RegularTriangulationTraits_3 Concept Reference Definition The concept RegularTriangulationTraits_3 is the first template parameter of the class CGAL::Regular_triangulation_3. It defines the geometric objects (points, segments...) forming the triangulation together with a few geometric predicates and constructions on these objects. We use here the same notation as in Section Regular Triangulation. To simplify notation, \(p\) will often denote in the sequel either the point \( p\in\mathbb{R}^3\) or the weighted point \( {p}^{(w)}=(p,w_p)\). Refines TriangulationTraits_3 Has models All models of the CGAL concept Kernel See alsoCGAL::Regular_triangulation_3 In addition to the requirements described for the traits class of CGAL::Triangulation_3, the geometric traits class of CGAL::Regular_triangulation_3 must fulfill the following requirements. Types typedef unspecified_type Line_3 The line type. typedef unspecified_type Object_3 The object type. typedef unspecified_type Plane_3 The plane type. typedef unspecified_type Ray_3 The ray type. typedef unspecified_type Weighted_point_3 The weighted point type. typedef unspecified_type Power_side_of_oriented_power_sphere_3 A predicate object, model of Kernel::PowerSideOfOrientedPowerSphere_3, that must provide the following function operators: typedef unspecified_type Compare_power_distance_3 A predicate object, model of Kernel::ComparePowerDistance_3, that must provide the function operator. typedef unspecified_type Construct_point_3 A constructor type, model of Kernel::ConstructPoint_3. typedef unspecified_type Construct_weighted_circumcenter_3 A constructor type, model of Kernel::ConstructWeightedCircumcenter_3. typedef unspecified_type Construct_object_3 A constructor object that must provide the function operators. typedef unspecified_type Construct_perpendicular_line_3 A constructor object that must provide the function operator. typedef unspecified_type Construct_plane_3 A constructor object that must provide the function operator. typedef unspecified_type Construct_ray_3 A constructor object that must provide the function operator. 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 operators. Operations Power_side_of_oriented_power_sphere_3 power_side_of_oriented_power_sphere_3_object () Compare_power_distance_3 compare_power_distance_3_object () Construct_point_3 construct_point_3_object () The following functions must be provided only if the member functions of CGAL::Regular_triangulation_3 returning elements of the dual diagram are called: Construct_weighted_circumcenter_3 construct_weighted_circumcenter_3_object () Construct_object_3 construct_object_3_object () Construct_perpendicular_line_3 construct_perpendicular_line_object () Construct_plane_3 construct_plane_3_object () Construct_ray_3 construct_ray_3_object () Member Typedef Documentation Compare_power_distance_3 typedef unspecified_type RegularTriangulationTraits_3::Compare_power_distance_3 A predicate object, model of Kernel::ComparePowerDistance_3, that must provide the function operator. Comparison_result operator()(Point_3 p, Weighted_point_3 q, Weighted_point_3 r), which compares the power distance between p and q to the power distance between p and r. NoteThis predicate is required if a call to nearest_power_vertex() or nearest_power_vertex_in_cell() is issued. Construct_object_3 typedef unspecified_type RegularTriangulationTraits_3::Construct_object_3 A constructor object that must provide the function operators. Object_3 operator()(Point_3 p), Object_3 operator()(Segment_3 s) and Object_3 operator()(Ray_3 r) that construct an object respectively from a point, a segment and a ray. NoteOnly required when the dual operations are used. Construct_perpendicular_line_3 typedef unspecified_type RegularTriangulationTraits_3::Construct_perpendicular_line_3 A constructor object that must provide the function operator. Line_3 operator()(Plane_3 pl, Point_3 p), which constructs the line perpendicular to pl passing through p. NoteOnly required when the dual operations are used. Construct_plane_3 typedef unspecified_type RegularTriangulationTraits_3::Construct_plane_3 A constructor object that must provide the function operator. Plane_3 operator()(Point_3 p, Point_3 q, Point_3 r), which