RegularTriangulationTraits_2 Concept Reference Definition The concept RegularTriangulationTraits_2 describes the requirements for the traits class of regular triangulations. It refines the concept TriangulationTraits_2 requiring the type CGAL::Weighted_point_2 and the power test predicate on those weighted points. A weighted point is basically a point augmented with a scalar weight. It can be seen as a circle when the weight is interpreted as a square radius. The power test on weighted points is the fundamental test to build regular triangulations as the side_of_oriented_circle test is the fundamental test of Delaunay triangulations. Refines TriangulationTraits_2 Has models All models of the CGAL concept Kernel See alsoCGAL::Regular_triangulation_2 Types typedef unspecified_type Weighted_point_2 The weighted point type. typedef unspecified_type Power_side_of_oriented_power_circle_2 A predicate object, model of PowerSideOfOrientedPowerCircle_2, that must provide operators for the power test applied to two, three, and four points. typedef unspecified_type Construct_point_2 A constructor type, model of Kernel::ConstructPoint_2. Optional Types The following type/predicate is required for a call to nearest_power_vertex(): typedef unspecified_type Compare_power_distance_2 A function object, model of Kernel::ComparePowerDistance_2 for comparing two power distances. The following types/predicates are required to compute duals: typedef unspecified_type Construct_weighted_circumcenter_2 A function object, model of Kernel::ConstructWeightedCircumcenter_2, that constructs the weighted circumcenter of three weighted points. typedef unspecified_type Construct_radical_axis_2 A function object, model of Kernel::ConstructRadicalAxis_2, that constructs the radical axis of two weighted points. Creation RegularTriangulationTraits_2 () Default constructor. RegularTriangulationTraits_2 (const RegularTriangulationTraits_2 &) Copy constructor. RegularTriangulationTraits_2 & operator= (const RegularTriangulationTraits_2 &) Assignment operator. Access to Predicate and Constructors Objects Power_side_of_oriented_power_circle_2 power_side_of_oriented_power_circle_2_object () Compare_power_distance_2 compare_power_distance_2_object () Construct_point_2 construct_point_2_object () Construct_weighted_circumcenter_2 construct_weighted_circumcenter_2_object () Construct_radical_axis_2 construct_radical_axis_2_object () Member Typedef Documentation Compare_power_distance_2 typedef unspecified_type RegularTriangulationTraits_2::Compare_power_distance_2 A function object, model of Kernel::ComparePowerDistance_2 for comparing two power distances. Must provide the operator: Comparison_result operator()(Point_2 p, Weighted_point_2 q, Weighted_point_2 r), which compares the power distance between p and q to the power distance between p and r. Construct_point_2 typedef unspecified_type RegularTriangulationTraits_2::Construct_point_2 A constructor type, model of Kernel::ConstructPoint_2. The operator() extracts the bare point from a weighted point. Point_2 operator() ( Weighted_point_2 p); Construct_radical_axis_2 typedef unspecified_type RegularTriangulationTraits_2::Construct_radical_axis_2 A function object, model of Kernel::ConstructRadicalAxis_2, that constructs the radical axis of two weighted points. Must provide the operator: Line_2 operator() ( Weighted_point_2 p, Weighted_point_2 q); Construct_weighted_circumcenter_2 typedef unspecified_type RegularTriangulationTraits_2::Construct_weighted_circumcenter_2 A function object, model of Kernel::ConstructWeightedCircumcenter_2, that constructs the weighted circumcenter of three weighted points. Must provide the operator: Point_2 operator() ( Weighted_point_2 p, Weighted_point_2 q, Weighted_point_2 r); Power_side_of_oriented_power_circle_2 typedef unspecified_type RegularTriangulationTraits_2::Power_side_of_oriented_power_circle_2 A predicate object, model of PowerSideOfOrientedPowerCircle_2, that must provide operators for the power test applied to two, three, and four points. Must provide the operators: Oriented_side operator() ( Weighted_point_2 p, Weighted_point_2 q, Weighted_point_2 r, Weighted_point_2 s) which is the power test for points p, q, r and s. Preconditionthe bare points corresponding to p, q, r are not collinear. Oriented_side operator() ( Weighted_point_2 p, Weighted_point_2 q, Weighted_point_2 r) which is the degenerate power test for collinear points p, q, r. Preconditionthe bare points corresponding to p, q, r are collinear and p != q. Oriented_side operator() ( Weighted_point_2 p, Weighted_point_2 q) which is the degenerate power test for weighted points p and q whose corresponding bare-points are identical. Preconditionthe bare points corresponding to p and q are identical. Weighted_point_2 typedef unspecified_type RegularTriangulationTraits_2::Weighted_point_2 The weighted point type. It has to be a model of the concept Kernel::WeightedPoint_2. NoteThe unweighted point type Point_2 is requested by the concept TriangulationTraits_2, which this concept refines. RegularTriangulationTraits_2 Generated by 1.17.0

Definition

The concept RegularTriangulationTraits_2 describes the requirements for the traits class of regular triangulations. It refines the concept TriangulationTraits_2 requiring the type CGAL::Weighted_point_2 and the power test predicate on those weighted points. A weighted point is basically a point augmented with a scalar weight. It can be seen as a circle when the weight is interpreted as a square radius. The power test on weighted points is the fundamental test to build regular triangulations as the side_of_oriented_circle test is the fundamental test of Delaunay triangulations.

