\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 2D Triangulation

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CGAL::Regular_triangulation_2< Traits, Tds > Class Template Reference
Triangulation Classes

`#include <CGAL/Regular_triangulation_2.h>`

## Inherits from

CGAL::Triangulation_2< Weighted_point_mapper_2< Traits >, Tds >.

## Definition

The class `Regular_triangulation_2` is designed to maintain the regular triangulation of a set of weighted points.

Let \( { PW} = \{(p_i, w_i), i = 1, \ldots , n \}\) be a set of weighted points where each \( p_i\) is a point and each \( w_i\) is a scalar called the weight of point \( p_i\). Alternatively, each weighted point \( (p_i, w_i)\) can be regarded as a two dimensional sphere with center \( p_i\) and radius \( r_i=\sqrt{w_i}\).

The power diagram of the set \( { PW}\) is a planar partition such that each cell corresponds to sphere \( (p_i, w_i)\) of \( { PW}\) and is the locus of points \( p\) whose power with respect to \( (p_i, w_i)\) is less than its power with respect to any other sphere \( (p_j, w_j)\) in \( { PW}\). The dual of this diagram is a triangulation whose domain covers the convex hull of the set \( { P}= \{ p_i, i = 1, \ldots , n \}\) of center points and whose vertices are a subset of \( { P}\). Such a triangulation is called a regular triangulation. The three points \( p_i, p_j\) and \( p_k\) of \( { P}\) form a triangle in the regular triangulation of \( { PW}\) iff there is a point \( p\) of the plane whose powers with respect to \( (p_i, w_i)\), \( (p_j, w_j)\) and \( (p_k, w_k)\) are equal and less than the power of \( p\) with respect to any other sphere in \( { PW}\).

Let us defined the power product of two weighted points \( (p_i, w_i)\) and \( (p_j, w_j)\) as:

\[ \Pi(p_i, w_i,p_j, w_j) = p_ip_j ^2 - w_i - w_j . \]

\( \Pi(p_i, w_i,p_j, 0)\) is simply the power of point \( p_j\) with respect to the sphere \( (p_i, w_i)\), and two weighted points are said to be orthogonal if their power product is null. The power circle of three weighted points \( (p_i, w_i)\), \( (p_j, w_j)\) and \( (p_k, w_k)\) is defined as the unique circle \( (\pi, \omega)\) orthogonal to \( (p_i, w_i)\), \( (p_j, w_j)\) and \( (p_k, w_k)\).

The regular triangulation of the sets \( { PW}\) satisfies the following regular property (which just reduces to the Delaunay property when all the weights are null): a triangle \( p_ip_jp_k\) of the regular triangulation of \( { PW}\) is such that the power product of any weighted point \( (p_l, w_l)\) of \( { PW}\) with the power circle of \( (p_i, w_i)\), \( (p_j, w_j)\) is \( (p_k, w_k)\) is positive or null. We call power test of the weighted point \( (p_l, w_l)\) with respect to the face \( p_ip_jp_k\), the predicates testing the sign of the power product of \( (p_l, w_l)\) with respect to the power circle of \( (p_i, w_i)\), \( (p_j, w_j)\) is \( (p_k, w_k)\). This power product is given by the following determinant

\[ \left| \begin{array}{cccc} 1 & x_i & y_i & x_i ^2 + y_i ^2 - w_i \\ 1 & x_j & y_j & x_j ^2 + y_j ^2 - w_j \\ 1 & x_k & y_k & x_k ^2 + y_k ^2 - w_k \\ 1 & x_l & y_l & x_l ^2 + y_l ^2 - w_l \end{array} \right| \]

A pair of neighboring faces \( p_ip_jp_k\) and \( p_ip_jp_l\) is said to be locally regular (with respect to the weights in \( { PW}\)) if the power test of \( (p_l,w_l)\) with respect to \( p_ip_jp_k\) is positive. A classical result of computational geometry establishes that a triangulation of the convex hull of \( { P}\) such that any pair of neighboring faces is regular with respect to \( { PW}\), is a regular triangulation of \( { PW}\).

Alternatively, the regular triangulation of the weighted points set \( { PW}\) can be obtained as the projection on the two dimensional plane of the convex hull of the set of three dimensional points \( { P'}= \{ (p_i,p_i ^2 - w_i ), i = 1, \ldots , n \}\).