constructs the plane passing through p, q and r. Preconditionp, q and r are non collinear. NoteOnly required when the dual operations are used. Construct_point_3 typedef unspecified_type RegularTriangulationTraits_3::Construct_point_3 A constructor type, model of Kernel::ConstructPoint_3. The operator() extracts the bare point from a weighted point. Point_3 operator() ( Weighted_point_3 p); Construct_ray_3 typedef unspecified_type RegularTriangulationTraits_3::Construct_ray_3 A constructor object that must provide the function operator. Ray_3 operator()(Point_3 p, Line_3 l), which constructs the ray starting at p with direction given by l. NoteOnly required when the dual operations are used. Construct_weighted_circumcenter_3 typedef unspecified_type RegularTriangulationTraits_3::Construct_weighted_circumcenter_3 A constructor type, model of Kernel::ConstructWeightedCircumcenter_3. The operator() constructs the bare point which is the center of the smallest orthogonal sphere to the input weighted points. Point_3 operator() ( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s); NoteOnly required when the dual operations are used. Power_side_of_bounded_power_sphere_3 typedef unspecified_type RegularTriangulationTraits_3::Power_side_of_bounded_power_sphere_3 A predicate object that must provide the function operators. Bounded_side operator()(Weighted_point_3 p, Weighted_point_3 t), which returns the sign of the power test of t with respect to the smallest sphere orthogonal to p (which is the sphere with center 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), which returns the sign of the power test of t with respect to the smallest sphere orthogonal to p and q, Bounded_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 t), which returns the sign of the power test of t with respect to the smallest sphere orthogonal to p, q, and r. Power_side_of_oriented_power_sphere_3 typedef unspecified_type RegularTriangulationTraits_3::Power_side_of_oriented_power_sphere_3 A predicate object, model of Kernel::PowerSideOfOrientedPowerSphere_3, that must provide the following 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), which performs the following: Let \( {z(p,q,r,s)}^{(w)}\) be the power sphere of the weighted points \( (p,q,r,s)\). Returns ON_ORIENTED_BOUNDARY if t is orthogonal to \( {z(p,q,r,s)}^{(w)}\), ON_NEGATIVE_SIDE if t lies outside the oriented sphere of center \( z(p,q,r,s)\) and radius \( \sqrt{ w_{z(p,q,r,s)}^2 + w_t^2 }\) (which is equivalent to \( \Pi({t}^{(w)},{z(p,q,r,s)}^{(w)}) >0\)), ON_POSITIVE_SIDE if t lies inside this oriented sphere. Preconditionp, q, r, s are not coplanar. Note that with this definition, if all the points have a weight equal to 0, then power_side_of_oriented_power_sphere_3(p,q,r,s,t) = side_of_oriented_sphere(p,q,r,s,t). Oriented_side operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 t), which has a definition analogous to the previous method, for coplanar points, with the power circle \( {z(p,q,r)}^{(w)}\). Preconditionp, q, r are not collinear and p, q, r, t are coplanar. If all the points have a weight equal to 0, then power_side_of_oriented_power_sphere_3(p,q,r,t) = side_of_oriented_circle(p,q,r,t). Oriented_side operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 t), which is the same for collinear points, where \( {z(p,q)}^{(w)}\) is the power segment of p and q. Preconditionp and q have different bare points, and p, q, t are collinear. If all points have a weight equal to 0, then power_side_of_oriented_power_sphere_3(p,q,t) gives the same answer as the kernel predicate s(p,q).has_on(t) would give, where s(p,q) denotes the segment with endpoints p and q. Oriented_side operator()( Weighted_point_3 p, Weighted_point_3 q), which is the same for equal bare points, then it returns the comparison of the weights (ON_POSITIVE_SIDE when q is heavier than p). Preconditionp and q have equal bare points. Weighted_point_3 typedef unspecified_type RegularTriangulationTraits_3::Weighted_point_3 The weighted point type. It has to be a model of the concept Kernel::WeightedPoint_3. NoteThe unweighted point type Point_3 is requested by the concept TriangulationTraits_3, which this concept refines. RegularTriangulationTraits_3 Generated by 1.17.0

