CGAL 4.12.1 - 3D Triangulations
DelaunayTriangulationTraits_3 Concept Reference

## Definition

The concept DelaunayTriangulationTraits_3 is the first template parameter of the class Delaunay_triangulation_3. It defines the geometric objects (points, segments...) forming the triangulation together with a few geometric predicates and constructions on these objects.

Refines:
TriangulationTraits_3
Has Models:

CGAL::Exact_predicates_inexact_constructions_kernel (recommended)

CGAL::Exact_predicates_exact_constructions_kernel (recommended for Voronoi)

In addition to the requirements described for the traits class of CGAL::Triangulation_3, the geometric traits class of a Delaunay triangulation must fulfill the following requirements:

## Types

typedef unspecified_type Line_3
The line type.

typedef unspecified_type Object_3
The object type.

typedef unspecified_type Ray_3
The ray type.

typedef unspecified_type Coplanar_side_of_bounded_circle_3
A predicate object that must provide the function operator. More...

typedef unspecified_type Side_of_oriented_sphere_3
A predicate object that must provide the function operator. More...

typedef unspecified_type Compare_distance_3
A predicate object that must provide the function operator. More...

When is_Gabriel functions are used, the traits class must in addition provide the following predicate object:

typedef unspecified_type Side_of_bounded_sphere_3
A predicate object that must provide the function operators. More...

In addition, only when the dual operations are used, the traits class must provide the following constructor objects:

typedef unspecified_type Construct_circumcenter_3
A constructor object that must provide the function operator. More...

typedef unspecified_type Construct_object_3
A constructor object that must provide the function operators. More...

typedef unspecified_type Construct_equidistant_line_3
A constructor object that must provide the function operator. More...

typedef unspecified_type Construct_ray_3
A constructor object that must provide the function operator. More...

## Operations

Coplanar_side_of_bounded_circle_3 coplanar_side_of_bounded_circle_3_object ()

Side_of_oriented_sphere_3 side_of_oriented_sphere_3_object ()

When using the Fast_location policy or the CGAL::Delaunay_triangulation_3::nearest_vertex() function, the traits must provide:

Compare_distance_3 compare_distance_3_object ()

The following functions must be provided only if the methods of Delaunay_triangulation_3 returning elements of the Voronoi diagram are instantiated:

Construct_circumcenter_3 construct_circumcenter_3_object ()

Construct_object_3 construct_object_3_object ()

Construct_equidistant_line_3 construct_equidistant_line_object ()

Construct_ray_3 construct_ray_3_object ()

## ◆ Compare_distance_3

A predicate object that must provide the function operator.

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

which compares the distance between p and q to the distance between p and r.

It is only needed when using the Fast_location policy or the nearest_vertex function.

## ◆ Construct_circumcenter_3

A constructor object that must provide the function operator.

Point_3 operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s),

which constructs the circumcenter of four points.

Precondition
p, q, r and s must be non coplanar.

It must also provide the function operator

Point_3 operator()(Point_3 p, Point_3 q, Point_3 r),

which constructs the circumcenter of three points.

Precondition
p, q and r must be non collinear.

## ◆ Construct_equidistant_line_3

A constructor object that must provide the function operator.

Line_3 operator()(Point_3 p1, Point_3 p2, Point_3 p3),

which constructs the line which is at the same distance from the three points.

Precondition
p1, p2 and p3 must be non collinear.

## ◆ Construct_object_3

A constructor object that must provide the function operators.

Object_3 operator()(Point_3 p),

Object_3 operator()(Segment_3 s) and

Object_3 operator()(Ray_3 r)

that construct an object respectively from a point, a segment and a ray.

## ◆ Construct_ray_3

A constructor object that must provide the function operator.

Ray_3 operator()(Point_3 p, Line_3 l),

which constructs the ray starting at p with direction given by l.

## ◆ Coplanar_side_of_bounded_circle_3

A predicate object that must provide the function operator.

Bounded_side operator()(Point p, Point q, Point r, Point s),

which determines the bounded side of the circle defined by p, q, and r on which s lies.

Precondition
p, q, r, and s are coplanar and p, q, and r are not collinear.

## ◆ Side_of_bounded_sphere_3

A predicate object that must provide the function operators.

Bounded_side operator()(Point_3 p, Point_3 q, Point_3 t),

which returns the position of the point t relative to the sphere that has pq as its diameter,

Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 t),

which returns the position of the point t relative to the sphere passing through p, q, and r and whose center is in the plane defined by these three points.

## ◆ Side_of_oriented_sphere_3

A predicate object that must provide the function operator.

Oriented_side operator()(Point p, Point q, Point r, Point s, Point t),

which determines on which side of the oriented sphere circumscribing p, q, r, s the point t lies.