#include <CGAL/Polyhedron_3.h>

Definition

A halfedge is an oriented edge between two vertices.

It is always paired with a halfedge pointing in the opposite direction. The Halfedge::opposite() member function returns this halfedge of opposite orientation. If a halfedge is incident to a facet the Halfedge::next() member function points to the successor halfedge around this facet. For border edges the Halfedge::next() member function points to the successor halfedge along the hole. For more than two border edges at a vertex, the next halfedge along a hole is not uniquely defined, but a consistent assignment of the next halfedge will be maintained in the data structure. An invariant is that successive assignments of the form h = h->next() cycle counterclockwise around the facet (or hole) and traverse all halfedges incident to this facet (or hole). A similar invariant is that successive assignments of the form h = h->next()->opposite() cycle clockwise around the vertex and traverse all halfedges incident to this vertex. Two circulators are provided for these circular orders.

The three classes Vertex, Halfedge, and Facet of the polyhedral surface. Member functions with shaded background are mandatory. The others are optionally supported.

The incidences encoded in Halfedge::opposite() and Halfedge::next() are available for each instantiation of polyhedral surfaces. The other incidences are optionally available as indicated with type tags. The Halfedge::prev() member function points to the preceding halfedge around the same facet. It is always available, though it might perform a search around the facet using the Halfedge::next() member function to find the previous halfedge if the underlying halfedge data structure does not provide an efficient Halfedge::prev() member function for halfedges. Handles to the incident vertex and facet are optionally stored.

The circulators are assignable to the Halfedge_handle. The circulators are bidirectional if the halfedge provided to the polyhedron with the Items template argument provides a member function Halfedge::prev(), otherwise they are of the forward category.

See also: CGAL::Polyhedron_3::Vertex; CGAL::Polyhedron_3::Facet; CGAL::Polyhedron_3<Traits>

Implementation

The member functions Halfedge::prev() and Halfedge::prev_on_vertex() work in constant time if Supports_halfedge_prev \( \equiv\) CGAL::Tag_true. Otherwise both methods search for the previous halfedge around the incident facet.

Types
typedef unspecified_type	Vertex
	type of incident vertices. More...

typedef unspecified_type	Facet
	type of incident facets. More...

typedef unspecified_type	Vertex_handle
	handle to vertex. More...

typedef unspecified_type	Halfedge_handle
	handle to halfedge. More...

typedef unspecified_type	Facet_handle
	handle to facet. More...

typedef unspecified_type	Halfedge_around_vertex_circulator
	circulator of halfedges around a vertex. More...

typedef unspecified_type	Halfedge_around_facet_circulator
	circulator of halfedges around a facet. More...

typedef unspecified_type	Vertex_const_handle

typedef unspecified_type	Halfedge_const_handle

typedef unspecified_type	Facet_const_handle

typedef unspecified_type	Halfedge_around_vertex_const_circulator

typedef unspecified_type	Halfedge_around_facet_const_circulator

typedef unspecified_type	Supports_halfedge_prev
	\( \equiv\) `CGAL::Tag_true` or `CGAL::Tag_false`. More...

typedef unspecified_type	Supports_halfedge_vertex
	\( \equiv\) `CGAL::Tag_true` or `CGAL::Tag_false`. More...

typedef unspecified_type	Supports_halfedge_face
	\( \equiv\) `CGAL::Tag_true` or `CGAL::Tag_false`. More...

Creation
	Halfedge ()
	default constructor. More...

Operations
Halfedge_handle	opposite ()

Halfedge_const_handle	opposite () const
	the opposite halfedge. More...

Halfedge_handle	next ()

Halfedge_const_handle	next () const
	the next halfedge around the facet. More...

Halfedge_handle	prev ()

Halfedge_const_handle	prev () const
	the previous halfedge around the facet. More...

Halfedge_handle	next_on_vertex ()

Halfedge_const_handle	next_on_vertex () const
	the next halfedge around the vertex (clockwise). More...

Halfedge_handle	prev_on_vertex ()

Halfedge_const_handle	prev_on_vertex () const
	the previous halfedge around the vertex (counterclockwise). More...

bool	is_border () const
	is true if `h` is a border halfedge. More...

bool	is_border_edge () const
	is true if this or the opposite halfedge is a border halfedge. More...

Halfedge_around_vertex_circulator	vertex_begin ()

Halfedge_around_vertex_const_circulator	vertex_begin () const
	circulator of halfedges around the vertex (clockwise), the vertex of the halfedges being `*this`. More...

