Modules
	CGAL::insert()

Functions
template<class GeomTraitsA , class GeomTraitsB , class GeomTraitsRes , class TopTraitsA , class TopTraitsB , class TopTraitsRes , class OverlayTraits >
void	CGAL::overlay (const Arrangement_2< GeomTraitsA, TopTraitsA > &arr1, const Arrangement_2< GeomTraitsB, TopTraitsB > &arr2, Arrangement_2< GeomTraitsRes, TopTraitsRes > &arr_res, OverlayTraits &ovl_tr)
	Computes the overlay of two arrangements `arr1` and `arr2`, and sets the output arrangement `res` to represent the overlaid arrangement.

template<typename Traits , typename Dcel1 , typename Dcel2 , typename ResDcel , typename OverlayTraits >
void	CGAL::overlay (const Arrangement_with_history_2< Traits, Dcel1 > &arr1, const Arrangement_with_history_2< Traits, Dcel2 > &arr2, Arrangement_with_history_2< Traits, ResDcel > &res, OverlayTraits &ovl_tr)
	Computes the overlay of two arrangements with history `arr1` and `arr2`, and sets the output arrangement with history `res` to represent the overlaid arrangement.

template<typename Traits , typename TopologyTraits , typename OutputIterator >
OutputIterator	CGAL::decompose (const Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &arr, OutputIterator oi)
	produces the symbolic vertical decomposition of a given arrangement.

template<typename Traits , typename Dcel , typename PointLocation >
Arrangement_2< Traits, Dcel >::Halfedge_handle	CGAL::insert_non_intersecting_curve (Arrangement_2< Traits, Dcel > &arr, const typename Traits::X_monotone_curve_2 &xc, const PointLocation &pl=walk_pl)
	Inserts a given \( x\)-monotone curve into a given arrangement, where the interior of the given curve is disjoint from all existing arrangement vertices and edges.

template<typename Traits , typename Dcel , InputIterator >
void	CGAL::insert_non_intersecting_curves (Arrangement_2< Traits, Dcel > &arr, InputIterator first, InputIterator last)
	Inserts a set of \( x\)-monotone curves in a given range into a given arrangement.

template<typename Traits , typename Dcel , typename PointLocation >
Arrangement_2< Traits, Dcel >::Vertex_handle	CGAL::insert_point (Arrangement_2< Traits, Dcel > &arr, const typename Traits::Point_2 &p, const PointLocation &pl=walk_pl)
	Inserts a given point into a given arrangement.

template<typename Traits , typename Dcel >
bool	CGAL::is_valid (const Arrangement_2< Traits, Dcel > &arr)
	Checks the validity of a given arrangement.

template<typename Traits , typename Dcel >
Arrangement_2< Traits, Dcel >::Face_handle	CGAL::remove_edge (Arrangement_2< Traits, Dcel > &arr, typename Arrangement_2< Traits, Dcel >::Halfedge_handle e)
	Removes an edge given by one of the twin halfedges that forms it, from a given arrangement.

template<typename Traits , typename Dcel >
bool	CGAL::remove_vertex (Arrangement_2< Traits, Dcel > &arr, typename Arrangement_2< Traits, Dcel >::Vertex_handle v)
	Attempts to removed a given vertex from a given arrangement.

template<typename GeometryTraits , typename TopologyTraits , typename Curve , typename PointLocation >
bool	CGAL::do_intersect (Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &arr, const Curve &c, const PointLocation &pl)
	Checks if a given curve or \(x\)-monotone curve intersects an existing arrangement's edges or vertices.

template<typename GeometryTraits , typename TopologyTraits , typename PointLocation >
Arrangement_on_surface_2< GeometryTraits, TopologyTraits >::Halfedge_handle	CGAL::insert_non_intersecting_curve (Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &arr, const typename Traits::X_monotone_curve_2 &xc, const PointLocation &pl=walk_pl)
	Inserts a given \( x\)-monotone curve into a given arrangement, where the interior of the given curve is disjoint from all existing arrangement vertices and edges.

template<typename GeometryTraits , typename TopologyTraits , typename InputIterator >
void	CGAL::insert_non_intersecting_curves (Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &arr, InputIterator first, InputIterator last)
	Inserts a set of \( x\)-monotone curves in a given range into a given arrangement.

