CGAL 6.2 - 3D Triangulations

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CGAL::Triangulation_3< Traits, TDS, SLDS > Class Template Reference
Reference Manual » Triangulation Classes

`#include <CGAL/Triangulation_3.h>`

## Inherits from

CGAL::Triangulation_utils_3.

Inherited by CGAL::Regular_triangulation_3< Traits, TDS, SLDS >.

## Definition

template<typename Traits, typename TDS, typename SLDS>
class CGAL::Triangulation_3< Traits, TDS, SLDS >

The class `Triangulation_3` represents a 3-dimensional tetrahedralization of points.

Template Parameters

  Traits | is the geometric traits class and must be a model of `TriangulationTraits_3`.
  TDS | is the triangulation data structure and must be a model of `TriangulationDataStructure_3`. `Default` may be used, with default type `Triangulation_data_structure_3<Triangulation_vertex_base_3<Traits>, Triangulation_cell_base_3<Traits> >`. Any custom type can be used instead of `Triangulation_vertex_base_3` and `Triangulation_cell_base_3`, provided that they are models of the concepts `TriangulationVertexBase_3` and `TriangulationCellBase_3`, respectively.
  SLDS | is an optional parameter to specify the type of the spatial lock data structure. It must be a model of the `SurjectiveLockDataStructure` concept, with `object` being a `Point` (as defined below). It is only used if the triangulation data structure used is concurrency-safe (i.e. when `TriangulationDataStructure_3::Concurrency_tag` is `Parallel_tag`). The default value is `Spatial_lock_grid_3<Tag_priority_blocking>` if the triangulation data structure is concurrency-safe, and `void` otherwise. In order to use concurrent operations, the user must provide a reference to a SLDS instance via the constructor or `Triangulation_3::set_lock_data_structure`.

Traversal of the Triangulation

The triangulation class provides several iterators and circulators that allow one to traverse it (completely or partially).

See also
  `CGAL::Delaunay_triangulation_3`
  `CGAL::Regular_triangulation_3`

Examples
  Triangulation_3/for_loop.cpp, Triangulation_3/simple_triangulation_3.cpp, and Triangulation_3/simplex.cpp.

## Public Types
enum | Locate_type { VERTEX =0 , EDGE , FACET , CELL , OUTSIDE_CONVEX_HULL , OUTSIDE_AFFINE_HULL }
| The enum `Locate_type` is defined by `Triangulation_3` to specify which case occurs when locating a point in the triangulation. More...

## Types
The class `Triangulation_3` defines the following types:
typedef Traits | Geom_traits
typedef TDS | Triangulation_data_structure
typedef SLDS | Lock_data_structure
typedef Triangulation_data_structure::Vertex::Point | Point
typedef Geom_traits::Segment_3 | Segment
typedef Geom_traits::Triangle_3 | Triangle
typedef Geom_traits::Tetrahedron_3 | Tetrahedron
Only vertices (0-faces) and cells (3-faces) are stored. Edges (1-faces) and facets (2-faces) are not explicitly represented and thus there are no corresponding classes (see Section Representation).
typedef Triangulation_data_structure::Vertex | Vertex
typedef Triangulation_data_structure::Cell | Cell
typedef Triangulation_data_structure::Facet | Facet
typedef Triangulation_data_structure::Edge | Edge
typedef Triangulation_data_structure::Concurrency_tag | Concurrency_tag
| Concurrency tag (from the TDS).
The vertices and faces of the triangulations are accessed through `handles`, `iterators` and `circulators`. A handle is a model of the `Handle` concept, and supports the two dereference operators and `operator->`. A circulator is a model of the concept `Circulator`. Iterators and circulators are bidirectional and non-mutable. The edges and facets of the triangulation can also be visited through iterators and circulators which are bidirectional and non-mutable. Iterators and circulators are convertible to the corresponding handles, thus the user can pass them directly as arguments to the functions. The handles are also model of the concepts `LessThanComparable` and `Hashable`, that is they can be used as keys in containers such as `std::map` and `boost::unordered_map`.
typedef Triangulation_data_structure::Vertex_handle | Vertex_handle
| handle to a vertex
typedef Triangulation_data_structure::Cell_handle | Cell_handle
| handle to a cell
typedef Triangulation_simplex_3< Self > | Simplex
| Reference to a simplex (vertex, edge, facet or cell) of the triangulation.
typedef Triangulation_data_structure::size_type | size_type
| Size type (an unsigned integral type)
typedef Triangulation_data_structure::difference_type | difference_type
| Difference type (a signed integral type)
typedef Triangulation_data_structure::Cell_iterator | All_cells_iterator
| iterator over cells
typedef Triangulation_data_structure::Facet_iterator | All_facets_iterator
| iterator over facets
typedef Triangulation_data_structure::Edge_iterator | All_edges_iterator
| iterator over edges
typedef Triangulation_data_structure::Vertex_iterator | All_vertices_iterator
| iterator over vertices
typedef unspecified_type | Finite_cells_iterator
| iterator over finite cells
typedef unspecified_type | Finite_facets_iterator
| iterator over finite facets
typedef unspecified_type | Finite_edges_iterator
| iterator over finite edges
typedef unspecified_type | Finite_vertices_iterator
| iterator over finite vertices
typedef unspecified_type | Point_iterator
| iterator over the points corresponding to the finite vertices of the triangulation.
typedef Triangulation_data_structure::Cell_circulator | Cell_circulator
| circulator over all cells incident to a given edge
typedef Triangulation_data_structure::Facet_circulator | Facet_circulator
| circulator over all facets incident to a given edge
typedef unspecified_type | Segment_cell_iterator
| iterator over cells intersected by a line segment.
typedef unspecified_type | Segment_simplex_iterator
| iterator over simplices intersected by a line segment.
In order to write C++ 11 `for`-loops we provide the following range types.
typedef Iterator_range< unspecified_type > | All_cell_handles
| range type for iterating over all cell handles (including infinite cells), with a nested type `iterator` that has as value type `Cell_handle`.
typedef Iterator_range< All_facets_iterator > | All_facets
| range type for iterating over facets.
typedef Iterator_range< All_edges_iterator > | All_edges
| range type for iterating over edges.
typedef Iterator_range< unspecified_type > | All_vertex_handles
| range type for iterating over all vertex handles, with a nested type `iterator` that has as value type `Vertex_handle`.
typedef Iterator_range< unspecified_type > | Finite_cell_handles
| range type for iterating over finite cell handles, with a nested type `iterator` that has as value type `Cell_handle`.
typedef Iterator_range< Finite_facets_iterator > | Finite_facets
| range type for iterating over finite facets.
typedef Iterator_range< Finite_edges_iterator > | Finite_edges
| range type for iterating over finite edges.
typedef Iterator_range< unspecified_type > | Finite_vertex_handles
| range type for iterating over finite vertex handles, with a nested type `iterator` that has as value type `Vertex_handle`.
typedef Iterator_range< unspecified_type > | Points
| range type for iterating over the points of the finite vertices.
typedef Iterator_range< unspecified_type > | Segment_traverser_cell_handles
| range type for iterating over the cells intersected by a line segment.
typedef Iterator_range< Segment_simplex_iterator > | Segment_traverser_simplices
| range type for iterating over the simplices intersected by a line segment.

## Creation
| Triangulation_3 (const Geom_traits &traits=Geom_traits(), Lock_data_structure *lock_ds=nullptr)
| Introduces a triangulation `t` having only one vertex which is the infinite vertex.
| Triangulation_3 (Lock_data_structure *lock_ds=nullptr, const Geom_traits &traits=Geom_traits())
| Same as the previous one, but with parameters in reverse order.
| Triangulation_3 (const Triangulation_3 &tr)
| Copy constructor.
template<class InputIterator >
| Triangulation_3 (InputIterator first, InputIterator last, const Geom_traits &traits=Geom_traits(), Lock_data_structure *lock_ds=nullptr)
| Equivalent to constructing an empty triangulation with the optional traits class argument and calling `insert(first,last)`.

## Assignment
Triangulation_3 & | operator= (const Triangulation_3 &tr)
| The triangulation `tr` is duplicated, and modifying the copy after the duplication does not modify the original.
void | swap (Triangulation_3 &tr)
| The triangulations `tr` and `t` are swapped.
void | clear ()
| Deletes all finite vertices and all cells of `t`.
template<class GT , class Tds1 , class Tds2 >
bool | operator== (const Triangulation_3< GT, Tds1 > &t1, const Triangulation_3< GT, Tds2 > &t2)
| Equality operator.
template<class GT , class Tds1 , class Tds2 >
bool | operator!= (const Triangulation_3< GT, Tds1 > &t1, const Triangulation_3< GT, Tds2 > &t2)
| The opposite of `operator==`.

## Access Functions
const Geom_traits & | geom_traits () const
| Returns a const reference to the geometric traits object.
const Triangulation_data_structure & | tds () const
| Returns a const reference to the triangulation data structure.
Triangulation_data_structure & | tds ()
| Returns a reference to the triangulation data structure.
int | dimension () const
| Returns the dimension of the affine hull.
size_type | number_of_vertices () const
| Returns the number of finite vertices.
size_type | number_of_cells () const
| Returns the number of cells or 0 if `t.dimension() < 3`.
Vertex_handle | infinite_vertex ()
| Returns the infinite vertex.
void | set_infinite_vertex (Vertex_handle v)
| This is an advanced function.
Cell_handle | infinite_cell () const
| Returns a cell incident to the infinite vertex.

