#include <CGAL/Constrained_triangulation_plus_2.h>

Inherits from

Tr.

Definition

template<typename Tr>
class CGAL::Constrained_triangulation_plus_2< Tr >

The class Constrained_triangulation_plus_2<Tr> provides a constrained triangulation with an additional data structure that keeps track of the input constraints and of their refinement in the triangulation.

The class Constrained_triangulation_plus_2<Tr> inherits from its template parameter Tr, which has to be instantiated by a constrained or constrained Delaunay triangulation. The intersection tag of the base class determines whether intersecting input constraints are supported or not. When intersections of input constraints are supported, the base class constructs a triangulation of the arrangement of the constraints, introducing new vertices at each proper intersection point.

The data structure maintains for each input constraint the sequence of vertices on this constraint. Note that there is not a one-to-one correspondence between an input constraint and the sequence of vertices. These vertices are either vertices of the input constraint or intersection points. Also consecutive identical points in the input constraint result in a single vertex in the sequence of vertices on this constraint. In case of an input constraint being degenerate to a point, this point is inserted but there will not be a zero length constraint.

Two consecutive vertices of a constraint form a subconstraint. A subconstraint is a pair of vertex handles and corresponds to a constrained edge of the triangulation, which is a pair of a face handle and an index.

The triangulation also enables the retrieval of the set of subconstraints of the triangulation (not ordered along constraints). It further enables the retrieval of the set of input constraints that induce a subconstraint. As it is straightforward to obtain a subconstraint from a constrained edge e, one can obtain the input constraints which induce e.

Template Parameters

Tr	must be either a `CGAL::Constrained_triangulation_2` or a `CGAL::Constrained_Delaunay_triangulation_2`

See also: CGAL::Constrained_triangulation_2<Traits,Tds>; CGAL::Constrained_Delaunay_triangulation_2<Traits,Tds>; ConstrainedTriangulationTraits_2; ConstrainedDelaunayTriangulationTraits_2

Examples: Triangulation_2/constrained_hierarchy_plus.cpp, Triangulation_2/constrained_plus.cpp, Triangulation_2/polylines_triangulation.cpp, and Triangulation_2/segment_soup_to_polylines.cpp.

Classes
class	Context
	A context enables the access to the vertices of a constraint that pass through a subconstraint. More...

Related Functions
(Note that these are not member functions.)
template<typename Tr >
std::ostream &	operator<< (std::ostream &os, const Constrained_triangulation_plus_2< Tr > &ctp)
	Writes the triangulation as for `Tr`, then writes one constraint per line, starting with the number of vertices and the indices of the vertices of the constraint.

template<typename Tr >
std::istream &	operator>> (std::istream &is, Constrained_triangulation_plus_2< Tr > &ctp)
	Reads a triangulation from stream `is` and assigns it to the triangulation.

Types
typedef Tr	Triangulation
	The triangulation base class.

typedef Tr::Intersection_tag	Intersection_tag
	The intersection tag as defined in `Tr`.

typedef unspecified_type	Constraint_id
	The identifier of a polyline constraint.

typedef unspecified_type	Constraint_iterator
	An iterator to visit all the input constraints.

typedef Iterator_range< Constraint_iterator >	Constraints
	A range type for iterating over all constraints.

typedef std::pair< Vertex_handle, Vertex_handle >	Subconstraint
	A subconstraint is a pair of vertices that correspond to an `Edge`.

typedef unspecified_type	Subconstraint_iterator
	An iterator to visit all the subconstraints of the triangulation.

typedef Iterator_range< Subconstraint_iterator >	Subconstraints
	A range type for iterating over all subconstraints.

typedef unspecified_type	Vertices_in_constraint_iterator
	An iterator on the vertices of the chain of subconstraints representing a constraint.

typedef unspecified_type	Vertices_in_constraint
	A range type for iterating over the vertices of the constraint.

typedef unspecified_type	Context_iterator
	An iterator on constraints enclosing a given subconstraint.

typedef Iterator_range< Context_iterator >	Contexts
	range type for iterating over contexts.

