The class Triangulation_3 represents a 3dimensional tetrahedralization of points.
#include <CGAL/Triangulation_3.h>
The first template argument must be a model of the TriangulationTraits_3 concept.
The second template argument must be a model of the TriangulationDataStructure_3 concept. It has the default value Triangulation_data_structure_3< Triangulation_vertex_base_3<TriangulationTraits_3>,Triangulation_cell_base_3<TriangulationTraits_3> >.
 
 
 


 


 


 


 

Only vertices ($$0faces) and cells ($$3faces) are stored. Edges ($$1faces) and facets ($$2faces) are not explicitly represented and thus there are no corresponding classes (see Section ).
 


 


 


 

The vertices and faces of the triangulations are accessed through handles, iterators and circulators. A handle is a type which supports the two dereference operators operator* and operator>. The Handle concept is documented in the support library. Iterators and circulators are bidirectional and nonmutable. The edges and facets of the triangulation can also be visited through iterators and circulators which are bidirectional and nonmutable.
Iterators and circulators are convertible to the corresponding handles, thus the user can pass them directly as arguments to the functions.
 
 handle to a vertex  
 
 handle to a cell  
 
 Size type (an unsigned integral type)  
 
 Difference type (a signed integral type)  
 
 iterator over cells  
 
 
iterator over facets  
 
 iterator over edges  
 
 
iterator over vertices 
 
iterator over finite cells
 
 
iterator over finite facets
 
 
iterator over finite edges
 
 
iterator over finite vertices
 
 
iterator over the points corresponding to the
finite vertices of the triangulation.

 
 circulator over all cells incident to a given edge  
 
 circulator over all facets incident to a given edge 
The triangulation class also defines the following enum type to specify which case occurs when locating a point in the triangulation.

 
Introduces a triangulation t having only one vertex which is the
infinite vertex.
 
 
Copy constructor. All vertices and faces are duplicated.
 
 
 
Introduces a triangulation t constructed by the repeated insertion
of the iterator range [first,last) of value type Point.


 
The triangulation tr is duplicated, and modifying the copy after the duplication does not modify the original. The previous triangulation held by t is deleted.  

 
The triangulations tr and t are swapped. t.swap(tr) should be preferred to t = tr or to t(tr) if tr is deleted after that. Indeed, there is no copy of cells and vertices, thus this method runs in constant time.  

 Deletes all finite vertices and all cells of t. 

 
Destructor. All vertices (including the infinite vertex) and cells are deleted.  
 

 
Equality operator. Returns true iff there exist a bijection between the vertices of t1 and those of t2 and a bijection between the cells of t1 and those of t2, which preserve the geometry of the triangulation, that is, the points of each corresponding pair of vertices are equal, and the tetrahedra corresponding to each pair of cells are equal (up to a permutation of their vertices).  
 

 
The opposite of operator==. 
 
 Returns a const reference to the geometric traits object.  
 
 Returns a const reference to the triangulation data structure. 
advanced 
 
 Returns a reference to the triangulation data structure. 
This method is mainly a help for users implementing their own triangulation algorithms.
advanced 
As previously said, the triangulation is a collection of cells that are either infinite or represent a finite tetrahedra, where an infinite cell is a cell incident to the infinite vertex. Similarly we call an edge (resp. facet) infinite if it is incident to the infinite vertex.
There is a method has_vertex in the cell class. The analogous methods for facets are defined here.

 
If v is a vertex of f, then j is the index of
v in the cell f.first, and the method returns true. Precondition: t.dimension()=3  

 
Same for facet (c,i). Computes the index j of v in c.  

 

 
Same as the first two methods, but these two methods do not return the index of the vertex. 
The following three methods test whether two facets have the same vertices.

 

 

 
For these three methods: Precondition: t.dimension()=3. 
The class Triangulation_3<TriangulationTraits_3,TriangulationDataStructure_3> provides two functions to locate a given point with respect to a triangulation. It provides also functions to test if a given point is inside a finite face or not. Note that the class Delaunay_triangulation_3 also provides a nearest_vertex() function.

 
If the point query lies inside the convex hull of the points, the cell
that contains the query in its interior is returned. If query lies on a
facet, an edge or on a vertex, one of the cells having query on
its boundary is returned. If the point query lies outside the convex hull of the points, an infinite cell with vertices $${ p, q, r, } is returned such that the tetrahedron $$( p, q, r, query ) is positively oriented (the rest of the triangulation lies on the other side of facet $$( p, q, r )). Note that locate works even in degenerate dimensions: in dimension 2 (resp. 1, 0) the Cell_handle returned is the one that represents the facet (resp. edge, vertex) containing the query point. The optional argument start is used as a starting place for the search.  

