template FT density_1d(const Bare_point& p, const Vertex_handle& v, const Sizing_field& sizing_field) const { FT s = sizing_field(p,v); CGAL_assertion(!is_zero(s)); // s^(d+2) return ( 1/(s*s*s) ); } /** * Return density_2d */ template FT density_2d(const Bare_point& p, const Vertex_handle& v, const Sizing_field& sizing_field) const { FT s = use_v ? sizing_field(p,v) : sizing_field(p); CGAL_assertion(!is_zero(s)); // s^(d+2) return ( 1/(s*s*s*s) ); } /** * Return density_3d */ template FT density_3d(const Bare_point& p, const Cell_handle& cell, const Sizing_field& sizing_field) const { FT s = sizing_field(p,cell); CGAL_assertion(!is_zero(s)); // s^(d+2) return ( 1/(s*s*s*s*s) ); } /** * Turns around the edge `edge` and adds the values computed from tets made by * `v` and the circumcenters of cells incident to `edge`. * * Note that this function abundantly uses dual() calls and using a cell base * which stores the circumcenter thus improves its efficiency. */ void turn_around_edge(const Vertex_handle& v, const Edge& edge, const C3T3& c3t3, Vector_3& move, FT& sum_masses, const Sizing_field& sizing_field) const { CGAL_precondition(c3t3.in_dimension(v) == 3); typedef typename Tr::Cell_circulator Cell_circulator; const Tr& tr = c3t3.triangulation(); typename GT::Construct_centroid_3 centroid = tr.geom_traits().construct_centroid_3_object(); typename GT::Construct_point_3 cp = tr.geom_traits().construct_point_3_object(); typename GT::Construct_tetrahedron_3 tetrahedron = tr.geom_traits().construct_tetrahedron_3_object(); typename GT::Construct_translated_point_3 translate = tr.geom_traits().construct_translated_point_3_object(); typename GT::Construct_vector_3 vector = tr.geom_traits().construct_vector_3_object(); typename GT::Compute_volume_3 volume = tr.geom_traits().compute_volume_3_object(); Cell_circulator current_cell = tr.incident_cells(edge); Cell_circulator done = current_cell; // a & b are fixed points const Weighted_point& wa = tr.point(v); const Bare_point& a = cp(wa); Bare_point b = tr.dual(current_cell); const Weighted_point& a_b = tr.point(current_cell, current_cell->index(v)); Vector_3 ba = Vector_3(cp(a_b), b); ++current_cell; CGAL_assertion(current_cell != done); // c & d are moving points Bare_point c = tr.dual(current_cell); const Weighted_point& a_c = tr.point(current_cell, current_cell->index(v)); Vector_3 ca = Vector_3(cp(a_c), c); ++current_cell; CGAL_assertion(current_cell != done); while ( current_cell != done ) { Bare_point d = tr.dual(current_cell); const Weighted_point& a_d = tr.point(current_cell, current_cell->index(v)); Vector_3 da = Vector_3(cp(a_d), d); Tetrahedron_3 tet = tetrahedron(a, translate(a, ba), translate(a, ca), translate(a, da)); Bare_point tet_centroid = centroid(tet); // Compute mass FT density = density_3d(tet_centroid, current_cell, sizing_field); FT abs_volume = CGAL::abs(volume(tet)); FT mass = abs_volume * density; move = move + mass * vector(a, tet_centroid); sum_masses += mass; ++current_cell; c = d; ca = da; } } }; } // end namespace Mesh_3 } //namespace CGAL #include #endif // CGAL_MESH_3_LLOYD_MOVE_H