SurfaceMeshTriangulation

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SurfaceMeshTriangulation_3

Definition

The concept SurfaceMeshTriangulation_3 describes the triangulation type used by the surface mesher make_surface_mesh to represent the three dimensional triangulation embedding the surface mesh. Thus, this concept describes the requirements for the triangulation type SurfaceMeshC2T3::Triangulation nested in the model of SurfaceMeshComplex2InTriangulation3 plugged as the template parameter SurfaceMeshC2T3 of make_surface_mesh. It also describes the requirements for the triangulation type plugged in the class Surface_mesh_complex_2_in_triangulation_3<Tr>.

Types

SurfaceMeshTriangulation_3::Point
	The point type. It must be DefaultConstructible, CopyConstructible and Assignable.

Vertices and cells of the triangulation are manipulated via handles, which support the two dereference operators operator* and operator->.

SurfaceMeshTriangulation_3::Vertex_handle
	Handle to a data representing a vertex. Vertex_handle must be a model of Handle and its value type must be model of TriangulationDataStructure_3::Vertex.
SurfaceMeshTriangulation_3::Cell_handle
	Handle to a data representing a cell. Cell_handle must be a model of Handle and its value type must be model of TriangulationDataStructure_3::Cell.

typedef CGAL::Triple<Cell_handle, int, int>
	Edge;	The edge type.
typedef std::pair<Cell_handle, int>
	Facet;	The facet type.

The following iterators allow one to visit all finite vertices, edges and facets of the triangulation.

SurfaceMeshTriangulation_3::Finite_vertices_iterator
	Iterator over finite vertices
SurfaceMeshTriangulation_3::Finite_edges_iterator
	Iterator over finite edges
SurfaceMeshTriangulation_3::Finite_facets_iterator
	Iterator over finite facets

SurfaceMeshTriangulation_3::Geom_traits
	The geometric traits class. Must be a model of DelaunayTriangulationTraits_3.

Creation

SurfaceMeshTriangulation_3 t;
	default constructor.

SurfaceMeshTriangulation_3 t ( tr);
	Copy constructor. All vertices and faces are duplicated.

Assignment

SurfaceMeshTriangulation_3 &	t = tr	The triangulation tr is duplicated, and modifying the copy after the duplication does not modify the original. The previous triangulation held by t is deleted.

void	t.clear ()	Deletes all finite vertices and all cells of t.

Access Functions

int	t.dimension ()	Returns the dimension of the affine hull.

DelaunayTriangulationTraits_3	t.geom_traits ()	Returns a const reference to a model of DelaunayTriangulationTraits_3.

Voronoi diagram

Object

t.dual ( Facet f)

Returns the dual of facet f, which is
in dimension 3: either a segment, if the two cells incident to f are finite, or a ray, if one of them is infinite;
in dimension 2: a point.

Queries

A point p is said to be in conflict with a cell c in dimension 3 (resp. a facet f in dimension 2) iff t.side_of_sphere(c, p) (resp. t.side_of_circle(f, p)) returns ON_BOUNDED_SIDE. The set of cells (resp. facets in dimension 2) which are in conflict with p is connected, and it forms a hole.

template <class OutputIteratorBoundaryFacets, class OutputIteratorCells, class OutputIteratorInternalFacets>

Triple<OutputIteratorBoundaryFacets, OutputIteratorCells, OutputIteratorInternalFacets>

t.find_conflicts ( Point p,
Cell_handle c,
OutputIteratorBoundaryFacets bfit,
OutputIteratorCells cit,
OutputIteratorInternalFacets ifit)

Computes the conflict hole induced by p. The starting cell (resp. facet) c must be in conflict. Then this function returns respectively in the output iterators:
- cit: the cells (resp. facets) in conflict.
- bfit: the facets (resp. edges) on the boundary, that is, the facets (resp. edges) (t, i) where the cell (resp. facet) t is in conflict, but t->neighbor(i) is not.
- ifit: the facets (resp. edges) inside the hole, that is, delimiting two cells (resp facets) in conflict.
Returns the Triple composed of the resulting output iterators.

The following iterators allow the user to visit facets, edges and vertices of the triangulation.

Finite_vertices_iterator

t.finite_vertices_begin ()

Starts at an arbitrary finite vertex. Then ++ and -- will iterate over finite vertices. Returns finite_vertices_end() when t.number_of_vertices()

=0.

Finite_vertices_iterator

t.finite_vertices_end ()

Past-the-end iterator

Finite_edges_iterator

t.finite_edges_begin ()

Starts at an arbitrary finite edge. Then ++ and -- will iterate over finite edges. Returns finite_edges_end() when t.dimension()

<1.