 
If query lies inside the affine hull of the points, the $$kface
(finite or infinite) that contains query in its interior is
returned, by means of the cell returned together with lt, which
is set to the locate type of the query (VERTEX, EDGE, FACET, CELL, or OUTSIDE_CONVEX_HULL if the cell is infinite and query
lies strictly in it) and two indices li and lj that
specify the $$kface of the cell containing query. If the $$kface is a cell, li and lj have no meaning; if it is a facet (resp. vertex), li gives the index of the facet (resp. vertex) and lj has no meaning; if it is and edge, li and lj give the indices of its vertices. If the point query lies outside the affine hull of the points, which can happen in case of degenerate dimensions, lt is set to OUTSIDE_AFFINE_HULL, and the cell returned has no meaning. As a particular case, if there is no finite vertex yet in the triangulation, lt is set to OUTSIDE_AFFINE_HULL and locate returns the default constructed handle. The optional argument start is used as a starting place for the search.  

 
Returns a value indicating on which side of the oriented boundary
of c the point p lies. More precisely, it returns:  ON_BOUNDED_SIDE if p is inside the cell. For an infinite cell this means that p lies strictly in the half space limited by its finite facet and not containing any other point of the triangulation.  ON_BOUNDARY if p on the boundary of the cell. For an infinite cell this means that p lies on the finite facet. Then lt together with li and lj give the precise location on the boundary. (See the descriptions of the locate methods.)  ON_UNBOUNDED_SIDE if p lies outside the cell. For an infinite cell this means that p does not satisfy either of the two previous conditions. Precondition: t.dimension() $$=3  

 
Returns a value indicating on which side of the oriented boundary
of f the point p lies:  ON_BOUNDED_SIDE if p is inside the facet. For an infinite facet this means that p lies strictly in the half plane limited by its finite edge and not containing any other point of the triangulation .  ON_BOUNDARY if p is on the boundary of the facet. For an infinite facet this means that p lies on the finite edge. lt, li and lj give the precise location of p on the boundary of the facet. li and lj refer to indices in the degenerate cell c representing f.  ON_UNBOUNDED_SIDE if p lies outside the facet. For an infinite facet this means that p does not satisfy either of the two previous conditions. Precondition: t.dimension() $$=2 and p lies in the plane containing the triangulation. f.second $$=3 (in dimension 2 there is only one facet per cell).  

 
Same as the previous method for the facet (c,3).  

 
Returns a value indicating on which side of the oriented boundary
of e the point p lies:  ON_BOUNDED_SIDE if p is inside the edge. For an infinite edge this means that p lies in the half line defined by the vertex and not containing any other point of the triangulation.  ON_BOUNDARY if p equals one of the vertices, li give the index of the vertex in the cell storing e  ON_UNBOUNDED_SIDE if p lies outside the edge. For an infinite edge this means that p lies on the other half line, which contains the other points of the triangulation. Precondition: t.dimension() $$=1 and p is collinear with the points of the triangulation. e.second $$=0 and e.third $$=1 (in dimension 1 there is only one edge per cell).  

 
Same as the previous method for edge $$(c,0,1). 
Two kinds of flips exist for a threedimensional triangulation. They are reciprocal. To be flipped, an edge must be incident to three tetrahedra. During the flip, these three tetrahedra disappear and two tetrahedra appear. Figure (left) shows the edge that is flipped as bold dashed, and one of its three incident facets is shaded. On the right, the facet shared by the two new tetrahedra is shaded.
Flips are possible only under the following conditions:
 the edge or facet to be flipped is not on the boundary of the convex
hull of the triangulation
 the five points involved are in convex position.
The following methods guarantee the validity of the resulting 3D triangulation.
Flips for a 2d triangulation are not implemented yet




 
Before flipping, these methods check that edge e=(c,i,j) is flippable (which is quite expensive). They return false or true according to this test.  

 

 
Should be preferred to the previous methods when the edge is
known to be flippable. Precondition: The edge is flippable.  




 
Before flipping, these methods check that facet f=(c,i) is flippable (which is quite expensive). They return false or true according to this test.  

 

 
Should be preferred to the previous methods when the facet is
known to be flippable. Precondition: The facet is flippable. 
The following operations are guaranteed to lead to a valid triangulation when they are applied on a valid triangulation.

