#include <CGAL/Arrangement_with_history_2.h>

Inherits from

Definition

An object arr of the class Arrangement_with_history_2 represents the planar subdivision induced by a set of input curves \( \cal C\). The arrangement is represented as a doubly-connected edge-list (Dcel). As is the case for the Arrangement_2<Traits,Dcel>, each Dcel vertex is associated with a point and each edge is associated with an \( x\)-monotone curve whose interior is disjoint from all other edges and vertices. Each such \( x\)-monotone curve is a subcurve of some \( C \in \cal C\) - or may represent an overlap among several curves in \( \cal C\).

The Arrangement_with_history_2 class-template extends the Arrangement_2 class-template by keeping an additional container of input curves representing \( \cal C\), and by maintaining a cross-mapping between these curves and the arrangement edges they induce. This way it is possible to determine the inducing curve(s) of each arrangement edge. This mapping also allows the traversal of input curves, and the traversal of edges induced by each curve.

The Arrangement_with_history_2 template has two parameters:

The Traits template-parameter should be instantiated with a model of the ArrangementTraits_2 concept. The traits class defines the Curve_2 type, which represents an input curve. It also defines the types of \( x\)-monotone curves and two-dimensional points, namely ArrangementTraits_2::X_monotone_curve_2 and ArrangementTraits_2::Point_2, respectively, and supports basic geometric predicates on them.
The Dcel template-parameter should be instantiated with a class that is a model of the ArrangementDcelWithRebind concept. The value of this parameter is by default Arr_default_dcel<Traits>.

See Also: ArrangementDcel; Arr_default_dcel<Traits>; ArrangementTraits_2; Arrangement_2<Traits,Dcel>; insertion functions; removal functions; overlaying arrangements

Examples:: Arrangement_on_surface_2/curve_history.cpp, Arrangement_on_surface_2/edge_manipulation_curve_history.cpp, and Arrangement_on_surface_2/io_curve_history.cpp.

Types
typedef Arrangement_with_history_2 < Traits_2, Dcel >	Self
	a private type used as an abbreviation of the `Arrangement_with_history_2` type hereafter.

typedef unspecified_type	Traits_2
	the traits class in use.

typedef Traits_2::Point_2	Point_2
	the point type, as defined by the traits class.

typedef Traits_2::X_monotone_curve_2	X_monotone_curve_2
	the \( x\)-monotone curve type, as defined by the traits class.

typedef Traits_2::Curve_2	Curve_2
	the curve type, as defined by the traits class.

In addition, the nested types `Vertex`, `Halfedge` and `Face` are defined, as well as all handle, iterator and circulator types, as defined by the `Arrangement_2` class-template.
typedef unspecified_type	Curve_handle
	a handle for an input curve.

typedef unspecified_type	Curve_iterator
	a bidirectional iterator over the curves that induce the arrangement. More...

typedef unspecified_type	Induced_edge_iterator
	an iterator over the edges induced by an input curve. More...

typedef unspecified_type	Originating_curve_iterator
	an iterator for the curves that originate a given arrangement edge. More...

Creation
	Arrangement_with_history_2 ()
	constructs an empty arrangement containing one unbounded face, which corresponds to the whole plane.

	Arrangement_with_history_2 (const Self &other)
	copy constructor.

	Arrangement_with_history_2 (Traits_2 *traits)
	constructs an empty arrangement that uses the given `traits` instance for performing the geometric predicates.

Assignment Methods
Self &	operator= (other)
	assignment operator.

void	assign (const Self &other)
	assigns the contents of another arrangement.

void	clear ()
	clears the arrangement.

Access Functions for Input Curves
See the `Arrangement_2` reference pages for the full list.
Size	number_of_curves () const
	returns the number of input curves that induce the arrangement.

Curve_iterator	curves_begin ()
	returns the begin-iterator of the curves inducing the arrangement.

Curve_iterator	curves_end ()
	returns the past-the-end iterator of the curves inducing the arrangement.

Size	number_of_induced_edges (Curve_handle ch) const
	returns the number of arrangement edges induced by the curve `ch`.

Induced_edge_iterator	induced_edges_begin (Curve_handle ch) const
	returns the begin-iterator of the edges induced by the curve `ch`.

Induced_edge_iterator	induced_edges_end (Curve_handle ch) const
	returns the past-the-end iterator of the edges induced by the curve `ch`.

Size	number_of_originating_curves (Halfedge_handle e) const
	returns the number of input curves that originate the edge `e`.

Originating_curve_iterator	originating_curves_begin (Halfedge_handle e) const
	returns the begin-iterator of the curves originating the edge `e`.

Originating_curve_iterator	originating_curves_end (Halfedge_handle e) const
	returns the past-the-end iterator of the curves originating the edge `e`.

Modifying Arrangement Edges
The following functions override their counterparts in the `Arrangement_2` class, as they also maintain the cross-relationships between the input curves and the edges they induce. See the `Arrangement_2` reference pages for the full list of functions for modifying arrangement vertices
Halfedge_handle	split_edge (Halfedge_handle e, const Point_2 &p)
	splits the edge `e` into two edges (more precisely, into two halfedge pairs), at a given split point `p`. More...

