CGAL 6.1.2 - 3D Mesh Generation: MeshTriangulationTraits_3 Concept Reference

CGAL 6.1.2 - 3D Mesh Generation

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MeshTriangulationTraits_3 Concept Reference
Reference Manual » Main Concepts

## Definition

The concept `MeshTriangulationTraits_3` describes the requirements for the geometric traits class of the underlying regular triangulation used during a mesh generation process.

Refines
  `RegularTriangulationTraits_3`

Has models
  All models of the CGAL concept `Kernel`

In addition to the requirements described for the traits class RegularTriangulationTraits_3, the geometric traits class of a regular triangulation used in mesh generation must fulfill the following requirements.

## Types
typedef unspecified_type | FT
| Numerical type.
typedef unspecified_type | Vector_3
| The vector type.
typedef unspecified_type | Sphere_3
| The sphere type.
typedef unspecified_type | Iso_cuboid_3
| The isocuboid type.
typedef unspecified_type | Aff_transformation_3
| The affine transformation type.
typedef unspecified_type | Do_intersect_3
| A predicate object that must provide the function operator:
typedef unspecified_type | Equal_3
| A predicate object that must provide the function operator:
typedef unspecified_type | Is_degenerate_3
| A predicate object that must provide the function operators:
typedef unspecified_type | Collinear_3
| A predicate object that must provide the function operator:
typedef unspecified_type | Compare_weighted_squared_radius_3
| A constructor object that must provide the function operators:
typedef unspecified_type | Compute_approximate_dihedral_angle_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Compute_area_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Compute_power_distance_to_power_sphere_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Compute_squared_area_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Compute_squared_distance_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Compute_squared_length_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Compute_squared_radius_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Compute_squared_radius_smallest_orthogonal_sphere_3
| A constructor object that must provide the function operators:
typedef unspecified_type | Compute_volume_3
| A constructor object that must provide the function operators:
typedef unspecified_type | Compute_weight_3
| A constructor object that must provide the function operators:
typedef unspecified_type | Construct_base_vector_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Construct_orthogonal_vector_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Construct_center_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Construct_weighted_circumcenter_3
| A constructor object that must provide the function operators:
typedef unspecified_type | Construct_centroid_3
| A constructor object that must provide the function operators:
typedef unspecified_type | Construct_midpoint_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Construct_normal_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Construct_sphere_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Construct_tetrahedron_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Construct_translated_point_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Construct_vector_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Construct_opposite_vector_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Construct_scaled_vector_3
| A constructor object that must provide the function operator:
typedef unspecified_type | Intersect_3
| A constructor object that must provide the function operators:

## Operations
The following functions give access to the predicate and construction objects:
Do_intersect_3 | do_intersect_3_object ()
Equal_3 | equal_3_object ()
Is_degenerate_3 | equal_3_object ()
Compare_weighted_squared_radius_3 | compare_weighted_squared_radius_3_object ()
Compute_approximate_dihedral_angle_3 | compute_approximate_dihedral_angle_3_object ()
Compute_area_3 | compute_area_3_object ()
Compute_power_distance_to_power_sphere_3 | compute_power_distance_to_power_sphere_3_object ()
Compute_squared_area_3 | compute_squared_area_3_object ()
Compute_squared_distance_3 | compute_squared_distance_3_object ()
Compute_squared_length_3 | compute_squared_length_3_object ()
Compute_squared_radius_3 | compute_squared_radius_3_object ()
Compute_squared_radius_smallest_orthogonal_sphere_3 | compute_squared_radius_smallest_orthogonal_sphere_3_object ()
Compute_volume_3 | compute_volume_3_object ()
Compute_weight_3 | compute_weight_3_object ()
Construct_base_vector_3 | construct_base_vector_3_object ()
Construct_orthogonal_vector_3 | construct_orthogonal_vector_3_object ()
Construct_center_3 | construct_center_3_object ()
Construct_midpoint_3 | construct_midpoint_3_object ()
Construct_normal_3 | construct_normal_3_object ()
Construct_sphere_3 | construct_sphere_3_object ()
Construct_tetrahedron_3 | construct_tetrahedron_3_object ()
Construct_translated_point_3 | construct_translated_point_3_object ()
Construct_vector_3 | construct_vector_3_object ()
Construct_scaled_vector_3 | construct_scaled_vector_3_object ()
Construct_opposite_vector_3 | construct_opposite_vector_3_object ()
Intersect_3 | intersect_3_object ()

## Member Typedef Documentation

## ◆ Collinear_3

typedef unspecified_type MeshTriangulationTraits_3::Collinear_3

A predicate object that must provide the function operator:

`bool operator()(Point_3 p, Point_3 q, Point_3 r)`

which returns `true` iff `p`, `q`, and `r` are collinear.

