CGAL 6.1.2 - 3D Simplicial Mesh Data Structures

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User Manual

Authors
  Pierre Alliez, Clément Jamin, Laurent Rineau, Stéphane Tayeb, Jane Tournois, Mariette Yvinec

Figure 62.1 A multi-domain tetrahedral mesh generated from a polyhedral surface with multiple surface patches. The complete triangulation is displayed, including cells that belong or do not belong to the mesh complex, surface patches, and feature edges.

# Mesh Complex

This package is devoted to the representation of 3-Dimensional simplicial mesh data structures.

A 3D simplicial complex is composed of points, line segments, triangles, tetrahedra, and their corresponding combinatorial description (namely vertices, edges, faces and cells). CGAL provides 3D triangulations, that describe both the geometry and connectivity of a 3D simplicial complex, implemented in the the 3D Triangulations and 3D Triangulation Data Structure packages.

We introduce the concept of mesh complex, that encodes extra information on top of a 3D triangulation to represent a valid simplicial complex. A mesh complex describes four subcomplexes of simplices of the support 3D triangulation, per dimension from 0 to 3:

- corner vertices (0D),
- feature edges (1D),
- surface facets (2D),
- domain cells (3D). See Figure 62.2.

The concept `MeshComplex_3InTriangulation_3` is a data structure devised to represent these three-dimensional complexes embedded in a `Triangulation_3`.

Figure 62.2 (Left and Middle) A multi-domain 3D mesh is represented by its tetrahedral cells, with subdomain indices, and surface indices (depicted with different colors here). (Right) The underlying `Triangulation_3` triangulates the whole convex hull of its vertices. The cells that lie outside the meshing domain are drawn in wire frame.

# Examples

## From Tetrahedron Soup to Triangulation_3

In the first example of this section, we build a random `Delaunay_triangulation_3` and use it to build a consistent though connectivity-free tetrahedron soup. The tetrahedron soup is then put back together in a `CGAL::Tetrahedral_remeshing::Remeshing_triangulation_3` before being set as the reference triangulation of a `Mesh_complex_3_in_triangulation_3`.

File SMDS_3/tetrahedron_soup_to_c3t3_example.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Delaunay_triangulation_3.h>
#include <CGAL/point_generators_3.h>
#include <CGAL/Tetrahedral_remeshing/Remeshing_triangulation_3.h>
#include <CGAL/Mesh_complex_3_in_triangulation_3.h>
#include <CGAL/tetrahedron_soup_to_triangulation_3.h>
#include <CGAL/IO/File_medit.h>
#include <vector>
#include <unordered_map>
using K = CGAL::Exact_predicates_inexact_constructions_kernel;
using DT3 = CGAL::Delaunay_triangulation_3<K>;
using Remeshing_triangulation = CGAL::Tetrahedral_remeshing::Remeshing_triangulation_3<K>;
using C3T3 = CGAL::Mesh_complex_3_in_triangulation_3<Remeshing_triangulation>;
using Point_3 = K::Point_3;
using Tetrahedron_3 = K::Tetrahedron_3;
using Vertex_handle = DT3::Vertex_handle;
using Subdomain_index = C3T3::Subdomain_index;
int main(int , char* [])
{
const int nbv = 100;
//a triangulation
DT3 delaunay;
std::unordered_map<Vertex_handle, int> v2i;
std::vector<DT3::Point> points(nbv);
std::vector<Tetrahedron_3> tetrahedra;
std::vector<std::array<int, 4> > tets_by_indices;
std::vector< Subdomain_index> subdomains;
//insert random points
CGAL::Random_points_in_cube_3<Point_3> randp(2.);
int i = 0;
while (i < nbv)
{
points[i] = *randp++;
Vertex_handle v = delaunay.insert(points[i]);
v2i[v] = i++;
}
tetrahedra.reserve(delaunay.number_of_finite_cells());
tets_by_indices.reserve(delaunay.number_of_finite_cells());
subdomains.reserve(delaunay.number_of_finite_cells());
for (DT3::Cell_handle c : delaunay.finite_cell_handles())
{
tetrahedra.push_back(delaunay.tetrahedron(c));
tets_by_indices.push_back( { v2i.at(c->vertex(0)),
v2i.at(c->vertex(1)),
v2i.at(c->vertex(2)),
v2i.at(c->vertex(3)) } );
subdomains.push_back(Subdomain_index(1));
}
//build triangulation from tetrahedra
Remeshing_triangulation tr
= CGAL::tetrahedron_soup_to_triangulation_3<Remeshing_triangulation>(tetrahedra);
//build triangulation from indices
Remeshing_triangulation tr2
= CGAL::tetrahedron_soup_to_triangulation_3<Remeshing_triangulation>(
points, tets_by_indices,
CGAL::parameters::subdomain_indices(std::cref(subdomains)));
//build a C3T3
C3T3 c3t3;
c3t3.set_triangulation(tr);
std::cout << "Number of corners: " << c3t3.number_of_corners() << std::endl;
std::cout << "Number of edges: " << c3t3.number_of_edges() << std::endl;
std::cout << "Number of facets: " << c3t3.number_of_facets() << std::endl;
std::cout << "Number of cells: " << c3t3.number_of_cells() << std::endl;
std::ofstream ofs("c3t3_output.mesh");
CGAL::IO::write_MEDIT(ofs, c3t3);
return EXIT_SUCCESS;
}
CGAL::Delaunay_triangulation_3
CGAL::Exact_predicates_inexact_constructions_kernel
CGAL::Mesh_complex_3_in_triangulation_3
A data structure to represent and maintain a 3D complex embedded in a 3D triangulation.
Definition: Mesh_complex_3_in_triangulation_3.h:190
CGAL::Tetrahedral_remeshing::Remeshing_triangulation_3
CGAL::IO::write_MEDIT
void write_MEDIT(std::ostream &os, const T3 &t3, const NamedParameters &np=parameters::default_values())
outputs a mesh complex to the medit (.mesh) file format.
Definition: File_medit.h:780

