CGAL 5.3 - 2D Triangulation
DelaunayTriangulationTraits_2 Concept Reference

## Definition

In addition to the requirements of the concept TriangulationTraits_2 the concept DelaunayTriangulationTraits_2 requires a predicate to check the empty circle property. The corresponding predicate type is called type Side_of_oriented_circle_2.

The additional types Line_2, Ray_2 and the constructor objects Construct_ray_2, Construct_circumcenter_2, Construct_bisector_2, Construct_midpoint are used to build the dual Voronoi diagram and are required only if the dual functions are called. The additional predicate type Compare_distance_2 is required if the method nearest_vertex() is used.

Refines:
TriangulationTraits_2
Has Models:

CGAL kernels

CGAL::Projection_traits_xy_3<K> (not for dual Voronoi functions)

CGAL::Projection_traits_yz_3<K> (not for dual Voronoi functions)

CGAL::Projection_traits_xz_3<K> (not for dual Voronoi functions)

TriangulationTraits_2

## Types

typedef unspecified_type Line_2
The line type. More...

typedef unspecified_type Ray_2
The type for ray. More...

typedef unspecified_type Side_of_oriented_circle_2
A function object to perform an incircle test for a point and three other points. More...

typedef unspecified_type Compare_distance_2
A function object to compare two distances for three points. More...

typedef unspecified_type Construct_circumcenter_2
A function object to construct the circumcenter of three points. More...

typedef unspecified_type Construct_bisector_2
A function object to construct the bisector of two points. More...

typedef unspecified_type Construct_ray_2
A function object to build a ray from a point and a line. More...

## Creation

DelaunayTriangulationTraits_2 ()
default constructor.

DelaunayTriangulationTraits_2 (DelaunayTriangulationTraits_2 dtt)
copy constructor

DelaunayTriangulationTraits_2 operator= (traits2)
Assignment operator.

Side_of_oriented_circle_2 side_of_oriented_circle_2_object ()

The following functions are required only if member functions of the Delaunay triangulation relative to the dual Voronoi diagram are called.

Compare_distance_2 compare_distance_2_object ()

Construct_circumcenter_2 construct_circumcenter_2_object ()

Construct_bisector_2 construct_bisector_2_object ()

Construct_ray_2 construct_ray_2_object ()

## ◆ Compare_distance_2

A function object to compare two distances for three points.

Provides the operator:

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

## ◆ Construct_bisector_2

A function object to construct the bisector of two points.

Provides the operator:

Line_2 operator()(Point_2 p, Point_2 q) which constructs the bisector line of points p and q. This type is required only if functions relative to the dual Voronoi diagram are called.

## ◆ Construct_circumcenter_2

A function object to construct the circumcenter of three points.

Provides the operator:

Point_2 operator()(Point_2 p, Point_2 q, Point_2 r) which returns the circumcenter of the three points p, q and r. This type is required only if functions relative to the dual Voronoi diagram are called.

## ◆ Construct_ray_2

A function object to build a ray from a point and a line.

Provides the operator:

Ray_2 operator() ( Point_2 p, Line_2 l);

## ◆ Line_2

The line type.

This type is required only if some dual functions are called.

## ◆ Ray_2

The type for ray.

This type is required only if some dual functions are called.

## ◆ Side_of_oriented_circle_2

A function object to perform an incircle test for a point and three other points.

Provides the operator:

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 p, q and r.