CGAL 5.3.1 - 3D Triangulations
TriangulationTraits_3 Concept Reference

## Definition

Refines:
SpatialSortingTraits_3

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.

Has Models:
All models of Kernel.
CGAL::Triangulation_3

## Types

typedef unspecified_type Point_3
The point type. More...

typedef unspecified_type Segment_3
The segment type.

typedef unspecified_type Triangle_3
The triangle type.

typedef unspecified_type Tetrahedron_3
The tetrahedron type.

typedef unspecified_type Construct_point_3
A constructor object that must provide the function operator. More...

typedef unspecified_type Construct_segment_3
A constructor object that must provide the function operator. More...

typedef unspecified_type Construct_triangle_3
A constructor object that must provide the function operator. More...

typedef unspecified_type Construct_tetrahedron_3
A constructor object that must provide the function operator. More...

typedef unspecified_type Compare_xyz_3
A predicate object that must provide the function operator. More...

typedef unspecified_type Coplanar_orientation_3
A predicate object that must provide the function operator. More...

typedef unspecified_type Orientation_3
A predicate object that must provide the function operator. More...

## Creation

Triangulation_traits_3 ()
Default constructor.

Triangulation_traits_3 (const Triangulation_traits_3 &tr)
Copy constructor.

## Operations

Construct_point_3 construct_point_3_object ()

Construct_segment_3 construct_segment_3_object ()

Construct_triangle_3 construct_triangle_3_object ()

Construct_tetrahedron_3 construct_tetrahedron_3_object ()

Compare_xyz_3 compare_xyz_3_object ()

Coplanar_orientation_3 coplanar_orientation_3_object ()

Orientation_3 orientation_3_object ()

## ◆ Compare_xyz_3

A predicate object that must provide the function operator.

Comparison_result operator()(Point_3 p, Point_3 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.

## ◆ Construct_point_3

A constructor object that must provide the function operator.

Point_3 operator()(Point_3 p),

which simply returns p.

Note
It is advised to return a const reference to p to avoid useless copies.
This peculiar requirement is necessary because CGAL::Triangulation_3 internally manipulates points with a Point type that is not always Point_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.

## ◆ 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.

## ◆ 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.

## ◆ Coplanar_orientation_3

A predicate object that must provide the function operator.

Orientation operator()(Point_3 p, Point_3 q, Point_3 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.

## ◆ Orientation_3

A predicate object that must provide the function operator.

Orientation operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 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.

## ◆ Point_3

The point type.

It must be DefaultConstructible, CopyConstructible and Assignable.