CGAL 6.1.1 - dD Triangulations

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DelaunayTriangulationTraits Concept Reference
Reference Manual » Concepts

## Definition

This concept describes the geometric types and predicates required to build a Delaunay triangulation. It corresponds to the first template parameter of the class `CGAL::Delaunay_triangulation<DelaunayTriangulationTraits_, TriangulationDataStructure_>`.

Refines
  `TriangulationTraits`

Has models
  `CGAL::Epick_d<Dim>`
  `CGAL::Epeck_d<Dim>`

See also
  `TriangulationTraits`

## Types
typedef unspecified_type | Side_of_oriented_sphere_d
| A predicate object that must provide the templated operator `template<typename ForwardIterator> Oriented_side operator()(ForwardIterator start, ForwardIterator end, const Point_d & p)`.
typedef unspecified_type | In_flat_side_of_oriented_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 Point_d & p)`.

## Creation
| DelaunayTriangulationTraits ()
| The default constructor (optional).

## Operations
The following methods permit access to the traits class's predicates and functors:
Side_of_oriented_sphere_d | side_of_oriented_sphere_d_object () const
In_flat_side_of_oriented_sphere_d | in_flat_side_of_oriented_sphere_d_object () const

## Member Typedef Documentation

## ◆ In_flat_side_of_oriented_sphere_d

typedef unspecified_type DelaunayTriangulationTraits::In_flat_side_of_oriented_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 Point_d & p)`.

The operator returns `ON_POSITIVE_SIDE`, `ON_NEGATIVE_SIDE` or `ON_ORIENTED_BOUNDARY` depending of the side of the query point `p` with respect to the sphere circumscribing the simplex defined by the points in range `[start,end)`. If the simplex is positively oriented according to `orient`, then the positive side of sphere corresponds geometrically to its bounded side. The points in range `[start,end)` and `p` are supposed to belong to the lower dimensional flat whose orientation is given by `orient`.

Precondition
  `std::distance(start,end)=k+1` where \( k\) is the number of points used to construct `orient`. The points in range `[start,end)` must be affinely independent, i.e., the simplex must not be flat. `p` must be in the flat generated by this simplex.

## ◆ Side_of_oriented_sphere_d

typedef unspecified_type DelaunayTriangulationTraits::Side_of_oriented_sphere_d

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

The operator returns `ON_POSITIVE_SIDE`, `ON_NEGATIVE_SIDE` or `ON_ORIENTED_BOUNDARY` depending of the side of the query point `p` with respect to the sphere circumscribing the simplex defined by the points in range `[start,end)`. If the simplex is positively oriented, then the positive side of sphere corresponds geometrically to its bounded side.

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

## Constructor & Destructor Documentation

## ◆ DelaunayTriangulationTraits()

DelaunayTriangulationTraits::DelaunayTriangulationTraits | ( |  | ) |

The default constructor (optional).

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