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CGAL::Periodic_3_Delaunay_triangulation_3<PT,TDS>

Definition

The class Periodic_3_Delaunay_triangulation_3 represents a Delaunay triangulation in three-dimensional periodic space.

#include <CGAL/Periodic_3_Delaunay_triangulation_3.h>

Parameters

The first template argument PT must be a model of the Periodic_3DelaunayTriangulationTraits_3 concept.

The second template argument TDS must be a model of the TriangulationDataStructure_3 concept with some additional functionality in cells and vertices. Its default value is Triangulation_data_structure_3<Triangulation_vertex_base_3<PT,Periodic_3_triangulation_ds_vertex_base_3<>>,Triangulation_cell_base_3<PT,Periodic_3_triangulation_ds_cell_base_3<>>>.

Inherits From

Periodic_3_triangulation_3<Periodic_3DelaunayTriangulationTraits_3,TriangulationDataStructure_3>

Creation

Periodic_3_Delaunay_triangulation_3<PT,TDS> dt ( Iso_cuboid domain = Iso_cuboid(0,0,0,1,1,1), Geom_traits traits = Geom_traits());
Creates an empty periodic Delaunay triangulation dt, with domain as original domain and possibly specifying a traits class traits.
Precondition: domain is a cube.


Periodic_3_Delaunay_triangulation_3<PT,TDS> dt ( Periodic_3_Delaunay_triangulation_3 dt1);
Copy constructor.


template < class InputIterator >
Periodic_3_Delaunay_triangulation_3<PT,TDS> dt ( InputIterator first,
InputIterator last,
Iso_cuboid domain = Iso_cuboid(0,0,0,1,1,1),
Geom_traits traits = Geom_traits());
Equivalent to constructing an empty triangulation with the optional domain and traits class arguments and calling insert(first,last).
Precondition: The value_type of first and last are Points lying inside domain and domain is a cube.

Operations

Insertion

The following methods insert points in the triangulation ensuring the empty sphere property of Delaunay triangulations. The inserted points need to lie in the original domain (see Section 37.1 of the user manual).

In the degenerate case when there are co-spherical points, the Delaunay triangulation is known not to be uniquely defined. In this case, Cgal chooses a particular Delaunay triangulation using a symbolic perturbation scheme [DT03].

Note that insertion of a new point can cause a switch from computing in 27-sheeted covering space to computing in 1-sheeted covering space, which invalidates some Vertex_handles and Cell_handles.

Vertex_handle dt.insert ( Point p, Cell_handle start = Cell_handle())
Inserts point p in the triangulation and returns the corresponding vertex. The optional argument start is used as a starting place for the point location.
Precondition: p lies in the original domain domain.

Vertex_handle dt.insert ( Point p, Locate_type lt, Cell_handle loc, int li, int lj)
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 Periodic_3_triangulation_3::locate().
Precondition: p lies in the original domain domain.

The following method allows one to insert several points. It returns the number of inserted points.

template < class InputIterator >
int dt.insert ( InputIterator first, InputIterator last, bool is_large_point_set = false)
Inserts the points in the iterator range [.first, last.). Returns the number of inserted points. This function uses spatial sorting (cf. chapter 61.2) and therefore is not guaranteed to insert the points following the order of InputIterator. If the third argument is_large_point_set is set to true a heuristic for optimizing the insertion of large point sets is applied.
Precondition: The value_type of first and last are Points lying inside domain.

Point moving

Vertex_handle dt.move_point ( Vertex_handle v, Point p)
Moves the point stored in v to p, while preserving the Delaunay property. This performs an action semantically equivalent to remove(v) followed by insert(p), but is supposedly faster when the point has not moved much. Returns the handle to the new vertex.
Precondition: p lies in the original domain domain.

Removal

When a vertex v is removed from a triangulation, all the cells incident to v must be removed, and the polyhedral region consisting of all the tetrahedra that are incident to v must be re-triangulated. So, the problem reduces to triangulating a polyhedral region, while preserving its boundary, or to compute a constrained triangulation. This is known to be sometimes impossible: the Schönhardt polyhedron cannot be triangulated [She98].

However, when dealing with Delaunay triangulations, the case of such polyhedra that cannot be re-triangulated cannot happen, so Cgal proposes a vertex removal.

void dt.remove ( Vertex_handle v) Removes the vertex v from the triangulation.

template < class InputIterator >
int dt.remove ( InputIterator first, InputIterator beyond)
Removes the vertices specified by the iterator range [first, beyond) of value type Vertex_handle. remove() is called over each element of the range. The number of vertices removed is returned.