## Non-Constant-Time Access Functions
As previously said, the triangulation is a collection of cells that are either infinite or represent a finite tetrahedra, where an infinite cell is a cell incident to the infinite vertex. Similarly we call an edge (resp. facet) `infinite` if it is incident to the infinite vertex.
size_type | number_of_facets () const
| The number of facets.
size_type | number_of_edges () const
| The number of edges.
size_type | number_of_finite_cells () const
| The number of finite cells.
size_type | number_of_finite_facets () const
| The number of finite facets.
size_type | number_of_finite_edges () const
| The number of finite edges.

## Geometric Access Functions
Tetrahedron | tetrahedron (Cell_handle c) const
| Returns the tetrahedron formed by the four vertices of `c`.
Triangle | triangle (Cell_handle c, int i) const
| Returns the triangle formed by the three vertices of facet `(c,i)`.
Triangle | triangle (const Facet &f) const
| Same as the previous method for facet `f`.
Segment | segment (const Edge &e) const
| Returns the line segment formed by the vertices of `e`.
Segment | segment (Cell_handle c, int i, int j) const
| Same as the previous method for edge `(c,i,j)`.
const Point & | point (Cell_handle c, int i) const
| Returns the point given by vertex `i` of cell `c`.
const Point & | point (Vertex_handle v) const
| Same as the previous method for vertex `v`.

## Tests for Finite and Infinite Vertices and Faces
bool | is_infinite (Vertex_handle v) const
| `true`, iff vertex `v` is the infinite vertex.
bool | is_infinite (Cell_handle c) const
| `true`, iff `c` is incident to the infinite vertex.
bool | is_infinite (Cell_handle c, int i) const
| `true`, iff the facet `i` of cell `c` is incident to the infinite vertex.
bool | is_infinite (const Facet &f) const
| `true` iff facet `f` is incident to the infinite vertex.
bool | is_infinite (Cell_handle c, int i, int j) const
| `true`, iff the edge `(i,j)` of cell `c` is incident to the infinite vertex.
bool | is_infinite (const Edge &e) const
| `true` iff edge `e` is incident to the infinite vertex.

## Queries
bool | is_vertex (const Point &p, Vertex_handle &v) const
| Tests whether `p` is a vertex of `t` by locating `p` in the triangulation.
bool | is_vertex (Vertex_handle v) const
| Tests whether `v` is a vertex of `t`.
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`.
bool | is_facet (Vertex_handle u, Vertex_handle v, Vertex_handle w, Cell_handle &c, int &i, int &j, int &k) const
| Tests whether `(u,v,w)` is a facet of `t`.
bool | is_cell (Cell_handle c) const
| Tests whether `c` is a cell of `t`.
bool | is_cell (Vertex_handle u, Vertex_handle v, Vertex_handle w, Vertex_handle x, Cell_handle &c, int &i, int &j, int &k, int &l) const
| Tests whether `(u,v,w,x)` is a cell of `t`.
bool | is_cell (Vertex_handle u, Vertex_handle v, Vertex_handle w, Vertex_handle x, Cell_handle &c) const
| Tests whether `(u,v,w,x)` is a cell of `t` and computes this cell `c`.
There is a method `has_vertex()` in the cell class. The analogous methods for facets are defined here.
bool | has_vertex (const Facet &f, Vertex_handle v, int &j) const
| If `v` is a vertex of `f`, then `j` is the index of `v` in the cell `f.first`, and the method returns `true`.
bool | has_vertex (Cell_handle c, int i, Vertex_handle v, int &j) const
| Same for facet `(c,i)`.
bool | has_vertex (const Facet &f, Vertex_handle v) const
bool | has_vertex (Cell_handle c, int i, Vertex_handle v) const
| Same as the first two methods, but these two methods do not return the index of the vertex.
The following three methods test whether two facets have the same vertices.
bool | are_equal (Cell_handle c, int i, Cell_handle n, int j) const
bool | are_equal (const Facet &f, const Facet &g) const
bool | are_equal (const Facet &f, Cell_handle n, int j) const
| For these three methods:

## Point Location
The class `Triangulation_3` provides two functions to locate a given point with respect to a triangulation. It provides also functions to test if a given point is inside a finite face or not. Note that the class `Delaunay_triangulation_3` also provides a `nearest_vertex()` function.
Cell_handle | locate (const Point &query, Cell_handle start=Cell_handle(), bool *could_lock_zone=nullptr) const
| If the point `query` lies inside the convex hull of the points, the cell that contains the query in its interior is returned.
Cell_handle | locate (const Point &query, Vertex_handle hint, bool *could_lock_zone=nullptr) const
| Same as above but uses `hint` as the starting place for the search.
Cell_handle | inexact_locate (const Point &query, Cell_handle start=Cell_handle()) const
| Same as `locate()` but uses inexact predicates.
Cell_handle | locate (const Point &query, Locate_type &lt, int &li, int &lj, Cell_handle start=Cell_handle(), bool *could_lock_zone=nullptr) 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`.
Cell_handle | locate (const Point &query, Locate_type &lt, int &li, int &lj, Vertex_handle hint, bool *could_lock_zone=nullptr) const
| Same as above but uses `hint` as the starting place for the search.
Bounded_side | side_of_cell (const Point &p, Cell_handle c, Locate_type &lt, int &li, int &lj) const
| Returns a value indicating on which side of the oriented boundary of `c` the point `p` lies.
Bounded_side | side_of_facet (const Point &p, const Facet &f, Locate_type &lt, int &li, int &lj) const
| Returns a value indicating on which side of the oriented boundary of `f` the point `p` lies:
Bounded_side | side_of_facet (const Point &p, Cell_handle c, Locate_type &lt, int &li, int &lj) const
| Same as the previous method for the facet `(c,3)`.
Bounded_side | side_of_edge (const Point &p, const Edge &e, Locate_type &lt, int &li) const
| Returns a value indicating on which side of the oriented boundary of `e` the point `p` lies:
Bounded_side | side_of_edge (const Point &p, Cell_handle c, Locate_type &lt, int &li) const
| Same as the previous method for edge \( (c,0,1)\).

## Flips
Two kinds of flips exist for a three-dimensional triangulation. They are reciprocal. To be flipped, an edge must be incident to three tetrahedra. During the flip, these three tetrahedra disappear and two tetrahedra appear. Figure 48.1 (left) shows the edge that is flipped as bold dashed, and one of its three incident facets is shaded. On the right, the facet shared by the two new tetrahedra is shaded. Flips are possible only under the following conditions: - the edge or facet to be flipped is not on the boundary of the convex hull of the triangulation - the five points involved are in convex position. Figure 48.1 Flips The following methods guarantee the validity of the resulting 3D triangulation. Flips for a 2d triangulation are not implemented yet.
bool | flip (Edge e)
bool | flip (Cell_handle c, int i, int j)
| Before flipping, these methods check that edge `e=(c,i,j)` is flippable (which is quite expensive).
void | flip_flippable (Edge e)
void | flip_flippable (Cell_handle c, int i, int j)
| Should be preferred to the previous methods when the edge is known to be flippable.
bool | flip (Facet f)
bool | flip (Cell_handle c, int i)
| Before flipping, these methods check that facet `f=(c,i)` is flippable (which is quite expensive).
void | flip_flippable (Facet f)
void | flip_flippable (Cell_handle c, int i)
| Should be preferred to the previous methods when the facet is known to be flippable.

## Insertions
The following operations are guaranteed to lead to a valid triangulation when they are applied on a valid triangulation.
Vertex_handle | insert (const Point &p, Cell_handle start=Cell_handle())
| Inserts the point `p` in the triangulation and returns the corresponding vertex.
Vertex_handle | insert (const Point &p, Vertex_handle hint)
| Same as above but uses `hint` as the starting place for the search.
Vertex_handle | insert (const Point &p, Locate_type lt, Cell_handle loc, int li, int lj)
| Inserts the point `p` in the triangulation and returns the corresponding vertex.
template<class PointInputIterator >
std::ptrdiff_t | insert (PointInputIterator first, PointInputIterator last)
| Inserts the points in the range `[first,last)` in the given order, and returns the number of inserted points.
template<class PointWithInfoInputIterator >
std::ptrdiff_t | insert (PointWithInfoInputIterator first, PointWithInfoInputIterator last)
| Inserts the points in the iterator range `[first,last)` in the given order, and returns the number of inserted points.
We also provide some other methods that can be used instead of `Triangulatation_3::insert()` when the place where the new point must be inserted is already known. They are also guaranteed to lead to a valid triangulation when they are applied on a valid triangulation.
Vertex_handle | insert_in_cell (const Point &p, Cell_handle c)
| Inserts the point `p` in the cell `c`.
Vertex_handle | insert_in_facet (const Point &p, const Facet &f)
| Inserts the point `p` in the facet `f`.
Vertex_handle | insert_in_facet (const Point &p, Cell_handle c, int i)
| As above, insertion in the facet `(c,i)`.
Vertex_handle | insert_in_edge (const Point &p, const Edge &e)
| Inserts `p` in the edge `e`.
Vertex_handle | insert_in_edge (const Point &p, Cell_handle c, int i, int j)
| As above, inserts `p` in the edge \( (i, j)\) of `c`.
Vertex_handle | insert_outside_convex_hull (const Point &p, Cell_handle c)
| The cell `c` must be an infinite cell containing `p`.
Vertex_handle | insert_outside_affine_hull (const Point &p)
| `p` is linked to all the points, and the infinite vertex is linked to all the points (including `p`) to triangulate the new infinite face, so that all the points now belong to the boundary of the convex hull.
template<class CellIt >
Vertex_handle | insert_in_hole (const Point &p, CellIt cell_begin, CellIt cell_end, Cell_handle begin, int i)
| Creates a new vertex by starring a hole.
template<class CellIt >
Vertex_handle | insert_in_hole (const Point &p, CellIt cell_begin, CellIt cell_end, Cell_handle begin, int i, Vertex_handle newv)
| Same as above, except that `newv` will be used as the new vertex, which must have been allocated previously with e.g. `create_vertex`.