Polyline Simplification
Advanced The polyline simplification algorithm described in Chapter Chapter_2D_Polyline_simplification operates on polyline constraints. The algorithm removes in each simplification step a vertex of a constraint and at the same time from the triangulation. The class `Constrained_triangulation_plus_2` stores for each constraint not only the sequence of vertices but also the original sequence of points at those vertices. As the `Vertices_in_constraint_iterator` enables the traversal of the current set of vertices, the `Points_in_constraint_iterator` enables the traversal of the points that were in the constraint before the simplification algorithm started. It enables the simplification algorithm to compute the error introduced by each simplification step: it is the distance of the current sequence (vertices) to the original sequence (points). Those stored points which do not correspond to a vertex can be removed afterward either for a single constraint or for all constraints. The simplification algorithm uses the following types and functions.
typedef unspecified_type	Points_in_constraint_iterator
	This is an advanced type.

Points_in_constraint_iterator	points_in_constraint_begin (Constraint_id cid) const
	This is an advanced function.

Points_in_constraint_iterator	points_in_constraint_end (Constraint_id cid) const
	This is an advanced function.

void	simplify (Vertices_in_constraint_iterator vicq)
	This is an advanced function.

size_type	remove_points_without_corresponding_vertex (Constraint_id cid)
	This is an advanced function.

void	remove_points_without_corresponding_vertex ()
	This is an advanced function.

Creation
	Constrained_triangulation_plus_2 (const Geom_traits &gt=Geom_traits())
	Introduces an empty triangulation.

	Constrained_triangulation_plus_2 (const Constrained_triangulation_plus_2 &ct)
	Copy constructor.

template<class ConstraintIterator >
	Constrained_triangulation_plus_2 (ConstraintIterator first, ConstraintIterator last, const Geom_traits &gt=Geom_traits())
	Introduces a constrained triangulation from the constraints in the range `[first,last)`.

Assignment
Constrained_triangulation_plus_2	operator= (const Constrained_triangulation_plus_2 &tr)
	Assignment.

void	swap (Constrained_triangulation_plus_2 tr)
	The triangulations `tr` and this triangulation are swapped.

Insertion and Removal
The class `Constrained_triangulation_plus_2` overwrites the following insertion and removal member functions for points and constraints.
Vertex_handle	insert (const Point &p, Face_handle start=Face_handle())
	inserts point `p` as a vertex of the triangulation.

Vertex_handle	insert (const Point &p, Locate_type lt, Face_handle loc, int li)
	inserts point `p` in the triangulation at the location given by `(lt,loc,i)`.

Vertex_handle	push_back (const Point &p)
	Equivalent to `insert(p)`.

template<class PointIterator >
size_type	insert (PointIterator first, PointIterator last)
	inserts the points in the range `[first,last)`.

Constraint_id	insert_constraint (Point a, Point b)
	inserts the constraint segment `ab` in the triangulation.

void	push_back (const std::pair< Point, Point > &c)
	inserts the constraint `c` in the triangulation.

Constraint_id	insert_constraint (Vertex_handle va, Vertex_handle vb)
	inserts a constraint whose endpoints are the vertices pointed by `va` and `vb` in the triangulation.

template<class PointIterator >
Constraint_id	insert_constraint (PointIterator first, PointIterator last, bool close=false)
	inserts a polyline defined by the points in the range `[first,last)` and returns the constraint id.

template<class ConstraintIterator >
std::size_t	insert_constraints (ConstraintIterator first, ConstraintIterator last)
	inserts the constraints in the range `[first,last)`.

template<class PointIterator , class IndicesIterator >
std::size_t	insert_constraints (PointIterator points_first, PointIterator points_last, IndicesIterator indices_first, IndicesIterator indices_last)
	Same as above except that each constraint is given as a pair of indices of the points in the range [points_first, points_last).

void	split_subconstraint_graph_into_constraints (const std::function< bool(Vertex_handle)> &is_terminal)
	splits into constraints the graph of subconstraints.

void	remove_constraint (Constraint_id cid)
	removes the constraint `cid`, without removing the points from the triangulation.

