CGAL 4.7 - 2D Triangulation: RegularTriangulationTraits_2 Concept Reference

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CGAL 4.7 - 2D Triangulation

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RegularTriangulationTraits_2 Concept Reference
Concepts

## Definition

The concept `RegularTriangulationTraits_2` describe the requirements for the traits class of regular triangulations. It refines the concept `TriangulationTraits_2` requiring the type `CGAL::Weighted_point` 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:

  `CGAL::Regular_triangulation_euclidean_traits_2<K,Weight>`

  `CGAL::Regular_triangulation_filtered_traits_2<FK>`

See Also
  `TriangulationTraits_2`
  `CGAL::Regular_triangulation_2<Traits,Tds>`

## Types

typedef unspecified_type | Bare_point
| Another name for the point type.
typedef unspecified_type | Weighted_point_2
| The weighted point type. More...
typedef unspecified_type | Power_test_2
| A function object which must provide operators for the power test applied to two, three, and four points. More...

## 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 for computing two power distances. More...
typedef unspecified_type | Construct_weighted_circumcenter_2
| A function object which constructs the weighted circumcenter of three weighted points. More...
typedef unspecified_type | Construct_radical_axis_2
| A function object constructs the radical axis of two weighted points. More...

## Creation

| RegularTriangulationTraits_2 ()
| Default constructor.
| RegularTriangulationTraits_2 (const RegularTriangulatioTraits_2 &)
| Copy constructor.
RegularTriangulationTraits_2 & | operator= (const RegularTriangulationTraits_2 &)
| Assignment operator.

## Access to Predicate and Constructors Objects

Power_test_2 | power_test_2_object ()
Compare_power_distance_2 | compare_power_distance_2_object ()
Construct_weighted_circumcenter_2 | construct_weighted_circumcenter_2_object ()
Construct_radical_axis_2 | construct_radical_axis_2_object ()

## Member Typedef Documentation

typedef unspecified_type RegularTriangulationTraits_2::Compare_power_distance_2

A function object for computing two power distances.

Must provide the operator:

`Comparison_result operator()(Bare_point 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`.

typedef unspecified_type RegularTriangulationTraits_2::Construct_radical_axis_2

A function object constructs the radical axis of two weighted points.

Must provide the operator:

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

typedef unspecified_type RegularTriangulationTraits_2::Construct_weighted_circumcenter_2

A function object which constructs the weighted circumcenter of three weighted points.

Must provide the operator:

`Bare_point operator() ( Weighted_point_2 p, Weighted_point_2 q, Weighted_point_2 r);`

typedef unspecified_type RegularTriangulationTraits_2::Power_test_2

A function object which 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 degenerated 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 degenerated 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.

typedef unspecified_type RegularTriangulationTraits_2::Weighted_point_2

The weighted point type.

It has to be a model of the concept `WeightedPoint`.