## Cell, Face, Edge and Vertex Iterators
The following iterators allow the user to visit cells, facets, edges and vertices of the triangulation. These iterators are non-mutable, bidirectional and their value types are respectively `Cell`, `Facet`, `Edge` and `Vertex`. They are all invalidated by any change in 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.
Finite_cells_iterator | finite_cells_begin () const
| Starts at an arbitrary finite cell.
Finite_cells_iterator | finite_cells_end () const
| Past-the-end iterator.
All_vertices_iterator | all_vertices_begin () const
| Starts at an arbitrary vertex.
All_vertices_iterator | all_vertices_end () const
| Past-the-end iterator.
All_edges_iterator | all_edges_begin () const
| Starts at an arbitrary edge.
All_edges_iterator | all_edges_end () const
| Past-the-end iterator.
All_facets_iterator | all_facets_begin () const
| Starts at an arbitrary facet.
All_facets_iterator | all_facets_end () const
| Past-the-end iterator.
All_cells_iterator | all_cells_begin () const
| Starts at an arbitrary cell.
All_cells_iterator | all_cells_end () const
| Past-the-end iterator.
Point_iterator | points_begin () const
| Iterates over the points of the triangulation.
Point_iterator | points_end () const
| Past-the-end iterator.

## Ranges
In order to write C++ 11 `for`-loops we provide a range type and member functions to generate ranges. Note that vertex and cell ranges are special. See Section Iterators and Ranges in the User Manual.
All_cell_handles | all_cell_handles () const
| returns a range of iterators over all cells (even the infinite cells).
All_facets | all_facets () const
| returns a range of iterators starting at an arbitrary facet.
All_edges | all_edges () const
| returns a range of iterators starting at an arbitrary edge.
All_vertex_handles | all_vertex_handles () const
| returns a range of iterators over all vertices (even the infinite one).
Finite_cell_handles | finite_cell_handles () const
| returns a range of iterators over finite cells.
Finite_facets | finite_facets () const
| returns a range of iterators starting at an arbitrary facet.
Finite_edges | finite_edges () const
| returns a range of iterators starting at an arbitrary edge.
Finite_vertex_handles | finite_vertex_handles () const
| returns a range of iterators over finite vertices.
Points | points () const
| returns a range of iterators over the points of finite vertices.
std::array< Vertex_handle, 2 > | vertices (const Edge &e) const
| returns an array containing the vertices of `e`, in the order of their indices `e.second` and `e.third` in the cell `e.first`.
std::array< Vertex_handle, 3 > | vertices (const Facet &f) const
| returns an array containing the vertices of `f`, in counterclockwise order on the face of `f.first` opposite to vertex `f.first->vertex(f.second)`.
std::array< Vertex_handle, 4 > | vertices (const Cell_handle c) const
| returns an array containing the vertices of `c`, in the same order as the indices in `c`.
Segment_traverser_cell_handles | segment_traverser_cell_handles () const
| returns a range of iterators over the cells intersected by a line segment
Segment_traverser_simplices | segment_traverser_simplices () const
| returns a range of iterators over the simplices intersected by a line segment

## Segment Cell Iterator
The triangulation defines an iterator that visits cells intersected by a line segment. Segment Cell Iterator iterates over a sequence of cells which union contains the segment `[s,t]`. The sequence of cells is "minimal" (removing any cell would make the union of the renaming cells not entirely containing `[s,t]`) and sorted along `[s,t]`. The "minimality" of the sequence implies that in degenerate cases, only one cell incident to the traversed simplex will be reported. The cells visited form a facet-connected region containing both source and target points of the line segment `[s,t]`. Each cell falls within one or more of the following categories: a finite cell whose interior is intersected by `[s,t]`. a finite cell with a facet `f` whose interior is intersected by `[s,t]` in a line segment. If such a cell is visited, its neighbor incident to `f` is not visited. a finite cell with an edge `e` whose interior is intersected by `[s,t]` in a line segment. If such a cell is visited, none of the other cells incident to `e` are visited. a finite cell with an edge `e` whose interior is intersected by `[s,t]` in a point. This cell forms a connected component together with the other cells incident to `e` that are visited. Exactly two of these visited cells also fall in category 1 or 2. a finite cell with a vertex `v` that is an endpoint of `[s,t]`. This cell also fits in either category 1 or 2. a finite cell with a vertex `v` that lies in the interior of `[s,t]`. This cell forms a connected component together with the other cells incident to `v` that are visited. Exactly two of these cells also fall in category 1 or 2. an infinite cell with a finite facet whose interior is intersected by the interior of `[s,t]`. an infinite cell with a finite edge `e` whose interior is intersected by the interior of `[s,t]`. If such a cell is visited, its infinite neighbor incident to `e` is not visited. Among the finite cells incident to `e` that are visited, exactly one also falls in category 1 or 2. an infinite cell with a finite vertex `v` that lies in the interior of `[s,t]`. If such a cell is visited, none of the other infinite cells incident to `v` are visited. Among the finite cells incident to `v` that are visited, exactly one also falls in category 1, 2, or 3. an infinite cell in the special case where the segment does not intersect any finite facet. In this case, exactly one infinite cell is visited. This cell shares a facet `f` with a finite cell `c` such that `f` is intersected by the line through the point `s` and the vertex of `c` opposite of `f`. Note that for categories 4 and 6, it is not predetermined which incident cells are visited. However, exactly two of the incident cells `c0,c1` visited also fall in category 1 or 2. The remaining incident cells visited make a facet-connected sequence connecting `c0` to `c1`. `Segment_cell_iterator` implements the concept `ForwardIterator` and is non-mutable. It is invalidated by any modification of one of the cells traversed. Its `value_type` is `Cell_handle`.
Segment_cell_iterator | segment_traverser_cells_begin (Vertex_handle vs, Vertex_handle vt) const
| returns the iterator that allows to visit the cells intersected by the line segment `[vs,vt]`.
Segment_cell_iterator | segment_traverser_cells_begin (const Point &ps, const Point &pt, Cell_handle hint=Cell_handle()) const
| returns the iterator that allows to visit the cells intersected by the line segment `[ps, pt]`.
Segment_cell_iterator | segment_traverser_cells_end () const
| returns the past-the-end iterator over the intersected cells.

## Segment Simplex Iterator
The triangulation defines an iterator that visits all the triangulation simplices (vertices, edges, facets and cells) intersected by a line segment. The iterator traverses a connected sequence of simplices - possibly of all dimensions - intersected by the line segment `[s, t]`. In the degenerate case where the query segment goes exactly through a vertex (or along an edge, or along a facet), only one of the cells incident to that vertex (or edge, or facet) is returned by the iterator, and not all of them. Each simplex falls within one or more of the following categories: a finite cell whose interior is intersected by `[s,t]`, a facet `f` whose interior is intersected by `[s,t]` in a point, a facet `f` whose interior is intersected by `[s,t]` in a line segment, an edge `e` whose interior is intersected by `[s,t]` in a point, an edge `e` whose interior is intersected by `[s,t]` in a line segment, a vertex `v` lying on `[s,t]`, an infinite cell with a finite facet whose interior is intersected by the interior of `[s,t]`, an infinite cell in the special case where the segment does not intersect any finite facet. In this case, exactly one infinite cell is visited. This cell shares a facet `f` with a finite cell `c` such that `f` is intersected by the line through the source of `[s,t]` and the vertex of `c` opposite of `f`. `Segment_simplex_iterator` implements the concept `ForwardIterator` and is non-mutable. It is invalidated by any modification of one of the cells traversed. Its `value_type` is `Triangulation_simplex_3`.
Segment_simplex_iterator | segment_traverser_simplices_begin (Vertex_handle vs, Vertex_handle vt) const
| returns the iterator that allows to visit the simplices intersected by the line segment `[vs,vt]`.
Segment_simplex_iterator | segment_traverser_simplices_begin (const Point &ps, const Point &pt, Cell_handle hint=Cell_handle()) const
| returns the iterator that allows to visit the simplices intersected by the line segment `[ps,pt]`.
Segment_simplex_iterator | segment_traverser_simplices_end () const
| returns the past-the-end iterator over the intersected simplices.

## Cell and Facet Circulators
The following circulators respectively visit all cells or all facets incident to a given edge. They are non-mutable and bidirectional. They are invalidated by any modification of one of the cells traversed.
Cell_circulator | incident_cells (Edge e) const
| Starts at an arbitrary cell incident to `e`.
Cell_circulator | incident_cells (Cell_handle c, int i, int j) const
| As above for edge `(i,j)` of `c`.
Cell_circulator | incident_cells (Edge e, Cell_handle start) const
| Starts at cell `start`.
Cell_circulator | incident_cells (Cell_handle c, int i, int j, Cell_handle start) const
| As above for edge `(i,j)` of `c`.
The following circulators on facets are defined only in dimension 3, though facets are defined also in dimension 2: there are only two facets sharing an edge in dimension 2.
Facet_circulator | incident_facets (Edge e) const
| Starts at an arbitrary facet incident to `e`.
Facet_circulator | incident_facets (Cell_handle c, int i, int j) const
| As above for edge `(i,j)` of `c`.
Facet_circulator | incident_facets (Edge e, Facet start) const
| Starts at facet `start`.
Facet_circulator | incident_facets (Edge e, Cell_handle start, int f) const
| Starts at facet of index `f` in `start`.
Facet_circulator | incident_facets (Cell_handle c, int i, int j, Facet start) const
| As above for edge `(i,j)` of `c`.
Facet_circulator | incident_facets (Cell_handle c, int i, int j, Cell_handle start, int f) const
| As above for edge `(i,j)` of `c` and facet `(start,f)`.