 
Inserts point p in the triangulation and returns the corresponding
vertex. If point p coincides with an already existing vertex, this vertex is returned and the triangulation remains unchanged. If point p lies in the convex hull of the points, it is added naturally: if it lies inside a cell, the cell is split into four cells, if it lies on a facet, the two incident cells are split into three cells, if it lies on an edge, all the cells incident to this edge are split into two cells. If point p is strictly outside the convex hull but in the affine hull, p is linked to all visible points on the convex hull to form the new triangulation. See Figure . If point p is outside the affine hull of the points, p is linked to all the points, and the dimension of the triangulation is incremented. All the points now belong to the boundary of the convex hull, so, the infinite vertex is linked to all the points to triangulate the new infinite face. See Figure . The optional argument start is used as a starting place for the search.  

 
Inserts point p in the triangulation and returns the corresponding vertex. Similar to the above insert() function, but takes as additional parameter the return values of a previous location query. See description of locate() above.  
 

 
Inserts the points in the range $$[.first,
last$$.). Returns the number of inserted points. Precondition: The value_type of first and last is Point. 
The previous methods are sufficient to build a whole triangulation. We also provide some other methods that can be used instead of insert(p) when the place where the new point p must be inserted is already known. They are also guaranteed to lead to a valid triangulation when they are applied on a valid triangulation.

 
Inserts point p in cell c. Cell c is split into 4
tetrahedra. Precondition: t.dimension() $$=3 and p lies strictly inside cell c.  

 
Inserts point p in facet f. In dimension 3, the 2
neighboring cells are split into 3 tetrahedra; in dimension 2, the facet
is split into 3 triangles. Precondition: t.dimension() $$ 2 and p lies strictly inside face f.  

 
As above, insertion in facet (c,i). Precondition: As above and $$i {0,1,2,3} in dimension 3, $$i = 3 in dimension 2.  

 
Inserts p in edge e. In dimension 3,
all the cells having this edge are split into 2 tetrahedra; in
dimension 2, the 2 neighboring facets are split into 2 triangles; in
dimension 1, the edge is split into 2 edges. Precondition: t.dimension() $$ 1 and p lies on edge e.  

 
As above, inserts p in edge $$(i, j) of c. Precondition: As above and $$i j. Moreover $$i,j {0,1,2,3} in dimension 3, $$i,j {0,1,2} in dimension 2, $$i,j {0,1} in dimension 1.  

 
The cell c must be an infinite cell containing p. Links p to all points in the triangulation that are visible from p. Updates consequently the infinite faces. See Figure . Precondition: t.dimension() $$>0, c, and the $$kface represented by c is infinite and contains t. 
Figure: insert_outside_convex_hull (2dimensional case).

 
p is linked to all the points, and the infinite vertex is linked
to all the points (including p) to triangulate the new infinite
face, so that all the points now belong to the boundary of the convex
hull. See Figure . This method can be used to insert the first point in an empty triangulation. Precondition: t.dimension() $$<3 and p lies outside the affine hull of the points. 
Figure: insert_outside_affine_hull (2dimensional case).
 

 
Creates a new vertex by starring a hole. It takes an iterator range
[cell_begin; cell_end[ of Cell_handles which specifies
a hole: a set of connected cells (resp. facets in dimension 2) which is
starshaped wrt p.
(begin, i) is a facet (resp. an edge) on the boundary of the hole,
that is, begin belongs to the set of cells (resp. facets) previously
described, and begin>neighbor(i) does not. Then this function deletes
all the cells (resp. facets) describing the hole, creates a new vertex
v, and for each facet (resp. edge) on the boundary of the hole, creates
a new cell (resp. facet) with v as vertex. Then v>set_point(p)
is called and v is returned. This operation is equivalent to calling tds().insert_in_hole(cell_begin, cell_end, begin, i); v>set_point(p). Precondition: t.dimension() $$ 2, the set of cells (resp. facets in dimension 2) is connected, its boundary is connected, and p lies inside the hole, which is starshaped wrt p. 
The triangulation class provides several iterators and circulators that allow one to traverse it (completely or partially).
The following iterators allow the user to visit cells, facets, edges and vertices of the triangulation. These iterators are nonmutable, bidirectional and their value types are respectively Cell, Facet, Edge and Vertex. They are all invalidated by any change in the triangulation.
 
