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CGAL 4.5 - dD Convex Hulls and Delaunay Triangulations
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DelaunayLiftedTraits_d Concept Reference

Definition

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

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.