## Traversal of the Incident Cells, Facets and Edges, and the Adjacent Vertices of a Given Vertex
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`.
bool | try_lock_and_get_incident_cells (Vertex_handle v, std::vector< Cell_handle > &cells) const
| Try to lock and copy the `Cell_handle`s of all cells incident to `v` into `cells`.
template<class OutputIterator >
OutputIterator | finite_incident_cells (Vertex_handle v, OutputIterator cells) const
| Copies the `Cell_handle`s of all finite cells incident to `v` to the output iterator `cells`.
template<class OutputIterator >
OutputIterator | incident_facets (Vertex_handle v, OutputIterator facets) const
| Copies all `Facet`s incident to `v` to the output iterator `facets`.
template<class OutputIterator >
OutputIterator | finite_incident_facets (Vertex_handle v, OutputIterator facets) const
| Copies all finite `Facet`s incident to `v` to the output iterator `facets`.
template<class OutputIterator >
OutputIterator | incident_edges (Vertex_handle v, OutputIterator edges) const
| Copies all `Edge`s incident to `v` to the output iterator `edges`.
template<class OutputIterator >
OutputIterator | finite_incident_edges (Vertex_handle v, OutputIterator edges) const
| Copies all finite `Edge`s incident to `v` to the output iterator `edges`.
template<class OutputIterator >
OutputIterator | adjacent_vertices (Vertex_handle v, OutputIterator vertices) const
| Copies the `Vertex_handle`s of all vertices adjacent to `v` to the output iterator `vertices`.
template<class OutputIterator >
OutputIterator | finite_adjacent_vertices (Vertex_handle v, OutputIterator vertices) const
| Copies the `Vertex_handle`s of all finite vertices adjacent to `v` to the output iterator `vertices`.
size_type | degree (Vertex_handle v) const
| Returns the degree of `v`, that is, the number of incident vertices.

## Traversal Between Adjacent Cells
int | mirror_index (Cell_handle c, int i) const
| Returns the index of `c` in its \( i^{th}\) neighbor.
Vertex_handle | mirror_vertex (Cell_handle c, int i) const
| Returns the vertex of the \( i^{th}\) neighbor of `c` that is opposite to `c`.
Facet | mirror_facet (Facet f) const
| Returns the same facet seen from the other adjacent cell.

## Checking
The responsibility of keeping a valid triangulation belongs to the user when using advanced operations allowing a direct manipulation of cells and vertices. We provide the user with the following methods to help debugging.
bool | is_valid (bool verbose=false) const
| This is a function for debugging purpose.
bool | is_valid (Cell_handle c, bool verbose=false) const
| This is a function for debugging purpose.

## I/O
The information in the `iostream` is: the dimension, the number of finite vertices, the non-combinatorial information about vertices (point, etc; note that the infinite vertex is numbered 0), the number of cells, the indices of the vertices of each cell, plus the non-combinatorial information about each cell, then the indices of the neighbors of each cell, where the index corresponds to the preceding list of cells. When dimension \( <\) 3, the same information is stored for faces of maximal dimension instead of cells.
istream & | operator>> (istream &is, Triangulation_3 &t)
| Reads the underlying combinatorial triangulation from `is` by calling the corresponding input operator of the triangulation data structure class (note that the infinite vertex is numbered 0), and the non-combinatorial information by calling the corresponding input operators of the vertex and the cell classes (such as point coordinates), which are provided by overloading the stream operators of the vertex and cell types.
ostream & | operator<< (ostream &os, const Triangulation_3 &t)
| Writes the triangulation `t` into `os`.
template<typename Tr_src , typename ConvertVertex , typename ConvertCell >
std::istream & | file_input (std::istream &is, ConvertVertex convert_vertex=ConvertVertex(), ConvertCell convert_cell=ConvertCell())
| The triangulation streamed in `is`, of original type `Tr_src`, is written into the triangulation.

## Concurrency
void | set_lock_data_structure (Lock_data_structure *lock_ds) const
| Set the pointer to the lock data structure.

## Additional Inherited Members
Static Public Member Functions inherited from CGAL::Triangulation_utils_3
static unsigned int | next_around_edge (unsigned int i, unsigned int j)
static int | vertex_triple_index (const int i, const int j)
static unsigned int | ccw (unsigned int i)
static unsigned int | cw (unsigned int i)

## Member Typedef Documentation

## ◆ Segment_cell_iterator

template<typename Traits , typename TDS , typename SLDS >

typedef unspecified_type CGAL::Triangulation_3< Traits, TDS, SLDS >::Segment_cell_iterator

iterator over cells intersected by a line segment.

`Segment_cell_iterator` implements the concept `ForwardIterator` and is non-mutable. Its value type is `Cell_handle`.

## ◆ Segment_simplex_iterator

template<typename Traits , typename TDS , typename SLDS >

typedef unspecified_type CGAL::Triangulation_3< Traits, TDS, SLDS >::Segment_simplex_iterator

iterator over simplices intersected by a line segment.

`Segment_simplex_iterator` implements the concept `ForwardIterator` and is non-mutable. Its value type is `Triangulation_simplex_3`.

## Member Enumeration Documentation

## ◆ Locate_type

template<typename Traits , typename TDS , typename SLDS >

enum CGAL::Triangulation_3::Locate_type

The enum `Locate_type` is defined by `Triangulation_3` to specify which case occurs when locating a point in the triangulation.

Enumerator
VERTEX |
EDGE |
FACET |
CELL |
OUTSIDE_CONVEX_HULL |
OUTSIDE_AFFINE_HULL |

## Constructor & Destructor Documentation

## ◆ Triangulation_3() [1/2]

template<typename Traits , typename TDS , typename SLDS >

CGAL::Triangulation_3< Traits, TDS, SLDS >::Triangulation_3 | ( | const Geom_traits & | traits = `Geom_traits()`,
|  | Lock_data_structure * | lock_ds = `nullptr`
| ) |  |

Introduces a triangulation `t` having only one vertex which is the infinite vertex.

`lock_ds` is an optional pointer to the lock data structure for parallel operations. It must be provided if concurrency is enabled.

## ◆ Triangulation_3() [2/2]

template<typename Traits , typename TDS , typename SLDS >

CGAL::Triangulation_3< Traits, TDS, SLDS >::Triangulation_3 | ( | const Triangulation_3< Traits, TDS, SLDS > & | tr | ) |

Copy constructor.

All vertices and faces are duplicated. The pointer to the lock data structure is not copied. Thus, the copy won't be concurrency-safe as long as the user has not call `Triangulation_3::set_lock_data_structure`.

## Member Function Documentation

## ◆ adjacent_vertices()

template<typename Traits , typename TDS , typename SLDS >
template<class OutputIterator >

OutputIterator CGAL::Triangulation_3< Traits, TDS, SLDS >::adjacent_vertices | ( | Vertex_handle | v,
|  | OutputIterator | vertices
| ) |  | const

Copies the `Vertex_handle`s of all vertices adjacent to `v` to the output iterator `vertices`.

If `t.dimension() < 0`, then do nothing. Returns the resulting output iterator.

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

## ◆ all_cell_handles()

template<typename Traits , typename TDS , typename SLDS >

All_cell_handles CGAL::Triangulation_3< Traits, TDS, SLDS >::all_cell_handles | ( |  | ) | const

returns a range of iterators over all cells (even the infinite cells).

Returns an empty range when `t.number_of_cells() == 0`.

Note
  While the value type of `All_cells_iterator` is `Cell`, the value type of `All_cell_handles::iterator` is `Cell_handle`.

## ◆ all_cells_begin()

template<typename Traits , typename TDS , typename SLDS >

All_cells_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::all_cells_begin | ( |  | ) | const

Starts at an arbitrary cell.

Iterates over all cells (even infinite ones). Returns `cells_end()` when `t.dimension() < 3`.

## ◆ all_edges()

template<typename Traits , typename TDS , typename SLDS >

All_edges CGAL::Triangulation_3< Traits, TDS, SLDS >::all_edges | ( |  | ) | const

returns a range of iterators starting at an arbitrary edge.

Returns an empty range when `t.dimension() < 2`.

## ◆ all_edges_begin()

template<typename Traits , typename TDS , typename SLDS >

All_edges_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::all_edges_begin | ( |  | ) | const

Starts at an arbitrary edge.

Iterates over all edges (even infinite ones). Returns `edges_end()` when `t.dimension() < 1`.

## ◆ all_facets()

template<typename Traits , typename TDS , typename SLDS >

All_facets CGAL::Triangulation_3< Traits, TDS, SLDS >::all_facets | ( |  | ) | const

returns a range of iterators starting at an arbitrary facet.

Returns an empty range when `t.dimension() < 2`.

## ◆ all_facets_begin()

template<typename Traits , typename TDS , typename SLDS >

All_facets_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::all_facets_begin | ( |  | ) | const

Starts at an arbitrary facet.

