RegularTriangulationTraits Concept Reference

Definition

This concept describes the geometric types and predicates required to build a regular triangulation. It corresponds to the first template parameter of the class CGAL::Regular_triangulation<RegularTriangulationTraits_, TriangulationDataStructure_>.

Refines:

TriangulationTraits

Has Models:

CGAL::Epick_d<Dim>

CGAL::Epeck_d<Dim>

See also

TriangulationTraits

Types
typedef unspecified_type	FT
	A number type that is a model for FieldNumberType.
typedef unspecified_type	Weighted_point_d
	The weighted point type.
typedef unspecified_type	Construct_point_d
	A function object that must provide the operator Point_d operator()(const Weighted_point_d & wp), returning wp without its weight.
typedef unspecified_type	Compute_weight_d
	A function object that must provide the operator FT operator()(const Weighted_point_d & wp), returning the weight of wp.
typedef unspecified_type	Power_side_of_power_sphere_d
	A predicate object that must provide the templated operator template<typename ForwardIterator> Oriented_side operator()(ForwardIterator start, ForwardIterator end, const Weighted_point_d & p). More...
typedef unspecified_type	In_flat_power_side_of_power_sphere_d
	A predicate object that must provide the templated operator template<typename ForwardIterator> Oriented_side operator()(Flat_orientation_d orient, ForwardIterator start, ForwardIterator end, const Weighted_point_d & p). More...

Creation
	RegularTriangulationTraits ()
	The default constructor (optional). More...

Operations
The following methods permit access to the traits class's predicates and functors:
Construct_point_d	construct_point_d_object () const
Compute_weight_d	compute_weight_d_object () const
Power_side_of_power_sphere_d	power_side_of_power_sphere_d_object () const
In_flat_power_side_of_power_sphere_d	in_flat_power_side_of_power_sphere_d_object () const

Member Typedef Documentation

In_flat_power_side_of_power_sphere_d

typedef unspecified_type RegularTriangulationTraits::In_flat_power_side_of_power_sphere_d

A predicate object that must provide the templated operator template<typename ForwardIterator> Oriented_side operator()(Flat_orientation_d orient, ForwardIterator start, ForwardIterator end, const Weighted_point_d & p).

The points in range [start,end) and p are supposed to belong to the lower-dimensional flat whose orientation is given by orient.

Let \( S \) be the power sphere of the weighted points in range [start,end) in this lower dimensional flat. The operator returns:

• ON_ORIENTED_BOUNDARY if p is orthogonal to \( S \),
• ON_NEGATIVE_SIDE if the power distance between p and \( S \) is positive.
• ON_POSITIVE_SIDE otherwise.

Precondition

std::distance(start,end)=k+1 where \( k\) is the number of points used to construct orient (dimension of the flat). The points in range [start,end) must be affinely independent. p must be in the flat generated by these points.

Power_side_of_power_sphere_d

typedef unspecified_type RegularTriangulationTraits::Power_side_of_power_sphere_d

A predicate object that must provide the templated operator template<typename ForwardIterator> Oriented_side operator()(ForwardIterator start, ForwardIterator end, const Weighted_point_d & p).

Let \( S \) be the power sphere of the weighted points in range [start,end). The operator returns:

• ON_ORIENTED_BOUNDARY if p is orthogonal to \( S \),
• ON_NEGATIVE_SIDE if the power distance between p and \( S \) is positive.
• ON_POSITIVE_SIDE otherwise.

Precondition

If Dimension is CGAL::Dimension_tag<D>, then std::distance(start,end)=D+1. The weighted points in range [start,end) must be affinely independent, i.e., the simplex must not be flat.

Constructor & Destructor Documentation

RegularTriangulationTraits()

RegularTriangulationTraits::RegularTriangulationTraits

(

)

The default constructor (optional).

This is not required when an instance of the traits is provided to the constructor of CGAL::Regular_triangulation.