template<typename GeometryTraits , typename TopologyTraits , typename PointLocation >
Arrangement_on_surface_2< GeometryTraits, TopologyTraits >::Vertex_handle	CGAL::insert_point (Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &arr, const typename Traits::Point_2 &p, const PointLocation &pl=walk_pl)
	Inserts a given point into a given arrangement.

template<typename GeometryTraits , typename TopologyTraits >
bool	CGAL::is_valid (const Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &arr)
	Checks the validity of a given arrangement.

template<typename GeometryTraits , typename TopologyTraits >
Arrangement_on_surface_2< GeometryTraits, TopologyTraits >::Face_handle	CGAL::remove_edge (Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &arr, typename Arrangement_on_surface_2< GeometryTraits, TopologyTraits >::Halfedge_handle e)
	Removes an edge given by one of the twin halfedges that forms it, from a given arrangement.

template<typename GeometryTraits , typename TopologyTraits >
bool	CGAL::remove_vertex (Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &arr, typename Arrangement_on_surface_2< GeometryTraits, TopologyTraits >::Vertex_handle v)
	Attempts to removed a given vertex from a given arrangement.

template<typename GeometryTraits , typename TopologyTraits , typename OutputIterator , typename PointLocation >
OutputIterator	CGAL::zone (Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &arr, const typename GeometryTraits::X_monotone_curve_2 &c, OutputIterator oi, const PointLocation &pl)
	computes the zone of the given \(x\)-monotone curve in a given arrangement.

template<typename GeometryTraits , typename TopologyTraits >
Size	CGAL::remove_curve (Arrangement_on_surface_with_history_2< GeometryTraits, TopologyTraits > &arr, typename Arrangement_on_surface_with_history_2< GeometryTraits, TopologyTraits >::Curve_handle ch)
	Removes a given curve from a given arrangement.

template<typename Traits , typename Dcel >
Size	CGAL::remove_curve (Arrangement_with_history_2< Traits, Dcel > &arr, typename Arrangement_with_history_2< Traits, Dcel >::Curve_handle ch)
	Removes a given curvespecified by its handle `ch`, from a given arrangement `arr`, deleting all the edges it induces.

Function Documentation

◆ decompose()

template<typename Traits , typename TopologyTraits , typename OutputIterator >

OutputIterator CGAL::decompose	(	const Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &	arr,
		OutputIterator	oi
	)

#include <CGAL/Arr_vertical_decomposition_2.h>

produces the symbolic vertical decomposition of a given arrangement.

More precisely, this function performs a batched vertical ray-shooting query from every arrangement vertex, and pairs each vertex with a pair of polymorphic objects, one corresponds to the arrangement feature that lies below it, and the other corresponds to the feature that lies above it.

The finite arrangement vertices and the features they "see", if exist, that are, the query results, are inserted in ascending \(xy\)-lexicographic order (of the query vertex) into an output container given through an output iterator. If the vertex is the top end-vertex of a vertical edge, we say that there is no feature below it; similarly, if it is the bottom end-vertex of a vertical edge, we say that there is no feature above it. Each feature, if exists, is represented by a discriminated union container that holds an object of one of the following types:

Arrangement_on_surface_2::Halfedge_const_handle, if the vertex is located above (or below) an edge. The given halfedge is always directed from right to left. In case there is no concrete edge below (or above) the vertex, and the arrangement is unbounded, then the object returned is a fictitious halfedge.
Arrangement_on_surface_2::Face_const_handle, in case there is no edge below (or above) the vertex, and the arrangement is bounded.
Arrangement_on_surface_2::Vertex_const_handle, in case the vertex is located vertically above (or below) another arrangement vertex.

The output of this function can be readily used for inserting vertical walls and physically decomposing the arrangement into pseudo-trapezoids.

Parameters

arr	The arrangement.
oi	The output iterator that points at the output container.

Returns: The past-the-end iterator of the output container.

Requirements

Precondition: Dereferencing oi must yield an object of type std::pair<Arrangement_on_surface_2::Vertex_const_handle, std::pair<std::optional<Type,std::optional<Type>>>, where Type is std::variant<Arrangement_on_surface_2::Vertex_const_handle, Arrangement_on_surface_2::Halfedge_const_handle, Arrangement_on_surface_2::Face_const_handle>.

