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 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.