Halfedge_around_facet_circulator	facet_begin ()

Halfedge_around_facet_const_circulator	facet_begin () const
	circulator of halfedges around the facet (counterclockwise). More...

std::size_t	vertex_degree () const
	the degree of the incident vertex, i.e., number of edges emanating from this vertex. More...

bool	is_bivalent () const
	returns `true` if the incident vertex has exactly two incident edges. More...

bool	is_trivalent () const
	returns `true` if the incident vertex has exactly three incident edges. More...

std::size_t	facet_degree () const
	the degree of the incident facet, i.e., number of edges on the boundary of this facet. More...

bool	is_triangle () const
	returns `true` if the incident facet is a triangle. More...

bool	is_quad () const
	returns `true` if the incident facet is a quadrilateral. More...

Operations available if Supports_halfedge_vertex is CGAL::Tag_true
Vertex_handle	vertex ()

Vertex_const_handle	vertex () const
	the incident vertex of the halfedge. More...

Operations available if Supports_halfedge_facet is CGAL::Tag_true
Facet_handle	facet ()

Facet_const_handle	facet () const
	the incident facet of the halfedge. More...

Member Typedef Documentation

◆ Facet

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Facet

type of incident facets.

◆ Facet_const_handle

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Facet_const_handle

◆ Facet_handle

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Facet_handle

handle to facet.

◆ Halfedge_around_facet_circulator

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Halfedge_around_facet_circulator

circulator of halfedges around a facet.

◆ Halfedge_around_facet_const_circulator

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Halfedge_around_facet_const_circulator

◆ Halfedge_around_vertex_circulator

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Halfedge_around_vertex_circulator

circulator of halfedges around a vertex.

◆ Halfedge_around_vertex_const_circulator

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Halfedge_around_vertex_const_circulator

◆ Halfedge_const_handle

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Halfedge_const_handle

◆ Halfedge_handle

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Halfedge_handle

handle to halfedge.

◆ Supports_halfedge_face

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Supports_halfedge_face

\( \equiv\) CGAL::Tag_true or CGAL::Tag_false.

◆ Supports_halfedge_prev

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Supports_halfedge_prev

\( \equiv\) CGAL::Tag_true or CGAL::Tag_false.

◆ Supports_halfedge_vertex

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Supports_halfedge_vertex

\( \equiv\) CGAL::Tag_true or CGAL::Tag_false.

◆ Vertex

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Vertex

type of incident vertices.

◆ Vertex_const_handle

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Vertex_const_handle

◆ Vertex_handle

template<typename Traits >

typedef unspecified_type CGAL::Polyhedron_3< Traits >::Halfedge::Vertex_handle

handle to vertex.

Constructor & Destructor Documentation

◆ Halfedge()

template<typename Traits >

CGAL::Polyhedron_3< Traits >::Halfedge::Halfedge ( )

default constructor.

Member Function Documentation

◆ facet() [1/2]

template<typename Traits >

Facet_handle CGAL::Polyhedron_3< Traits >::Halfedge::facet ( )

◆ facet() [2/2]

template<typename Traits >

Facet_const_handle CGAL::Polyhedron_3< Traits >::Halfedge::facet ( ) const

the incident facet of the halfedge.

If the hafedge is a border halfedge the result is default construction of the handle.

◆ facet_begin() [1/2]

template<typename Traits >

Halfedge_around_facet_circulator CGAL::Polyhedron_3< Traits >::Halfedge::facet_begin ( )

◆ facet_begin() [2/2]

template<typename Traits >

Halfedge_around_facet_const_circulator CGAL::Polyhedron_3< Traits >::Halfedge::facet_begin ( ) const

circulator of halfedges around the facet (counterclockwise).

◆ facet_degree()

template<typename Traits >

std::size_t CGAL::Polyhedron_3< Traits >::Halfedge::facet_degree ( ) const

the degree of the incident facet, i.e., number of edges on the boundary of this facet.

◆ is_bivalent()

template<typename Traits >

bool CGAL::Polyhedron_3< Traits >::Halfedge::is_bivalent ( ) const

returns true if the incident vertex has exactly two incident edges.

◆ is_border()

template<typename Traits >

bool CGAL::Polyhedron_3< Traits >::Halfedge::is_border ( ) const

is true if h is a border halfedge.

◆ is_border_edge()

template<typename Traits >

bool CGAL::Polyhedron_3< Traits >::Halfedge::is_border_edge ( ) const

is true if this or the opposite halfedge is a border halfedge.

◆ is_quad()

template<typename Traits >

bool CGAL::Polyhedron_3< Traits >::Halfedge::is_quad ( ) const

returns true if the incident facet is a quadrilateral.

◆ is_triangle()

template<typename Traits >

bool CGAL::Polyhedron_3< Traits >::Halfedge::is_triangle ( ) const

returns true if the incident facet is a triangle.

◆ is_trivalent()

template<typename Traits >

bool CGAL::Polyhedron_3< Traits >::Halfedge::is_trivalent ( ) const

returns true if the incident vertex has exactly three incident edges.