Refines

TriangulationTraits_2

Has models

All models of the CGAL concept Kernel

See also

CGAL::Regular_triangulation_2

Types
typedef unspecified_type	Weighted_point_2
	The weighted point type.
typedef unspecified_type	Power_side_of_oriented_power_circle_2
	A predicate object, model of PowerSideOfOrientedPowerCircle_2, that must provide operators for the power test applied to two, three, and four points.
typedef unspecified_type	Construct_point_2
	A constructor type, model of Kernel::ConstructPoint_2.

Optional Types
The following type/predicate is required for a call to nearest_power_vertex():
typedef unspecified_type	Compare_power_distance_2
	A function object, model of Kernel::ComparePowerDistance_2 for comparing two power distances.
The following types/predicates are required to compute duals:
typedef unspecified_type	Construct_weighted_circumcenter_2
	A function object, model of Kernel::ConstructWeightedCircumcenter_2, that constructs the weighted circumcenter of three weighted points.
typedef unspecified_type	Construct_radical_axis_2
	A function object, model of Kernel::ConstructRadicalAxis_2, that constructs the radical axis of two weighted points.

Creation
	RegularTriangulationTraits_2 ()
	Default constructor.
	RegularTriangulationTraits_2 (const RegularTriangulationTraits_2 &)
	Copy constructor.
RegularTriangulationTraits_2 &	operator= (const RegularTriangulationTraits_2 &)
	Assignment operator.

Access to Predicate and Constructors Objects
Power_side_of_oriented_power_circle_2	power_side_of_oriented_power_circle_2_object ()
Compare_power_distance_2	compare_power_distance_2_object ()
Construct_point_2	construct_point_2_object ()
Construct_weighted_circumcenter_2	construct_weighted_circumcenter_2_object ()
Construct_radical_axis_2	construct_radical_axis_2_object ()

Member Typedef Documentation

Compare_power_distance_2

typedef unspecified_type RegularTriangulationTraits_2::Compare_power_distance_2

A function object, model of Kernel::ComparePowerDistance_2 for comparing two power distances.

Must provide the operator:

Comparison_result operator()(Point_2 p, Weighted_point_2 q, Weighted_point_2 r), which compares the power distance between p and q to the power distance between p and r.

Construct_point_2

typedef unspecified_type RegularTriangulationTraits_2::Construct_point_2

A constructor type, model of Kernel::ConstructPoint_2.

The operator() extracts the bare point from a weighted point.

Point_2 operator() ( Weighted_point_2 p);

Construct_radical_axis_2

typedef unspecified_type RegularTriangulationTraits_2::Construct_radical_axis_2

A function object, model of Kernel::ConstructRadicalAxis_2, that constructs the radical axis of two weighted points.

Must provide the operator:

Line_2 operator() ( Weighted_point_2 p, Weighted_point_2 q);

Construct_weighted_circumcenter_2

typedef unspecified_type RegularTriangulationTraits_2::Construct_weighted_circumcenter_2

A function object, model of Kernel::ConstructWeightedCircumcenter_2, that constructs the weighted circumcenter of three weighted points.

Must provide the operator:

Point_2 operator() ( Weighted_point_2 p, Weighted_point_2 q, Weighted_point_2 r);

Power_side_of_oriented_power_circle_2

typedef unspecified_type RegularTriangulationTraits_2::Power_side_of_oriented_power_circle_2

A predicate object, model of PowerSideOfOrientedPowerCircle_2, that must provide operators for the power test applied to two, three, and four points.

Must provide the operators:

• Oriented_side operator() ( Weighted_point_2 p, Weighted_point_2 q, Weighted_point_2 r, Weighted_point_2 s) which is the power test for points p, q, r and s.

  Precondition

  the bare points corresponding to p, q, r are not collinear.

• Oriented_side operator() ( Weighted_point_2 p, Weighted_point_2 q, Weighted_point_2 r) which is the degenerate power test for collinear points p, q, r.

  Precondition

  the bare points corresponding to p, q, r are collinear and p != q.

• Oriented_side operator() ( Weighted_point_2 p, Weighted_point_2 q) which is the degenerate power test for weighted points p and q whose corresponding bare-points are identical.

  Precondition

  the bare points corresponding to p and q are identical.

Weighted_point_2

typedef unspecified_type RegularTriangulationTraits_2::Weighted_point_2

The weighted point type.

It has to be a model of the concept Kernel::WeightedPoint_2.

Note

The unweighted point type Point_2 is requested by the concept TriangulationTraits_2, which this concept refines.