Iterates over all facets (even infinite ones). Returns `facets_end()` when `t.dimension() < 2`.

## ◆ all_vertex_handles()

template<typename Traits , typename TDS , typename SLDS >

All_vertex_handles CGAL::Triangulation_3< Traits, TDS, SLDS >::all_vertex_handles | ( |  | ) | const

returns a range of iterators over all vertices (even the infinite one).

Note
  While the value type of `All_vertices_iterator` is `Vertex`, the value type of `All_vertex_handles::iterator` is `Vertex_handle`.

## ◆ all_vertices_begin()

template<typename Traits , typename TDS , typename SLDS >

All_vertices_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::all_vertices_begin | ( |  | ) | const

Starts at an arbitrary vertex.

Iterates over all vertices (even the infinite one).

## ◆ are_equal()

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::are_equal | ( | const Facet & | f,
|  | Cell_handle | n,
|  | int | j
| ) |  | const

For these three methods:

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

## ◆ degree()

template<typename Traits , typename TDS , typename SLDS >

size_type CGAL::Triangulation_3< Traits, TDS, SLDS >::degree | ( | Vertex_handle | v | ) | const

Returns the degree of `v`, that is, the number of incident vertices.

The infinite vertex is counted.

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

## ◆ finite_adjacent_vertices()

template<typename Traits , typename TDS , typename SLDS >
template<class OutputIterator >

OutputIterator CGAL::Triangulation_3< Traits, TDS, SLDS >::finite_adjacent_vertices | ( | Vertex_handle | v,
|  | OutputIterator | vertices
| ) |  | const

Copies the `Vertex_handle`s of all finite vertices adjacent to `v` to the output iterator `vertices`.

If `t.dimension() < 0`, then do nothing. Returns the resulting output iterator.

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

## ◆ finite_cell_handles()

template<typename Traits , typename TDS , typename SLDS >

Finite_cell_handles CGAL::Triangulation_3< Traits, TDS, SLDS >::finite_cell_handles | ( |  | ) | const

returns a range of iterators over finite cells.

Returns an empty range when `t.number_of_cells() == 0`.

Note
  While the value type of `Finite_cells_iterator` is `Cell`, the value type of `Finite_cell_handles::iterator` is `Cell_handle`.

## ◆ finite_cells_begin()

template<typename Traits , typename TDS , typename SLDS >

Finite_cells_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::finite_cells_begin | ( |  | ) | const

Starts at an arbitrary finite cell.

Then `++` and `--` will iterate over finite cells. Returns `finite_cells_end()` when `t.dimension() < 3`.

## ◆ finite_edges()

template<typename Traits , typename TDS , typename SLDS >

Finite_edges CGAL::Triangulation_3< Traits, TDS, SLDS >::finite_edges | ( |  | ) | const

returns a range of iterators starting at an arbitrary edge.

Returns an empty range when `t.dimension() < 2`.

## ◆ finite_edges_begin()

template<typename Traits , typename TDS , typename SLDS >

Finite_edges_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::finite_edges_begin | ( |  | ) | const

Starts at an arbitrary finite edge.

Then `++` and `--` will iterate over finite edges.

## ◆ finite_facets()

template<typename Traits , typename TDS , typename SLDS >

Finite_facets CGAL::Triangulation_3< Traits, TDS, SLDS >::finite_facets | ( |  | ) | const

returns a range of iterators starting at an arbitrary facet.

Returns an empty range when `t.dimension() < 2`.

## ◆ finite_facets_begin()

template<typename Traits , typename TDS , typename SLDS >

Finite_facets_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::finite_facets_begin | ( |  | ) | const

Starts at an arbitrary finite facet.

Then `++` and `--` will iterate over finite facets. Returns `finite_facets_end()` when `t.dimension() < 2`.

## ◆ finite_incident_cells()

template<typename Traits , typename TDS , typename SLDS >
template<class OutputIterator >

OutputIterator CGAL::Triangulation_3< Traits, TDS, SLDS >::finite_incident_cells | ( | Vertex_handle | v,
|  | OutputIterator | cells
| ) |  | const

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

Returns the resulting output iterator.

Precondition
  `t.dimension() == 3`
  `v != Vertex_handle()` and `t.is_vertex(v)`.

## ◆ finite_incident_edges()

template<typename Traits , typename TDS , typename SLDS >
template<class OutputIterator >

OutputIterator CGAL::Triangulation_3< Traits, TDS, SLDS >::finite_incident_edges | ( | Vertex_handle | v,
|  | OutputIterator | edges
| ) |  | const

Copies all finite `Edge`s incident to `v` to the output iterator `edges`.

Returns the resulting output iterator.

Precondition
  `t.dimension() > 0`
  `v != Vertex_handle()` and `t.is_vertex(v)`.

## ◆ finite_incident_facets()

template<typename Traits , typename TDS , typename SLDS >
template<class OutputIterator >

OutputIterator CGAL::Triangulation_3< Traits, TDS, SLDS >::finite_incident_facets | ( | Vertex_handle | v,
|  | OutputIterator | facets
| ) |  | const

Copies all finite `Facet`s incident to `v` to the output iterator `facets`.

Returns the resulting output iterator.

Precondition
  `t.dimension() > 1`
  `v != Vertex_handle()` and `t.is_vertex(v)`.

## ◆ finite_vertex_handles()

template<typename Traits , typename TDS , typename SLDS >

Finite_vertex_handles CGAL::Triangulation_3< Traits, TDS, SLDS >::finite_vertex_handles | ( |  | ) | const

returns a range of iterators over finite vertices.

Note
  While the value type of `Finite_vertices_iterator` is `Vertex`, the value type of `Finite_vertex_handles::iterator` is `Vertex_handle`.

## ◆ finite_vertices_begin()

template<typename Traits , typename TDS , typename SLDS >

Finite_vertices_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::finite_vertices_begin | ( |  | ) | const

Starts at an arbitrary finite vertex.

Then `++` and `--` will iterate over finite vertices.

## ◆ flip() [1/2]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::flip | ( | Cell_handle | c,
|  | int | i
| ) |  |

Before flipping, these methods check that facet `f=(c,i)` is flippable (which is quite expensive).

They return `false` or `true` according to this test.

## ◆ flip() [2/2]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::flip | ( | Cell_handle | c,
|  | int | i,
|  | int | j
| ) |  |

Before flipping, these methods check that edge `e=(c,i,j)` is flippable (which is quite expensive).

They return `false` or `true` according to this test.

## ◆ flip_flippable() [1/2]

template<typename Traits , typename TDS , typename SLDS >

void CGAL::Triangulation_3< Traits, TDS, SLDS >::flip_flippable | ( | Cell_handle | c,
|  | int | i
| ) |  |

Should be preferred to the previous methods when the facet is known to be flippable.

Precondition
  The facet is flippable.

## ◆ flip_flippable() [2/2]

template<typename Traits , typename TDS , typename SLDS >

void CGAL::Triangulation_3< Traits, TDS, SLDS >::flip_flippable | ( | Cell_handle | c,
|  | int | i,
|  | int | j
| ) |  |

Should be preferred to the previous methods when the edge is known to be flippable.

Precondition
  The edge is flippable.

## ◆ has_vertex() [1/2]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::has_vertex | ( | Cell_handle | c,
|  | int | i,
|  | Vertex_handle | v,
|  | int & | j
| ) |  | const

Same for facet `(c,i)`.

Computes the index `j` of `v` in `c`.

## ◆ has_vertex() [2/2]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::has_vertex | ( | const Facet & | f,
|  | Vertex_handle | v,
|  | int & | j
| ) |  | const

If `v` is a vertex of `f`, then `j` is the index of `v` in the cell `f.first`, and the method returns `true`.

Precondition
  `t.dimension() == 3`

## ◆ incident_cells() [1/3]

template<typename Traits , typename TDS , typename SLDS >

Cell_circulator CGAL::Triangulation_3< Traits, TDS, SLDS >::incident_cells | ( | Edge | e | ) | const

Starts at an arbitrary cell incident to `e`.

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

## ◆ incident_cells() [2/3]

template<typename Traits , typename TDS , typename SLDS >

Cell_circulator CGAL::Triangulation_3< Traits, TDS, SLDS >::incident_cells | ( | Edge | e,
|  | Cell_handle | start
| ) |  | const

Starts at cell `start`.

Precondition
  `t.dimension() == 3` and `start` is incident to `e`.

## ◆ incident_cells() [3/3]

template<typename Traits , typename TDS , typename SLDS >
template<class OutputIterator >

OutputIterator CGAL::Triangulation_3< Traits, TDS, SLDS >::incident_cells | ( | Vertex_handle | v,
|  | OutputIterator | cells
| ) |  | const

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

Returns the resulting output iterator.

Precondition
  `t.dimension() == 3`
  `v != Vertex_handle()` and `t.is_vertex(v)`.

## ◆ incident_edges()

template<typename Traits , typename TDS , typename SLDS >
template<class OutputIterator >

OutputIterator CGAL::Triangulation_3< Traits, TDS, SLDS >::incident_edges | ( | Vertex_handle | v,
|  | OutputIterator | edges
| ) |  | const

Copies all `Edge`s incident to `v` to the output iterator `edges`.

Returns the resulting output iterator.

Precondition
  `t.dimension() > 0`
  `v != Vertex_handle()` and `t.is_vertex(v)`.

## ◆ incident_facets() [1/3]

template<typename Traits , typename TDS , typename SLDS >

Facet_circulator CGAL::Triangulation_3< Traits, TDS, SLDS >::incident_facets | ( | Edge | e | ) | const

Starts at an arbitrary facet incident to `e`.