Definition

The concept RegularTriangulationTraits_3 is the first template parameter of the class CGAL::Regular_triangulation_3. It defines the geometric objects (points, segments...) forming the triangulation together with a few geometric predicates and constructions on these objects.

We use here the same notation as in Section Regular Triangulation. To simplify notation, \(p\) will often denote in the sequel either the point \( p\in\mathbb{R}^3\) or the weighted point \( {p}^{(w)}=(p,w_p)\).

Refines

TriangulationTraits_3

Has models

All models of the CGAL concept Kernel

See also

CGAL::Regular_triangulation_3

In addition to the requirements described for the traits class of CGAL::Triangulation_3, the geometric traits class of CGAL::Regular_triangulation_3 must fulfill the following requirements.

Types
typedef unspecified_type	Line_3
	The line type.
typedef unspecified_type	Object_3
	The object type.
typedef unspecified_type	Plane_3
	The plane type.
typedef unspecified_type	Ray_3
	The ray type.
typedef unspecified_type	Weighted_point_3
	The weighted point type.
typedef unspecified_type	Power_side_of_oriented_power_sphere_3
	A predicate object, model of Kernel::PowerSideOfOrientedPowerSphere_3, that must provide the following function operators:
typedef unspecified_type	Compare_power_distance_3
	A predicate object, model of Kernel::ComparePowerDistance_3, that must provide the function operator.
typedef unspecified_type	Construct_point_3
	A constructor type, model of Kernel::ConstructPoint_3.
typedef unspecified_type	Construct_weighted_circumcenter_3
	A constructor type, model of Kernel::ConstructWeightedCircumcenter_3.
typedef unspecified_type	Construct_object_3
	A constructor object that must provide the function operators.
typedef unspecified_type	Construct_perpendicular_line_3
	A constructor object that must provide the function operator.
typedef unspecified_type	Construct_plane_3
	A constructor object that must provide the function operator.
typedef unspecified_type	Construct_ray_3
	A constructor object that must provide the function operator.
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 operators.

Operations
Power_side_of_oriented_power_sphere_3	power_side_of_oriented_power_sphere_3_object ()
Compare_power_distance_3	compare_power_distance_3_object ()
Construct_point_3	construct_point_3_object ()
The following functions must be provided only if the member functions of CGAL::Regular_triangulation_3 returning elements of the dual diagram are called:
Construct_weighted_circumcenter_3	construct_weighted_circumcenter_3_object ()
Construct_object_3	construct_object_3_object ()
Construct_perpendicular_line_3	construct_perpendicular_line_object ()
Construct_plane_3	construct_plane_3_object ()
Construct_ray_3	construct_ray_3_object ()

Member Typedef Documentation

Compare_power_distance_3

typedef unspecified_type RegularTriangulationTraits_3::Compare_power_distance_3

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

Comparison_result operator()(Point_3 p, Weighted_point_3 q, Weighted_point_3 r),

which compares the power distance between p and q to the power distance between p and r.

Note

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

Construct_object_3

typedef unspecified_type RegularTriangulationTraits_3::Construct_object_3

A constructor object that must provide the function operators.

Object_3 operator()(Point_3 p),

Object_3 operator()(Segment_3 s) and

Object_3 operator()(Ray_3 r)

that construct an object respectively from a point, a segment and a ray.

Note

Only required when the dual operations are used.

Construct_perpendicular_line_3

typedef unspecified_type RegularTriangulationTraits_3::Construct_perpendicular_line_3

A constructor object that must provide the function operator.

Line_3 operator()(Plane_3 pl, Point_3 p),

which constructs the line perpendicular to pl passing through p.

Note

Only required when the dual operations are used.

Construct_plane_3

typedef unspecified_type RegularTriangulationTraits_3::Construct_plane_3

A constructor object that must provide the function operator.

Plane_3 operator()(Point_3 p, Point_3 q, Point_3 r),

which constructs the plane passing through p, q and r.

Precondition

p, q and r are non collinear.

Note

Only required when the dual operations are used.

