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

CGAL 4.8 - 3D Periodic Triangulations

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Periodic_3TriangulationTraits_3 Concept Reference
Concepts

## Definition

The concept `Periodic_3TriangulationTraits_3` is the first template parameter of the class `Periodic_3_triangulation_3`. It refines the concept `TriangulationTraits_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 `TriangulationTraits_3` can be used directly. For efficiency reasons we maintain for each functor the version without offsets.

Refines:
  TriangulationTraits_3

Has Models:
  CGAL::Periodic_3_triangulation_traits_3

In addition to the requirements described for the traits class TriangulationTraits_3, the geometric traits class of a Periodic triangulation must fulfill the following requirements.

## 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...
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...

## 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 ()
Construct_segment_3 | construct_segment_3_object ()
Construct_triangle_3 | construct_triangle_3_object ()
Construct_tetrahedron_3 | construct_tetrahedron_3_object ()

## Member Typedef Documentation

typedef unspecified_type Periodic_3TriangulationTraits_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_3TriangulationTraits_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_3TriangulationTraits_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_3TriangulationTraits_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_3TriangulationTraits_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_3TriangulationTraits_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_3TriangulationTraits_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_3TriangulationTraits_3::Periodic_3_offset_3

The offset type.

It must be a model of the concept `Periodic_3Offset_3`.

typedef unspecified_type Periodic_3TriangulationTraits_3::Point_3

The point type.

It must be a model of `Kernel::Point_3`.

typedef unspecified_type Periodic_3TriangulationTraits_3::Segment_3

A segment type.

It must be a model of `Kernel::Segment_3`.

typedef unspecified_type Periodic_3TriangulationTraits_3::Tetrahedron_3

A tetrahedron type.

It must be a model of `Kernel::Tetrahedron_3`.

typedef unspecified_type Periodic_3TriangulationTraits_3::Triangle_3

A triangle type.

It must be a model of `Kernel::Triangle_3`.

typedef unspecified_type Periodic_3TriangulationTraits_3::Vector_3

The vector type.

It must be a model of `Kernel::Vector_3`.

## Member Function Documentation

void Periodic_3TriangulationTraits_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.