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CGAL 4.8 - 2D Periodic Triangulations

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Periodic_2DelaunayTriangulationTraits_2 Concept Reference
Concepts

## Definition

The concept `Periodic_2DelaunayTriangulationTraits_2` is the first template parameter of the class `Periodic_2_Delaunay_triangulation_2`. It refines the concept `Periodic_2TriangulationTraits_2` and `DelaunayTriangulationTraits_2` from the CGAL 2D Triangulation Reference package. It redefines the geometric objects, predicates and constructions to work with point-offset pairs. In most cases the offsets will be (0,0) and the predicates from `DelaunayTriangulationTraits_2` can be used directly. For efficiency reasons we maintain for each functor the version without offsets.

Refines:
  DelaunayTriangulationTraits_2 and Periodic_2TriangulationTraits_2

In addition to the requirements of the concept Periodic_2TriangulationTraits_2, the concept Periodic_2DelaunayTriangulationTraits_2 provides a predicate to check the empty circle property. The corresponding predicate type is called type Periodic_2DelaunayTriangulationTraits_2::Side_of_oriented_circle_2.

The additional constructor object Periodic_2DelaunayTriangulationTraits_2::Construct_circumcenter_2 is used to build the dual Voronoi diagram and are required only if the dual functions are called. The additional predicate type Periodic_2DelaunayTriangulationTraits_2::Compare_distance_2 is required if calls to nearest_vertex(..) are issued.

Has Models:

  `CGAL::Periodic_2_Delaunay_triangulation_traits_2<Traits, Offset>`

  `CGAL::Periodic_2_triangulation_traits_2<Traits, Offset>`, which implements additional the Delaunay predicates as well if the template parameter Traits is a model of `DelaunayTriangulationTraits_2`.

See Also
  `DelaunayTriangulationTraits_2`

## Types
typedef unspecified_type | Side_of_oriented_circle_2
| Predicate object. More...
typedef unspecified_type | Construct_circumcenter_2
| Constructor object. More...
typedef unspecified_type | Compare_distance_2
| Predicate type. More...

## Predicate functions
Side_of_oriented_circle_2 | side_of_oriented_circle_2_object ()
| Required only if `side_of_oriented_circle` is called called.
Construct_circumcenter_2 | construct_circumcenter_2_object ()
| Required only if `circumcenter` is called.
Compare_distance_2 | compare_distance_2_object ()
| Required only if `compare_distance` is called.

## Member Typedef Documentation

typedef unspecified_type Periodic_2DelaunayTriangulationTraits_2::Compare_distance_2

Predicate type.

Provides the operators:

`Comparison_result operator()(Point_2 p, Point_2 q, Point_2 r)` which returns `SMALLER`, `EQUAL` or `LARGER` according to the distance between p and q being smaller, equal or larger than the distance between p and r.

`Comparison_result operator()(Point_2 p, Point_2 q, Point_2 r, Periodic_2_offset_2 o_p, Periodic_2_offset_2 o_q Periodic_2_offset_2 o_r)` which returns `SMALLER`, `EQUAL` or `LARGER` according to the distance between `(p, o_p)`, and `(q, o_q)` being smaller, equal or larger than the distance between `(p, o_p)` and `(r, o_r)`.

This type is only require if `nearest_vertex` queries are issued.

typedef unspecified_type Periodic_2DelaunayTriangulationTraits_2::Construct_circumcenter_2

Constructor object.

Provides the operators:

`Point operator()(Point p, Point q, Point r)`

which returns the circumcenter of the three points `p, q` and `r`.

`Point operator()(Point p, Point q, Point r, Periodic_2_offset_2 o_p, Periodic_2_offset_2 o_q Periodic_2_offset_2 o_r)`

which returns the circumcenter of the three points `(p, o_p), (q, o_q)` and `(r, o_r)`.

This type is required only if the function `Point circumcenter(Face_handle f)`is called.

typedef unspecified_type Periodic_2DelaunayTriangulationTraits_2::Side_of_oriented_circle_2

Predicate object.

Must provide the operators

`Oriented_side operator()(Point p, Point q, Point r, Point s)` which takes four points \( p, q, r, s\) as arguments and returns `ON_POSITIVE_SIDE`, `ON_NEGATIVE_SIDE` or, `ON_ORIENTED_BOUNDARY` according to the position of points `s` with respect to the oriented circle through through \( p,q\) and \( r\) and

`Oriented_side operator()( Point p, Point q, Point r, Point s, Periodic_2_offset_2 o_p, Periodic_2_offset_2 o_q Periodic_2_offset_2 o_r, Periodic_2_offset_2 o_s)` which takes four points \( (p, o_p), (q, o_q), (r, o_r), (s, o_s)\) as arguments and returns `ON_POSITIVE_SIDE`, `ON_NEGATIVE_SIDE` or, `ON_ORIENTED_BOUNDARY` according to the position of points `(s, o_s)` with respect to the oriented circle through through `(p, o_p), (q, o_q)` and `(r, o_r)`.

This type is required only if the function `side_of_oriented_circle(Face_handle f, Point p)` is called.