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CGAL 4.3 - Halfedge Data Structures

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HalfedgeDS Concept Reference
Concepts

## Definition

The concept of a halfedge data structure (abbreviated as `HalfedgeDS`, or `HDS` for template parameters) defines an edge-centered data structure capable of maintaining incidence information of vertices, edges, and faces, for example for planar maps or polyhedral surfaces. It is a combinatorial data structure, geometric interpretation is added by classes built on top of the halfedge data structure.

The data structure defined here is known as the FE-structure [9], as halfedges [6], [2] or as the doubly connected edge list (DCEL) [3], although the original reference for the DCEL [7] describes a different data structure. The halfedge data structure can also be seen as one of the variants of the quad-edge data structure [4]. In general, the quad-edge data can represent non-orientable 2-manifolds, but the variant here is restricted to orientable 2-manifolds only. An overview and comparison of these different data structures together with a thorough description of the design implemented here can be found in [5].

Each edge is represented by two halfedges with opposite orientations. Each halfedge can store a reference to an incident face and an incident vertex. For each face and each vertex an incident halfedge is stored. Reduced variants of the halfedge data structure can omit some of these incidences, for example the reference to halfedges in vertices or the storage of vertices at all. See Figure figureOptionalMethods for the incidences, the mandatory and optional member functions possible for vertices, halfedges, and faces.

The three classes Vertex, Halfedge, and Face of the halfedge data structure. Member functions with shaded background are mandatory. The others are optionally supported.
A `HalfedgeDS` organizes the internal storage of its items. Examples are a list-based or a vector-based storage. The `HalfedgeDS` exhibits most of the characteristics of the container class used internally, for example the iterator category. A vector resizes automatically when a new item exceeds the reserved space. Since resizing is an expensive operation for a `HalfedgeDS` in general and only possible in a well defined state of the data structure (no dangling handles), it must be called explicitly in advance for a `HalfedgeDS` before inserting new items beyond the current capacity. Classes built on top of a `HalfedgeDS` are advised to call the `reserve()` member function before creating new items.

Parameters

A `HalfedgeDS` is a class template and will be used as argument for other class templates, for example `CGAL::Polyhedron_3`. The template parameters to instantiate the `HalfedgeDS` will be provided by this other class template. Therefore, the three template parameters and their meaning are mandatory. We distinguish between the template `HalfedgeDS` and an instantiation of it.

`Traits` is a traits class that will be passed to the item types in `Items`. It will not be used in `HalfedgeDS` itself. `Items` is a model of the `HalfedgeDSItems` concept. `Alloc` is an allocator that fulfills all requirements of allocators for STL container classes. The `rebind` mechanism from `Alloc` will be used to create appropriate allocators internally. A default argument is mandatory for `Alloc`, for example, the macro `CGAL_ALLOCATOR(int)` from the `<CGAL/memory.h>` header file can be used as default allocator.

Has Models:

  CGAL::HalfedgeDS_default

  CGAL::HalfedgeDS_list

  CGAL::HalfedgeDS_vector

See Also
  `HalfedgeDSItems`
  `CGAL::Polyhedron_3<Traits>`
  `CGAL::HalfedgeDS_vertex_base<Refs>`
  `CGAL::HalfedgeDS_halfedge_base<Refs>`
  `CGAL::HalfedgeDS_face_base<Refs>`
  `CGAL::HalfedgeDS_items_decorator<HDS>`
  `CGAL::HalfedgeDS_decorator<HDS>`
  `CGAL::HalfedgeDS_const_decorator<HDS>`

## Types
typedef unspecified_type | Items
| model of `HalfedgeDSItems` concept.
typedef unspecified_type | size_type
| size type.
typedef unspecified_type | difference_type
| difference type.
typedef unspecified_type | iterator_category
| iterator category for all iterators.
typedef unspecified_type | allocator_type
| allocator type `Alloc`.
typedef unspecified_type | Vertex
| model of `HalfedgeDSVertex` concept.
typedef unspecified_type | Halfedge
| model of `HalfedgeDSHalfedge` concept.
typedef unspecified_type | Face
| model of `HalfedgeDSFace` concept.

