CGAL 5.1.2 - 2D Triangulation Data Structure
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The concept TriangulationDataStructure_2
describes the requirements for the second template parameter of the basic triangulation class Triangulation_2<Traits,Tds>
and of all other 2D triangulation classes.
The concept can be seen as a container for the faces and vertices of the triangulation. The concept TriangulationDataStructure_2
includes two sub-concepts TriangulationDataStructure_2::Vertex
and TriangulationDataStructure_2::Face
.
The TriangulationDataStructure_2
maintains incidence and adjacency relations among vertices and faces.
Each triangular face gives access to its three incident vertices and to its three adjacent faces. Each vertex gives access to one of its incident faces and through that face to the circular list of its incident faces.
The three vertices of a face are indexed with 0, 1, and 2. The neighbors of a face are also indexed with 0, 1, and 2 in such a way that the neighbor indexed by i
is opposite to the vertex with the same index.
Each edge has two implicit representations : the edge of a face f
which is opposed to the vertex indexed i
, can be represented as well as an edge of the neighbor(i)
of f
. See Figure 40.1.
The triangulation data structure is responsible for the combinatorial integrity of the triangulation. This means that the triangulation data structure allows to perform some combinatorial operations on the triangulation and guarantees the maintenance on proper incidence and adjacency relations among the vertices and faces. The term combinatorial operations means that those operations are purely topological and do not depend on the geometric embedding. Insertion of a new vertex in a given face, or in a given edge, suppression of a vertex of degree three, flip of two edges are examples of combinatorial operations.
I/O
The information output in the iostream
is: the dimension, the number of (finite) vertices, the number of (finite) faces. Then comes for each vertex, the non combinatorial information stored in that vertex if any. Then comes for each faces, the indices of its vertices and the non combinatorial information (if any) stored in this face. Then comes for each face again the indices of the neighboring faces. The index of an item (vertex of face) the rank of this item in the output order. When dimension \( <\) 2, the same information is output for faces of maximal dimension instead of faces.
Concepts | |
concept | Face |
The concept TriangulationDataStructure_2::Face describes the types used to store the faces face class of a TriangulationDataStructure_2 . A TriangulationDataStructure_2::Face stores three handles to its three vertices and three handles to its three neighbors. The vertices are indexed 0,1, and 2 in counterclockwise order. The neighbor indexed i lies opposite to vertex i . More... | |
concept | Vertex |
The concept TriangulationDataStructure_2::Vertex describes the type used by a TriangulationDataStructure_2 to store the vertices. More... | |
Types | |
typedef unspecified_type | size_type |
Size type (unsigned integral type) | |
typedef unspecified_type | difference_type |
Difference type (signed integral type) | |
typedef unspecified_type | Vertex |
The vertex type, requirements for this type are described in concept TriangulationDataStructure_2::Vertex . | |
typedef unspecified_type | Face |
The face type, requirements for this type are described in concept TriangulationDataStructure_2::Face . | |
typedef unspecified_type | Vertex_handle |
Handle to a vertex. More... | |
typedef unspecified_type | Face_handle |
Handle to a face. More... | |
typedef std::pair< Face_handle, int > | Edge |
The edge type. More... | |
Iterators and Circulators | |
The iterators allow one to visit all the vertices, edges and faces of a triangulation data structure. The circulators allow to visit all the edges or faces incident to a given vertex and all the vertices adjacent to a given vertex. The iterators and circulators are bidirectional and non mutable, and they are convertible to the corresponding handles, thus they can be passed directly as argument to the functions expecting a handle. A face circulator is invalidated by any modification of the face it points to. An edge circulator is invalidated by any modification of any of the two faces incident to the edge pointed to. A vertex circulator that turns around vertex | |
typedef unspecified_type | Face_iterator |
typedef unspecified_type | Edge_iterator |
typedef unspecified_type | Vertex_iterator |
typedef unspecified_type | Face_circulator |
