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.

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. More...

typedef unspecified_type	Compare_weighted_squared_radius_3
	A predicate object that must provide the function operators: More...

typedef unspecified_type	Compare_power_distance_3
	A predicate object, model of `ComparePowerDistance_3`, that must provide the function operator. More...

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: More...

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. More...

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. More...

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

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.

typedef unspecified_type Periodic_3RegularTriangulationTraits_3::Compare_weighted_squared_radius_3

A predicate object that must provide the function operators:

Orientation operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s, FT w),

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

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

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.

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.

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).

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.

Definition

Operations

Member Typedef Documentation