\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 3D Triangulations

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DelaunayTriangulationTraits_3 Concept Reference
Concepts

## 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)

  CGAL::Filtered_kernel

  CGAL::Cartesian

  CGAL::Simple_cartesian

  CGAL::Homogeneous

  CGAL::Simple_homogeneous

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...
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
The following functions give access to the predicate and construction objects:
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 ()

## Member Typedef Documentation

typedef unspecified_type DelaunayTriangulationTraits_3::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.

typedef unspecified_type DelaunayTriangulationTraits_3::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.

typedef unspecified_type DelaunayTriangulationTraits_3::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.

typedef unspecified_type DelaunayTriangulationTraits_3::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.

typedef unspecified_type DelaunayTriangulationTraits_3::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`.

typedef unspecified_type DelaunayTriangulationTraits_3::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.

typedef unspecified_type DelaunayTriangulationTraits_3::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.