## ◆ Compare_weighted_squared_radius_3

typedef unspecified_type MeshTriangulationTraits_3::Compare_weighted_squared_radius_3

A constructor object that must provide the function operators:

`bool operator()(Point_3 p, FT w)`

`bool operator()(Point_3 p, Point_3 q, FT w)`

`bool operator()(Point_3 p, Point_3 q, Point_3 r, FT w)`

`bool operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, FT w)`

which compare the weight of the smallest sphere orthogonal to the input weighted point(s) with the input weight.

## ◆ Compute_approximate_dihedral_angle_3

typedef unspecified_type MeshTriangulationTraits_3::Compute_approximate_dihedral_angle_3

A constructor object that must provide the function operator:

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

which returns an approximation of the signed dihedral angle in the tetrahedron `pqrs` of edge `pq`. The sign is negative if `orientation(p,q,r,s)` is `CGAL::NEGATIVE` and positive otherwise. The angle is given in degrees.

## ◆ Compute_area_3

typedef unspecified_type MeshTriangulationTraits_3::Compute_area_3

A constructor object that must provide the function operator:

`FT operator()(Point_3 p, Point_3 q, Point_3 r)`

which returns the area of the triangle `p`, `q`, `r`.

## ◆ Compute_power_distance_to_power_sphere_3

typedef unspecified_type MeshTriangulationTraits_3::Compute_power_distance_to_power_sphere_3

A constructor object that must provide the function operator:

`FT operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Point_3 t)`

which returns the squared radius of the sphere centered in `t` and orthogonal to the sphere orthogonal to `p`, `q`, `r`, and `s`.

## ◆ Compute_squared_area_3

typedef unspecified_type MeshTriangulationTraits_3::Compute_squared_area_3

A constructor object that must provide the function operator:

`FT operator()(Triangle_3 t)`

which returns the squared area of the triangle `t`.

## ◆ Compute_squared_distance_3

typedef unspecified_type MeshTriangulationTraits_3::Compute_squared_distance_3

A constructor object that must provide the function operator:

`FT operator()(Point_3 p, Point_3 q)`

which returns the squared distance between two points `p` and `q`.

## ◆ Compute_squared_length_3

typedef unspecified_type MeshTriangulationTraits_3::Compute_squared_length_3

A constructor object that must provide the function operator:

`FT operator()(Vector_3 v)`

which returns the squared length of `v`.

## ◆ Compute_squared_radius_3

typedef unspecified_type MeshTriangulationTraits_3::Compute_squared_radius_3

A constructor object that must provide the function operator:

`FT operator()(Sphere_3 s)`

which returns the squared radius of the sphere `s`.

## ◆ Compute_squared_radius_smallest_orthogonal_sphere_3

typedef unspecified_type MeshTriangulationTraits_3::Compute_squared_radius_smallest_orthogonal_sphere_3

A constructor object that must provide the function operators:

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

`FT operator()(Point_3 p, Point_3 q, Point_3 r)`

`FT operator()(Point_3 p, Point_3 q)`

which returns the squared radius of the smallest sphere orthogonal to the argument(s).

## ◆ Compute_volume_3

typedef unspecified_type MeshTriangulationTraits_3::Compute_volume_3

A constructor object that must provide the function operators:

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

`FT operator()(Tetrahedron_3 t)`

which respectively return the volume of the tetrahedron formed by the points `p`, `q`, `r`, and `s` and of the tetrahedron `t`.

## ◆ Compute_weight_3

typedef unspecified_type MeshTriangulationTraits_3::Compute_weight_3

A constructor object that must provide the function operators:

`FT operator()(Weighted_point_3 p)`

which returns the weight of the weighted point `p`.

## ◆ Construct_base_vector_3

typedef unspecified_type MeshTriangulationTraits_3::Construct_base_vector_3

A constructor object that must provide the function operator:

`Vector_3 operator()(Plane_3 p, int index)`,

which returns:

- when `index == 1`: a vector `b1` that is orthogonal to the normal `n` to plane `h`;
- when `index == 2`: a vector `b2` that is orthogonal to `n` and `b1` and such that for an arbitrary point `p` on the plane `h`, the orientation of `p`, `p + b1`, `p + b2`, and `p + n` is positive.

## ◆ Construct_center_3

typedef unspecified_type MeshTriangulationTraits_3::Construct_center_3

A constructor object that must provide the function operator:

`Point_3 operator()(Sphere_3 q)`,

which returns the center of the sphere `s`.