 
Starts at an arbitrary finite vertex. Then ++ and  will iterate over finite vertices. Returns finite_vertices_end() when t.number_of_vertices() $$=0.  
 
 
Pasttheend iterator  
 
 
Starts at an arbitrary finite edge. Then ++ and  will iterate over finite edges. Returns finite_edges_end() when t.dimension() $$<1.  
 
 
Pasttheend iterator  
 
 
Starts at an arbitrary finite facet. Then ++ and  will iterate over finite facets. Returns finite_facets_end() when t.dimension() $$<2.  
 
 
Pasttheend iterator  
 
 
Starts at an arbitrary finite cell. Then ++ and  will iterate over finite cells. Returns finite_cells_end() when t.dimension() $$<3.  
 
 
Pasttheend iterator  
 
 
Starts at an arbitrary vertex. Iterates over all vertices (even the infinite one). Returns vertices_end() when t.number_of_vertices() $$=0.  
 
 
Pasttheend iterator  

 
Starts at an arbitrary edge. Iterates over all edges (even infinite ones). Returns edges_end() when t.dimension() $$<1.  

 Pasttheend iterator 
 
 
Starts at an arbitrary facet. Iterates over all facets (even infinite ones). Returns facets_end() when t.dimension() $$<2.  
 
 
Pasttheend iterator  

 
Starts at an arbitrary cell. Iterates over all cells (even infinite ones). Returns cells_end() when t.dimension() $$<3.  

 Pasttheend iterator 

 Iterates over the points of the triangulation. 

 Pasttheend iterator 
The following circulators respectively visit all cells or all facets incident to a given edge. They are nonmutable and bidirectional. They are invalidated by any modification of one of the cells traversed.

 
Starts at an arbitrary cell incident to e. Precondition: t.dimension() $$=3.  

 
As above for edge (i,j) of c.  

 
Starts at cell start. Precondition: t.dimension() $$=3 and start is incident to e.  

 
As above for edge (i,j) of c. 
The following circulators on facets are defined only in dimension 3, though facets are defined also in dimension 2: there are only two facets sharing an edge in dimension 2.

 
Starts at an arbitrary facet incident to e. Precondition: t.dimension() $$=3  

 
As above for edge (i,j) of c.  

 
Starts at facet start. Precondition: start is incident to e.  

 
Starts at facet of index f in start.  

 
As above for edge (i,j) of c.  

 
As above for edge (i,j) of c and facet (start,f). 
 

 
Copies the Cell_handles of all cells incident to v to the output
iterator cells. If t.dimension() $$<3, then do nothing.
Returns the resulting output iterator. Precondition: v $$ Vertex_handle(), t.is_vertex(v).  
 

 
Copies the Vertex_handles of all vertices incident to v to the
output iterator vertices. If t.dimension() $$<2, then do
nothing. Returns the resulting output iterator. Precondition: v $$ Vertex_handle(), t.is_vertex(v).  

 
Returns the degree of a vertex, that is, the number of incident vertices.
The infinite vertex is counted. Precondition: v $$ Vertex_handle(), t.is_vertex(v). 
advanced 

 
Checks the combinatorial validity of the triangulation. Checks also the
validity of its geometric embedding (see
Section ). When verbose is set to true, messages describing the first invalidity encountered are printed.  

 
Checks the combinatorial validity of the cell by calling the
is_valid method of the TriangulationDataStructure_3 cell class. Also checks the
geometric validity of c, if c is finite. (See
Section .) When verbose is set to true, messages are printed to give a precise indication of the kind of invalidity encountered. 
advanced 
CGAL provides an interface to Geomview for a 3Dtriangulation. See the chapter on Geomview in the Support Library manual. #include <CGAL/IO/Triangulation_geomview_ostream_3.h>

 
Reads the underlying combinatorial triangulation from is by calling the corresponding input operator of the triangulation data structure class, and the noncombinatorial information by calling the corresponding input operators of the vertex and the cell classes. Assigns the resulting triangulation to t.  

 
Writes the triangulation t into os. 
The information in the iostream is: the dimension, the number of finite vertices, the noncombinatorial information about vertices (point, etc), the number of cells, the indices of the vertices of each cell, plus the noncombinatorial information about each cell, then the indices of the neighbors of each cell, where the index corresponds to the preceding list of cells. When dimension $$< 3, the same information is stored for faces of maximal dimension instead of cells.
TriangulationDataStructure_3::Vertex
TriangulationDataStructure_3::Cell