## DelaunayTriangulationTraits_3

### 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

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

### Types

DelaunayTriangulationTraits_3::Line_3
The line type.

DelaunayTriangulationTraits_3::Object_3
The object type.

DelaunayTriangulationTraits_3::Ray_3
The ray 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.

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.

The following additional predicate is required only when the Triangulation_hierarchy_3 is used on top of Delaunay_triangulation_3:

 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.

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

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.

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.

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.

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.

### Operations

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

 Coplanar_side_of_bounded_circle_3 traits.coplanar_side_of_bounded_circle_3_object () Side_of_oriented_sphere_3 traits.side_of_oriented_sphere_3_object ()

When using the triangulation hierarchy, the traits must provide:

 Compare_distance_3 traits.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 traits.construct_circumcenter_3_object () Construct_object_3 traits.construct_object_3_object () Construct_perpendicular_line_3 traits.construct_perpendicular_line_object () Construct_plane_3 traits.construct_plane_3_object () Construct_ray_3 traits.construct_ray_3_object ()