\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.5.2 - 3D Periodic Triangulations
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Periodic_3DelaunayTriangulationTraits_3 Concept Reference
Concepts

Definition

The concept Periodic_3DelaunayTriangulationTraits_3 is the first template parameter of the classes Periodic_3_Delaunay_triangulation_3 and Periodic_3_triangulation_3. It refines the concept DelaunayTriangulationTraits_3 from the CGAL 3D Triangulations Reference package. It redefines the geometric objects, predicates and constructions to work with point-offset pairs. In most cases the offsets will be (0,0,0) and the predicates from DelaunayTriangulationTraits_3 can be used directly. For efficiency reasons we maintain for each functor the version without offsets.

Refines:
DelaunayTriangulationTraits_3
Has Models:
CGAL::Periodic_3_triangulation_traits_3
See Also
DelaunayTriangulationTraits_3

In addition to the requirements described for the traits class DelaunayTriangulationTraits_3, the geometric traits class of a Periodic Delaunay triangulation must fulfill the following requirements.

Note
The optional types must be provided in any case, however they can be replaced by dummy types if the respective functions are not used.

Types

typedef unspecified_type Point_3
 The point type. More...
 
typedef unspecified_type Vector_3
 The vector type. More...
 
typedef unspecified_type Periodic_3_offset_3
 The offset type. More...
 
typedef unspecified_type Iso_cuboid_3
 A type representing an axis-aligned cuboid. More...
 

The following three types represent geometric primitives in \( \mathbb R^3\).

They are required to provide functions converting primitives from \( \mathbb T_c^3\) to \( \mathbb R^3\), i.e. constructing representatives in \( \mathbb R^3\).

typedef unspecified_type Segment_3
 A segment type. More...
 
typedef unspecified_type Triangle_3
 A triangle type. More...
 
typedef unspecified_type Tetrahedron_3
 A tetrahedron type. More...
 
typedef unspecified_type Compare_xyz_3
 A predicate object that must provide the function operators. More...
 
typedef unspecified_type Orientation_3
 A predicate object that must provide the function operators. More...
 
typedef unspecified_type Side_of_oriented_sphere_3
 A predicate object that must provide the function operators. More...
 
typedef unspecified_type Compare_distance_3
 A predicate object that must provide the function operators. More...
 

In addition, only when vertex removal is used, the traits class must provide the following predicate objects

typedef unspecified_type Coplanar_orientation_3
 A predicate object that must provide the function operators. More...
 
typedef unspecified_type Coplanar_side_of_bounded_circle_3
 A predicate object that must provide the function operators. More...
 

In addition, only when is_Gabriel is used, the traits class must provide the following predicate object:

typedef unspecified_type Side_of_bounded_sphere_3
 A predicate object that must provide the function operators. More...
 

Note that the traits must provide exact constructions in order to guarantee exactness of the following construction functors.

typedef unspecified_type Construct_point_3
 A constructor object that must provide the function operator. More...
 
typedef unspecified_type Construct_segment_3
 A constructor object that must provide the function operators. More...
 
typedef unspecified_type Construct_triangle_3
 A constructor object that must provide the function operators. More...
 
typedef unspecified_type Construct_tetrahedron_3
 A constructor object that must provide the function operators. More...
 

In addition, only when the dual operations are used, the traits class must provide the following constructor object:

typedef unspecified_type Construct_circumcenter_3
 A constructor object that must provide the function operators. More...
 

Creation

 Periodic_3_triangulation_traits_3 ()
 Default constructor.
 
 Periodic_3_triangulation_traits_3 (const Periodic_triangulation_traits_3 &tr)
 Copy constructor.
 

Access Functions

void set_domain (Iso_cuboid_3 domain)
 Set the size of the fundamental domain. More...
 

Operations

The following functions give access to the predicate and construction objects:

Compare_xyz_3 compare_xyz_3_object ()
 
Orientation_3 orientation_3_object ()
 
Side_of_oriented_sphere_3 side_of_oriented_sphere_3_object ()
 
Compare_distance_3 compare_distance_3_object ()
 

The following functions must be provided if vertex removal is used; otherwise dummy functions can be provided.