Halfedge_handle	merge_edge (Halfedge_handle e1, Halfedge_handle e2)
	merges the edges represented by `e1` and `e2` into a single edge. More...

Face_handle	remove_edge (Halfedge_handle e, bool remove_source=true, bool remove_target=true)
	removes the edge `e` from the arrangement. More...

Additional Inherited Members
Public Types inherited from CGAL::Arrangement_2< Traits, Dcel >
typedef Arrangement_2 < Traits_2, Dcel >	Self
	a private type used as an abbreviation of the `Arrangement_2` type hereafter.

typedef Traits	Traits_2
	the traits class in use.

typedef Traits_2::Point_2	Point_2
	the point type, as defined by the traits class.

typedef Traits_2::X_monotone_curve_2	X_monotone_curve_2
	the \( x\)-monotone curve type, as defined by the traits class.

typedef Dcel::Size	Size
	the size type (equivalent to `size_t`).

typedef unspecified_type	Vertex_handle
	a handle for an arrangement vertex.

typedef unspecified_type	Halfedge_handle
	a handle for a halfedge. More...

typedef unspecified_type	Face_handle
	a handle for an arrangement face.

typedef unspecified_type	Vertex_iterator
	a bidirectional iterator over the vertices of the arrangement. More...

typedef unspecified_type	Halfedge_iterator
	a bidirectional iterator over the halfedges of the arrangement. More...

typedef unspecified_type	Edge_iterator
	a bidirectional iterator over the edges of the arrangement. More...

typedef unspecified_type	Face_iterator
	a bidirectional iterator over the faces of arrangement. More...

typedef unspecified_type	Unbounded_face_iterator
	a bidirectional iterator over the unbounded faces of arrangement. More...

typedef unspecified_type	Halfedge_around_vertex_circulator
	a bidirectional circulator over the halfedges that have a given vertex as their target. More...

typedef unspecified_type	Ccb_halfedge_circulator
	a bidirectional circulator over the halfedges of a CCB (connected component of the boundary). More...

typedef unspecified_type	Hole_iterator
	a bidirectional iterator over the holes (i.e., inner CCBs) contained inside a given face. More...

typedef unspecified_type	Isolated_vertex_iterator
	a bidirectional iterator over the isolated vertices contained inside a given face. More...

Public Member Functions inherited from CGAL::Arrangement_2< Traits, Dcel >
	Arrangement_2 ()
	constructs an empty arrangement containing one unbounded face, which corresponds to the entire plane.

	Arrangement_2 (const Self &other)
	copy constructor.

	Arrangement_2 (const Traits_2 *traits)
	constructs an empty arrangement that uses the given `traits` instance for performing the geometric predicates.

Self &	operator= (other)
	assignment operator.

void	assign (const Self &other)
	assigns the contents of another arrangement.

void	clear ()
	clears the arrangement.

Traits_2 *	get_traits ()
	returns the traits object used by the arrangement instance. More...

bool	is_empty () const
	determines whether the arrangement is empty (contains only the unbounded face, with no vertices or edges).

Size	number_of_vertices () const
	returns the number of vertices in the arrangement.

Size	number_of_isolated_vertices () const
	returns the total number of isolated vertices in the arrangement.

Vertex_iterator	vertices_begin ()
	returns the begin-iterator of the vertices in the arrangement.

Vertex_iterator	vertices_end ()
	returns the past-the-end iterator of the vertices in the arrangement.

unspecified_type	vertex_handles ()
	returns a range over handles of the arrangement vertices .

Size	number_of_vertices_at_infinity () const
	returns the number of arrangement vertices that lie at infinity and are not associated with valid points. More...

Size	number_of_halfedges () const
	returns the number of halfedges in the arrangement.

Halfedge_iterator	halfedges_begin ()
	returns the begin-iterator of the halfedges in the arrangement.

Halfedge_iterator	halfedges_end ()
	returns the past-the-end iterator of the halfedges in the arrangement.

unspecified_type	halfedge_handles ()
	returns a range over handles of the arrangement halfedges .

Size	number_of_edges () const
	returns the number of edges in the arrangement (equivalent to `arr.number_of_halfedges() / 2`).

Edge_iterator	edges_begin ()
	returns the begin-iterator of the edges in the arrangement.

Edge_iterator	edges_end ()
	returns the past-the-end iterator of the edges in the arrangement.

unspecified_type	edge_handles ()
	returns a range over handles of the arrangement edges .

Face_handle	unbounded_face ()
	returns a handle for an unbounded face of the arrangement. More...

Size	number_of_faces () const
	returns the number of faces in the arrangement.

Face_iterator	faces_begin ()
	returns the begin-iterator of the faces in the arrangement.

Face_iterator	faces_end ()
	returns the past-the-end iterator of the faces in the arrangement.

unspecified_type	face_handles ()
	returns a range over handles of the arrangement faces .

Size	number_of_unbounded_faces () const
	returns the number of unbounded faces in the arrangement. More...