The vertices of the regular triangulation of a set of weighted points \( { PW}\) form only a subset of the set of center points of \( { PW}\). Therefore the insertion of a weighted point in a regular triangulation does not necessarily imply the creation of a new vertex. If the new inserted point does not appear as a vertex in the regular triangulation, it is said to be hidden.

Hidden points are stored in special vertices called hidden vertices. A hidden point is considered as hidden by the facet of the triangulation where its point component is located : in fact, the hidden point can appear as vertex of the triangulation only if this facet is removed. Each face of a regular triangulation stores the list of hidden vertices whose points are located in the facet. When a facet is removed, points hidden by this facet are reinserted in the triangulation.

Template Parameters

  Traits | is the geometric traits parameter and must be a model of the concept `RegularTriangulationTraits_2`. The concept `RegularTriangulationTraits_2` refines the concept `TriangulationTraits_2` by adding the type `Weighted_point_2` to describe weighted points and the type `Power_test_2` to perform power tests on weighted points.
  Tds | must be a model of `TriangulationDataStructure_2`. The face base of a regular triangulation has to be a model of the concept `RegularTriangulationFaceBase_2`. while the vertex base class has to be a model of `RegularTriangulationVertexBase_2`. CGAL provides a default instantiation for the `Tds` parameter by the class `Triangulation_data_structure_2 < Regular_triangulation_vertex_base_2<Traits>, Regular_triangulation_face_base_2<Traits> >`.

See Also
  `CGAL::Triangulation_2<Traits,Tds>`
  `TriangulationDataStructure_2`
  `RegularTriangulationTraits_2`
  `RegularTriangulationFaceBase_2`
  `RegularTriangulationVertexBase_2`
  `CGAL::Regular_triangulation_face_base_2<Traits>`
  `CGAL::Regular_triangulation_vertex_base_2<Traits>`

Examples:
  Triangulation_2/info_insert_with_pair_iterator_regular_2.cpp, and Triangulation_2/regular.cpp.

## Types
typedef Traits::Distance | Distance
typedef Traits::Line | Line
typedef Traits::Ray | Ray
typedef Traits::Bare_point | Bare_point
typedef Traits::Weighted_point | Weighted_point
typedef unspecified_type | All_vertices_iterator
| An iterator that allows to enumerate the vertices that are not hidden.
typedef unspecified_type | Finite_vertices_iterator
| An iterator that allows to enumerate the finite vertices that are not hidden.
typedef unspecified_type | Hidden_vertices_iterator
| An iterator that allows to enumerate the hidden vertices.

## Creation
| Regular_triangulation_2 (const Traits &gt=Traits())
| Introduces an empty regular triangulation.
| Regular_triangulation_2 (const Regular_triangulation_2 &rt)
| Copy constructor.
template<class InputIterator >
Regular_triangulation_2 < Traits, Tds > | Regular_triangulation_2 (InputIterator first, InputIterator last, Traits gt=Traits())
| Equivalent to constructing an empty triangulation with the optional traits class argument and calling insert(first,last).

## Insertion and Removal
Vertex_handle | insert (const Weighted_point &p, Face_handle f=Face_handle())
| inserts weighted point `p` in the regular triangulation. More...
Vertex_handle | insert (const Weighted_point &p, Locate_type lt, Face_handle loc, int li)
| insert a weighted point `p` whose bare-point is assumed to be located in `lt,loc,li`. More...
Vertex_handle | push_back (const Point &p)
| Equivalent to `insert(p)`.
template<class InputIterator >
std::ptrdiff_t | insert (InputIterator first, InputIterator last)
| inserts the weighted points in the range `[first,last)`. More...
template<class WeightedPointWithInfoInputIterator >
std::ptrdiff_t | insert (WeightedPointWithInfoInputIterator first, WeightedPointWithInfoInputIterator last)
| inserts the weighted points in the range `[first,last)`. More...
void | remove (Vertex_handle v)
| removes the vertex from the triangulation.

