## TriangulationTraits_3

### Definition

The concept TriangulationTraits_3 is the first template parameter of the class Triangulation_3. It defines the geometric objects (points, segments, triangles and tetrahedra) forming the triangulation together with a few geometric predicates and constructions on these objects : lexicographical comparison, orientation in case of coplanar points and orientation in space.

### Types

 TriangulationTraits_3::Point_3 The point type. It must be DefaultConstructible, CopyConstructible and Assignable. TriangulationTraits_3::Segment_3 The segment type. TriangulationTraits_3::Tetrahedron_3 The tetrahedron type. TriangulationTraits_3::Triangle_3 The triangle type. TriangulationTraits_3::Construct_segment_3 A constructor object that must provide the function operator Segment_3 operator()(Point_3 p, Point_3 q), which constructs a segment from two points. TriangulationTraits_3::Construct_triangle_3 A constructor object that must provide the function operator Triangle_3 operator()(Point_3 p, Point_3 q, Point_3 r ), which constructs a triangle from three points. TriangulationTraits_3::Construct_tetrahedron_3 A constructor object that must provide the function operator Tetrahedron_3 operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s), which constructs a tetrahedron from four points. TriangulationTraits_3::Compare_xyz_3 A predicate object that must provide the function operator Comparison_result operator()(Point p, Point q), which returns EQUAL if the two points are equal. Otherwise it must return a consistent order for any two points chosen in a same line. TriangulationTraits_3::Coplanar_orientation_3 A predicate object that must provide the function operator Orientation operator()(Point p, Point q, Point r), which returns COLLINEAR if the points are collinear. Otherwise it must return a consistent orientation for any three points chosen in a same plane. TriangulationTraits_3::Orientation_3 A predicate object that must provide the function operator Orientation operator()(Point p, Point q, Point r, Point s), which returns POSITIVE, if s lies on the positive side of the oriented plane h defined by p, q, and r, returns NEGATIVE if s lies on the negative side of h, and returns COPLANAR if s lies on h.

### Creation

 TriangulationTraits_3 traits; Default constructor. TriangulationTraits_3 traits ( Triangulation_traits_3 tr); Copy constructor.

### Operations

 Construct_tetrahedron_3 traits.construct_tetrahedron_3_object () Construct_triangle_3 traits.construct_triangle_3_object () Construct_segment_3 traits.construct_segment_3_object () Compare_xyz_3 traits.compare_xyz_3_object () Coplanar_orientation_3 traits.coplanar_orientation_3_object () Orientation_3 traits.orientation_3_object ()