typedef unspecified_type | Edge_circulator |
typedef unspecified_type | Vertex_circulator |
Creation | |
TriangulationDataStructure_2 () | |
Default constructor. | |
TriangulationDataStructure_2 (const TriangulationDataStructure_2 &tds1) | |
Copy constructor, performing a deep copy, that is all vertices and faces are duplicated. | |
TriangulationDataStructure_2 & | operator= (const TriangulationDataStructure_2 &tds1) |
Assignment. More... | |
Vertex_handle | copy_tds (const TriangulationDataStructure_2 &tds1, Vertex_handle v=Vertex_handle()) |
tds1 is copied into the triangulation data structure. More... | |
template<class TDS_src , class ConvertVertex , class ConvertFace > | |
Vertex_handle tds | copy_tds (const TDS_src &tds_src, typename TDS_src::Vertex_handle v, const ConvertVertex &convert_vertex, const ConvertFace &convert_face) |
tds_src is copied into this . More... | |
void | swap (TriangulationDataStructure_2 &tds1) |
Swaps the triangulation data structure and tds1 . More... | |
void | clear () |
Deletes all faces and all finite vertices. | |
Access Functions | |
int | dimension () const |
This is an advanced function. More... | |
size_type | number_of_vertices () const |
returns the number of vertices in the triangulation data structure. | |
size_type | number_of_faces () const |
returns the number of two dimensional faces in the triangulation data structure. | |
size_type | number_of_edges () const |
returns the number of edges in the triangulation data structure. | |
size_type | number_of_full_dim_faces () const |
returns the number of full dimensional faces, i.e. faces of dimension equal to the dimension of the triangulation data structure. More... | |
Setting | |
void | set_dimension (int n) |
sets the dimension. | |
Queries | |
bool | is_vertex (Vertex_handle v) const |
returns true if v is a vertex of the triangulation data structure. | |
bool | is_edge (Face_handle fh, int i) const |
returns true if (fh,i) is an edge of the triangulation data structure. More... | |
bool | is_edge (Vertex_handle va, Vertex_handle vb) const |
returns true if (va, vb) is an edge of the triangulation data structure. | |
bool | is_edge (Vertex_handle va, Vertex_handle vb, Face_handle &fr, int &i) const |
returns true if (va, vb) is an edge of the triangulation data structure. More... | |
bool | is_face (Face_handle fh) const |
returns true if fh is a face of the triangulation data structure. More... | |
bool | is_face (Vertex_handle v1, Vertex_handle v2, Vertex_handle v3) const |
true if there is a face having v1 , v2 , and v3 as vertices. | |
bool | is_face (Vertex_handle v1, Vertex_handle v2, Vertex_handle v3, Face_handle &fr) const |
true if there is a face having v1 , v2 , and v3 as vertices. More... | |
Traversing the Triangulation | |
Face_iterator | faces_begin () const |
visits all faces. | |
Face_iterator | faces_end () const |
Vertex_iterator | vertices_begin () const |
visits all vertices. | |
Vertex_iterator | vertices_end () const |
Edge_iterator | edges_begin () const |
visits all edges. | |
Edge_iterator | edges_end () const |
Vertex_circulator | incident_vertices (Vertex_handle v, Face_handle f=Face_handle()) const |
Edge_circulator | incident_edges (Vertex_handle v, Face_handle f=Face_handle()) const |
Face_circulator | incident_faces (Vertex_handle v, Face_handle f=Face_handle()) const |
Vertex_handle | mirror_vertex (Face_handle f, int i) const |
returns vertex of f->neighbor(i) . | |
int | mirror_index (Face_handle f, int i) const |
returns the index of f as a neighbor of f->neighbor(i) . | |
Edge | mirror_edge (Edge e) const |
returns the same edge seen from the other adjacent face. | |
Modifiers | |
void | flip (Face_handle f, int i) |
exchanges the edge incident to f and f->neighbor(i) with the other diagonal of the quadrilateral formed by f and f->neighbor(i) . More... | |
Vertex_handle | insert_first () |
creates the first vertex and returns a handle to it. | |
Vertex_handle | insert_second () |
creates the second vertex and returns a handle to it. | |
Vertex_handle | insert_in_edge (Face_handle f, int i) |
adds a vertex v splitting edge i of face f . More... | |
Vertex_handle | insert_in_face (Face_handle f) |
adds a vertex v splitting face f in three. More... | |
Vertex_handle | insert_dim_up (Vertex_handle w, bool orient=true) |
adds a vertex v , increasing by one the dimension of the triangulation data structure. More... | |
void | remove_degree_3 (Vertex_handle v, Face_handle f=Face_handle()) |
removes a vertex of degree 3. More... | |
void | remove_second (Vertex_handle v) |