Precondition
  `t.dimension() == 3`

## ◆ incident_facets() [2/3]

template<typename Traits , typename TDS , typename SLDS >

Facet_circulator CGAL::Triangulation_3< Traits, TDS, SLDS >::incident_facets | ( | Edge | e,
|  | Facet | start
| ) |  | const

Starts at facet `start`.

Precondition
  `start` is incident to `e`.

## ◆ incident_facets() [3/3]

template<typename Traits , typename TDS , typename SLDS >
template<class OutputIterator >

OutputIterator CGAL::Triangulation_3< Traits, TDS, SLDS >::incident_facets | ( | Vertex_handle | v,
|  | OutputIterator | facets
| ) |  | const

Copies all `Facet`s incident to `v` to the output iterator `facets`.

Returns the resulting output iterator.

Precondition
  `t.dimension() > 1`
  `v != Vertex_handle()` and `t.is_vertex(v)`.

## ◆ inexact_locate()

template<typename Traits , typename TDS , typename SLDS >

Cell_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::inexact_locate | ( | const Point & | query,
|  | Cell_handle | start = `Cell_handle()`
| ) |  | const

Same as `locate()` but uses inexact predicates.

This function returns a handle on a cell that is a good approximation of the exact location of `query`, while being faster. Note that it may return a handle on a cell whose interior does not contain `query`. When the triangulation has dimension smaller than 3, `start` is returned.

Note that this function is available only if the cartesian coordinates of `query` are accessible with functions `x()`, `y()` and `z()`.

## ◆ insert() [1/3]

template<typename Traits , typename TDS , typename SLDS >

Vertex_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::insert | ( | const Point & | p,
|  | Cell_handle | start = `Cell_handle()`
| ) |  |

Inserts the point `p` in the triangulation and returns the corresponding vertex.

If point `p` coincides with an already existing vertex, this vertex is returned and the triangulation remains unchanged.

If point `p` lies in the convex hull of the points, it is added naturally: if it lies inside a cell, the cell is split into four cells, if it lies on a facet, the two incident cells are split into three cells, if it lies on an edge, all the cells incident to this edge are split into two cells.

If point `p` is strictly outside the convex hull but in the affine hull, `p` is linked to all visible points on the convex hull to form the new triangulation. See Figure Triangulation3figinsert_outside_convex_hull.

If point `p` is outside the affine hull of the points, `p` is linked to all the points, and the dimension of the triangulation is incremented. All the points now belong to the boundary of the convex hull, so, the infinite vertex is linked to all the points to triangulate the new infinite face. See Figure Triangulation3figinsert_outside_affine_hull. The optional argument `start` is used as a starting place for the search.

## ◆ insert() [2/3]

template<typename Traits , typename TDS , typename SLDS >

Vertex_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::insert | ( | const Point & | p,
|  | Locate_type | lt,
|  | Cell_handle | loc,
|  | int | li,
|  | int | lj
| ) |  |

Inserts the point `p` in the triangulation and returns the corresponding vertex.

Similar to the above `insert()` function, but takes as additional parameter the return values of a previous location query. See description of locate() above.

## ◆ insert() [3/3]

template<typename Traits , typename TDS , typename SLDS >
template<class PointWithInfoInputIterator >

std::ptrdiff_t CGAL::Triangulation_3< Traits, TDS, SLDS >::insert | ( | PointWithInfoInputIterator | first,
|  | PointWithInfoInputIterator | last
| ) |  |

Inserts the points in the iterator range `[first,last)` in the given order, and returns the number of inserted points.

Given a pair `(p,i)`, the vertex `v` storing `p` also stores `i`, that is `v.point() == p` and `v.info() == i`. If several pairs have the same point, only one vertex is created, and one of the objects of type `Vertex::Info` will be stored in the vertex.

Precondition
  `Vertex` must be model of the concept `TriangulationVertexBaseWithInfo_3`.

Template Parameters

  PointWithInfoInputIterator | must be an input iterator with the value type `std::pair<Point,Vertex::Info>`.

## ◆ insert_in_cell()

template<typename Traits , typename TDS , typename SLDS >

Vertex_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::insert_in_cell | ( | const Point & | p,
|  | Cell_handle | c
| ) |  |

Inserts the point `p` in the cell `c`.

The cell `c` is split into 4 tetrahedra.

Precondition
  `t.dimension() == 3` and `p` lies strictly inside cell `c`.

## ◆ insert_in_edge() [1/2]

template<typename Traits , typename TDS , typename SLDS >

Vertex_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::insert_in_edge | ( | const Point & | p,
|  | Cell_handle | c,
|  | int | i,
|  | int | j
| ) |  |

As above, inserts `p` in the edge \( (i, j)\) of `c`.

Precondition
  As above and \( i\neq j\). Moreover \( i,j \in\{0,1,2,3\}\) in dimension 3, \( i,j \in\{0,1,2\}\) in dimension 2, \( i,j \in\{0,1\}\) in dimension 1.

## ◆ insert_in_edge() [2/2]

template<typename Traits , typename TDS , typename SLDS >

Vertex_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::insert_in_edge | ( | const Point & | p,
|  | const Edge & | e
| ) |  |

Inserts `p` in the edge `e`.

In dimension 3, all the cells having this edge are split into 2 tetrahedra; in dimension 2, the 2 neighboring facets are split into 2 triangles; in dimension 1, the edge is split into 2 edges.

Precondition
  `t.dimension()` \( \geq1\) and `p` lies on edge `e`.

## ◆ insert_in_facet() [1/2]

template<typename Traits , typename TDS , typename SLDS >

Vertex_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::insert_in_facet | ( | const Point & | p,
|  | Cell_handle | c,
|  | int | i
| ) |  |

As above, insertion in the facet `(c,i)`.

Precondition
  As above and \( i \in\{0,1,2,3\}\) in dimension 3, \( i = 3\) in dimension 2.

## ◆ insert_in_facet() [2/2]

template<typename Traits , typename TDS , typename SLDS >

Vertex_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::insert_in_facet | ( | const Point & | p,
|  | const Facet & | f
| ) |  |

Inserts the point `p` in the facet `f`.

In dimension 3, the 2 neighboring cells are split into 3 tetrahedra; in dimension 2, the facet is split into 3 triangles.

Precondition
  `t.dimension()` \( \geq2\) and `p` lies strictly inside face `f`.

## ◆ insert_in_hole()

template<typename Traits , typename TDS , typename SLDS >
template<class CellIt >

Vertex_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::insert_in_hole | ( | const 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.

This operation is equivalent to calling `tds().insert_in_hole(cell_begin, cell_end, begin, i); v->set_point(p)`.

Precondition
  `t.dimension()` \( \geq2\), 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`.

## ◆ insert_outside_affine_hull()

template<typename Traits , typename TDS , typename SLDS >

Vertex_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::insert_outside_affine_hull | ( | const Point & | p | ) |

`p` is linked to all the points, and the infinite vertex is linked to all the points (including `p`) to triangulate the new infinite face, so that all the points now belong to the boundary of the convex hull.

See Figure Triangulation3figinsert_outside_affine_hull.

This method can be used to insert the first point in an empty triangulation.

Precondition
  `t.dimension() < 3` and `p` lies outside the affine hull of the points.

insert_outside_affine_hull() (2-dimensional case)

## ◆ insert_outside_convex_hull()

template<typename Traits , typename TDS , typename SLDS >

Vertex_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::insert_outside_convex_hull | ( | const Point & | p,
|  | Cell_handle | c
| ) |  |

The cell `c` must be an infinite cell containing `p`.

Links `p` to all points in the triangulation that are visible from `p`. Updates consequently the infinite faces. See Figure Triangulation3figinsert_outside_convex_hull.

Precondition
  `t.dimension() > 0`, `c`, and the \( k\)-face represented by `c` is infinite and contains `t`.

insert_outside_convex_hull() (2-dimensional case)

## ◆ is_cell() [1/2]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::is_cell | ( | Vertex_handle | u,
|  | Vertex_handle | v,
|  | Vertex_handle | w,
|  | Vertex_handle | x,
|  | Cell_handle & | c
| ) |  | const

Tests whether `(u,v,w,x)` is a cell of `t` and computes this cell `c`.

Precondition
  `u`, `v`, `w` and `x` are vertices of `t`.

## ◆ is_cell() [2/2]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::is_cell | ( | Vertex_handle | u,
|  | Vertex_handle | v,
|  | Vertex_handle | w,
|  | Vertex_handle | x,
|  | Cell_handle & | c,
|  | int & | i,
|  | int & | j,
|  | int & | k,
|  | int & | l
| ) |  | const

Tests whether `(u,v,w,x)` is a cell of `t`.

If the cell `c` is found, the method computes the indices `i`, `j`, `k` and `l` of the vertices `u`, `v`, `w` and `x` in `c`, in this order.

Precondition
  `u`, `v`, `w` and `x` are vertices of `t`.

## ◆ is_edge()

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::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`.

## ◆ is_facet()

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::is_facet | ( | Vertex_handle | u,
|  | Vertex_handle | v,
|  | Vertex_handle | w,
|  | Cell_handle & | c,
|  | int & | i,
|  | int & | j,
|  | int & | k
| ) |  | const

Tests whether `(u,v,w)` is a facet of `t`.

If the facet is found, it computes a cell `c` having this facet and the indices `i`, `j` and `k` of the vertices `u`, `v` and `w` in `c`, in this order.