Access
Constraint_iterator	constraints_begin () const
	returns a `Constraint_iterator` that points at the first constraint of the triangulation.

Constraint_iterator	constraints_end () const
	returns the past-the-end iterator of the constraints of the triangulation.

Constraints	constraints () const
	returns a range of constraints.

Subconstraint_iterator	subconstraints_begin () const
	returns a `Subconstraint_iterator` pointing at the first subconstraint of the triangulation.

Subconstraint_iterator	subconstraints_end () const
	returns the past-the-end iterator of the subconstraints of the triangulation.

Subconstraints	subconstraints () const
	returns a range of subconstraints.

int	number_of_enclosing_constraints (Vertex_handle va, Vertex_handle vb) const
	returns the number of constraints enclosing the subconstraint `(va,vb)`.

Context	context (Vertex_handle va, Vertex_handle vb) const
	returns the `Context` relative to one of the constraints enclosing the subconstraint `(va,vb)`.

Context_iterator	contexts_begin (Vertex_handle va, Vertex_handle vb) const
	returns an iterator pointing at the first `Context` of the sequence of contexts corresponding to the constraints enclosing the subconstraint `(va,vb)`.

Context_iterator	contexts_end (Vertex_handle va, Vertex_handle vb) const
	returns an iterator past the end `Context` of the sequence of contexts corresponding to the constraints enclosing the subconstraint `(va,vb)`.

Contexts	contexts (Vertex_handle va, Vertex_handle vb) const
	returns a range of contexts.

Vertices_in_constraint_iterator	vertices_in_constraint_begin (Constraint_id cid) const
	returns an iterator on the first vertex on the constraint `cid`.

Vertices_in_constraint_iterator	vertices_in_constraint_end (Constraint_id cid) const
	returns an iterator past the last vertex on the constraint `cid`.

Vertices_in_constraint	vertices_in_constraint (Constraint_id cid) const
	returns a range of the vertices on the constraint `cid`.

Member Typedef Documentation

◆ Constraint_id

template<typename Tr >

typedef unspecified_type CGAL::Constrained_triangulation_plus_2< Tr >::Constraint_id

The identifier of a polyline constraint.

The class is model of Assignable, CopyConstructible, DefaultConstructible, LessThanComparable and EqualityComparable.

A default constructed Constraint_id is a singular value that can not be the ID of a constraint.

◆ Constraint_iterator

template<typename Tr >

typedef unspecified_type CGAL::Constrained_triangulation_plus_2< Tr >::Constraint_iterator

An iterator to visit all the input constraints.

The order of visit is undefined. The value type of this iterator is Constraint_id.

◆ Context_iterator

template<typename Tr >

typedef unspecified_type CGAL::Constrained_triangulation_plus_2< Tr >::Context_iterator

An iterator on constraints enclosing a given subconstraint.

The value type of this iterator is Context.

◆ Points_in_constraint_iterator

template<typename Tr >

typedef unspecified_type CGAL::Constrained_triangulation_plus_2< Tr >::Points_in_constraint_iterator

This is an advanced type.

Advanced

An iterator on the points of the original constraint before simplification steps are applied. The value type of this iterator is Point. A Vertices_in_constraint_iterator can be converted into a Points_in_constraint_iterator, but not the other way around.

◆ Subconstraint_iterator

template<typename Tr >

typedef unspecified_type CGAL::Constrained_triangulation_plus_2< Tr >::Subconstraint_iterator

An iterator to visit all the subconstraints of the triangulation.

The order of visit is undefined. The value type of this iterator is std::pair<Subconstraint,std::list<Context>*> corresponding to the vertices of the subconstraint.