Examples: Arrangement_on_surface_2/bounded_vertical_decomposition.cpp.

◆ do_intersect()

template<typename GeometryTraits , typename TopologyTraits , typename Curve , typename PointLocation >

bool CGAL::do_intersect	(	Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &	arr,
		const Curve &	c,
		const PointLocation &	pl
	)

#include <CGAL/Arrangement_on_surface_2.h>

Checks if a given curve or \(x\)-monotone curve intersects an existing arrangement's edges or vertices.

If the give curve is not an \(x\)-monotone curve then the function subdivides the given curve into \( x\)-monotone subcurves and isolated vertices . Each subcurve is in turn checked for intersection. The function uses the zone algorithm to check if the curve intersects the arrangement. First, the curve's left endpoint is located. Then, its zone is computed starting from its left endpoint location. The zone computation terminates when an intersection with an arrangement's edge/vertex is found or when the right endpoint is reached.

A given point-location object is used for locating the left endpoint of the given curve in the existing arrangement. By default, the function uses the "walk along line" point-location strategy - namely an instance of the class Arr_walk_along_line_point_location<Arrangement_on_surface_2<GeometryTraits, TopologyTraits> >.

Checks if the given curve or \( x\)-monotone curve c intersects edges or vertices of the existing arrangement arr.

Precondition: If provided, pl must be attached to the given arrangement arr.

Requirements

If c is \( x\)-monotone then the instantiated GeometryTraits class must model the ArrangementXMonotoneTraits_2 concept. If c is a curve then the instantiated GeometryTraits class must model the ArrangementTraits_2 concept. That is, it should define the Curve_2 type, and support its subdivision into \( x\)-monotone subcurves (and perhaps isolated points).
The point-location object pl, must model the ArrangementPointLocation_2 concept.

◆ insert_non_intersecting_curve() [1/2]

template<typename Traits , typename Dcel , typename PointLocation >

Arrangement_2< Traits, Dcel >::Halfedge_handle CGAL::insert_non_intersecting_curve	(	Arrangement_2< Traits, Dcel > &	arr,
		const typename Traits::X_monotone_curve_2 &	xc,
		const PointLocation &	pl = `walk_pl`
	)

#include <CGAL/Arrangement_2.h>

Inserts a given \( x\)-monotone curve into a given arrangement, where the interior of the given curve is disjoint from all existing arrangement vertices and edges.

Under this assumption, it is possible to locate the endpoints of the given curve in the arrangement, and use one of the specialized insertion member-functions of the arrangement according to the results. The insertion operations creates a single new edge, that is, two twin halfedges, and the function returns a handle for the one directed lexicographically in increasing order (from left to right).

A given point-location object is used for answering the two point-location queries on the given curve endpoints. By default, the function uses the "walk along line" point-location strategy - namely, an instance of the class Arr_walk_along_line_point_location<Arrangement_2<Traits,Dcel> >.

Precondition: If provided, pl must be attached to the given arrangement arr.

Requirements

The instantiated Traits class must model the restricted ArrangementBasicTraits_2 concept, as no intersections are computed.
The point-location object pl must model the ArrangementPointLocation_2 concept.

◆ insert_non_intersecting_curve() [2/2]

template<typename GeometryTraits , typename TopologyTraits , typename PointLocation >

Arrangement_on_surface_2< GeometryTraits, TopologyTraits >::Halfedge_handle CGAL::insert_non_intersecting_curve	(	Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &	arr,
		const typename Traits::X_monotone_curve_2 &	xc,
		const PointLocation &	pl = `walk_pl`
	)

#include <CGAL/Arrangement_on_surface_2.h>

Inserts a given \( x\)-monotone curve into a given arrangement, where the interior of the given curve is disjoint from all existing arrangement vertices and edges.

Under this assumption, it is possible to locate the endpoints of the given curve in the arrangement, and use one of the specialized insertion member-functions of the arrangement according to the results. The insertion operations creates a single new edge, that is, two twin halfedges, and the function returns a handle for the one directed lexicographically in increasing order (from left to right).

A given point-location object is used for answering the two point-location queries on the given curve endpoints. By default, the function uses the "walk along line" point-location strategy - namely, an instance of the class Arr_walk_along_line_point_location<Arrangement_on_surface_2<GeometryTraits, TopologyTraits> >.