Unbounded_face_iterator	unbounded_faces_begin ()
	returns the begin-iterator of the unbounded faces in the arrangement.

Unbounded_face_iterator	unbounded_faces_end ()
	returns the past-the-end iterator of the unbounded faces in the arrangement.

Face_handle	fictitious_face ()
	returns a handle to the fictitious face of the arrangement. More...

Vertex_handle	non_const_handle (Vertex_const_handle v)
	casts the given constant vertex handle to an equivalent mutable handle.

Halfedge_handle	non_const_handle (Halfedge_const_handle e)
	casts the given constant halfedge handle to an equivalent mutable handle.

Face_handle	non_const_handle (Face_const_handle f)
	casts the given constant face handle to an equivalent mutable handle.

Vertex_handle	insert_in_face_interior (const Point_2 &p, Face_handle f)
	inserts the point `p` into the arrangement as an isolated vertex in the interior of the face `f` and returns a handle for the newly created vertex. More...

Halfedge_handle	insert_in_face_interior (const X_monotone_curve_2 &c, Face_handle f)
	inserts the curve `c` that is entirely contained in the interior of a given face `f`. More...

Halfedge_handle	insert_from_left_vertex (const X_monotone_curve_2 &c, Vertex_handle v)
	inserts the curve `c` into the arrangement, such that its left endpoint corresponds to a given arrangement vertex. More...

Halfedge_handle	insert_from_right_vertex (const X_monotone_curve_2 &c, Vertex_handle v)
	inserts the curve `c` into the arrangement, such that its right endpoint corresponds to a given arrangement vertex. More...

Halfedge_handle	insert_at_vertices (const X_monotone_curve_2 &c, Vertex_handle v1, Vertex_handle v2)
	inserts the curve `c` into the arrangement, such that both `c`'s endpoints correspond to existing arrangement vertices, given by `v1` and `v2`. More...

Halfedge_handle	insert_in_face_interior (const X_monotone_curve_2 &c, Halfedge_handle fict_pred1, Halfedge_handle fict_pred2=Halfedge_handle())
	inserts an unbounded curve `c` into the arrangement, such that `c` is entirely contained within a single unbounded face of the arrangement. More...

Halfedge_handle	insert_from_left_vertex (const X_monotone_curve_2 &c, Halfedge_handle pred)
	inserts the curve `c` into the arrangement, such that its left endpoint corresponds to a given arrangement vertex. More...

Halfedge_handle	insert_from_left_vertex (const X_monotone_curve_2 &c, Halfedge_handle pred, Halfedge_handle fict_pred)
	inserts an unbounded curve `c` into the arrangement, such that its left endpoint is bounded and corresponds to a given arrangement vertex. More...

Halfedge_handle	insert_from_right_vertex (const X_monotone_curve_2 &c, Halfedge_handle pred)
	inserts the curve `c` into the arrangement, such that its right endpoint corresponds to a given arrangement vertex. More...

Halfedge_handle	insert_from_right_vertex (const X_monotone_curve_2 &c, Halfedge_handle pred, Halfedge_handle fict_pred)
	inserts an unbounded curve `c` into the arrangement, such that its right endpoint is bounded and corresponds to a given arrangement vertex. More...

Halfedge_handle	insert_at_vertices (const X_monotone_curve_2 &c, Halfedge_handle pred1, Vertex_handle v2)
	inserts the curve `c` into the arrangement, such that both `c`'s endpoints correspond to existing arrangement vertices, given by `pred1->target()` and `v2`. More...

Halfedge_handle	insert_at_vertices (const X_monotone_curve_2 &c, Halfedge_handle pred1, Halfedge_handle pred2)
	inserts the curve `c` into the arrangement, such that both `c`'s endpoints correspond to existing arrangement vertices, given by `pred1->target()` and `pred2->target()`. More...

Vertex_handle	modify_vertex (Vertex_handle v, const Point_2 &p)
	sets `p` to be the point associated with the vertex `v`. More...

Face_handle	remove_isolated_vertex (Vertex_handle v)
	removes the isolated vertex `v` from the arrangement. More...

Halfedge_handle	modify_edge (Halfedge_handle e, const X_monotone_curve_2 &c)
	sets `c` to be the \( x\)-monotone curve associated with the edge `e`. More...

Halfedge_handle	split_edge (Halfedge_handle e, const X_monotone_curve_2 &c1, const X_monotone_curve_2 &c2)
	splits the edge `e` into two edges (more precisely, into two halfedge pairs), associated with the given subcurves `c1` and `c2`, and creates a vertex that corresponds to the split point. More...

Halfedge_handle	merge_edge (Halfedge_handle e1, Halfedge_handle e2, const X_monotone_curve_2 &c)
	merges the edges represented by `e1` and `e2` into a single edge, associated with the given merged curve `c`. More...

Face_handle	remove_edge (Halfedge_handle e, bool remove_source=true, bool remove_target=true)
	removes the edge `e` from the arrangement. More...

bool	is_valid () const
	returns `true` if `arr` represents a valid instance of `Arrangement_2`. More...