◆ Vertices_in_constraint_iterator

template<typename Tr >

typedef unspecified_type CGAL::Constrained_triangulation_plus_2< Tr >::Vertices_in_constraint_iterator

An iterator on the vertices of the chain of subconstraints representing a constraint.

The value type of this iterator is Vertex_handle.

Constructor & Destructor Documentation

◆ Constrained_triangulation_plus_2()

template<typename Tr >

template<class ConstraintIterator >

CGAL::Constrained_triangulation_plus_2< Tr >::Constrained_triangulation_plus_2	(	ConstraintIterator	first,
		ConstraintIterator	last,
		const Geom_traits &	gt = `Geom_traits()`
	)

Introduces a constrained triangulation from the constraints in the range [first,last).

Template Parameters

ConstraintIterator must be an InputIterator with the value type std::pair<Point,Point> or Segment.

Member Function Documentation

◆ context()

template<typename Tr >

Context CGAL::Constrained_triangulation_plus_2< Tr >::context	(	Vertex_handle	va,
		Vertex_handle	vb
	)		const

returns the Context relative to one of the constraints enclosing the subconstraint (va,vb).

Precondition: va and vb refer to the vertices of a constrained edge of the triangulation.

◆ contexts_begin()

template<typename Tr >

Context_iterator CGAL::Constrained_triangulation_plus_2< Tr >::contexts_begin	(	Vertex_handle	va,
		Vertex_handle	vb
	)		const

returns an iterator pointing at the first Context of the sequence of contexts corresponding to the constraints enclosing the subconstraint (va,vb).

Precondition: va and vb refer to the vertices of a constrained edge of the triangulation.

◆ contexts_end()

template<typename Tr >

Context_iterator CGAL::Constrained_triangulation_plus_2< Tr >::contexts_end	(	Vertex_handle	va,
		Vertex_handle	vb
	)		const

returns an iterator past the end Context of the sequence of contexts corresponding to the constraints enclosing the subconstraint (va,vb).

Precondition: va and vb refer to the vertices of a constrained edge of the triangulation.

◆ insert() [1/2]

template<typename Tr >

Vertex_handle CGAL::Constrained_triangulation_plus_2< Tr >::insert	(	const Point &	p,
		Locate_type	lt,
		Face_handle	loc,
		int	li
	)

inserts point p in the triangulation at the location given by (lt,loc,i).

See also: Triangulation_2::locate()

◆ insert() [2/2]

template<typename Tr >

template<class PointIterator >

size_type CGAL::Constrained_triangulation_plus_2< Tr >::insert	(	PointIterator	first,
		PointIterator	last
	)

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

Returns the number of inserted points.

Template Parameters

PointIterator must be an InputIterator with the value type Point.

◆ insert_constraint() [1/3]

template<typename Tr >

Constraint_id CGAL::Constrained_triangulation_plus_2< Tr >::insert_constraint	(	Point	a,
		Point	b
	)

inserts the constraint segment ab in the triangulation.

If the two points are equal the point is inserted but no constraint, and a default constructed Constraint_id is returned.

Examples: Triangulation_2/constrained_hierarchy_plus.cpp, Triangulation_2/constrained_plus.cpp, Triangulation_2/polylines_triangulation.cpp, and Triangulation_2/segment_soup_to_polylines.cpp.

◆ insert_constraint() [2/3]

template<typename Tr >

template<class PointIterator >

Constraint_id CGAL::Constrained_triangulation_plus_2< Tr >::insert_constraint	(	PointIterator	first,
		PointIterator	last,
		bool	close = `false`
	)

inserts a polyline defined by the points in the range [first,last) and returns the constraint id.

The polyline is considered as a closed curve if the first and last point are equal or if close == true. This enables for example passing the vertex range of a Polygon_2. When traversing the vertices of a closed polyline constraint with a Vertices_in_constraint_iterator the first and last vertex are the same. In case the range is empty a default constructed Constraint_id is returned. In case the range contains only one point or all points are equal the point is inserted but no constraint, and a default constructed Constraint_id is returned.