## ◆ Construct_centroid_3

typedef unspecified_type MeshTriangulationTraits_3::Construct_centroid_3

A constructor object that must provide the function operators:

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

`Point_3 operator()(Tetrahedron_3 t)`,

which respectively return the centroid of the points `p`, `q`, and `r`, and the centroid of the tetrahedron `t`.

## ◆ Construct_midpoint_3

typedef unspecified_type MeshTriangulationTraits_3::Construct_midpoint_3

A constructor object that must provide the function operator:

`Point_3 operator()(Point_3 p, Point_3 q)`,

which returns the midpoint of the segment pq.

## ◆ Construct_normal_3

typedef unspecified_type MeshTriangulationTraits_3::Construct_normal_3

A constructor object that must provide the function operator:

`Vector_3 operator()(Point_3 p, Point_3 q, Point_3 r)`,

which returns the normal of the vectors `q-p` and `r-p`.

## ◆ Construct_opposite_vector_3

typedef unspecified_type MeshTriangulationTraits_3::Construct_opposite_vector_3

A constructor object that must provide the function operator:

`Vector_3 operator()(Vector_3 v)`

which returns `-v`.

## ◆ Construct_orthogonal_vector_3

typedef unspecified_type MeshTriangulationTraits_3::Construct_orthogonal_vector_3

A constructor object that must provide the function operator:

`Vector_3 operator()(Plane_3 p)`,

which returns a vector that is orthogonal to the plane `p` and directed to the positive side of `p`.

## ◆ Construct_scaled_vector_3

typedef unspecified_type MeshTriangulationTraits_3::Construct_scaled_vector_3

A constructor object that must provide the function operator:

`Vector_3 operator()(Vector_3 v, FT scale)`

which returns the vector `v` scaled by a factor `scale`.

## ◆ Construct_sphere_3

typedef unspecified_type MeshTriangulationTraits_3::Construct_sphere_3

A constructor object that must provide the function operator:

`Sphere_3 operator()(Point_3 p, Point_3 q)`,

which returns the smallest sphere which passes through the points p and q.

## ◆ Construct_tetrahedron_3

typedef unspecified_type MeshTriangulationTraits_3::Construct_tetrahedron_3

A constructor object that must provide the function operator:

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

which returns the tetrahedron with vertices `p`, `q`, `r`, and `s`.

## ◆ Construct_translated_point_3

typedef unspecified_type MeshTriangulationTraits_3::Construct_translated_point_3

A constructor object that must provide the function operator:

`Point_3 operator()(Point_3 p, Vector_3 v)`,

which returns the point obtained by translating `p` by the vector `v`.

## ◆ Construct_vector_3

typedef unspecified_type MeshTriangulationTraits_3::Construct_vector_3

A constructor object that must provide the function operator:

`Vector_3 operator()(Point_3 p, Point_3 q)`,

which returns a vector from two points.

## ◆ Construct_weighted_circumcenter_3

typedef unspecified_type MeshTriangulationTraits_3::Construct_weighted_circumcenter_3

A constructor object that must provide the function operators:

`Point_3 operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s)`,

`Point_3 operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s)`,

`Point_3 operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s)`,

which return the center of the smallest orthogonal sphere to the input weighted points.

## ◆ Do_intersect_3

typedef unspecified_type MeshTriangulationTraits_3::Do_intersect_3

A predicate object that must provide the function operator:

`bool operator()(Sphere_3 s1, Sphere_3 s2)`

which returns `true` iff the spheres `s1` and `s2` intersect.

## ◆ Equal_3

typedef unspecified_type MeshTriangulationTraits_3::Equal_3

A predicate object that must provide the function operator:

`bool operator()(Point_3 p, Point_3 q)`

which returns `true` iff `p` and `q` are equal.

## ◆ FT

typedef unspecified_type MeshTriangulationTraits_3::FT

Numerical type.

Must be a model of `FieldNumberType` and `FieldWithSqrt`, and constructible from a `double`.

## ◆ Intersect_3

typedef unspecified_type MeshTriangulationTraits_3::Intersect_3

A constructor object that must provide the function operators:

`std::optional< std::variant< T... > > operator()(Segment_3 s, Plane_3 p)`

`std::optional< std::variant< T... > > operator()(Ray_3 r, Iso_cuboid i)`

`std::optional< std::variant< T... > > operator()(Segment_3 s, Iso_cuboid i)`

which returns the intersection region of two geometrical objects.

## ◆ Is_degenerate_3

typedef unspecified_type MeshTriangulationTraits_3::Is_degenerate_3

A predicate object that must provide the function operators:

`bool operator()(Segment_3 s)`

`bool operator()(Ray_3 s)`

which return `true` iff the object is degenerate.