Precondition
  `u`, `v` and `w` are vertices of `t`.

## ◆ is_infinite() [1/5]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::is_infinite | ( | Cell_handle | c | ) | const

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

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

## ◆ is_infinite() [2/5]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::is_infinite | ( | Cell_handle | c,
|  | int | i
| ) |  | const

`true`, iff the facet `i` of cell `c` is incident to the infinite vertex.

Precondition
  `t.dimension()` \( \geq2\) and \( i\in\{0,1,2,3\}\) in dimension 3, \( i=3\) in dimension 2.

## ◆ is_infinite() [3/5]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::is_infinite | ( | Cell_handle | c,
|  | int | i,
|  | int | j
| ) |  | const

`true`, iff the edge `(i,j)` of cell `c` is incident to the infinite vertex.

Precondition
  `t.dimension()` \( \geq1\) and \( i\neq j\). Moreover \( i,j \in\{0,1,2,3\}\) in dimension 3, \( i,j \in\{0,1,2\}\) in dimension 2, \( i,j \in\{0,1\}\) in dimension 1.

## ◆ is_infinite() [4/5]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::is_infinite | ( | const Edge & | e | ) | const

`true` iff edge `e` is incident to the infinite vertex.

Precondition
  `t.dimension()` \( \geq1\).

## ◆ is_infinite() [5/5]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::is_infinite | ( | const Facet & | f | ) | const

`true` iff facet `f` is incident to the infinite vertex.

Precondition
  `t.dimension()` \( \geq2\).

## ◆ is_valid() [1/2]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::is_valid | ( | bool | verbose = `false` | ) | const

This is a function for debugging purpose.

Debugging Support

Checks the combinatorial validity of the triangulation. Checks also the validity of its geometric embedding (see Section Representation). When `verbose` is set to `true`, messages describing the first invalidity encountered are printed.

## ◆ is_valid() [2/2]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::is_valid | ( | Cell_handle | c,
|  | bool | verbose = `false`
| ) |  | const

This is a function for debugging purpose.

Debugging Support

Checks the combinatorial validity of the cell by calling the `is_valid` method of the cell class. Also checks the geometric validity of `c`, if `c` is finite. (See Section Representation.)

When `verbose` is set to `true`, messages are printed to give a precise indication of the kind of invalidity encountered.

## ◆ is_vertex()

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::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.

## ◆ locate() [1/2]

template<typename Traits , typename TDS , typename SLDS >

Cell_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::locate | ( | const Point & | query,
|  | Cell_handle | start = `Cell_handle()`,
|  | bool * | could_lock_zone = `nullptr`
| ) |  | 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.

The optional argument `could_lock_zone` is used by the concurrency-safe version of the triangulation. When the pointer is not null, the locate will try to lock all the cells along the walk. If it succeeds, `*could_lock_zone` is `true`, otherwise it is false. In any case, the locked cells are not unlocked by `locate`, leaving this choice to the user.

## ◆ locate() [2/2]

template<typename Traits , typename TDS , typename SLDS >

Cell_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::locate | ( | const Point & | query,
|  | Locate_type & | lt,
|  | int & | li,
|  | int & | lj,
|  | Cell_handle | start = `Cell_handle()`,
|  | bool * | could_lock_zone = `nullptr`
| ) |  | 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.

The optional argument `could_lock_zone` is used by the concurrency-safe version of the triangulation. When the pointer is not null, the locate will try to lock all the cells along the walk. If it succeeds, `*could_lock_zone` is `true`, otherwise it is false. In any case, the locked cells are not unlocked by `locate`, leaving this choice to the user.

## ◆ mirror_index()

template<typename Traits , typename TDS , typename SLDS >

int CGAL::Triangulation_3< Traits, TDS, SLDS >::mirror_index | ( | Cell_handle | c,
|  | int | i
| ) |  | const

Returns the index of `c` in its \( i^{th}\) neighbor.

Precondition
  \( i \in\{0, 1, 2, 3\}\).

## ◆ mirror_vertex()

template<typename Traits , typename TDS , typename SLDS >

Vertex_handle CGAL::Triangulation_3< Traits, TDS, SLDS >::mirror_vertex | ( | Cell_handle | c,
|  | int | i
| ) |  | const

Returns the vertex of the \( i^{th}\) neighbor of `c` that is opposite to `c`.

Precondition
  \( i \in\{0, 1, 2, 3\}\).

## ◆ number_of_edges()

template<typename Traits , typename TDS , typename SLDS >

size_type CGAL::Triangulation_3< Traits, TDS, SLDS >::number_of_edges | ( |  | ) | const

The number of edges.

Returns 0 if `t.dimension() < 1`.

## ◆ number_of_facets()

template<typename Traits , typename TDS , typename SLDS >

size_type CGAL::Triangulation_3< Traits, TDS, SLDS >::number_of_facets | ( |  | ) | const

The number of facets.

Returns 0 if `t.dimension() < 2`.

## ◆ number_of_finite_cells()

template<typename Traits , typename TDS , typename SLDS >

size_type CGAL::Triangulation_3< Traits, TDS, SLDS >::number_of_finite_cells | ( |  | ) | const

The number of finite cells.

Returns 0 if `t.dimension() < 3`.

## ◆ number_of_finite_edges()

template<typename Traits , typename TDS , typename SLDS >

size_type CGAL::Triangulation_3< Traits, TDS, SLDS >::number_of_finite_edges | ( |  | ) | const

The number of finite edges.

Returns 0 if `t.dimension() < 1`.

## ◆ number_of_finite_facets()

template<typename Traits , typename TDS , typename SLDS >

size_type CGAL::Triangulation_3< Traits, TDS, SLDS >::number_of_finite_facets | ( |  | ) | const

The number of finite facets.

Returns 0 if `t.dimension() < 2`.

## ◆ operator=()

template<typename Traits , typename TDS , typename SLDS >

Triangulation_3 & CGAL::Triangulation_3< Traits, TDS, SLDS >::operator= | ( | const Triangulation_3< Traits, TDS, SLDS > & | 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.

## ◆ operator==()

template<typename Traits , typename TDS , typename SLDS >
template<class GT , class Tds1 , class Tds2 >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::operator== | ( | const Triangulation_3< GT, Tds1 > & | t1,
|  | const Triangulation_3< GT, Tds2 > & | t2
| ) |  |

Equality operator.

Returns `true` iff there exist a bijection between the vertices of `t1` and those of `t2` and a bijection between the cells of `t1` and those of `t2`, which preserve the geometry of the triangulation, that is, the points of each corresponding pair of vertices are equal, and the tetrahedra corresponding to each pair of cells are equal (up to a permutation of their vertices).

## ◆ point() [1/2]

template<typename Traits , typename TDS , typename SLDS >

const Point & CGAL::Triangulation_3< Traits, TDS, SLDS >::point | ( | Cell_handle | c,
|  | int | i
| ) |  | const

Returns the point given by vertex `i` of cell `c`.

Precondition
  `t.dimension()` \( \geq0\) and \( i \in\{0,1,2,3\}\) in dimension 3, \( i \in\{0,1,2\}\) in dimension 2, \( i \in\{0,1\}\) in dimension 1, \( i = 0\) in dimension 0, and the vertex is finite.

## ◆ point() [2/2]

template<typename Traits , typename TDS , typename SLDS >

const Point & CGAL::Triangulation_3< Traits, TDS, SLDS >::point | ( | Vertex_handle | v | ) | const

Same as the previous method for vertex `v`.

Precondition
  `t.dimension()` \( \geq0\) and the vertex is finite.

## ◆ segment() [1/2]

template<typename Traits , typename TDS , typename SLDS >

Segment CGAL::Triangulation_3< Traits, TDS, SLDS >::segment | ( | Cell_handle | c,
|  | int | i,
|  | int | j
| ) |  | const

Same as the previous method for edge `(c,i,j)`.

Precondition
  As above and \( i\neq j\). Moreover \( i,j \in\{0,1,2,3\}\) in dimension 3, \( i,j \in\{0,1,2\}\) in dimension 2, \( i,j \in\{0,1\}\) in dimension 1, and the edge is finite.

## ◆ segment() [2/2]

template<typename Traits , typename TDS , typename SLDS >

Segment CGAL::Triangulation_3< Traits, TDS, SLDS >::segment | ( | const Edge & | e | ) | const

Returns the line segment formed by the vertices of `e`.

Precondition
  `t.dimension()` \( \geq1\) and `e` is finite.

## ◆ segment_traverser_cells_begin() [1/2]

template<typename Traits , typename TDS , typename SLDS >

Segment_cell_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::segment_traverser_cells_begin | ( | const Point & | ps,
|  | const Point & | pt,
|  | Cell_handle | hint = `Cell_handle()`
| ) |  | const

returns the iterator that allows to visit the cells intersected by the line segment `[ps, pt]`.

If `[ps,pt]` entirely lies outside the convex hull, the iterator visits exactly one infinite cell.

The initial value of the iterator is the cell containing `ps`. If more than one cell contains `ps` (e.g. if `ps` lies on a vertex), the initial value is the cell intersected by the interior of the line segment `[ps,pt]`. If `ps` lies outside the convex hull and `pt` inside the convex full, the initial value is the infinite cell which finite facet is intersected by the interior of `[ps,pt]`.

The first cell containing `pt` is the last valid value of the iterator. It is followed by `segment_traverser_cells_end()`.

The optional argument `hint` can reduce the time to construct the iterator if it is geometrically close to `ps`.

