## 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.

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.