Definition

The concept SurfaceMeshComplex_2InTriangulation_3 describes a data structure designed to represent a two dimensional pure complex embedded in a three dimensional triangulation.

A complex is a set \( C\) of faces such that:

any subface of a face in \( C\) is a face of \( C\)
two faces of \( C\) are disjoint or share a common subface

The complex is two dimensional, if its faces have dimension at most two. It is pure if any face in the complex is a subface of some face of maximal dimension. Thus, a two dimensional pure complex is a set of facets together with their edges and vertices. A two dimensional pure complex embedded in a three dimensional triangulation is a subset of the facets of this triangulation, together with their edges and vertices.

The concept SurfaceMeshComplex_2InTriangulation_3 is particularly suited to handle surface meshes obtained as the restriction to a surface of a three dimensional Delaunay triangulation. A model of this concept is a type to be plugged as first template parameter in the function template CGAL::make_surface_mesh().

Has models
: CGAL::Surface_mesh_complex_2_in_triangulation_3<Tr>

See also: CGAL::make_surface_mesh()

Types
`SurfaceMeshComplex_2InTriangulation_3` provides the following types.
enum	Face_status { NOT_IN_COMPLEX , BOUNDARY , REGULAR , SINGULAR }
	A type to describe the status of a face (facet, edge, or vertex) with respect to the 2D pure complex. More...

typedef unspecified_type	Triangulation
	The type of the embedding 3D triangulation.

typedef Triangulation::Vertex_handle	Vertex_handle
	The type of the embedding triangulation vertex handles.

typedef Triangulation::Cell_handle	Cell_handle
	The type of the embedding triangulation cell handles.

typedef Triangulation::Facet	Facet
	The type of the embedding triangulation facets.

typedef Triangulation::Edge	Edge
	The type of the embedding triangulation edges.

typedef Triangulation::size_type	size_type
	Size type (an unsigned integral type)

typedef unspecified_type	Facet_iterator
	An iterator type to visit the facets of the 2D complex.

typedef unspecified_type	Edge_iterator
	An iterator type to visit the edges of the 2D complex.

typedef unspecified_type	Vertex_iterator
	An iterator type to visit vertices of the 2D complex.

typedef unspecified_type	Boundary_edges_iterator
	An iterator type to visit the boundary edges of the 2D complex.

Creation
	SurfaceMeshComplex_2InTriangulation_3 (Triangulation &t3)
	Builds an empty 2D complex embedded in the triangulation `t3`

Member access
/**
Triangulation &	triangulation ()
	Returns the reference to the triangulation.

Modifications
void	add_to_complex (Facet f)
	Adds facet `f` to the 2D complex.

void	add_to_complex (Cell_handle c, int i)
	Adds facet `(c,i)` to the 2D complex.

void	remove_from_complex (Facet f)
	Removes facet `f` from the 2D complex.

void	remove_from_complex (Cell_handle c, int i)
	Removes facet `(c,i)` from the 2D complex.

Queries
Queries on the status of individual face with respect to the 2D complex.
size_type	number_of_facets () const
	Returns the number of facets that belong to the 2D complex.

Face_status	face_status (Facet f)
	Returns the status of the facet `f` with respect to the 2D complex.

Face_status	face_status (Cell_handle c, int i)
	Returns the status of the facet `(c,i)` with respect to the 2D complex.

Face_status	face_status (Edge e)
	Returns the status of edge `e` in the 2D complex.

Face_status	face_status (Cell_handle c, int i, int j)
	Returns the status of edge `(c,i,j)` in the 2D complex.

Face_status	face_status (Vertex_handle v)
	Returns the status of vertex `v` in the 2D complex.

bool	is_in_complex (Facet f)
	Returns `true`, if the facet `f` belongs to the 2D complex.

bool	is_in_complex (Cell_handle c, int i)
	Returns `true`, if the facet `(c,i)` belongs to the 2D complex.

bool	is_in_complex (Edge e)
	Returns `true`, if the edge `e` belongs to the 2D complex.

bool	is_in_complex (Cell_handle c, int i, int j)
	Returns `true`, if the edge `(c,i,j)` belongs to the 2D complex.

bool	is_in_complex (Vertex_handle v)
	Returns `true`, if the vertex `v` belongs to the 2D complex.

bool	is_regular_or_boundary_for_vertices (Vertex_handle v)
	Returns true if the status of vertex `v` is `REGULAR` or `BOUNDARY`.