Construct_point_3

typedef unspecified_type RegularTriangulationTraits_3::Construct_point_3

A constructor type, model of Kernel::ConstructPoint_3.

The operator() extracts the bare point from a weighted point.

Point_3 operator() ( Weighted_point_3 p);

Construct_ray_3

typedef unspecified_type RegularTriangulationTraits_3::Construct_ray_3

A constructor object that must provide the function operator.

Ray_3 operator()(Point_3 p, Line_3 l),

which constructs the ray starting at p with direction given by l.

Note

Only required when the dual operations are used.

Construct_weighted_circumcenter_3

typedef unspecified_type RegularTriangulationTraits_3::Construct_weighted_circumcenter_3

A constructor type, model of Kernel::ConstructWeightedCircumcenter_3.

The operator() constructs the bare point which is the center of the smallest orthogonal sphere to the input weighted points.

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

Note

Only required when the dual operations are used.

Power_side_of_bounded_power_sphere_3

typedef unspecified_type RegularTriangulationTraits_3::Power_side_of_bounded_power_sphere_3

A predicate object that must provide the function operators.

Bounded_side operator()(Weighted_point_3 p, Weighted_point_3 t),

which returns the sign of the power test of t with respect to the smallest sphere orthogonal to p (which is the sphere with center 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),

which returns the sign of the power test of t with respect to the smallest sphere orthogonal to p and q,

Bounded_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 t),

which returns the sign of the power test of t with respect to the smallest sphere orthogonal to p, q, and r.

Power_side_of_oriented_power_sphere_3

typedef unspecified_type RegularTriangulationTraits_3::Power_side_of_oriented_power_sphere_3

A predicate object, model of Kernel::PowerSideOfOrientedPowerSphere_3, that must provide the following 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),

which performs the following:

Let \( {z(p,q,r,s)}^{(w)}\) be the power sphere of the weighted points \( (p,q,r,s)\). Returns

• ON_ORIENTED_BOUNDARY if t is orthogonal to \( {z(p,q,r,s)}^{(w)}\),
• ON_NEGATIVE_SIDE if t lies outside the oriented sphere of center \( z(p,q,r,s)\) and radius \( \sqrt{ w_{z(p,q,r,s)}^2 + w_t^2 }\) (which is equivalent to \( \Pi({t}^{(w)},{z(p,q,r,s)}^{(w)}) >0\)),
• ON_POSITIVE_SIDE if t lies inside this oriented sphere.

Precondition

p, q, r, s are not coplanar. Note that with this definition, if all the points have a weight equal to 0, then power_side_of_oriented_power_sphere_3(p,q,r,s,t) = side_of_oriented_sphere(p,q,r,s,t).

---

Oriented_side operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 t),

which has a definition analogous to the previous method, for coplanar points, with the power circle \( {z(p,q,r)}^{(w)}\).

Precondition

p, q, r are not collinear and p, q, r, t are coplanar. If all the points have a weight equal to 0, then power_side_of_oriented_power_sphere_3(p,q,r,t) = side_of_oriented_circle(p,q,r,t).

---

Oriented_side operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 t),

which is the same for collinear points, where \( {z(p,q)}^{(w)}\) is the power segment of p and q.

Precondition

p and q have different bare points, and p, q, t are collinear. If all points have a weight equal to 0, then power_side_of_oriented_power_sphere_3(p,q,t) gives the same answer as the kernel predicate s(p,q).has_on(t) would give, where s(p,q) denotes the segment with endpoints p and q.

---

Oriented_side operator()( Weighted_point_3 p, Weighted_point_3 q),

which is the same for equal bare points, then it returns the comparison of the weights (ON_POSITIVE_SIDE when q is heavier than p).

Precondition

p and q have equal bare points.

Weighted_point_3

typedef unspecified_type RegularTriangulationTraits_3::Weighted_point_3

The weighted point type.

It has to be a model of the concept Kernel::WeightedPoint_3.

Note

The unweighted point type Point_3 is requested by the concept TriangulationTraits_3, which this concept refines.