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

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.