removes the before last vertex. | |
void | remove_first (Vertex_handle v) |
removes the last vertex. | |
void | remove_dim_down (Vertex_handle v) |
removes vertex v incident to all other vertices and decreases by one the dimension of the triangulation data structure. More... | |
void | dim_down (Face_handle f, int i) |
must be called when the displacement of a vertex decreases the dimension of the triangulation data structure. More... | |
Advanced Modifiers | |
The following modifiers are required for convenience of the advanced user. They do not guarantee the combinatorial validity of the resulting triangulation data structure. | |
template<class FaceIt > | |
Vertex_handle | insert_in_hole (FaceIt face_begin, FaceIt face_end) |
creates a new vertex v and uses it to star a hole. More... | |
template<class FaceIt > | |
void | insert_in_hole (Vertex_handle new_v, FaceIt face_begin, FaceIt face_end) |
same as above, except that new_v will be used as the new vertex, which must have been allocated previously, for example with create_vertex . | |
template<class EdgeIt > | |
Vertex_handle | star_hole (EdgeIt edge_begin, EdgeIt edge_end) |
creates a new vertex v and use it to star the hole whose boundary is described by the sequence of edges [edge_begin, edge_end) . More... | |
template<class EdgeIt , class FaceIt > | |
Vertex_handle | star_hole (EdgeIt edge_begin, EdgeIt edge_end, FaceIt face_begin, FaceIt face_end) |
same as above, except that, to build the new faces, the algorithm first recycles faces in the sequence [face_begin, face_end) and create new ones when the sequence is exhausted. | |
template<class EdgeIt > | |
void | star_hole (Vertex_handle v, EdgeIt edge_begin, EdgeIt edge_end) |
uses vertex v to star the hole whose boundary is described by the sequence of edges [edge_begin, edge_end) . | |
template<class EdgeIt , class FaceIt > | |
void | star_hole (Vertex_handle v, EdgeIt edge_begin, EdgeIt edge_end, FaceIt face_begin, FaceIt face_end) |
same as above, recycling faces in the sequence [face_begin, face_end) . | |
void | make_hole (Vertex_handle v, List_edges &hole) |
removes the vertex v, and store in hole the list of edges on the boundary of the hole. | |
Vertex_handle | create_vertex () |
adds a new vertex. | |
Face_handle | create_face (Face_handle f1, int i1, Face_handle f2, int i2, Face_handle f3, int i3) |
adds a face which is the neighbor i1 of f1 , i2 of f2 and i3 of f3 . | |
Face_handle | create_face (Face_handle f1, int i1, Face_handle f2, int i2) |
adds a face which is the neighbor i1 of f1 , and the neighbor i2 of f2 . | |
Face_handle | create_face (Face_handle f1, int i1, Vertex_handle v) |
adds a face which is the neighbor i1 of f1 , and has v as vertex. | |
Face_handle | create_face (Vertex_handle v1, Vertex_handle v2, Vertex_handle v3) |
adds a face with vertices v1 , v2 and v3 . | |
Face_handle | create_face (Vertex_handle v1, Vertex_handle v2, Vertex_handle v3, Face_handle f1, Face_handle f2, Face_handle f3) |
adds a face with vertices v1 , v2 and v3 , and neighbors f1 , f2 , f3 . | |
Face_handle | create_face () |
adds a face whose vertices and neighbors are set to Vertex_handle() and Face_handle() . | |
void | delete_face (Face_handle) |
deletes a face. | |
void | delete_vertex (Vertex_handle) |
deletes a vertex. | |
Miscellaneous | |
int | ccw (int i) const |
returns \( i+1\) modulo 3, with \( 0\leq i \leq2\). | |
int | cw (int i) const |
returns \( i+2\) modulo 3, with \( 0\leq i \leq2\). | |
bool | is_valid () |
checks the combinatorial validity of the triangulation data structure: call the is_valid() member function for each vertex and each face, checks the number of vertices and the Euler relation between numbers of vertices, faces and edges. | |
size_type | degree (Vertex_handle v) const |
Returns the degree of v in the triangulation data structure. | |
void | file_output (ostream &os, Vertex_handle v=Vertex_handle(), bool skip_first=false) |
writes the triangulation data structure into the stream os . More... | |
Vertex_handle | file_input (istream &is, bool skip_first=false) |
inputs the triangulation data structure from file and returns a handle to the first input vertex. More... | |
istream & | operator>> (istream &is, TriangulationDataStructure_3 &tds) |
reads a combinatorial triangulation data structure from is and assigns it to tthe triangulation data structure. | |
ostream & | operator<< (ostream &os, const TriangulationDataStructure_3 &tds) |
writes tds into the stream os . | |
typedef std::pair<Face_handle,int> TriangulationDataStructure_2::Edge |
The edge type.
