\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.6 - 3D Triangulations
 All Classes Namespaces Files Functions Variables Typedefs Enumerations Enumerator Groups Pages
List of all members
CGAL::Regular_triangulation_euclidean_traits_3< K, Weight > Class Template Reference
Traits Classes

#include <CGAL/Regular_triangulation_euclidean_traits_3.h>

Inherits from

K.

Definition

The class Regular_triangulation_euclidean_traits_3 is designed as a default traits class for the class Regular_triangulation_3<RegularTriangulationTraits_3,TriangulationDataStructure_3>.

It provides Weighted_point_3, a class for weighted points, which derives from the three dimensional point class K::Point_3.

Template Parameters
Kmust be a model of the Kernel concept.
Weightis optional. If is it not provided, K::RT will be used.

The class is a model of the concept RegularTriangulationTraits_3 but it also contains predicates and constructors on weighted points that are not required in the concept RegularTriangulationTraits_3.

Note that filtered predicates are automatically used if the Boolean Has_filtered_predicates in the kernel provided as template parameter of that class is set to true.

Is Model Of:
RegularTriangulationTraits_3

Operations

The following functions give access to the predicate and constructor functors.

Examples:
Triangulation_3/parallel_insertion_and_removal_in_regular_3.cpp, and Triangulation_3/regular_3.cpp.

Types

typedef K::Point_3 Bare_point
 The type for point \( p\) of a weighted point \( {p}^{(w)}=(p,w_p)\).
 
typedef Weighted_point
< Bare_point, Weight > 		Weighted_point_3
 The type for weighted points.
 

Types for Predicate Functors

typedef unspecified_type Power_test_3
 A predicate type for power test. More...
 
typedef unspecified_type Compare_power_distance_3
 A predicate type to compare power distance. More...
 
typedef unspecified_type Compare_weighted_squared_radius_3
 A predicate type. More...
 
typedef unspecified_type In_smallest_orthogonal_sphere_3
 A predicate type. More...
 
typedef unspecified_type Side_of_bounded_orthogonal_sphere_3
 A predicate type. More...
 
typedef unspecified_type Does_simplex_intersect_dual_support_3
 A predicate type. More...
 

Types for Constructor Functors

typedef unspecified_type Construct_weighted_circumcenter_3
 A constructor type. More...
 
typedef unspecified_type Compute_power_product_3
 A functor type. More...
 
typedef unspecified_type Compute_squared_radius_smallest_orthogonal_sphere_3
 A functor type. More...
 
typedef unspecified_type Compute_critical_squared_radius_3
 A functor type. More...
 

Operations

Power_test_3 power_test_3_object ()
 
Compare_power_distance_3 compare_power_distance_3_object ()
 
Compare_weighted_squared_radius_3 compare_weighted_squared_radius_3_object ()
 
In_smallest_orthogonal_sphere_3 in_smallest_orthogonal_sphere_3_object ()
 
Side_of_bounded_orthogonal_sphere_3 side_of_bounded_orthogonal_sphere_3_object ()
 
Does_simplex_intersect_dual_support_3 does_simplex_intersect_dual_support_3_object ()
 
Construct_weighted_circumcenter_3 construct_weighted_circumcenter_3_object ()
 
Compute_power_product_3 compute_power_product_3_object ()
 
Compute_squared_radius_smallest_orthogonal_sphere_3 compute_squared_radius_smallest_orthogonal_sphere_3_object ()
 
Compute_critical_squared_radius_3 compute_critical_squared_radius_3_object ()
 

Member Typedef Documentation

template<typename K, typename Weight>
typedef unspecified_type CGAL::Regular_triangulation_euclidean_traits_3< K, Weight >::Compare_power_distance_3

A predicate type to compare power distance.

Belongs to the RegularTriangulationTraits_3 concept.

template<typename K, typename Weight>
typedef unspecified_type CGAL::Regular_triangulation_euclidean_traits_3< K, Weight >::Compare_weighted_squared_radius_3

A predicate type.

The operator() takes weighted point(s) as arguments, together with one weight. It compares the weight of the smallest sphere orthogonal to the weighted points with the input weight.