Template Parameters

PointIterator must be an InputIterator with the value type Point.

◆ insert_constraint() [3/3]

template<typename Tr >

Constraint_id CGAL::Constrained_triangulation_plus_2< Tr >::insert_constraint	(	Vertex_handle	va,
		Vertex_handle	vb
	)

inserts a constraint whose endpoints are the vertices pointed by va and vb in the triangulation.

If the two vertex handles are equal no constraint is inserted, and a default constructed Constraint_id is returned.

◆ insert_constraints() [1/2]

template<typename Tr >

template<class ConstraintIterator >

std::size_t CGAL::Constrained_triangulation_plus_2< Tr >::insert_constraints	(	ConstraintIterator	first,
		ConstraintIterator	last
	)

inserts the constraints in the range [first,last).

Note that this function is not guaranteed to insert the constraints following the order of ConstraintIterator, as spatial_sort() is used to improve efficiency. More precisely, all endpoints are inserted prior to the segments and according to the order provided by the spatial sort. Once endpoints have been inserted, the segments are inserted in the order of the input iterator, using the vertex handles of its endpoints. In case the constraints are degenerate the points are inserted, but no constraints.

Template Parameters

ConstraintIterator must be an InputIterator with the value type std::pair<Point,Point> or Segment.

Returns: the number of inserted points.

◆ insert_constraints() [2/2]

template<typename Tr >

template<class PointIterator , class IndicesIterator >

std::size_t CGAL::Constrained_triangulation_plus_2< Tr >::insert_constraints	(	PointIterator	points_first,
		PointIterator	points_last,
		IndicesIterator	indices_first,
		IndicesIterator	indices_last
	)

Same as above except that each constraint is given as a pair of indices of the points in the range [points_first, points_last).

The indices must go from 0 to std::distance(points_first, points_last)

Template Parameters

PointIterator	is an `InputIterator` with the value type `Point`.
IndicesIterator	is an `InputIterator` with `std::pair<Int, Int>` where `Int` is an integral type implicitly convertible to `std::size_t`

Note: points are inserted even if they are not endpoint of a constraint.

Returns: the number of inserted points.

◆ number_of_enclosing_constraints()

template<typename Tr >

int CGAL::Constrained_triangulation_plus_2< Tr >::number_of_enclosing_constraints	(	Vertex_handle	va,
		Vertex_handle	vb
	)		const

returns the number of constraints enclosing the subconstraint (va,vb).

Precondition: va and vb refer to the vertices of a constrained edge of the triangulation.

◆ operator=()

template<typename Tr >

Constrained_triangulation_plus_2 CGAL::Constrained_triangulation_plus_2< Tr >::operator= ( const Constrained_triangulation_plus_2< Tr > & tr )

Assignment.

All the vertices and faces are duplicated. The bidirectional mapping between constraints and subconstraints is also duplicated.

◆ points_in_constraint_begin()

template<typename Tr >

Points_in_constraint_iterator CGAL::Constrained_triangulation_plus_2< Tr >::points_in_constraint_begin ( Constraint_id cid ) const

This is an advanced function.

Advanced

Returns an iterator to the first point on the constraint before any simplification step.

◆ points_in_constraint_end()

template<typename Tr >

Points_in_constraint_iterator CGAL::Constrained_triangulation_plus_2< Tr >::points_in_constraint_end ( Constraint_id cid ) const

This is an advanced function.

Advanced

Returns an iterator past the last point on the constraint before any simplification step.

◆ push_back()

template<typename Tr >

void CGAL::Constrained_triangulation_plus_2< Tr >::push_back ( const std::pair< Point, Point > & c )

inserts the constraint c in the triangulation.

If the two points are equal the point is inserted but no constraint, and a default constructed Constraint_id is returned.

