CGAL 6.2 - 3D Periodic Triangulations

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Periodic_3RegularTriangulationTraits_3 Concept Reference
Reference Manual » Concepts

## Definition

The concept `Periodic_3RegularTriangulationTraits_3` is the first template parameter of the class `CGAL::Periodic_3_regular_triangulation_3`. It refines the concept `RegularTriangulationTraits_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 `RegularTriangulationTraits_3` can be used directly. For efficiency reasons we maintain for each functor the version without offsets.

Refines
  `Periodic_3TriangulationTraits_3`
  `RegularTriangulationTraits_3`

Has models
  `CGAL::Periodic_3_regular_triangulation_traits_3`

In addition to the requirements described for the traits class `RegularTriangulationTraits_3`, the geometric traits class of a periodic regular 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 | Power_side_of_oriented_power_sphere_3
| A predicate object that must provide the function operators.
typedef unspecified_type | Compare_squared_distance_3
| A predicate object that must provide the function operators:
typedef unspecified_type | Compare_weighted_squared_radius_3
| A predicate object that must provide the function operators:
typedef unspecified_type | Compare_power_distance_3
| A predicate object, model of `ComparePowerDistance_3`, that must provide the function operator.
When vertex removal is used, the traits class must in addition provide the following predicate object
typedef unspecified_type | Coplanar_orientation_3
| A predicate object that must provide the function operators:
When `is_Gabriel` functions are used, the traits class must in addition provide the following predicate object:
typedef unspecified_type | Power_side_of_bounded_power_sphere_3
| A predicate object that must provide the function operator.
When the dual operations are used, the traits class must in addition provide the following constructor object:
typedef unspecified_type | Construct_weighted_circumcenter_3
| A constructor object that must provide the function operator.

## Operations
The following functions give access to the predicate and construction objects:
Power_side_of_oriented_power_sphere_3 | power_side_of_oriented_power_sphere_3_object ()
Compare_weighted_squared_radius_3 | compare_weighted_squared_radius_3_object ()
The following function must be provided if vertex removal is used; otherwise dummy functions can be provided.
Coplanar_orientation_3 | coplanar_3_orientation_3_object ()
The following function must be provided only if the methods of `Periodic_3_regular_triangulation_3` returning elements of the Voronoi diagram are used; otherwise a dummy function can be provided.
Construct_weighted_circumcenter_3 | construct_weighted_circumcenter_3_object ()

## Member Typedef Documentation

## ◆ Compare_power_distance_3

typedef unspecified_type Periodic_3RegularTriangulationTraits_3::Compare_power_distance_3

A predicate object, model of `ComparePowerDistance_3`, that must provide the function operator.

`Comparison_result operator()(Point_3 p, Weighted_point_3 q, Weighted_point_3 r, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r)`,

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

Note
  This predicate is required if a call to `nearest_power_vertex()` or `nearest_power_vertex_in_cell()` is issued.

## ◆ Compare_squared_distance_3

typedef unspecified_type Periodic_3RegularTriangulationTraits_3::Compare_squared_distance_3

A predicate object that must provide the function operators:

`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 compares the squared distance between `(p, o_p)` and `(q, o_q)` and the squared distance between `(r, o_r)` and `(s, o_s)`, and returns `SMALLER`, `EQUAL`, or `LARGER`.

Precondition
  `p`, `q`, `r`, and `s` lie inside the domain.

## ◆ Compare_weighted_squared_radius_3

typedef unspecified_type Periodic_3RegularTriangulationTraits_3::Compare_weighted_squared_radius_3

A predicate object that must provide the function operators:

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

and

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

which compares the weight of the smallest sphere orthogonal to the input weighted points with the input weight `w` and returns `SMALLER`, `EQUAL`, or `LARGER`.

Precondition
  `p`, `q`, `r`, and `s` lie inside the domain.

## ◆ Construct_weighted_circumcenter_3

typedef unspecified_type Periodic_3RegularTriangulationTraits_3::Construct_weighted_circumcenter_3

A constructor object that must provide the function operator.

`Weighted_point_3 operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_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 weighted circumcenter of four point-offset pairs.

Precondition
  `p`, `q`, `r`, `s` lie inside the domain. `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.

## ◆ Coplanar_orientation_3

typedef unspecified_type Periodic_3RegularTriangulationTraits_3::Coplanar_orientation_3

A predicate object that must provide the function operators:

`Orientation operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_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()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r)`,

which is the same for point-offset pairs.

Precondition
  `p`, `q`, `r` lie inside the domain.

## ◆ Power_side_of_bounded_power_sphere_3

typedef unspecified_type Periodic_3RegularTriangulationTraits_3::Power_side_of_bounded_power_sphere_3

A predicate object that must provide the function operator.

`Bounded_side operator()(Weighted_point_3 p, Weighted_point_3 t, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_t)`,

which returns the sign of the power test of `(t,o_t)` with respect to the smallest sphere orthogonal to `(p,o_p)` (which is the sphere with center `(p,o_p)` and squared radius `-w_p` with `w_p` the weight of `p`),

`Bounded_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 t, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_t)`,

which returns the sign of the power test of `(t,o_t)` with respect to the smallest sphere orthogonal to `(p,o_p)` and `(q,o_q)`,

`Bounded_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_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 sign of the power test of `(t,o_t)` with respect to the smallest sphere orthogonal to `(p,o_p)`, `(q,o_q)`, and `(r,o_r)`.

## ◆ Power_side_of_oriented_power_sphere_3

typedef unspecified_type Periodic_3RegularTriangulationTraits_3::Power_side_of_oriented_power_sphere_3

A predicate object that must provide the function operators.

`Oriented_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s, Weighted_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 the position of the point-offset pair `(t,o_t)` with respect to the power sphere of the point-offset pairs `(p,o_p), (q,o_q), (r,o_r), (s,o_s)`.

Precondition
  `p`, `q`, `r`, `s`, `t` lie inside the domain and `p, q, r, s` are not coplanar.

When vertex removal is used, the predicate must in addition provide the function operators

`Oriented_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_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_t)`,

which has a definition similar to the previous method, for coplanar points, with the power circle of `p,q,r`.

Precondition
  `p`, `q`, `r`, `t` lie inside the domain, `p, q, r` are not collinear, and `(p,o_p), (q,o_q), (r,o_r), (t,o_t)` are coplanar.

`Oriented_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 t, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_t)`,

which is the same for collinear points, and the power segment of `(p,o_p)` and `(q,o_q)`,

Precondition
  `p`, `q`, `t` lie inside the domain, `p` and `q` have different Bare_points, and `(p,o_p), (q,o_q), (t,o_t)` are collinear.

`Oriented_side operator()(Weighted_point_3 p, Weighted_point_3 q, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q)`,

which is the same for equal points, that is when `(p,o_p)` and `(q,o_q)` have equal coordinates, then it returns the comparison of the weights.

Precondition
  `p` and `q` lie inside the domain and have equal Bare_points.