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CGAL 4.8 - 3D Surface Mesh Generation

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SurfaceMeshTriangulation_3 Concept Reference
Concepts

## Definition

The concept `SurfaceMeshTriangulation_3` describes the triangulation type used by the surface mesher `CGAL::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 `SurfaceMeshComplex_2InTriangulation_3` plugged as the template parameter `SurfaceMeshC2T3` of `CGAL::make_surface_mesh()`. It also describes the requirements for the triangulation type plugged in the class `CGAL::Surface_mesh_complex_2_in_triangulation_3<Tr>`.

Has Models:
  Any 3D Delaunay triangulation class of CGAL

See Also
  `CGAL::Triangulation_3<TriangulationTraits_3,TriangulationDataStructure_3>`
  `CGAL::Delaunay_triangulation_3<DelaunayTriangulationTraits_3,TriangulationDataStructure_3>`
  `SurfaceMeshComplex_2InTriangulation_3`
  `CGAL::Surface_mesh_complex_2_in_triangulation_3<Tr>`
  `CGAL::make_surface_mesh()`

## Types
Vertices and cells of the triangulation are manipulated via handles, which support the two dereference operators and `operator->`. The following iterators allow one to visit all finite vertices, edges and facets of the triangulation.
typedef unspecified_type | Point
| The point type. More...
typedef unspecified_type | Vertex_handle
| Handle to a data representing a vertex. More...
typedef unspecified_type | Cell_handle
| Handle to a data representing a cell. More...
typedef CGAL::Triple < Cell_handle, int, int > | Edge
| The edge type.
typedef std::pair< Cell_handle, int > | Facet
| The facet type.
typedef unspecified_type | Finite_vertices_iterator
| Iterator over finite vertices.
typedef unspecified_type | Finite_edges_iterator
| Iterator over finite edges.
typedef unspecified_type | Finite_facets_iterator
| Iterator over finite facets.
typedef unspecified_type | Geom_traits
| The geometric traits class. More...

## Creation
| SurfaceMeshTriangulation_3 ()
| Default constructor.
| SurfaceMeshTriangulation_3 (SurfaceMeshTriangulation_3 tr)
| Copy constructor. More...

## Assignment
SurfaceMeshTriangulation_3 & | operator= (const SurfaceMeshTriangulation_3 &tr)
| The triangulation `tr` is duplicated, and modifying the copy after the duplication does not modify the original. More...
void | clear ()
| Deletes all finite vertices and all cells of `t`.

## Access Functions
int | dimension () const
| Returns the dimension of the affine hull.
const DelaunayTriangulationTraits_3 & | geom_traits () const
| Returns a const reference to a model of `DelaunayTriangulationTraits_3`.

## Voronoi diagram
Object | dual (Facet f) const
| Returns the dual of facet `f`. More...

## 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 > | find_conflicts (Point p, Cell_handle c, OutputIteratorBoundaryFacets bfit, OutputIteratorCells cit, OutputIteratorInternalFacets ifit)
| Computes the conflict hole induced by `p`. More...
The following iterators allow the user to visit facets, edges and vertices of the triangulation.
Finite_vertices_iterator | finite_vertices_begin () const
| Starts at an arbitrary finite vertex.
Finite_vertices_iterator | finite_vertices_end () const
| Past-the-end iterator.
Finite_edges_iterator | finite_edges_begin () const
| Starts at an arbitrary finite edge.
Finite_edges_iterator | finite_edges_end () const
| Past-the-end iterator.
Finite_facets_iterator | finite_facets_begin () const
| Starts at an arbitrary finite facet.
Finite_facets_iterator | finite_facets_end () const
| Past-the-end iterator.
template<class OutputIterator >
OutputIterator | incident_cells (Vertex_handle v, OutputIterator cells) const
| Copies the `Cell_handle`s of all cells incident to `v` to the output iterator `cells`. More...
template<class OutputIterator >
OutputIterator | incident_cells (Vertex_handle v, OutputIterator cells) const
| Copies the `Cell_handle`s of all cells incident to `v` to the output iterator `cells`. More...
bool | is_vertex (const Point &p, Vertex_handle &v) const
| Tests whether `p` is a vertex of `t` by locating `p` in the triangulation. More...
bool | is_edge (Vertex_handle u, Vertex_handle v, Cell_handle &c, int &i, int &j) const
| Tests whether `(u,v)` is an edge of `t`. More...
bool | is_infinite (const Vertex_handle v) const
| Returns `true`, iff vertex `v` is the infinite vertex.
bool | is_infinite (const Cell_handle c) const
| Returns `true`, iff `c` is incident to the infinite vertex. More...
Facet | mirror_facet (Facet f) const
| Returns the same facet seen from the other adjacent cell.
int | vertex_triple_index (const int i, const int j)
| Return the indexes of the `j`th vertex of the facet of a cell opposite to vertex `i`.