## Handles and Iterators
The following handles and iterators have appropriate non-mutable counterparts, i.e., `const_handle` and `const_iterator`. The mutable types are assignable to their non-mutable counterparts. The iterators are assignable to the respective handle types. Wherever the handles appear in function parameter lists, the corresponding iterators can be used as well. Note The handle types must have a default constructor that creates a unique and always the same handle value. It will be used in analogy to `NULL` for pointers.
typedef unspecified_type | Vertex_handle
| handle to vertex.
typedef unspecified_type | Halfedge_handle
| handle to halfedge.
typedef unspecified_type | Face_handle
| handle to face.
typedef unspecified_type | Vertex_iterator
| iterator over all vertices.
typedef unspecified_type | Halfedge_iterator
| iterator over all halfedges.
typedef unspecified_type | Face_iterator
| iterator over all faces.
typedef unspecified_type | Supports_vertex_halfedge
| `Vertex::halfedge()`.
typedef unspecified_type | Supports_halfedge_prev
| `Halfedge::prev()`.
typedef unspecified_type | Supports_halfedge_vertex
| `Halfedge::vertex()`.
typedef unspecified_type | Supports_halfedge_face
| `Halfedge::face()`.
typedef unspecified_type | Supports_face_halfedge
| `Face::halfedge()`.
typedef unspecified_type | Supports_removal
| removal of individual elements.

## Static Member Functions
Advanced When writing an items type, such as a user defined vertex, certain functions need to create a handle but knowing only a pointer, for example, the `this`-pointer. The following static member functions of `HalfedgeDS` create such a corresponding handle for an item type from a pointer. This conversion encapsulates possible adjustments for hidden data members in the true item type, such as linked-list pointers. Note that the user provides item types with the `Items` template argument, which may differ from the `Vertex`, `Halfedge`, and `Face` types defined in `HalfedgeDS`. If they differ, they are derived from the user provided item types. We denote the user item types with `Vertex_base`, `Halfedge_base`, and `Face_base` in the following. The fully qualified name for `Vertex_base` would be for example - assuming that the type `Self` refers to the instantiated `HalfedgeDS` - typedef typename Items::template Vertex_wrapper<Self,Traits> Vertex_wrapper; typedef typename Vertex_wrapper::Vertex Vertex_base; Implementing these functions relies on the fundamental assumption that an iterator (or handle) of the internally used container class can be constructed from a pointer of a contained item only. This is true and controlled by us for `CGAL::In_place_list`. It is true for the `std::vector` of major STL distributions, but not necessarily guaranteed. We might switch to an internal implementation if need arises.
static Vertex_handle | vertex_handle (Vertex_base *v)
static Vertex_const_handle | vertex_handle (const Vertex_base *v)
static Halfedge_handle | halfedge_handle (Halfedge_base *h)
static Halfedge_const_handle | halfedge_handle (const Halfedge_base *h)
static Face_handle | face_handle (Face_base *f)
static Face_const_handle | face_handle (const Face_items *f)

## Creation
| HalfedgeDS ()
| empty halfedge data structure.
| HalfedgeDS (size_type v, size_type h, size_type f)
| storage reserved for `v` vertices, `h` halfedges, and `f` faces.
| HalfedgeDS (const HalfedgeDS< Traits, Items, Alloc > &hds2)
| copy constructor. More...
HalfedgeDS< Traits, Items, Alloc > & | operator= (const HalfedgeDS< Traits, Items, Alloc > &hds2)
| assignment operator. More...
void | reserve (size_type v, size_type h, size_type f)
| reserves storage for `v` vertices, `h` halfedges, and `f` faces. More...

## Access Member Functions
Size | size_of_vertices () const
| number of vertices.
Size | size_of_halfedges () const
| number of halfedges.
Size | size_of_faces () const
| number of faces.
Size | capacity_of_vertices () const
| space reserved for vertices.
Size | capacity_of_halfedges () const
| space reserved for halfedges.
Size | capacity_of_faces () const
| space reserved for faces.
size_t | bytes () const
| bytes used for the halfedge data structure.
size_t | bytes_reserved () const
| bytes reserved for the halfedge data structure.
allocator_type | get_allocator () const
| allocator object.
Vertex_iterator | vertices_begin ()
| iterator over all vertices.
Vertex_iterator | vertices_end ()
Halfedge_iterator | halfedges_begin ()
| iterator over all halfedges
Halfedge_iterator | halfedges_end ()
Face_iterator | faces_begin ()
| iterator over all faces.
Face_iterator | faces_end ()