The Edge(f,i)
is edge common to faces f
and f.neighbor(i)
. It is also the edge joining the vertices vertex(cw(i))
and vertex(ccw(i))
of f
.
Handle to a face.
Handle to a vertex.
Vertex_handle TriangulationDataStructure_2::copy_tds | ( | const TriangulationDataStructure_2 & | tds1, |
Vertex_handle | v = Vertex_handle() |
||
) |
tds1
is copied into the triangulation data structure.
If v != Vertex_handle()
, the vertex of the triangulation data structure corresponding to v
is returned, otherwise Vertex_handle()
is returned.
v
is a vertex of tds1
. Vertex_handle tds TriangulationDataStructure_2::copy_tds | ( | const TDS_src & | tds_src, |
typename TDS_src::Vertex_handle | v, | ||
const ConvertVertex & | convert_vertex, | ||
const ConvertFace & | convert_face | ||
) |
tds_src
is copied into this
.
As the vertex and face types might be different and incompatible, the creation of new faces and vertices is made thanks to the functors convert_vertex
and convert_face
, that convert vertex and face types. For each vertex v_src
in tds_src
, the corresponding vertex v_tgt
in this
is a copy of the vertex returned by convert_vertex(v_src)
. The same operations are done for faces with the functor convert_face. If v != TDS_src::Vertex_handle()
, a handle to the vertex created in this
that is the copy of v
is returned, otherwise Vertex_handle()
is returned.
ConvertVertex
must provide two operator()'s that are responsible for converting the source vertex v_src
into the target vertex:Vertex operator()(const TDS_src::Vertex& v_src);
This operator is used to create the vertex from v_src
.void operator()(const TDS_src::Vertex& v_src, Vertex& v_tgt);
This operator is meant to be used in case heavy data should transferred to v_tgt
.f_src
into the target face:Face operator()(const TDS_src::Face& f_src);
This operator is used to create the face from f_src
.void operator()(const TDS_src::Face& f_src, Face& f_tgt);
This operator is meant to be used in case heavy data should transferred to f_tgt
.v
is a vertex of tds_src
or is Vertex_handle()
. void TriangulationDataStructure_2::dim_down | ( | Face_handle | f, |
int | i | ||
) |
must be called when the displacement of a vertex decreases the dimension of the triangulation data structure.
The link of a vertex v
is formed by the edges disjoint from v
that are included in the faces incident to v
. When the link of v = f->vertex(i)
contains all the other vertices of the two-dimensional triangulation data structure ( \( \mathbb{S}^2\)), dim_down()
crushes the two-dimensional data-structure ( \( \mathbb{S}^2\)) onto the one-dimensional data structure ( \( \mathbb{S}^1\)) formed by the link of v
augmented with the vertex v
itself; this one is placed on the edge (f, i)
(see Figure figtdsdim_down_2).
dimension()
must be equal to 2
, the degree of f->vertex(i)
must be equal to the total number of vertices minus 1.int TriangulationDataStructure_2::dimension | ( | ) | const |
This is an advanced function.
returns the dimension of the triangulation data structure.
Vertex_handle TriangulationDataStructure_2::file_input | ( | istream & | is, |
bool | skip_first = false |
||
) |
inputs the triangulation data structure from file and returns a handle to the first input vertex.
If skip_first
is true, it is assumed that the first vertex has been omitted when output.
void TriangulationDataStructure_2::file_output | ( | ostream & | os, |
Vertex_handle | v = Vertex_handle() , |
||
bool | skip_first = false |
||
) |
writes the triangulation data structure into the stream os
.
If v
is not Vertex_handle()
, vertex v
is output first or skipped if skip_first
is true.
void TriangulationDataStructure_2::flip | ( | Face_handle | f, |
int | i | ||
) |
exchanges the edge incident to f
and f->neighbor(i)
with the other diagonal of the quadrilateral formed by f
and f->neighbor(i)
.
Edge_circulator TriangulationDataStructure_2::incident_edges | ( | Vertex_handle | v, |
Face_handle | f = Face_handle() |
||
) | const |
f
is given, it has to be a face of incident to v
and the circulator begins with the edge (f,cw(i))
of f
if i
is the index of v
in f
. Face_circulator TriangulationDataStructure_2::incident_faces | ( | Vertex_handle | v, |
Face_handle | f = Face_handle() |
||
) | const |
f
is given, it has to be a face of incident to v
and the circulator begins with the face f
. Vertex_circulator TriangulationDataStructure_2::incident_vertices | ( | Vertex_handle | v, |
Face_handle | f = Face_handle() |
||
) | const |
f
is given, it has to be incident to be a face incident to v
and the circulator begins with the vertex f->vertex(ccw(i))
if i
is the index of v
in f
. Vertex_handle TriangulationDataStructure_2::insert_dim_up | ( | Vertex_handle | w, |
bool | orient = true |
||
) |
adds a vertex v
, increasing by one the dimension of the triangulation data structure.