## Point location
Cell_handle | locate (const Point &query, Cell_handle start=Cell_handle()) const
| If the point `query` lies inside the convex hull of the points, the cell that contains the query in its interior is returned. More...
Cell_handle | locate (const Point &query, Locate_type &lt, int &li, int &lj, Cell_handle start=Cell_handle()) const
| 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`. More...
template<class CellIt >
Vertex_handle | insert_in_hole (Point p, CellIt cell_begin, CellIt cell_end, Cell_handle begin, int i)
| Creates a new vertex by starring a hole. More...

## Member Typedef Documentation

typedef unspecified_type 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 unspecified_type SurfaceMeshTriangulation_3::Geom_traits

The geometric traits class.

Must be a model of `DelaunayTriangulationTraits_3`.

typedef unspecified_type SurfaceMeshTriangulation_3::Point

The point type.

It must be DefaultConstructible, CopyConstructible and Assignable.

typedef unspecified_type 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`.

## Constructor & Destructor Documentation

SurfaceMeshTriangulation_3::SurfaceMeshTriangulation_3 | ( | SurfaceMeshTriangulation_3 | tr) |  |

Copy constructor.

All vertices and faces are duplicated.

## Member Function Documentation

Object SurfaceMeshTriangulation_3::dual | ( | Facet | f) |  | const

Returns the dual of facet `f`.

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.

template<class OutputIteratorBoundaryFacets , class OutputIteratorCells , class OutputIteratorInternalFacets >

Triple<OutputIteratorBoundaryFacets, OutputIteratorCells, OutputIteratorInternalFacets> SurfaceMeshTriangulation_3::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.

template<class OutputIterator >

OutputIterator SurfaceMeshTriangulation_3::incident_cells | ( | Vertex_handle | v,
|  | OutputIterator | cells
| ) |  | const

Copies the `Cell_handle`s of all cells incident to `v` to the output iterator `cells`.

If `t.dimension() < 3`, then do nothing. Returns the resulting output iterator.

Precondition
  `v != Vertex_handle()`, `t.is_vertex(v)`.

template<class OutputIterator >

OutputIterator SurfaceMeshTriangulation_3::incident_cells | ( | Vertex_handle | v,
|  | OutputIterator | cells
| ) |  | const

Copies the `Cell_handle`s of all cells incident to `v` to the output iterator `cells`.

If `t.dimension() < 3`, then do nothing. Returns the resulting output iterator.

template<class CellIt >

Vertex_handle SurfaceMeshTriangulation_3::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_handle`s 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`.

bool SurfaceMeshTriangulation_3::is_edge | ( | Vertex_handle | u,
|  | Vertex_handle | v,
|  | Cell_handle & | c,
|  | int & | i,
|  | int & | j
| ) |  | const

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 SurfaceMeshTriangulation_3::is_infinite | ( | const Cell_handle | c) |  | const

Returns `true`, iff `c` is incident to the infinite vertex.

Precondition
  `t.dimension() == 3`.

bool SurfaceMeshTriangulation_3::is_vertex | ( | const Point & | p,
|  | Vertex_handle & | v
| ) |  | const

Tests whether `p` is a vertex of `t` by locating `p` in the triangulation.

If `p` is found, the associated vertex `v` is given.

Cell_handle SurfaceMeshTriangulation_3::locate | ( | const Point & | query,
|  | Cell_handle | start = `Cell_handle()`
| ) |  | const

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 \( \{ p, q, r, \infty\}\) 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 SurfaceMeshTriangulation_3::locate | ( | const Point & | query,
|  | Locate_type & | lt,
|  | int & | li,
|  | int & | lj,
|  | Cell_handle | start = `Cell_handle()`
| ) |  | const

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.

SurfaceMeshTriangulation_3& SurfaceMeshTriangulation_3::operator= | ( | const SurfaceMeshTriangulation_3 & | 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.