◆ remove_points_without_corresponding_vertex() [1/2]

template<typename Tr >

void CGAL::Constrained_triangulation_plus_2< Tr >::remove_points_without_corresponding_vertex ( )

This is an advanced function.

Advanced

Removes all original points that correspond to vertices in the constraints which have been removed by the simplify() function.

◆ remove_points_without_corresponding_vertex() [2/2]

template<typename Tr >

size_type CGAL::Constrained_triangulation_plus_2< Tr >::remove_points_without_corresponding_vertex ( Constraint_id cid )

This is an advanced function.

Advanced

Removes the original points that correspond to vertices in the constraint cid which have been removed by the simplify() function.

◆ simplify()

template<typename Tr >

void CGAL::Constrained_triangulation_plus_2< Tr >::simplify ( Vertices_in_constraint_iterator vicq )

This is an advanced function.

Advanced

Removes the vertex at vicq from the constraint and the triangulation. The point of that vertex remains stored in the sequence of original points of the constraint until remove_points_without_corresponding_vertex(Constraint_id) or remove_points_without_corresponding_vertex() is called.

The polyline simplification algorithm described in Chapter Chapter_2D_Polyline_simplification operates on polyline constraints and applies simplify() to vertices in constraints based on a cost and stop function.

Precondition: Each vertex of the triangulation must be either a vertex of a constraint or a vertex at the intersection of constraints.; vicq must neither be the first nor the last vertex on a constraint.; The vertex referred by vicq is not contained in any other constraint.; Let vip and vir be defined as vip = std::prev(vicq) and vir = std::next(vicr).; The line segment between *vicp->point() and *vicr->point() must not intersect any constraint.

◆ split_subconstraint_graph_into_constraints()

template<typename Tr >

void CGAL::Constrained_triangulation_plus_2< Tr >::split_subconstraint_graph_into_constraints ( const std::function< bool(Vertex_handle)> & is_terminal )

splits into constraints the graph of subconstraints.

Consider the graph g={V,E} where V is the set of vertices of the triangulation and E is the set of all subconstraints of all constraints of the triangulation.

This function splits into polylines the graph g at vertices of degree greater than 2 and at vertices for which is_terminal(v)==true.

Each computed polyline is stored as a constraint of the triangulation.

Warning: all existing constraints will be discarded.

Parameters

is_terminal An optional function returning true if the vertex v of degree 2 is a polyline endpoint and false otherwise. If omitted, a function always returning false will be used, that is no degree 2 vertex will be considered as a polyline endpoint.

See also: split_graph_into_polylines()

◆ swap()

template<typename Tr >

void CGAL::Constrained_triangulation_plus_2< Tr >::swap ( Constrained_triangulation_plus_2< Tr > tr )

The triangulations tr and this triangulation are swapped.

This operation should be preferred to the assignment or to the copy constructor if tr is deleted after that.

Inherits from

Definition

Classes

Related Functions

Types

Polyline Simplification

Creation

Assignment

Insertion and Removal

Access

Member Typedef Documentation

◆ Constraint_id

◆ Constraint_iterator

◆ Context_iterator

◆ Points_in_constraint_iterator

◆ Subconstraint_iterator

◆ Vertices_in_constraint_iterator

Constructor & Destructor Documentation

◆ Constrained_triangulation_plus_2()

Member Function Documentation

◆ context()

◆ contexts_begin()

◆ contexts_end()

◆ insert() [1/2]

◆ insert() [2/2]

◆ insert_constraint() [1/3]

◆ insert_constraint() [2/3]

◆ insert_constraint() [3/3]

◆ insert_constraints() [1/2]

◆ insert_constraints() [2/2]

◆ number_of_enclosing_constraints()

◆ operator=()

◆ points_in_constraint_begin()

◆ points_in_constraint_end()

◆ push_back()

◆ remove_points_without_corresponding_vertex() [1/2]

◆ remove_points_without_corresponding_vertex() [2/2]

◆ simplify()

◆ split_subconstraint_graph_into_constraints()

◆ swap()