CGAL 5.1.2 - Halfedge Data Structures
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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.
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.
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., 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.
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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 The following static member functions of 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 | |
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 | |
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 The three | |
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 | |
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... | |
HalfedgeDS< Traits, Items, Alloc >::HalfedgeDS | ( | const HalfedgeDS< Traits, Items, Alloc > & | hds2 | ) |
copy constructor.
hds2
contains no dangling handles. 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.
normalize_border()
has been called and no halfedge insertion or removal and no change in border status of the halfedges have occurred since then. 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
.
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
.
h->opposite()
denotes a halfedge. 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.
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.
HalfedgeDS<Traits,Items,Alloc>& HalfedgeDS< Traits, Items, Alloc >::operator= | ( | const HalfedgeDS< Traits, Items, Alloc > & | hds2 | ) |
assignment operator.
hds2
contains no dangling handles. 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.
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.
normalize_border()
has been called and no halfedge insertion or removal and no change in border status of the halfedges have occurred since then. 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.
normalize_border()
has been called and no halfedge insertion or removal and no change in border status of the halfedges have occurred since then. 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.