Precondition: If provided, pl must be attached to the given arrangement arr.

Requirements

The instantiated Traits class must model the restricted ArrangementBasicTraits_2 concept, as no intersections are computed.
The point-location object pl must model the ArrangementPointLocation_2 concept.

◆ insert_non_intersecting_curves() [1/2]

template<typename Traits , typename Dcel , InputIterator >

void CGAL::insert_non_intersecting_curves	(	Arrangement_2< Traits, Dcel > &	arr,
		InputIterator	first,
		InputIterator	last
	)

#include <CGAL/Arrangement_2.h>

Inserts a set of \( x\)-monotone curves in a given range into a given arrangement.

The insertion is performed in an aggregated manner, using the sweep-line algorithm. The input curves should be pairwise disjoint in their interior and pairwise interior-disjoint from all existing arrangement vertices and edges.

Requirements

The instantiated Traits class must model the ArrangementBasicTraits_2 concept, as no intersections are computed.
InputIterator::value_type must be Traits::X_monotone_curve_2

◆ insert_non_intersecting_curves() [2/2]

template<typename GeometryTraits , typename TopologyTraits , typename InputIterator >

void CGAL::insert_non_intersecting_curves	(	Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &	arr,
		InputIterator	first,
		InputIterator	last
	)

#include <CGAL/Arrangement_on_surface_2.h>

Inserts a set of \( x\)-monotone curves in a given range into a given arrangement.

The insertion is performed in an aggregated manner, using the sweep-line algorithm. The input curves should be pairwise disjoint in their interior and pairwise interior-disjoint from all existing arrangement vertices and edges.

Requirements

The instantiated Traits class must model the ArrangementBasicTraits_2 concept, as no intersections are computed.
InputIterator::value_type must be Traits::X_monotone_curve_2

◆ insert_point() [1/2]

template<typename Traits , typename Dcel , typename PointLocation >

Arrangement_2< Traits, Dcel >::Vertex_handle CGAL::insert_point	(	Arrangement_2< Traits, Dcel > &	arr,
		const typename Traits::Point_2 &	p,
		const PointLocation &	pl = `walk_pl`
	)

#include <CGAL/Arrangement_2.h>

Inserts a given point into a given arrangement.

It uses a given point-location object to locate the given point in the given arrangement. If the point coincides with an existing vertex, there is nothing left to do; if it lies on an edge, the edge is split at the point. Otherwise, the point is contained inside a face, and is inserted as an isolated vertex inside this face. By default, the function uses the "walk along line" point-location strategy - namely, an instance of the class Arr_walk_along_line_point_location<Arrangement_2<Traits,Dcel> >. In either case, the function returns a handle for the vertex associated with the point.

Precondition: If provided, pl must be attached to the given arrangement arr.

Requirements

The instantiated Traits class must model the ArrangementXMonotoneTraits_2 concept. Not all expressions listed by this concept are required. In fact the traits class must model the ArrangementBasicTraits_2 concept, and support the splitting functionality.
The point-location object pl, must model the ArrangementPointLocation_2 concept.

◆ insert_point() [2/2]

template<typename GeometryTraits , typename TopologyTraits , typename PointLocation >

Arrangement_on_surface_2< GeometryTraits, TopologyTraits >::Vertex_handle CGAL::insert_point	(	Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &	arr,
		const typename Traits::Point_2 &	p,
		const PointLocation &	pl = `walk_pl`
	)

#include <CGAL/Arrangement_on_surface_2.h>

Inserts a given point into a given arrangement.

It uses a given point-location object to locate the given point in the given arrangement. If the point coincides with an existing vertex, there is nothing left to do; if it lies on an edge, the edge is split at the point. Otherwise, the point is contained inside a face, and is inserted as an isolated vertex inside this face. By default, the function uses the "walk along line" point-location strategy - namely, an instance of the class Arr_walk_along_line_point_location<Arrangement_on_surface_2<GeometryTraits, TopologyTraits> >. In either case, the function returns a handle for the vertex associated with the point.

Precondition: If provided, pl must be attached to the given arrangement arr.