## Queries
template<class OutputItFaces , class OutputItBoundaryEdges , class OutputItHiddenVertices >
CGAL::Triple< OutputItFaces, OutputItBoundaryEdges, OutputItHiddenVertices > | get_conflicts_and_boundary_and_hidden_vertices (const Weighted_point &p, OutputItFaces fit, OutputItBoundaryEdges eit, OutputItHiddenVertices vit, Face_handle start) const
| outputs the faces, boundary edges, and hidden vertices of the conflict zone of point `p` to output iterators. More...
template<class OutputItFaces , class OutputItBoundaryEdges >
std::pair< OutputItFaces, OutputItBoundaryEdges > | get_conflicts_and_boundary (const Weighted_point &p, OutputItFaces fit, OutputItBoundaryEdges eit, Face_handle start) const
| outputs the faces and boundary edges of the conflict zone of point `p` to output iterators. More...
template<class OutputItFaces , class OutputItHiddenVertices >
std::pair< OutputItFaces, OutputItHiddenVertices > | get_conflicts_and_hidden_vertices (const Weighted_point &p, OutputItFaces fit, OutputItHiddenVertices vit, Face_handle start) const
| outputs the faces and hidden vertices of the conflict zone of point `p` to output iterators. More...
template<class OutputItBoundaryEdges , class OutputItHiddenVertices >
std::pair < OutputItBoundaryEdges, OutputItHiddenVertices > | get_boundary_of_conflicts_and_hidden_vertices (const Weighted_point &p, OutputItBoundaryEdges eit, OutputItHiddenVertices vit, Face_handle start) const
| outputs the boundary edges and hidden vertices of the conflict zone of point `p` to output iterators. More...
template<class OutputItFaces >
OutputItFaces | get_conflicts (const Point &p, OutputItFaces fit, Face_handle start) const
| outputs the faces of the conflict zone of point `p` to output iterators. More...
template<class OutputItBoundaryEdges >
OutputItBoundaryEdges | get_boundary_of_conflicts (const Point &p, OutputItBoundaryEdges eit, Face_handle start) const
| outputs the boundary edges of the conflict zone of `p` in counterclockwise order where each edge is described through the incident face which is not in conflict with `p`. More...
template<class OutputItHiddenVertices >
OutputItHiddenVertices | get_hidden_vertices (const Point &p, OutputItHiddenVertices vit, Face_handle start) const
| outputs the hidden vertices of the conflict zone of `p` into an output iterator. More...
Vertex_handle | nearest_power_vertex (Bare_point p) const
| Returns the vertex of the triangulation which is nearest to `p` with respect to the power distance. More...

## Access Functions
int | number_of_vertices () const
| returns the number of finite vertices that are not hidden.
int | number_of_hidden_vertices () const
| returns the number of hidden vertices.
Hidden_vertices_iterator | hidden_vertices_begin () const
| starts at an arbitrary hidden vertex.
Hidden_vertices_iterator | hidden_vertices_end () const
| past the end iterator for the sequence of hidden vertices.
Finite_vertices_iterator | finite_vertices_begin () const
| starts at an arbitrary unhidden finite vertex
Finite_vertices_iterator | finite_vertices_end () const
| Past-the-end iterator.
All_vertices_iterator | all_vertices_end () const
| starts at an arbitrary unhidden vertex.
All_vertices_iterator | all_vertices_begin () const
| past the end iterator.

## Dual Power Diagram
The following member functions provide the elements of the dual power diagram.
Point | weighted_circumcenter (const Face_handle &f) const
| returns the center of the circle orthogonal to the three weighted points corresponding to the vertices of face `f`. More...
Point | dual (const Face_handle &f) const
| same as weighted_circumcenter.
Object | dual (const Edge &e) const
| If both incident faces are finite, returns a segment whose endpoints are the duals of each incident face. More...
Object | dual (const Edge_circulator &ec) const
| Idem.
Object | dual (const Edge_iterator &ei) const
| Idem.
template<class Stream >
Stream & | draw_dual (Stream &ps)
| output the dual power diagram to stream `ps`.

## Predicates
Oriented_side | power_test (Face_handle f, const Weighted_point &p) const
| Returns the power test of `p` with respect to the power circle associated with `f`.

## Miscellaneous
bool | is_valid (bool verbose=false, int level=0) const
| Tests the validity of the triangulation as a `Triangulation_2` and additionally test the regularity of the triangulation. More...

## Additional Inherited Members
Public Member Functions inherited from CGAL::Triangulation_cw_ccw_2
| Triangulation_cw_ccw_2 ()
| default constructor.
int | ccw (const int i) const
| returns the index of the neighbor or vertex that is next to the neighbor or vertex with index `i` in counterclockwise order around a face.
int | cw (const int i) const
| returns the index of the neighbor or vertex that is next to the neighbor or vertex with index `i` in counterclockwise order around a face.