## Input/Output Example

The example below illustrates how to use the IO functions for reading and writing a triangulation with the `Medit` file format (See [1] for a comprehensive description of this file format.).

File SMDS_3/c3t3_example.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Triangulation_3.h>
#include <CGAL/Triangulation_data_structure_3.h>
#include <CGAL/Simplicial_mesh_cell_base_3.h>
#include <CGAL/Simplicial_mesh_vertex_base_3.h>
#include <CGAL/Mesh_complex_3_in_triangulation_3.h>
#include <CGAL/tetrahedral_remeshing.h>
#include <CGAL/tags.h>
#include <CGAL/IO/File_medit.h>
#include <fstream>
using K = CGAL::Exact_predicates_inexact_constructions_kernel;
using Subdomain_index = int;
using Surface_patch_index = unsigned char;
using Curve_index = char;
using Corner_index = short;
using Cb = CGAL::Simplicial_mesh_cell_base_3<K, Subdomain_index, Surface_patch_index>;
using Vb = CGAL::Simplicial_mesh_vertex_base_3<K, Subdomain_index, Surface_patch_index,
Curve_index, Corner_index>;
using Tds = CGAL::Triangulation_data_structure_3<Vb, Cb, CGAL::Sequential_tag>;
using Triangulation = CGAL::Triangulation_3<K, Tds>;
using C3t3 = CGAL::Mesh_complex_3_in_triangulation_3<Triangulation>;
int main(int argc, char* argv[])
{
std::cout.precision(17);
std::cerr.precision(17);
std::string filename = (argc > 1) ? std::string(argv[1])
: CGAL::data_file_path("meshes/elephant.mesh");
Triangulation tr;
std::ifstream is(filename, std::ios_base::in);
if(!CGAL::IO::read_MEDIT(is, tr))
{
std::cerr << "Failed to read" << std::endl;
return EXIT_FAILURE;
}
// [call a remeshing algorithm]
std::ofstream os("after_remeshing.mesh");
CGAL::IO::write_MEDIT(os, tr, CGAL::parameters::all_vertices(true));
os.close();
Triangulation tr2;
std::ifstream is2("after_remeshing.mesh");
if(!CGAL::IO::read_MEDIT(is2, tr2))
{
std::cerr << "Failed to read (#2)" << std::endl;
return EXIT_FAILURE;
}
std::cout << "Done" << std::endl;
return EXIT_SUCCESS;
}
CGAL::Simplicial_mesh_cell_base_3
The class Simplicial_mesh_cell_base_3 is a model of the concept SimplicialMeshCellBase_3.
Definition: Simplicial_mesh_cell_base_3.h:76
CGAL::Simplicial_mesh_vertex_base_3
The class Simplicial_mesh_vertex_base_3 is a model of the concept SimplicialMeshVertexBase_3.
Definition: Simplicial_mesh_vertex_base_3.h:95
CGAL::Triangulation_3
CGAL::Triangulation_data_structure_3
CGAL::IO::read_MEDIT
bool read_MEDIT(std::istream &in, T3 &t3, const NamedParameters &np=parameters::default_values())
reads a mesh complex written in the medit (.mesh) file format.
Definition: File_medit.h:910

## More Examples In Other Packages

The `Mesh_complex_3_in_triangulation_3` is widely used in the 3D Mesh Generation package. Many more examples can be found in its Examples section.

The package Tetrahedral Remeshing also makes use of the `Mesh_complex_3_in_triangulation_3` since it serves as a post-processing for tetrahedral mesh generation. Some examples can be found in the Examples section.

# Implementation History

The code of the `MeshComplex_3InTriangulation_3` and its variants were initially part of the package 3D Mesh Generation. With the meshing and remeshing processes becoming more versatile, it was moved to its own package in the release 5.6 of CGAL.