## Insertion
Note that the vertex-related and the face-related member functions may not be provided for a `HalfedgeDS` that does not support vertices or faces respectively.
Vertex_handle | vertices_push_back (const Vertex &v)
| appends a copy of `v` to the halfedge data structure. More...
Halfedge_handle | edges_push_back (const Halfedge &h, const Halfedge &g)
| appends a copy of `h` and a copy of `g` to the halfedge data structure and makes them opposite to each other. More...
Halfedge_handle | edges_push_back (const Halfedge &h)
| appends a copy of `h` and a copy of `h->opposite()` to the halfedge data structure and makes them opposite to each other. More...
Face_handle | faces_push_back (const Face &f)
| appends a copy of `f` to the halfedge data structure. More...

## Removal
Erasing single elements is optional and indicated with the type tag `Supports_removal`. The three `pop_back()` and the `clear()` member functions are mandatory. If vertices or faces are not supported for a `HalfedgeDS` the three `pop_back()` and the `clear()` member functions must be provided as null operations.
void | vertices_pop_front ()
| removes the first vertex if vertices are supported and `Supports_removal` \( \equiv\) `CGAL::Tag_true`.
void | vertices_pop_back ()
| removes the last vertex.
void | vertices_erase (Vertex_handle v)
| removes the vertex `v` if vertices are supported and `Supports_removal` \( \equiv\) `CGAL::Tag_true`.
void | vertices_erase (Vertex_handle first, Vertex_handle last)
| removes the range of vertices `[first,last)` if vertices are supported and `Supports_removal` \( \equiv\) `CGAL::Tag_true`.
void | edges_pop_front ()
| removes the first two halfedges if `Supports_removal` \( \equiv\) `CGAL::Tag_true`.
void | edges_pop_back ()
| removes the last two halfedges.
void | edges_erase (Halfedge_handle h)
| removes the pair of halfedges `h` and `h->opposite()` if `Supports_removal` \( \equiv\) `CGAL::Tag_true`.
void | edges_erase (Halfedge_handle first, Halfedge_handle last)
| removes the range of edges `[first},last)` if `Supports_removal` \( \equiv\) `CGAL::Tag_true`.
void | faces_pop_front ()
| removes the first face if faces are supported and `Supports_removal` \( \equiv\) `CGAL::Tag_true`.
void | faces_pop_back ()
| removes the last face.
void | faces_erase (Face_handle f)
| removes the face `f` if faces are supported and `Supports_removal` \( \equiv\) `CGAL::Tag_true`.
void | faces_erase (Face_handle first, Face_handle last)
| removes the range of faces `[first,last)` if faces are supported and `Supports_removal` \( \equiv\) `CGAL::Tag_true`.
void | vertices_clear ()
| removes all vertices.
void | edges_clear ()
| removes all halfedges.
void | faces_clear ()
| removes all faces.
void | clear ()
| removes all elements.

## Operations with Border Halfedges
Advanced The following notion of border halfedges is particular useful where the halfedge data structure is used to model surfaces with boundary, i.e., surfaces with missing faces or open regions. Halfedges incident to an open region are called border halfedges. A halfedge is a border edge if the halfedge itself or its opposite halfedge is a border halfedge. The only requirement to work with border halfedges is that the `Halfedge` class provides a member function `HalfedgeDSHalfedge::is_border()` returning a `bool`. Usually, the halfedge data structure supports faces and the value of the default constructor of the face handle will indicate a border halfedge, but this may not be the only possibility. The `HalfedgeDSHalfedge::is_border()` predicate divides the edges into two classes, the border edges and the non-border edges. The following normalization reorganizes the sequential storage of the edges such that the non-border edges precede the border edges, and that for each border edge the latter of the two halfedges is a border halfedge (the first one might be a border halfedge too). The normalization stores the number of border halfedges, as well as the halfedge iterator where the border edges start at, within the halfedge data structure. These values will be invalid after further halfedge insertions or removals and changes in the border status of a halfedge. There is no automatic update required.
void | normalize_border ()
| sorts halfedges such that the non-border edges precede the border edges. More...
Size | size_of_border_halfedges () const
| number of border halfedges. More...
Size | size_of_border_edges () const
| number of border edges. More...
Halfedge_iterator | border_halfedges_begin ()
| halfedge iterator starting with the border edges. More...