## Member Function Documentation

template<typename Traits , typename Tds >

Object CGAL::Regular_triangulation_2< Traits, Tds >::dual | ( | const Edge & | e) |  | const

If both incident faces are finite, returns a segment whose endpoints are the duals of each incident face.

If only one incident face is finite, returns a ray whose endpoint is the dual of the finite incident face and supported by the line which is the bisector of the edge's endpoints. If both incident faces are infinite, returns the line which is the bisector of the edge's endpoints otherwise.

template<typename Traits , typename Tds >
template<class OutputItBoundaryEdges >

OutputItBoundaryEdges CGAL::Regular_triangulation_2< Traits, Tds >::get_boundary_of_conflicts | ( | const Point & | p,
|  | OutputItBoundaryEdges | eit,
|  | Face_handle | start
| ) |  | const

outputs the boundary edges of the conflict zone of `p` in counterclockwise order where each edge is described through the incident face which is not in conflict with `p`.

The function returns the resulting output iterator.

template<typename Traits , typename Tds >
template<class OutputItBoundaryEdges , class OutputItHiddenVertices >

std::pair<OutputItBoundaryEdges,OutputItHiddenVertices> CGAL::Regular_triangulation_2< Traits, Tds >::get_boundary_of_conflicts_and_hidden_vertices | ( | const Weighted_point & | p,
|  | OutputItBoundaryEdges | eit,
|  | OutputItHiddenVertices | vit,
|  | Face_handle | start
| ) |  | const

outputs the boundary edges and hidden vertices of the conflict zone of point `p` to output iterators.

See `get_conflicts_and_boundary_and_hidden_vertices()` for details. The function returns in a `std::pair` the resulting output iterators.

template<typename Traits , typename Tds >
template<class OutputItFaces >

OutputItFaces CGAL::Regular_triangulation_2< Traits, Tds >::get_conflicts | ( | const Point & | p,
|  | OutputItFaces | fit,
|  | Face_handle | start
| ) |  | const

outputs the faces of the conflict zone of point `p` to output iterators.

The function returns the resulting output iterator.

Precondition
  `dimension()==2`.

template<typename Traits , typename Tds >
template<class OutputItFaces , class OutputItBoundaryEdges >

std::pair<OutputItFaces,OutputItBoundaryEdges> CGAL::Regular_triangulation_2< Traits, Tds >::get_conflicts_and_boundary | ( | const Weighted_point & | p,
|  | OutputItFaces | fit,
|  | OutputItBoundaryEdges | eit,
|  | Face_handle | start
| ) |  | const

outputs the faces and boundary edges of the conflict zone of point `p` to output iterators.

See `get_conflicts_and_boundary_and_hidden_vertices()` for details.

The function returns in a `std::pair` the resulting output iterators.

Precondition
  `dimension()==2`.

template<typename Traits , typename Tds >
template<class OutputItFaces , class OutputItBoundaryEdges , class OutputItHiddenVertices >

CGAL::Triple<OutputItFaces,OutputItBoundaryEdges,OutputItHiddenVertices> CGAL::Regular_triangulation_2< Traits, Tds >::get_conflicts_and_boundary_and_hidden_vertices | ( | const Weighted_point & | p,
|  | OutputItFaces | fit,
|  | OutputItBoundaryEdges | eit,
|  | OutputItHiddenVertices | vit,
|  | Face_handle | start
| ) |  | const

outputs the faces, boundary edges, and hidden vertices of the conflict zone of point `p` to output iterators.

Template Parameters

  OutputItFaces | is an output iterator with `Face_handle` as value type.
  OutputItBoundaryEdges | is an output iterator with `Edge` as value type.
  OutputItHiddenVertices | is an output iterator with `Vertex_handle` as value type.

This member function outputs in the container pointed to by `fit` the faces which are in conflict with point `p`, i.e., the faces whose power circles have negative power wrt. `p`. It outputs in the container pointed to by `eit` the boundary of the zone in conflict with `p`. It inserts the vertices that would be hidden by `p` into the container pointed to by `vit`. The boundary edges of the conflict zone are output in counter-clockwise order and each edge is described through its incident face which is not in conflict with `p`. The function returns in a `CGAL::Triple` the resulting output iterators.

Precondition
  `dimension()==2`.

template<typename Traits , typename Tds >
template<class OutputItFaces , class OutputItHiddenVertices >

std::pair<OutputItFaces,OutputItHiddenVertices> CGAL::Regular_triangulation_2< Traits, Tds >::get_conflicts_and_hidden_vertices | ( | const Weighted_point & | p,
|  | OutputItFaces | fit,
|  | OutputItHiddenVertices | vit,
|  | Face_handle | start
| ) |  | const

outputs the faces and hidden vertices of the conflict zone of point `p` to output iterators.

