\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 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`. It refines the concept `DelaunayTriangulationTraits_3` from the CGAL 3D Triangulations. 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:

  Periodic_3TriangulationTraits_3

  DelaunayTriangulationTraits_3

Has Models:
  CGAL::Periodic_3_Delaunay_triangulation_traits_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.

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...
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_Delaunay_triangulation_traits_3 ()
| Default constructor.
| Periodic_3_Delaunay_triangulation_traits_3 (const Periodic_triangulation_traits_3 &tr)
| Copy constructor.

## Operations
The following functions give access to the predicate and construction objects:
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 ()
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::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::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::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.