Requirements

The instantiated Traits class must model the ArrangementXMonotoneTraits_2 concept. Not all expressions listed by this concept are required. In fact the traits class must model the ArrangementBasicTraits_2 concept, and support the splitting functionality.
The point-location object pl, must model the ArrangementPointLocation_2 concept.

◆ is_valid() [1/2]

template<typename Traits , typename Dcel >

bool CGAL::is_valid ( const Arrangement_2< Traits, Dcel > & arr )

#include <CGAL/Arrangement_2.h>

Checks the validity of a given arrangement.

Invokes the member function arr.is_valid() to verify the topological correctness of the arrangement. Then it performs additional validity tests. It checks that all \( x\)-monotone curves associated with arrangement edges are pairwise disjoint in their interior. Then it makes sure that all holes and all isolated vertices are located within the proper arrangement faces. Note that the test carried out by this function may take a considerable amount of time; it is recommended to be used only for debugging purposes.

Requirements

The instantiated traits class must model the concept ArranagmentXMonotoneTraits_2.

◆ is_valid() [2/2]

template<typename GeometryTraits , typename TopologyTraits >

bool CGAL::is_valid ( const Arrangement_on_surface_2< GeometryTraits, TopologyTraits > & arr )

#include <CGAL/Arrangement_on_surface_2.h>

Checks the validity of a given arrangement.

Invokes the member function arr.is_valid() to verify the topological correctness of the arrangement. Then it performs additional validity tests. It checks that all \( x\)-monotone curves associated with arrangement edges are pairwise disjoint in their interior. Then it makes sure that all holes and all isolated vertices are located within the proper arrangement faces. Note that the test carried out by this function may take a considerable amount of time; it is recommended to be used only for debugging purposes.

Requirements

The instantiated traits class must model the concept ArranagmentXMonotoneTraits_2.

◆ overlay() [1/2]

template<class GeomTraitsA , class GeomTraitsB , class GeomTraitsRes , class TopTraitsA , class TopTraitsB , class TopTraitsRes , class OverlayTraits >

void CGAL::overlay	(	const Arrangement_2< GeomTraitsA, TopTraitsA > &	arr1,
		const Arrangement_2< GeomTraitsB, TopTraitsB > &	arr2,
		Arrangement_2< GeomTraitsRes, TopTraitsRes > &	arr_res,
		OverlayTraits &	ovl_tr
	)

#include <CGAL/Arr_overlay_2.h>

Computes the overlay of two arrangements arr1 and arr2, and sets the output arrangement res to represent the overlaid arrangement.

Computes the overlay of two input arrangement objects, and returns the overlaid arrangement. All three arrangements can be instantiated with different geometric traits classes and different DCEL classes (encapsulated in the various topology-traits classes). The geometry traits of the resulting arrangement is used to construct the resulting arrangement. This means that all the types (e.g., Traits::Point_2, Traits::Curve_2, and Traits::Point_2) of both input arrangements have to be convertible to the types in the resulting arrangement. A given overlay-traits object is used to properly construct the overlaid DCEL that represents the resulting arrangement.

Precondition: res does not refer to either arr1 or arr2 (that is, "self overlay" is not supported).; The overlay-traits object ovl_tr must model the OverlayTraits concept, which is able to construct records of the ResDcel class on the basis of the Dcel1 and Dcel2 records that induce them.

See also: OverlayTraits

Examples: Arrangement_on_surface_2/face_extension_overlay.cpp, Arrangement_on_surface_2/overlay.cpp, Arrangement_on_surface_2/overlay_color.cpp, and Arrangement_on_surface_2/overlay_unbounded.cpp.

◆ overlay() [2/2]

template<typename Traits , typename Dcel1 , typename Dcel2 , typename ResDcel , typename OverlayTraits >

void CGAL::overlay	(	const Arrangement_with_history_2< Traits, Dcel1 > &	arr1,
		const Arrangement_with_history_2< Traits, Dcel2 > &	arr2,
		Arrangement_with_history_2< Traits, ResDcel > &	res,
		OverlayTraits &	ovl_tr
	)

#include <CGAL/Arr_overlay_2.h>

Computes the overlay of two arrangements with history arr1 and arr2, and sets the output arrangement with history res to represent the overlaid arrangement.

The function also constructs a consolidated set of curves that induce res.

