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CGAL 4.8 - dD Convex Hulls and Delaunay Triangulations

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DelaunayLiftedTraits_d Concept Reference
Concepts

## Definition

Deprecated:
  This package is deprecated since the version 4.6 of CGAL. The package dD Triangulations should be used instead.

Requirements of the second traits class to be used with the class `CGAL::Delaunay_d`.

Has Models:

  `CGAL::Cartesian_d<FT,LA>`

  `CGAL::Homogeneous_d<RT,LA>`

## Types
typedef unspecified_type | Point_d
| the dD point type on which the Delaunay algorithm operates
typedef unspecified_type | Hyperplane_d
| a dD plane
typedef unspecified_type | Vector_d
| a dD vector
typedef unspecified_type | Ray_d
| a dD ray
typedef unspecified_type | RT
| a arithmetic ring type
typedef unspecified_type | Construct_vector_d
| Function object type that provides `Vector_d operator()(int d, CGAL::Null_vector)`, which constructs and returns the null vector.
typedef unspecified_type | Construct_hyperplane_d
| Function object type that provides `Hyperplane_d operator()(ForwardIterator first, ForwardIterator last, Point_d p, CGAL::Oriented_side side)`, which constructs and returns a hyperplane passing through the points in `tuple[first,last)` and oriented such that `p` is on the side `side` of the returned hyperplane. More...
typedef unspecified_type | Vector_to_point_d
| Function object type that provides `Point_d operator()(Vector_d v)`, which constructs and returns the point defined by \( 0+v\).
typedef unspecified_type | Point_to_vector_d
| Function object type that provides `Vector_d operator()(Point_d v)`, which constructs and returns the vector defined by \( p-0\).
typedef unspecified_type | Orientation_d
| Function object type that provides `Orientation operator()(ForwardIterator first, ForwardIterator last)`, which determines the orientation of the points `tuple[first,last)`.
typedef unspecified_type | Orthogonal_vector_d
| Function object type that provides `Vector_d operator()(Hyperplane_d h)`, which constructs and returns a vector orthogonal to `h` and pointing from the boundary into its positive halfspace.
typedef unspecified_type | Oriented_side_d
| Predicate object type that provides `Oriented_side operator()(Hyperplane_d h, Point_d p)`, which determines the oriented side of `p` with respect to `h`.
typedef unspecified_type | Has_on_positive_side_d
| Predicate object type that provides `bool operator()(Hyperplane_d h, Point_d p)`, which return true iff `p` lies in the positive halfspace determined by `h`.
typedef unspecified_type | Affinely_independent_d
| Predicate object type that provides `bool operator()(ForwardIterator first, ForwardIterator last)`, which determines if the points `tuple[first,last)` are affinely independent.
typedef unspecified_type | Contained_in_simplex_d
| Predicate object type that provides `bool operator()(ForwardIterator first, ForwardIterator last, Point_d p)`, which determines if `p` is contained in the closed simplex defined by the points in `tuple[first,last)`.
typedef unspecified_type | Contained_in_affined_hull_d
| Predicate object type that provides `bool operator()(ForwardIterator first, ForwardIterator last, Point_d p)`, which determines if `p` is contained in the affine hull of the points in `tuple[first,last)`.
typedef unspecified_type | Intersect_d
| Predicate object type that provides `Object operator()(Ray_d r, Hyperplane_d h)`, which determines if `r` and `h` intersect and returns the corresponding polymorphic object.
The previous requirements are all identical to the requirements of concept `ConvexHullTraits_d`. The Delaunay class adds the following requirements.
typedef unspecified_type | Project_along_d_axis_d
| Predicate object type that provides `DelaunayTraits_d::Point_d operator()(Point_d p)`, which determines the \( d-1\)-dimensional point from the \( d\)-dimensional point \( p\) while discarding the last coordinate.
typedef unspecified_type | Lift_to_paraboloid_d
| Predicate object type that provides `Point_d operator()(DelaunayTraits_d::Point_d p)`, which determines the \( d\)-dimensional point from the \( d-1\)-dimensional point \( p\) while lifting it to the paraboloid of revolution.
typedef unspecified_type | Component_accessor_d
| Predicate object type that provides `RT homogeneous(Vector_d v,int i)` and `int dimension(Vector_d v)`, where the former determines the \( i\)th coordinate of \( v\) and the latter the dimension of \( v\).

## Creation
| DelaunayLiftedTraits_d ()
| A default constructor and copy constructor is required.

## Operations
For each of the above function and predicate object types, `Func_obj_type`, a function must exist with the name `func_obj_type_object` that creates an instance of the function or predicate object type. For example:
Construct_vector_d | construct_vector_d_object ()

## Member Typedef Documentation

typedef unspecified_type DelaunayLiftedTraits_d::Construct_hyperplane_d

Function object type that provides `Hyperplane_d operator()(ForwardIterator first, ForwardIterator last, Point_d p, CGAL::Oriented_side side)`, which constructs and returns a hyperplane passing through the points in `tuple[first,last)` and oriented such that `p` is on the side `side` of the returned hyperplane.

When `side==ON_ORIENTED_BOUNDARY` then any hyperplane containing the tuple is returned.