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## 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

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

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 ()

### Has Models

CGAL::Exact_predicates_inexact_constructions_kernel (recommended)
CGAL::Exact_predicates_exact_constructions_kernel
CGAL::Filtered_kernel
CGAL::Cartesian
CGAL::Simple_cartesian
CGAL::Homogeneous
CGAL::Simple_homogeneous

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The CGAL Project . Tue, December 21, 2004 .