## Constructor & Destructor Documentation

template<typename Traits , typename Items , typename Alloc >

HalfedgeDS< Traits, Items, Alloc >::HalfedgeDS | ( | const HalfedgeDS< Traits, Items, Alloc > & | hds2) |  |

copy constructor.

Precondition
  `hds2` contains no dangling handles.

## Member Function Documentation

template<typename Traits , typename Items , typename Alloc >

Halfedge_iterator HalfedgeDS< Traits, Items, Alloc >::border_halfedges_begin | ( | ) |  |

halfedge iterator starting with the border edges.

The range [`halfedges_begin(), border_halfedges_begin()`) denotes all non-border edges. The range [`border_halfedges_begin(), halfedges_end()`) denotes all border edges.

Precondition
  `normalize_border()` has been called and no halfedge insertion or removal and no change in border status of the halfedges have occurred since then.

template<typename Traits , typename Items , typename Alloc >

Halfedge_handle HalfedgeDS< Traits, Items, Alloc >::edges_push_back | ( | const Halfedge & | h,
|  | const Halfedge & | g
| ) |  |

appends a copy of `h` and a copy of `g` to the halfedge data structure and makes them opposite to each other.

Returns a handle of the copy of `h`.

template<typename Traits , typename Items , typename Alloc >

Halfedge_handle HalfedgeDS< Traits, Items, Alloc >::edges_push_back | ( | const Halfedge & | h) |  |

appends a copy of `h` and a copy of `h->opposite()` to the halfedge data structure and makes them opposite to each other.

Returns a handle of the copy of `h`.

Precondition
  `h->opposite()` denotes a halfedge.

template<typename Traits , typename Items , typename Alloc >

Face_handle HalfedgeDS< Traits, Items, Alloc >::faces_push_back | ( | const Face & | f) |  |

appends a copy of `f` to the halfedge data structure.

Returns a handle of the new face.

template<typename Traits , typename Items , typename Alloc >

void HalfedgeDS< Traits, Items, Alloc >::normalize_border | ( | ) |  |

sorts halfedges such that the non-border edges precede the border edges.

For each border edge that is incident to a face, the halfedge iterator will reference the halfedge incident to the face right before the halfedge incident to the open region.

template<typename Traits , typename Items , typename Alloc >

HalfedgeDS<Traits,Items,Alloc>& HalfedgeDS< Traits, Items, Alloc >::operator= | ( | const HalfedgeDS< Traits, Items, Alloc > & | hds2) |  |

assignment operator.

Precondition
  `hds2` contains no dangling handles.

template<typename Traits , typename Items , typename Alloc >

void HalfedgeDS< Traits, Items, Alloc >::reserve | ( | size_type | v,
|  | size_type | h,
|  | size_type | f
| ) |  |

reserves storage for `v` vertices, `h` halfedges, and `f` faces.

If all capacities are already greater or equal than the requested sizes nothing happens. Otherwise, the halfedge data structure will be resized and all handles, iterators and circulators are invalid.

Precondition
  If resizing is necessary the halfedge data structure contains no dangling handles.

template<typename Traits , typename Items , typename Alloc >

Size HalfedgeDS< Traits, Items, Alloc >::size_of_border_edges | ( | ) |  | const

number of border edges.

If `size_of_border_edges()` is equal to `size_of_border_halfedges()` all border edges are incident to a face on one side and to an open region on the other side.

Precondition
  `normalize_border()` has been called and no halfedge insertion or removal and no change in border status of the halfedges have occurred since then.

template<typename Traits , typename Items , typename Alloc >

Size HalfedgeDS< Traits, Items, Alloc >::size_of_border_halfedges | ( | ) |  | const

number of border halfedges.

An edge with no incident face counts as two border halfedges.

Precondition
  `normalize_border()` has been called and no halfedge insertion or removal and no change in border status of the halfedges have occurred since then.

template<typename Traits , typename Items , typename Alloc >

Vertex_handle HalfedgeDS< Traits, Items, Alloc >::vertices_push_back | ( | const Vertex & | v) |  |

appends a copy of `v` to the halfedge data structure.

Returns a handle of the new vertex.