Comparison_result operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s, FT w) ;

Comparison_result operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, FT w) ;

Comparison_result operator()( Weighted_point_3 p, Weighted_point_3 q, FT w) ;

Comparison_result operator()( Weighted_point_3 p, FT w) ;

template<typename K, typename Weight>
typedef unspecified_type CGAL::Regular_triangulation_euclidean_traits_3< K, Weight >::Compute_critical_squared_radius_3

A functor type.

The operator() takes weighted points as arguments and computes the squared radius of the sphere centered in the last point and orthogonal to the other weighted points. The last argument is a weighted point but its weight does not matter. This construction is ad hoc for pumping slivers. For robustness issue, a predicate to compare critical squared radii for a given last point should be needed.

FT operator() ( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s, Weighted_point_3 t);

template<typename K, typename Weight>
typedef unspecified_type CGAL::Regular_triangulation_euclidean_traits_3< K, Weight >::Compute_power_product_3

A functor type.

The operator() computes the power distance between its arguments.

FT operator() ( Weighted_point_3 p, Weighted_point_3 q) ;

template<typename K, typename Weight>
typedef unspecified_type CGAL::Regular_triangulation_euclidean_traits_3< K, Weight >::Compute_squared_radius_smallest_orthogonal_sphere_3

A functor type.

The operator() computes the squared radius of the smallest sphere orthogonal to the argument(s).

FT operator() ( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s);

FT operator() ( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r);

FT operator() ( Weighted_point_3 p, Weighted_point_3 q);

FT operator() ( Weighted_point_3 p);

template<typename K, typename Weight>
typedef unspecified_type CGAL::Regular_triangulation_euclidean_traits_3< K, Weight >::Construct_weighted_circumcenter_3

A constructor type.

The operator() constructs the bare point which is the center of the smallest orthogonal sphere to the input weighted points.

Bare_point operator() ( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s);

Bare_point operator() ( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r);

Bare_point operator() ( Weighted_point_3 p, Weighted_point_3 q);

template<typename K, typename Weight>
typedef unspecified_type CGAL::Regular_triangulation_euclidean_traits_3< K, Weight >::Does_simplex_intersect_dual_support_3

A predicate type.

The operator() takes weighted points as arguments, considers the subspace of points with equal power distance with respect to its arguments and the intersection of this subspace with the affine hull of the bare points associated to the arguments. The operator() returns ON_BOUNDED_SIDE, ON_BOUNDARY or ON_UNBOUNDED_SIDE according to the position of this intersection with respect to the simplex formed by the bare points. This predicate is useful for flow computations.

Bounded_side operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s) ;

Bounded_side operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r) ;

Bounded_side operator()( Weighted_point_3 p, Weighted_point_3 q) ;

template<typename K, typename Weight>
typedef unspecified_type CGAL::Regular_triangulation_euclidean_traits_3< K, Weight >::In_smallest_orthogonal_sphere_3

A predicate type.

The operator() takes weighted points as arguments and returns the sign of the power distance of the last one with respect to the smallest sphere orthogonal to the others.

Sign operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s, Weighted_point_3 t) ;

Sign operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s) ;

Sign operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r) ;

Sign operator()( Weighted_point_3 p, Weighted_point_3 q) ;

template<typename K, typename Weight>
typedef unspecified_type CGAL::Regular_triangulation_euclidean_traits_3< K, Weight >::Power_test_3

A predicate type for power test.

Belongs to the RegularTriangulationTraits_3 concept.

template<typename K, typename Weight>
typedef unspecified_type CGAL::Regular_triangulation_euclidean_traits_3< K, Weight >::Side_of_bounded_orthogonal_sphere_3

A predicate type.

The operator() is similar to the operator() of In_smallest_orthogonal_sphere_3 except that the returned type is not a Sign but belongs to the enum Bounded_side (A NEGATIVE, ZERO and POSITIVE) corresponding respectively to ON_BOUNDED_SIDE, ON_BOUNDARY and ON_UNBOUNDED_SIDE)).

Bounded_side operator() ( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s, Weighted_point_3 t) ;

Bounded_side operator() ( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s) ;

Bounded_side operator() ( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r) ;