Coplanar_orientation_3 coplanar_3_orientation_3_object ()
 
Coplanar_side_of_bounded_circle_3 coplanar_side_of_bounded_circle_3_object ()
 

The following function must be provided only if the is_Gabriel methods of Periodic_3_Delaunay_triangulation_3 are used; otherwise a dummy function can be provided.

Side_of_bounded_sphere_3 side_of_bounded_sphere_3_object ()
 
Construct_segment_3 construct_segment_3_object ()
 
Construct_triangle_3 construct_triangle_3_object ()
 
Construct_tetrahedron_3 construct_tetrahedron_3_object ()
 

The following function must be provided only if the methods of Periodic_3_Delaunay_triangulation_3 returning elements of the Voronoi diagram are used; otherwise a dummy function can be provided:

Construct_circumcenter_3 construct_circumcenter_3_object ()
 

Member Typedef Documentation

typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Compare_distance_3

A predicate object that must provide the function operators.

Comparison_result operator()(Point_3 p, Point_3 q, Point_3 r),

which compares the distance between p and q to the distance between p and r and

Comparison_result operator()(Point_3 p, Point_3 q, Point_3 r, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r),

which compares the distance between (p,o_p) and (q,o_q) to the distance between (p,o_p) and (r,o_r).

Precondition
p, q, r lie inside the domain.
typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Compare_xyz_3

A predicate object that must provide the function operators.

Comparison_result operator()(Point_3 p, Point_3 q),

which returns EQUAL if the two points are equal and

Comparison_result operator()(Point_3 p, Point_3 q, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q),

which returns EQUAL if the two point-offset pairs are equal. Otherwise it must return a consistent order for any two points chosen in a same line.

Precondition
p, q lie inside the domain.
typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Construct_circumcenter_3

A constructor object that must provide the function operators.

Point_3 operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s),

which constructs the circumcenter of four points and

Point_3 operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s),

which constructs the circumcenter of four point-offset pairs.

Precondition
p, q, r and s as well as (p,o_p), (q,o_q), (r,o_r) and (s,o_s) must be non coplanar. p, q, r, s lie inside the domain.
typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Construct_point_3

A constructor object that must provide the function operator.

Point_3 operator()(Point_3 p, Periodic_3_offset_3 o_p),

which constructs a point from a point-offset pair.

Precondition
p lies inside the domain.
typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Construct_segment_3

A constructor object that must provide the function operators.

Segment_3 operator()(Point_3 p, Point_3 q),

which constructs a segment from two points and

Segment_3 operator()(Point_3 p, Point_3 q, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q),

which constructs a segment from two point-offset pairs.

Precondition
p, q lie inside the domain.
typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Construct_tetrahedron_3

A constructor object that must provide the function operators.

Tetrahedron_3 operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s),

which constructs a tetrahedron from four points and

Tetrahedron_3 operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s),

which constructs a tetrahedron from four point-offset pairs.

Precondition
p, q, r, s lie inside the domain.
typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Construct_triangle_3

A constructor object that must provide the function operators.

Triangle_3 operator()(Point_3 p, Point_3 q, Point_3 r ),

which constructs a triangle from three points and

Triangle_3 operator()(Point_3 p, Point_3 q, Point_3 r, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r),

which constructs a triangle from three point-offset pairs.

Precondition
p, q, r lie inside the domain.
typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Coplanar_orientation_3

A predicate object that must provide the function operators.

Orientation operator()(Point_3 p, Point_3 q, Point_3 r),

which returns COLLINEAR, if the points are collinear; otherwise it must return a consistent orientation for any three points chosen in a same plane and

Orientation operator()(Point_3 p, Point_3 q, Point_3 r Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r),

which returns COLLINEAR, if the point-offset pairs are collinear; otherwise it must return a consistent orientation for any three point-offset pairs chosen in a same plane.

Precondition
p, q, r lie inside the domain.
typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Coplanar_side_of_bounded_circle_3

A predicate object that must provide the function operators.

Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s),

which determines the bounded side of the circle defined by p, q, and r on which the point s lies and

Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s),

which determines the bounded side of the circle defined by (p,o_p), (q,o_q), and (r,o_r) on which the point-offset pair (s,o_s) lies.

Precondition
p,q,r, and s are coplanar and p,q, and r are not collinear, (p,o_p),(q,o_q),(r,o_r), and (s,o_s) are coplanar and (p,o_p),(q,o_q), and (r,o_r) are not collinear, respectively, and p, q, r, s, t lie inside the domain.
typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Iso_cuboid_3

A type representing an axis-aligned cuboid.

It must be a model of Kernel::Iso_cuboid_3.

typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Orientation_3

A predicate object that must provide the function operators.

Orientation operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s),

which returns POSITIVE, if s lies on the positive side of the oriented plane h defined by p, q, and r, returns NEGATIVE if s lies on the negative side of h, and returns COPLANAR if s lies on h and

Orientation operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s),

which returns POSITIVE, if the point-offset pair (s,o_s) lies on the positive side of the oriented plane h defined by (p,o_p), (q,o_q), and (r,o_r), returns NEGATIVE if (s,o_s) lies on the negative side of h, and returns COPLANAR if (s,o_s) lies on h.

Precondition
p, q, r, s lie inside the domain.
typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Periodic_3_offset_3

The offset type.

It must be a model of the concept Periodic_3Offset_3.

typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Point_3

The point type.

It must be a model of Kernel::Point_3.

typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Segment_3

A segment type.

It must be a model of Kernel::Segment_3.

typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Side_of_bounded_sphere_3

A predicate object that must provide the function operators.

Bounded_side operator()(Point_3 p, Point_3 q, Point_3 t),

which returns the position of the point t relative to the sphere that has pq as its diameter,

Bounded_side operator()(Point_3 p, Point_3 q, Point_3 t, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_t),

which returns the position of the point-offset pair (t,o_t) relative to the sphere that has (p,o_p)(q,o_q) as its diameter,

Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 t),

which returns the position of the point t relative to the sphere passing through p, q, and r and whose center is in the plane defined by these three points,

Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 t, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_q),

which returns the position of the point-offset pair (t,o_t) relative to the sphere passing through (p,o_p), (q,o_q), and (r,o_r) and whose center is in the plane defined by these three point-offset pairs,

Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Point_3 t),

which returns the relative position of point t to the sphere defined by p, q, r, and s; the order of the points p, q, r, and s does not matter, and

Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Point_3 t, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s, Periodic_3_offset_3 o_q),

which returns the relative position of the point-offset pair (t,o_t) to the sphere defined by (p,o_p), (q,o_q), (r,o_r), and (s,o_s); the order of the point-offset pairs (p,o_p), (q,o_q), (r,o_r), and (s,o_s) does not matter.

Precondition
p, q, r, and s are not coplanar, (p,o_p), (q,o_q), (r,o_r), and (s,o_s) are not coplanar, p, q, r, s, t lie inside the domain.
typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Side_of_oriented_sphere_3

A predicate object that must provide the function operators.

Oriented_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Point_3 t),

which determines on which side of the oriented sphere circumscribing p, q, r, s the point t lies and

Oriented_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Point_3 t, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s, Periodic_3_offset_3 o_t),

which determines on which side of the oriented sphere circumscribing (p,o_p), (q,o_q), (r,o_r), (s,o_s) the point-offset pair (t,o_t) lies.

Precondition
p, q, r, s, t lie inside the domain.
typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Tetrahedron_3

A tetrahedron type.

It must be a model of Kernel::Tetrahedron_3.

typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Triangle_3

A triangle type.

It must be a model of Kernel::Triangle_3.

typedef unspecified_type Periodic_3DelaunayTriangulationTraits_3::Vector_3

The vector type.

It must be a model of Kernel::Vector_3.

Member Function Documentation

void Periodic_3DelaunayTriangulationTraits_3::set_domain ( Iso_cuboid_3  domain)

Set the size of the fundamental domain.

This is necessary to evaluate predicates correctly.

Precondition
domain represents a cube.