Vertex v
and the existing vertex w
are linked to all the vertices of the triangulation data structure. The Boolean orient
decides the final orientation of all faces. A handle to vertex v
is returned.
Vertex_handle TriangulationDataStructure_2::insert_in_edge | ( | Face_handle | f, |
int | i | ||
) |
adds a vertex v
splitting edge i
of face f
.
Return a handle to v
.
Vertex_handle TriangulationDataStructure_2::insert_in_face | ( | Face_handle | f | ) |
adds a vertex v
splitting face f
in three.
Face f
is modified, two new faces are created. Return a handle to v
Vertex_handle TriangulationDataStructure_2::insert_in_hole | ( | FaceIt | face_begin, |
FaceIt | face_end | ||
) |
creates a new vertex v
and uses it to star a hole.
Given a set of faces 'F' describing a simply connected hole (i.e., a topological disk), the function deletes all the faces in F
, creates a new vertex v
and for each edge on the boundary of the hole creates a new face with v
as a vertex. The input is an iterator range [face_begin, face_end[
of Face_handle
s over the connected faces in F
. The handle to the new vertex v
is returned.
tds.dimension() = 2
and the set of faces has the topology of a disk. bool TriangulationDataStructure_2::is_edge | ( | Face_handle | fh, |
int | i | ||
) | const |
returns true
if (fh,i)
is an edge of the triangulation data structure.
Returns false
when dimension() < 1
.
bool TriangulationDataStructure_2::is_edge | ( | Vertex_handle | va, |
Vertex_handle | vb, | ||
Face_handle & | fr, | ||
int & | i | ||
) | const |
returns true
if (va, vb)
is an edge of the triangulation data structure.
In addition, if true is returned fr
and i
are set such that the pair (fr,i)
is the description of the ordered edge (va, vb)
.
bool TriangulationDataStructure_2::is_face | ( | Face_handle | fh | ) | const |
returns true
if fh
is a face of the triangulation data structure.
Returns false
when dimension() < 2
.
bool TriangulationDataStructure_2::is_face | ( | Vertex_handle | v1, |
Vertex_handle | v2, | ||
Vertex_handle | v3, | ||
Face_handle & | fr | ||
) | const |
true
if there is a face having v1
, v2
, and v3
as vertices.
In addition, if true
is returned, fr
is a handle to the face with v1
, v2
and v3
as vertices.
size_type TriangulationDataStructure_2::number_of_full_dim_faces | ( | ) | const |
returns the number of full dimensional faces, i.e. faces of dimension equal to the dimension of the triangulation data structure.
This is the actual number of faces stored in the triangulation data structure.
TriangulationDataStructure_2& TriangulationDataStructure_2::operator= | ( | const TriangulationDataStructure_2 & | tds1 | ) |
Assignment.
All the vertices and faces of tds1
are duplicated in the triangulation data structure. Former faces and vertices of the triangulation data structure , if any, are deleted.
void TriangulationDataStructure_2::remove_degree_3 | ( | Vertex_handle | v, |
Face_handle | f = Face_handle() |
||
) |
removes a vertex of degree 3.
Two of the incident faces are destroyed, the third one is modified. If parameter f
is specified, it has to be a face incident to v
and will be the modified face.
v
is a finite vertex with degree 3 and, if specified, face f
is incident to v
.void TriangulationDataStructure_2::remove_dim_down | ( | Vertex_handle | v | ) |
removes vertex v
incident to all other vertices and decreases by one the dimension of the triangulation data structure.
Vertex_handle TriangulationDataStructure_2::star_hole | ( | EdgeIt | edge_begin, |
EdgeIt | edge_end | ||
) |
creates a new vertex v
and use it to star the hole whose boundary is described by the sequence of edges [edge_begin, edge_end)
.
Returns a handle to the vertex.
void TriangulationDataStructure_2::swap | ( | TriangulationDataStructure_2 & | tds1 | ) |
Swaps the triangulation data structure and tds1
.
Should be preferred to an assignment or copy constructor when tds1
is deleted after that.