Queries

Bounded_side dt.side_of_sphere ( Cell_handle c, Point p, Offset off = Offset(0,0,0))
Returns a value indicating on which side of the circumscribed sphere of c the point-offset pair (p,off) lies. More precisely, it returns:
- ON_BOUNDED_SIDE if (p,off) is inside the sphere.
- ON_BOUNDARY if (p,off) on the boundary of the sphere.
- ON_UNBOUNDED_SIDE if (p,off) lies outside the sphere.
Precondition: p lies in the original domain domain.

Vertex_handle dt.nearest_vertex ( Point p, Cell_handle c = Cell_handle())
Returns any nearest vertex to the point p, or the default constructed handle if the triangulation is empty. The optional argument c is a hint specifying where to start the search. It always returns a vertex corresponding to a point inside domain even if computing in a multiply sheeted covering space.
Precondition: c is a cell of dt and p lies in the original domain domain.

Vertex_handle dt.nearest_vertex_in_cell ( Cell_handle c, Point p, Offset off = Offset(0,0,0))
Returns the vertex of the cell c that is nearest to the point-offset pair (p,off).
Precondition: p lies in the original domain domain.

A point-offset pair (p,off) is said to be in conflict with a cell c iff dt.side_of_sphere(c, p, off) returns ON_BOUNDED_SIDE. The set of cells that are in conflict with (p,off) is star-shaped.

template <class OutputIteratorBoundaryFacets, class OutputIteratorCells, class OutputIteratorInternalFacets>
Triple<OutputIteratorBoundaryFacets, OutputIteratorCells, OutputIteratorInternalFacets>
dt.find_conflicts ( Point p,
Cell_handle c,
OutputIteratorBoundaryFacets bfit,
OutputIteratorCells cit,
OutputIteratorInternalFacets ifit)
Computes the conflict hole induced by p. The starting cell c must be in conflict. Then this function returns respectively in the output iterators:
- cit: the cells in conflict.
- bfit: the facets on the boundary, that is, the facets (t, i) where the cell t is in conflict, but t->neighbor(i) is not.
- ifit: the facets inside the hole, that is, delimiting two cells in conflict.
Returns the pair composed of the resulting output iterators.
Precondition: c is in conflict with p and p lies in the original domain domain.

template <class OutputIterator>
OutputIterator dt.vertices_in_conflict ( Point p, Cell_handle c, OutputIterator res)
Similar to find_conflicts(), but reports the vertices which are on the boundary of the conflict hole of p, in the output iterator res. Returns the resulting output iterator.
Precondition: c is in conflict with p and p lies in the original domain domain.

A face (cell, facet or edge) is said to be a Gabriel face iff its smallest circumscribing sphere do not enclose any vertex of the triangulation. Any Gabriel face belongs to the Delaunay triangulation, but the reciprocal is not true. The following member functions test the Gabriel property of Delaunay faces.

bool dt.is_Gabriel ( Cell_handle c, int i)
bool dt.is_Gabriel ( Cell_handle c, int i, int j)
bool dt.is_Gabriel ( Facet f)
bool dt.is_Gabriel ( Edge e)

Voronoi diagram

Cgal offers several functions to display the Voronoi diagram of a set of points in 3D.

Note that a traits class providing exact constructions should be used in order to guarantee the computation of the Voronoi diagram (as opposed to computing the triangulation only, which requires only exact predicates).

Point dt.dual ( Cell_handle c) Returns the representative of the circumcenter of the four vertices of c that lies in the original domain domain.

Periodic_segment dt.dual ( Facet f) Returns the dual of facet f, which is a periodic segment.

Periodic_segment dt.dual ( Cell_handle c, int i) same as the previous method for facet (c,i).
Precondition: i ∈ {0,1,2,3}

template <class Stream>
Stream & dt.draw_dual ( Stream & os) Sends the set of duals to all the facets of dt into os.

Checking

bool dt.is_valid ( bool verbose = false)
Checks the combinatorial validity of the triangulation and the validity of its geometric embedding (see Section 37.2). Also checks that all the circumscribing spheres of cells are empty.
When verbose is set to true, messages describing the first invalidity encountered are printed.

bool dt.is_valid ( Cell_handle c, bool verbose = false)
Checks the combinatorial and geometric validity of the cell (see Section 37.2). Also checks that the circumscribing sphere of cells is empty.
When verbose is set to true, messages are printed to give a precise indication of the kind of invalidity encountered.

These methods are mainly a debugging help for the users of advanced features.

See Also

Periodic_3_triangulation_3

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CGAL Open Source Project. Release 3.5. 1 October 2009.