See `get_conflicts_and_boundary_and_hidden_vertices()` for details. The function returns in a `std::pair` the resulting output iterators.

Precondition
  `dimension()==2`.

template<typename Traits , typename Tds >
template<class OutputItHiddenVertices >

OutputItHiddenVertices CGAL::Regular_triangulation_2< Traits, Tds >::get_hidden_vertices | ( | const Point & | p,
|  | OutputItHiddenVertices | vit,
|  | Face_handle | start
| ) |  | const

outputs the hidden vertices of the conflict zone of `p` into an output iterator.

The function returns the resulting output iterator.

template<typename Traits , typename Tds >

Vertex_handle CGAL::Regular_triangulation_2< Traits, Tds >::insert | ( | const Weighted_point & | p,
|  | Face_handle | f = `Face_handle()`
| ) |  |

inserts weighted point `p` in the regular triangulation.

If the point `p` does not appear as a vertex of the triangulation, the returned vertex is a hidden vertex. If given the parameter `f` is used as an hint for the place to start the location process of point `p`.

template<typename Traits , typename Tds >

Vertex_handle CGAL::Regular_triangulation_2< Traits, Tds >::insert | ( | const Weighted_point & | p,
|  | Locate_type | lt,
|  | Face_handle | loc,
|  | int | li
| ) |  |

insert a weighted point `p` whose bare-point is assumed to be located in `lt,loc,li`.

See the description of member function `Triangulation_2::locate()`.

template<typename Traits , typename Tds >
template<class InputIterator >

std::ptrdiff_t CGAL::Regular_triangulation_2< Traits, Tds >::insert | ( | InputIterator | first,
|  | InputIterator | last
| ) |  |

inserts the weighted points in the range `[first,last)`.

It returns the difference of the number of vertices between after and before the insertions (it may be negative due to hidden points). Note that this function is not guaranteed to insert the weighted points following the order of `InputIterator`, as `spatial_sort()` is used to improve efficiency.

Template Parameters

  InputIterator | must be an input iterator with the value type `Weighted_point`.

template<typename Traits , typename Tds >
template<class WeightedPointWithInfoInputIterator >

std::ptrdiff_t CGAL::Regular_triangulation_2< Traits, Tds >::insert | ( | WeightedPointWithInfoInputIterator | first,
|  | WeightedPointWithInfoInputIterator | last
| ) |  |

inserts the weighted points in the range `[first,last)`.

It returns the difference of the number of vertices between after and before the insertions (it may be negative due to hidden points). Note that this function is not guaranteed to insert the weighted points following the order of `WeightedPointWithInfoInputIterator`, as `spatial_sort` is used to improve efficiency. 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, one of the objects of type `Vertex::Info` will be stored in the vertex.

Precondition
  `Vertex` must be model of the concept `TriangulationVertexBaseWithInfo_2`.

Template Parameters

  WeightedPointWithInfoInputIterator | must be an input iterator with value type `std::pair<Weighted_point,Vertex::Info>`.

template<typename Traits , typename Tds >

bool CGAL::Regular_triangulation_2< Traits, Tds >::is_valid | ( | bool | verbose = `false`,
|  | int | level = `0`
| ) |  | const

Tests the validity of the triangulation as a `Triangulation_2` and additionally test the regularity of the triangulation.

This method is useful to debug regular triangulation algorithms implemented by the user.

template<typename Traits , typename Tds >

Vertex_handle CGAL::Regular_triangulation_2< Traits, Tds >::nearest_power_vertex | ( | Bare_point | p) |  | const

Returns the vertex of the triangulation which is nearest to `p` with respect to the power distance.

This means that the power of the query point `p` with respect to the weighted point in the nearest vertex is smaller than the power of `p` with respect to the weighted point in any other vertex. Ties are broken arbitrarily. The default constructed handle is returned if the triangulation is empty.

template<typename Traits , typename Tds >

Point CGAL::Regular_triangulation_2< Traits, Tds >::weighted_circumcenter | ( | const Face_handle & | f) |  | const

returns the center of the circle orthogonal to the three weighted points corresponding to the vertices of face `f`.

Precondition
  `f` is not infinite.