Finite_edges_iterator

t.finite_edges_end ()

Past-the-end iterator

Finite_facets_iterator

t.finite_facets_begin ()

Starts at an arbitrary finite facet. Then ++ and -- will iterate over finite facets. Returns finite_facets_end() when t.dimension()

<2.

Finite_facets_iterator

t.finite_facets_end ()

Past-the-end iterator

template <class OutputIterator>

OutputIterator

t.incident_cells ( Vertex_handle v, OutputIterator cells)

Copies the Cell_handles of all cells incident to v to the output iterator cells. If t.dimension()

<3, then do nothing. Returns the resulting output iterator.

Precondition:

Vertex_handle(), t.is_vertex(v).

template <class OutputIterator>

OutputIterator

t.incident_cells ( Vertex_handle v, OutputIterator cells)

Copies the Cell_handles of all cells incident to v to the output iterator cells. If t.dimension()

<3, then do nothing. Returns the resulting output iterator.

bool

t.is_vertex ( Point p, Vertex_handle & v)

Tests whether p is a vertex of t by locating p in the triangulation. If p is found, the associated vertex v is given.

bool

t.is_edge ( Vertex_handle u, Vertex_handle v, Cell_handle & c, int & i, int & j)

Tests whether (u,v) is an edge of t. If the edge is found, it gives a cell c having this edge and the indices i and j of the vertices u and v in c, in this order.

Precondition:

u and v are vertices of t.

bool

t.is_infinite ( const Vertex_handle v)

true, iff vertex v is the infinite vertex.

bool

t.is_infinite ( const Cell_handle c)

true, iff c is incident to the infinite vertex.

Precondition:

t.dimension()

=3.

Facet

t.mirror_facet ( Facet f)

Returns the same facet viewed from the other adjacent cell.

int

t.vertex_triple_index ( const int i, const int j)

Return the indexes of the jth vertex of the facet of a cell opposite to vertex i.

Point location

Cell_handle

t.locate ( Point query, Cell_handle start = Cell_handle())

If the point query lies inside the convex hull of the points, the cell that contains the query in its interior is returned. If query lies on a facet, an edge or on a vertex, one of the cells having query on its boundary is returned.
If the point query lies outside the convex hull of the points, an infinite cell with vertices

{ p, q, r, } is returned such that the tetrahedron

( p, q, r, query ) is positively oriented (the rest of the triangulation lies on the other side of facet

( p, q, r )).
Note that locate works even in degenerate dimensions: in dimension 2 (resp. 1, 0) the Cell_handle returned is the one that represents the facet (resp. edge, vertex) containing the query point.
The optional argument start is used as a starting place for the search.

Cell_handle

t.locate ( Point query, Locate_type & lt, int & li, int & lj, Cell_handle start = Cell_handle())

If query lies inside the affine hull of the points, the

k-face (finite or infinite) that contains query in its interior is returned, by means of the cell returned together with lt, which is set to the locate type of the query (VERTEX, EDGE, FACET, CELL, or OUTSIDE_CONVEX_HULL if the cell is infinite and query lies strictly in it) and two indices li and lj that specify the

k-face of the cell containing query.
If the

k-face is a cell, li and lj have no meaning; if it is a facet (resp. vertex), li gives the index of the facet (resp. vertex) and lj has no meaning; if it is and edge, li and lj give the indices of its vertices.
If the point query lies outside the affine hull of the points, which can happen in case of degenerate dimensions, lt is set to OUTSIDE_AFFINE_HULL, and the cell returned has no meaning. As a particular case, if there is no finite vertex yet in the triangulation, lt is set to OUTSIDE_AFFINE_HULL and locate returns the default constructed handle.
The optional argument start is used as a starting place for the search.

template <class CellIt>

Vertex_handle

t.insert_in_hole ( Point p, CellIt cell_begin, CellIt cell_end, Cell_handle begin, int i)

Creates a new vertex by starring a hole. It takes an iterator range [cell_begin; cell_end[ of Cell_handles which specifies a hole: a set of connected cells (resp. facets in dimension 2) which is star-shaped wrt p. (begin, i) is a facet (resp. an edge) on the boundary of the hole, that is, begin belongs to the set of cells (resp. facets) previously described, and begin->neighbor(i) does not. Then this function deletes all the cells (resp. facets) describing the hole, creates a new vertex v, and for each facet (resp. edge) on the boundary of the hole, creates a new cell (resp. facet) with v as vertex. Then v->set_point(p) is called and v is returned.

Precondition:

t.dimension()

2, the set of cells (resp. facets in dimension 2) is connected, its boundary is connected, and p lies inside the hole, which is star-shaped wrt p.

Has Models

Any 3D Delaunay triangulation class of CGAL