◆ next() [1/2]

template<typename Traits >

Halfedge_handle CGAL::Polyhedron_3< Traits >::Halfedge::next ( )

◆ next() [2/2]

template<typename Traits >

Halfedge_const_handle CGAL::Polyhedron_3< Traits >::Halfedge::next ( ) const

the next halfedge around the facet.

◆ next_on_vertex() [1/2]

template<typename Traits >

Halfedge_handle CGAL::Polyhedron_3< Traits >::Halfedge::next_on_vertex ( )

◆ next_on_vertex() [2/2]

template<typename Traits >

Halfedge_const_handle CGAL::Polyhedron_3< Traits >::Halfedge::next_on_vertex ( ) const

the next halfedge around the vertex (clockwise).

Is equal to h.next()->opposite().

◆ opposite() [1/2]

template<typename Traits >

Halfedge_handle CGAL::Polyhedron_3< Traits >::Halfedge::opposite ( )

◆ opposite() [2/2]

template<typename Traits >

Halfedge_const_handle CGAL::Polyhedron_3< Traits >::Halfedge::opposite ( ) const

the opposite halfedge.

◆ prev() [1/2]

template<typename Traits >

Halfedge_handle CGAL::Polyhedron_3< Traits >::Halfedge::prev ( )

◆ prev() [2/2]

template<typename Traits >

Halfedge_const_handle CGAL::Polyhedron_3< Traits >::Halfedge::prev ( ) const

the previous halfedge around the facet.

◆ prev_on_vertex() [1/2]

template<typename Traits >

Halfedge_handle CGAL::Polyhedron_3< Traits >::Halfedge::prev_on_vertex ( )

◆ prev_on_vertex() [2/2]

template<typename Traits >

Halfedge_const_handle CGAL::Polyhedron_3< Traits >::Halfedge::prev_on_vertex ( ) const

the previous halfedge around the vertex (counterclockwise).

Is equal to h.opposite()->prev().

◆ vertex() [1/2]

template<typename Traits >

Vertex_handle CGAL::Polyhedron_3< Traits >::Halfedge::vertex ( )

◆ vertex() [2/2]

template<typename Traits >

Vertex_const_handle CGAL::Polyhedron_3< Traits >::Halfedge::vertex ( ) const

the incident vertex of the halfedge.

◆ vertex_begin() [1/2]

template<typename Traits >

Halfedge_around_vertex_circulator CGAL::Polyhedron_3< Traits >::Halfedge::vertex_begin ( )

◆ vertex_begin() [2/2]

template<typename Traits >

Halfedge_around_vertex_const_circulator CGAL::Polyhedron_3< Traits >::Halfedge::vertex_begin ( ) const

circulator of halfedges around the vertex (clockwise), the vertex of the halfedges being *this.

◆ vertex_degree()

template<typename Traits >

std::size_t CGAL::Polyhedron_3< Traits >::Halfedge::vertex_degree ( ) const

the degree of the incident vertex, i.e., number of edges emanating from this vertex.

Definition

Types

Creation

Operations

Operations available if Supports_halfedge_vertex is CGAL::Tag_true

Operations available if Supports_halfedge_facet is CGAL::Tag_true

Member Typedef Documentation

◆ Facet

◆ Facet_const_handle

◆ Facet_handle

◆ Halfedge_around_facet_circulator

◆ Halfedge_around_facet_const_circulator

◆ Halfedge_around_vertex_circulator

◆ Halfedge_around_vertex_const_circulator

◆ Halfedge_const_handle

◆ Halfedge_handle

◆ Supports_halfedge_face

◆ Supports_halfedge_prev

◆ Supports_halfedge_vertex

◆ Vertex

◆ Vertex_const_handle

◆ Vertex_handle

Constructor & Destructor Documentation

◆ Halfedge()

Member Function Documentation

◆ facet() [1/2]

◆ facet() [2/2]

◆ facet_begin() [1/2]

◆ facet_begin() [2/2]

◆ facet_degree()

◆ is_bivalent()

◆ is_border()

◆ is_border_edge()

◆ is_quad()

◆ is_triangle()

◆ is_trivalent()

◆ next() [1/2]

◆ next() [2/2]

◆ next_on_vertex() [1/2]

◆ next_on_vertex() [2/2]

◆ opposite() [1/2]

◆ opposite() [2/2]

◆ prev() [1/2]

◆ prev() [2/2]

◆ prev_on_vertex() [1/2]

◆ prev_on_vertex() [2/2]

◆ vertex() [1/2]

◆ vertex() [2/2]

◆ vertex_begin() [1/2]

◆ vertex_begin() [2/2]

◆ vertex_degree()