Computes the overlay of two input arrangement objects, and returns the overlaid arrangement. All three arrangements can be instantiated with different geometric traits classes and different DCEL classes (encapsulated in the various topology-traits classes). The geometry traits of the resulting arrangement is used to construct the resulting arrangement. This means that all the types (e.g., Traits::Point_2, Traits::Curve_2, and Traits::Point_2) of both input arrangements have to be convertible to the types in the resulting arrangement. A given overlay-traits object is used to properly construct the overlaid DCEL that represents the resulting arrangement.

Precondition: res does not refer to either arr1 or arr2 (that is, "self overlay" is not supported).; The overlay-traits object ovl_tr must model the OverlayTraits concept, which is able to construct records of the ResDcel class on the basis of the Dcel1 and Dcel2 records that induce them.

See also: OverlayTraits

◆ remove_curve() [1/2]

template<typename GeometryTraits , typename TopologyTraits >

Size CGAL::remove_curve	(	Arrangement_on_surface_with_history_2< GeometryTraits, TopologyTraits > &	arr,
		typename Arrangement_on_surface_with_history_2< GeometryTraits, TopologyTraits >::Curve_handle	ch
	)

#include <CGAL/Arrangement_on_surface_with_history_2.h>

Removes a given curve from a given arrangement.

The curve is specified by its handle ch, from the arrangement arr, by deleting all the edges it induces. The function returns the number of deleted edges.

◆ remove_curve() [2/2]

template<typename Traits , typename Dcel >

Size CGAL::remove_curve	(	Arrangement_with_history_2< Traits, Dcel > &	arr,
		typename Arrangement_with_history_2< Traits, Dcel >::Curve_handle	ch
	)

#include <CGAL/Arrangement_with_history_2.h>

Removes a given curvespecified by its handle ch, from a given arrangement arr, deleting all the edges it induces.

The function returns the number of deleted edges.

◆ remove_edge() [1/2]

template<typename Traits , typename Dcel >

Arrangement_2< Traits, Dcel >::Face_handle CGAL::remove_edge	(	Arrangement_2< Traits, Dcel > &	arr,
		typename Arrangement_2< Traits, Dcel >::Halfedge_handle	e
	)

#include <CGAL/Arrangement_2.h>

Removes an edge given by one of the twin halfedges that forms it, from a given arrangement.

Once the edge is removed, if the vertices associated with its endpoints become isolated, they are removed as well. The call remove_edge(arr, e) is equivalent to the call arr.remove_edge (e, true, true). However, this free function requires that Traits be a model of the refined concept ArrangementXMonotoneTraits_2, which requires merge operations on \( x\)-monotone curves. If one of the end-vertices of the given edge becomes redundant after the edge is removed (see remove_vertex() for the definition of a redundant vertex), it is removed, and its incident edges are merged. If the edge-removal operation causes two faces to merge, the merged face is returned. Otherwise, the face to which the edge was incident before the removal is returned.

Requirements

The instantiated traits class must model the concept ArrangementXMonotoneTraits_2.

◆ remove_edge() [2/2]

template<typename GeometryTraits , typename TopologyTraits >

Arrangement_on_surface_2< GeometryTraits, TopologyTraits >::Face_handle CGAL::remove_edge	(	Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &	arr,
		typename Arrangement_on_surface_2< GeometryTraits, TopologyTraits >::Halfedge_handle	e
	)

#include <CGAL/Arrangement_on_surface_2.h>

Removes an edge given by one of the twin halfedges that forms it, from a given arrangement.

Once the edge is removed, if the vertices associated with its endpoints become isolated, they are removed as well. The call remove_edge(arr, e) is equivalent to the call arr.remove_edge (e, true, true). However, this free function requires that Traits be a model of the refined concept ArrangementXMonotoneTraits_2, which requires merge operations on \( x\)-monotone curves. If one of the end-vertices of the given edge becomes redundant after the edge is removed (see remove_vertex() for the definition of a redundant vertex), it is removed, and its incident edges are merged. If the edge-removal operation causes two faces to merge, the merged face is returned. Otherwise, the face to which the edge was incident before the removal is returned.

Requirements

The instantiated traits class must model the concept ArrangementXMonotoneTraits_2.