Precondition
  `ps` and `pt` must be different points.
  `t.dimension() >= 2`. If the dimension is 2, both `ps` and `pt` must lie in the affine hull.

## ◆ segment_traverser_cells_begin() [2/2]

template<typename Traits , typename TDS , typename SLDS >

Segment_cell_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::segment_traverser_cells_begin | ( | Vertex_handle | vs,
|  | Vertex_handle | vt
| ) |  | const

returns the iterator that allows to visit the cells intersected by the line segment `[vs,vt]`.

The initial value of the iterator is the cell containing `vs` and intersected by the line segment `[vs,vt]` in its interior.

The first cell incident to `vt` is the last valid value of the iterator. It is followed by `segment_traverser_cells_end()`.

Precondition
  `vs` and `vt` must be different vertices and neither can be the infinite vertex.
  `t.dimension() >= 2`

## ◆ segment_traverser_cells_end()

template<typename Traits , typename TDS , typename SLDS >

Segment_cell_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::segment_traverser_cells_end | ( |  | ) | const

returns the past-the-end iterator over the intersected cells.

This iterator cannot be dereferenced. It indicates when the `Segment_cell_iterator` has passed the target.

Precondition
  `t.dimension() >= 2`

## ◆ segment_traverser_simplices_begin() [1/2]

template<typename Traits , typename TDS , typename SLDS >

Segment_simplex_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::segment_traverser_simplices_begin | ( | const Point & | ps,
|  | const Point & | pt,
|  | Cell_handle | hint = `Cell_handle()`
| ) |  | const

returns the iterator that allows to visit the simplices intersected by the line segment `[ps,pt]`.

If `[ps,pt]` entirely lies outside the convex hull, the iterator visits exactly one infinite cell.

The initial value of the iterator is the lowest dimension simplex containing `ps`.

The iterator remains valid until the first simplex containing `pt` is passed.

The optional argument `hint` can reduce the time to construct the iterator if it is close to `ps`.

Precondition
  `ps` and `pt` must be different points.
  `t.dimension() >= 2`. If the dimension is 2, both `ps` and `pt` must lie in the affine hull.

## ◆ segment_traverser_simplices_begin() [2/2]

template<typename Traits , typename TDS , typename SLDS >

Segment_simplex_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::segment_traverser_simplices_begin | ( | Vertex_handle | vs,
|  | Vertex_handle | vt
| ) |  | const

returns the iterator that allows to visit the simplices intersected by the line segment `[vs,vt]`.

The initial value of the iterator is `vs`. The iterator remains valid until `vt` is passed.

Precondition
  `vs` and `vt` must be different vertices and neither can be the infinite vertex.
  `t.dimension() >= 2`

## ◆ segment_traverser_simplices_end()

template<typename Traits , typename TDS , typename SLDS >

Segment_simplex_iterator CGAL::Triangulation_3< Traits, TDS, SLDS >::segment_traverser_simplices_end | ( |  | ) | const

returns the past-the-end iterator over the intersected simplices.

This iterator cannot be dereferenced. It indicates when the `Segment_simplex_iterator` has passed the target.

Precondition
  `t.dimension() >= 2`

## ◆ set_infinite_vertex()

template<typename Traits , typename TDS , typename SLDS >

void CGAL::Triangulation_3< Traits, TDS, SLDS >::set_infinite_vertex | ( | Vertex_handle | v | ) |

This is an advanced function.

Advanced

This method is meant to be used only if you have done a low-level operation on the underlying tds that invalidated the infinite vertex. Sets the infinite vertex.

## ◆ side_of_cell()

template<typename Traits , typename TDS , typename SLDS >

Bounded_side CGAL::Triangulation_3< Traits, TDS, SLDS >::side_of_cell | ( | const Point & | p,
|  | Cell_handle | c,
|  | Locate_type & | lt,
|  | int & | li,
|  | int & | lj
| ) |  | const

Returns a value indicating on which side of the oriented boundary of `c` the point `p` lies.

More precisely, it returns:

- `ON_BOUNDED_SIDE` if `p` is inside the cell. For an infinite cell this means that `p` lies strictly in the half space limited by its finite facet and not containing any other point of the triangulation.
- `ON_BOUNDARY` if p on the boundary of the cell. For an infinite cell this means that `p` lies on the finite facet. Then `lt` together with `li` and `lj` give the precise location on the boundary. (See the descriptions of the locate methods.)
- `ON_UNBOUNDED_SIDE` if `p` lies outside the cell. For an infinite cell this means that `p` does not satisfy either of the two previous conditions.
  Precondition
    `t.dimension() == 3`

## ◆ side_of_edge()

template<typename Traits , typename TDS , typename SLDS >

Bounded_side CGAL::Triangulation_3< Traits, TDS, SLDS >::side_of_edge | ( | const Point & | p,
|  | const Edge & | e,
|  | Locate_type & | lt,
|  | int & | li
| ) |  | const

Returns a value indicating on which side of the oriented boundary of `e` the point `p` lies:

- `ON_BOUNDED_SIDE` if `p` is inside the edge. For an infinite edge this means that `p` lies in the half line defined by the vertex and not containing any other point of the triangulation.
- `ON_BOUNDARY` if `p` equals one of the vertices, `li` give the index of the vertex in the cell storing `e`
- `ON_UNBOUNDED_SIDE` if `p` lies outside the edge. For an infinite edge this means that `p` lies on the other half line, which contains the other points of the triangulation.
  Precondition
    `t.dimension() == 1` and `p` is collinear with the points of the triangulation. `e.second == 0` and `e.third` \( =1\) (in dimension 1 there is only one edge per cell).

## ◆ side_of_facet()

template<typename Traits , typename TDS , typename SLDS >

Bounded_side CGAL::Triangulation_3< Traits, TDS, SLDS >::side_of_facet | ( | const Point & | p,
|  | const Facet & | f,
|  | Locate_type & | lt,
|  | int & | li,
|  | int & | lj
| ) |  | const

Returns a value indicating on which side of the oriented boundary of `f` the point `p` lies:

- `ON_BOUNDED_SIDE` if `p` is inside the facet. For an infinite facet this means that `p` lies strictly in the half plane limited by its finite edge and not containing any other point of the triangulation .
- `ON_BOUNDARY` if `p` is on the boundary of the facet. For an infinite facet this means that `p` lies on the finite edge. `lt`, `li` and `lj` give the precise location of `p` on the boundary of the facet. `li` and `lj` refer to indices in the degenerate cell `c` representing `f`.
- `ON_UNBOUNDED_SIDE` if `p` lies outside the facet. For an infinite facet this means that `p` does not satisfy either of the two previous conditions.

Precondition
  `t.dimension() == 2` and `p` lies in the plane containing the triangulation. `f.second` \( =3\) (in dimension 2 there is only one facet per cell).

## ◆ swap()

template<typename Traits , typename TDS , typename SLDS >

void CGAL::Triangulation_3< Traits, TDS, SLDS >::swap | ( | Triangulation_3< Traits, TDS, SLDS > & | tr | ) |

The triangulations `tr` and `t` are swapped.

`t.swap(tr)` should be preferred to `t = tr` or to `t(tr)` if `tr` is deleted after that. Indeed, there is no copy of cells and vertices, thus this method runs in constant time.

## ◆ tds()

template<typename Traits , typename TDS , typename SLDS >

Triangulation_data_structure & CGAL::Triangulation_3< Traits, TDS, SLDS >::tds | ( |  | ) |

Returns a reference to the triangulation data structure.

Advanced

This method is mainly a help for users implementing their own triangulation algorithms. The responsibility of keeping a valid triangulation belongs to the user when using advanced operations allowing a direct manipulation of the `tds`.

## ◆ tetrahedron()

template<typename Traits , typename TDS , typename SLDS >

Tetrahedron CGAL::Triangulation_3< Traits, TDS, SLDS >::tetrahedron | ( | Cell_handle | c | ) | const

Returns the tetrahedron formed by the four vertices of `c`.

Precondition
  `t.dimension() == 3` and the cell is finite.

## ◆ triangle() [1/2]

template<typename Traits , typename TDS , typename SLDS >

Triangle CGAL::Triangulation_3< Traits, TDS, SLDS >::triangle | ( | Cell_handle | c,
|  | int | i
| ) |  | const

Returns the triangle formed by the three vertices of facet `(c,i)`.

The triangle is oriented so that its normal points to the inside of cell `c`.

Precondition
  `t.dimension()` \( \geq2\) and \( i \in\{0,1,2,3\}\) in dimension 3, \( i = 3\) in dimension 2, and the facet is finite.

## ◆ triangle() [2/2]

template<typename Traits , typename TDS , typename SLDS >

Triangle CGAL::Triangulation_3< Traits, TDS, SLDS >::triangle | ( | const Facet & | f | ) | const

Same as the previous method for facet `f`.

Precondition
  `t.dimension()` \( \geq2\) and the facet is finite.

## ◆ try_lock_and_get_incident_cells()

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Triangulation_3< Traits, TDS, SLDS >::try_lock_and_get_incident_cells | ( | Vertex_handle | v,
|  | std::vector< Cell_handle > & | cells
| ) |  | const

Try to lock and copy the `Cell_handle`s of all cells incident to `v` into `cells`.

Returns `true` in case of success. Otherwise, `cells` is emptied and the function returns `false`. In any case, the locked cells are not unlocked by `try_lock_and_get_incident_cells()`, leaving this choice to the user.

Precondition
  `t.dimension() == 3`
  `v != Vertex_handle()` and `t.is_vertex(v)`.