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 the CGAL concept Kernel

See also

CGAL::Triangulation_3

Types
typedef unspecified_type	Point_3
	The point type.
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.
typedef unspecified_type	Construct_segment_3
	A constructor object that must provide the function operator.
typedef unspecified_type	Construct_triangle_3
	A constructor object that must provide the function operator.
typedef unspecified_type	Construct_tetrahedron_3
	A constructor object that must provide the function operator.
typedef unspecified_type	Compare_xyz_3
	A predicate object that must provide the function operator.
typedef unspecified_type	Coplanar_orientation_3
	A predicate object that must provide the function operator.
typedef unspecified_type	Orientation_3
	A predicate object that must provide the function operator.

Creation
	Triangulation_traits_3 ()
	Default constructor.
	Triangulation_traits_3 (const Triangulation_traits_3 &tr)
	Copy constructor.

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

Member Typedef Documentation

Compare_xyz_3

typedef unspecified_type TriangulationTraits_3::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

typedef unspecified_type TriangulationTraits_3::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

typedef unspecified_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.

Construct_tetrahedron_3

typedef unspecified_type 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.

Construct_triangle_3

typedef unspecified_type 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.

Coplanar_orientation_3

typedef unspecified_type TriangulationTraits_3::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

typedef unspecified_type TriangulationTraits_3::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

typedef unspecified_type TriangulationTraits_3::Point_3

The point type.

It must be DefaultConstructible, CopyConstructible and Assignable.