◆ remove_vertex() [1/2]

template<typename Traits , typename Dcel >

bool CGAL::remove_vertex	(	Arrangement_2< Traits, Dcel > &	arr,
		typename Arrangement_2< Traits, Dcel >::Vertex_handle	v
	)

#include <CGAL/Arrangement_2.h>

Attempts to removed a given vertex from a given arrangement.

The vertex can be removed if it is either an isolated vertex, (and has no incident edge,) or if it is a redundant vertex. That is, it has exactly two incident edges, whose associated curves can be merged to form a single \( x\)-monotone curve. The function returns a boolean value that indicates whether it succeeded removing the vertex from the arrangement.

Requirements

The instantiated Traits class must model the ArrangementXMonotoneTraits_2 concept. Not all expressions listed by this concept are required. In fact the traits class must model the ArrangementBasicTraits_2 concept and support the merging functionality.

◆ remove_vertex() [2/2]

template<typename GeometryTraits , typename TopologyTraits >

bool CGAL::remove_vertex	(	Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &	arr,
		typename Arrangement_on_surface_2< GeometryTraits, TopologyTraits >::Vertex_handle	v
	)

#include <CGAL/Arrangement_on_surface_2.h>

Attempts to removed a given vertex from a given arrangement.

The vertex can be removed if it is either an isolated vertex, (and has no incident edge,) or if it is a redundant vertex. That is, it has exactly two incident edges, whose associated curves can be merged to form a single \( x\)-monotone curve. The function returns a boolean value that indicates whether it succeeded removing the vertex from the arrangement.

Requirements

The instantiated Traits class must model the ArrangementXMonotoneTraits_2 concept. Not all expressions listed by this concept are required. In fact the traits class must model the ArrangementBasicTraits_2 concept and support the merging functionality.

◆ zone()

template<typename GeometryTraits , typename TopologyTraits , typename OutputIterator , typename PointLocation >

OutputIterator CGAL::zone	(	Arrangement_on_surface_2< GeometryTraits, TopologyTraits > &	arr,
		const typename GeometryTraits::X_monotone_curve_2 &	c,
		OutputIterator	oi,
		const PointLocation &	pl
	)

#include <CGAL/Arrangement_on_surface_2.h>

computes the zone of the given \(x\)-monotone curve in a given arrangement.

More precisely, this function finds the arrangement vertices, edges ,and faces that the given \(x\)-monotone curve intersects, and inserts them in the order they are discovered when traversing the \(x\)-monotone curve from left to right into an output contaiuner given through an output iterator. An object in the resulting zone is represented by a discriminated union container that holds a vertex handle, halfedge handle, or a face handle.

A given point-location object is used for answering point-location queries during the insertion process. By default, the function uses the "walk along line" point-location strategy, namely, an instance of the class Arr_walk_along_line_point_location<Arrangement_on_surface_2<GeometryTraits, TopologyTraits>>.

Parameters

arr	The given arrangement.
c	The \(x\)-monotone curve.
oi	The output iterator that points at the output container.
pl	The point-location object.

Returns: The past-the-end iterator of the output container.

Precondition: If provided, pl must be attached to the given arrangement arr.; The instantiated GeometryTraits class must model the ArrangementXMonotoneTraits_2 concept.; The point-location object pl, must model the ArrangementPointLocation_2 concept.; Dereferencing oi must yield a polymorphic object of type std::variant<Arrangement_on_surface_2::Vertex_handle, Arrangement_on_surface_2::Halfedge_handle, Arrangement_on_surface_2::Face_handle>.

Requirements

Modules

Functions

Function Documentation

◆ decompose()

◆ do_intersect()

◆ insert_non_intersecting_curve() [1/2]

◆ insert_non_intersecting_curve() [2/2]

◆ insert_non_intersecting_curves() [1/2]

◆ insert_non_intersecting_curves() [2/2]

◆ insert_point() [1/2]

◆ insert_point() [2/2]

◆ is_valid() [1/2]

◆ is_valid() [2/2]

◆ overlay() [1/2]

◆ overlay() [2/2]

◆ remove_curve() [1/2]

◆ remove_curve() [2/2]

◆ remove_edge() [1/2]

◆ remove_edge() [2/2]

◆ remove_vertex() [1/2]

◆ remove_vertex() [2/2]

◆ zone()