Traversal of the complex
The data structure provides iterators to visit the facets, edges and vertices of the complex. All those iterators are bidirectional and non mutable.
Facet_iterator	facets_begin ()
	Returns an iterator with value type `Facet` to visit the facets of the 2D complex.

Facet_iterator	facets_end ()
	Returns the past the end iterator for the above iterator.

Edge_iterator	edges_begin ()
	Returns an iterator with value type `Edge` to visit the edges of the 2D complex which are not isolated.

Edge_iterator	edges_end ()
	Returns the past the end iterator for the above iterator.

Boundary_edges_iterator	boundary_edges_begin ()
	Returns an iterator with value type `Edge` to visit the boundary edges of the complex.

Boundary_edges_iterator	boundary_edges_end ()
	Returns the past the end iterator for the above iterator.

Vertex_iterator	vertices_begin ()
	Returns an iterator with value type `Vertex_handle` to visit the vertices of the 2D complex.

Vertex_iterator	vertices_end ()
	Returns the past the end iterator for the above iterator.

template<class OutputIterator >
OutputIterator	incident_facets (Vertex_handle v, OutputIterator facets)
	Copies the facets of the complex incident to `v` to the output iterator `facets`.

The following function is the basic function to walk on the 2D complex
Facet	neighbor (Facet f, int j)
	Returns the facet of the complex which is the neighbor of the facet `f` opposite to the vertex with index `j` of `f`.

Facet	neighbor (Cell_handle c, int i, int j)
	Returns the facet of the complex which is the neighbor of the facet `f` opposite to the vertex with index `j` of `f`.

Member Typedef Documentation

◆ Triangulation

typedef unspecified_type SurfaceMeshComplex_2InTriangulation_3::Triangulation

The type of the embedding 3D triangulation.

Must be a model of SurfaceMeshTriangulation_3.

Member Enumeration Documentation

◆ Face_status

enum SurfaceMeshComplex_2InTriangulation_3::Face_status

A type to describe the status of a face (facet, edge, or vertex) with respect to the 2D pure complex.

A NOT_IN_COMPLEX face does not belong to the 2D complex. Facets can only be NOT_IN_COMPLEX or REGULAR depending on whether they belong to the 2D complex on not. Edges and vertices can be NOT_IN_COMPLEX, BOUNDARY, REGULAR or SINGULAR. An edge in the complex is BOUNDARY, REGULAR, or SINGULAR, if it is incident to respectively 1, 2, or 3 or more facets in the complex. The status of a vertex is determined by the adjacency graph of the facets of the 2D complex incident to that vertex. The vertex of the 2D complex is BOUNDARY, if this adjacency graph is a simple path, it is REGULAR, if the adjacency graph is cyclic, and SINGULAR in any other case.

Enumerator
NOT_IN_COMPLEX
BOUNDARY
REGULAR
SINGULAR

Member Function Documentation

◆ incident_facets()

template<class OutputIterator >

OutputIterator SurfaceMeshComplex_2InTriangulation_3::incident_facets	(	Vertex_handle	v,
		OutputIterator	facets
	)

Copies the facets of the complex incident to v to the output iterator facets.

Returns the resulting output iterator.

Precondition: c2t3.triangulation().dimension() == 3, v != Vertex_handle(), c2t3.triangulation().is_vertex(v).

◆ is_regular_or_boundary_for_vertices()

bool SurfaceMeshComplex_2InTriangulation_3::is_regular_or_boundary_for_vertices ( Vertex_handle v )

Returns true if the status of vertex v is REGULAR or BOUNDARY.

Precondition: All the edges of the complex incident to v are REGULAR or BOUNDARY.

◆ neighbor() [1/2]

Facet SurfaceMeshComplex_2InTriangulation_3::neighbor	(	Cell_handle	c,
		int	i,
		int	j
	)

Returns the facet of the complex which is the neighbor of the facet f opposite to the vertex with index j of f.

See above.

◆ neighbor() [2/2]

Facet SurfaceMeshComplex_2InTriangulation_3::neighbor	(	Facet	f,
		int	j
	)

Returns the facet of the complex which is the neighbor of the facet f opposite to the vertex with index j of f.

The vertices of the facet f = (cell c, i) are numbered (0,1,2) (according to the vertex_triple_index(i,j) member function of Triangulation_3) in such a way that facet f is oriented by the outward normal of tetraedra c. If there is no such neighbor, or if the edge is singular the functions returns Facet().

Definition

Types

Creation

Member access

Modifications

Queries

Traversal of the complex