MeshTriangulationTraits_3 Concept Reference

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 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. More...
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: More...
typedef unspecified_type	Equal_3
	A predicate object that must provide the function operator: More...
typedef unspecified_type	Is_degenerate_3
	A predicate object that must provide the function operators: More...
typedef unspecified_type	Compare_weighted_squared_radius_3
	A constructor object that must provide the function operators: More...
typedef unspecified_type	Compute_approximate_dihedral_angle_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Compute_area_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Compute_power_distance_to_power_sphere_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Compute_squared_area_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Compute_squared_distance_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Compute_squared_length_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Compute_squared_radius_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Compute_squared_radius_smallest_orthogonal_sphere_3
	A constructor object that must provide the function operators: More...
typedef unspecified_type	Compute_volume_3
	A constructor object that must provide the function operators: More...
typedef unspecified_type	Compute_weight_3
	A constructor object that must provide the function operators: More...
typedef unspecified_type	Construct_base_vector_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Construct_orthogonal_vector_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Construct_center_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Construct_centroid_3
	A constructor object that must provide the function operators: More...
typedef unspecified_type	Construct_midpoint_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Construct_normal_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Construct_sphere_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Construct_tetrahedron_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Construct_translated_point_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Construct_vector_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Construct_opposite_vector_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Construct_scaled_vector_3
	A constructor object that must provide the function operator: More...
typedef unspecified_type	Intersect_3
	A constructor object that must provide the function operators: More...

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

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.

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:

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

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

boost::optional< boost::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.