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CGAL 4.4 - 3D Triangulations
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RegularTriangulationTraits_3 Concept Reference
Concepts

Definition

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

Refines:
TriangulationTraits_3

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

Has Models:
CGAL::Regular_triangulation_euclidean_traits_3

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.
 

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 \( {p}^{(w)}=(p,w_p)\).

typedef unspecified_type Weighted_point_3
 The weighted point type.
 
typedef unspecified_type Bare_point
 The (un-weighted) point type.
 
typedef unspecified_type Power_test_3
 A predicate object which must provide the following function operators: More...
 
typedef unspecified_type Compare_power_distance_3
 A predicate object that must provide the function operator. More...
 
typedef unspecified_type Construct_weighted_circumcenter_3
 A constructor type. More...
 
typedef unspecified_type Construct_object_3
 A constructor object that must provide the function operators. More...
 
typedef unspecified_type Construct_perpendicular_line_3
 A constructor object that must provide the function operator. More...
 
typedef unspecified_type Construct_plane_3
 A constructor object that must provide the function operator. More...
 
typedef unspecified_type Construct_ray_3
 A constructor object that must provide the function operator. More...
 

Operations

Power_test_3 power_test_3_object ()
 

/*! The following functions must be provided only if the member functions of 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

typedef unspecified_type RegularTriangulationTraits_3::Compare_power_distance_3

A predicate object 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.
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.
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.
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.
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.
typedef unspecified_type RegularTriangulationTraits_3::Construct_weighted_circumcenter_3

A constructor type.

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

Bare_point 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.
typedef unspecified_type RegularTriangulationTraits_3::Power_test_3

A predicate object which 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_test_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 an 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_test_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_test_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 points, that is when p and q have equal coordinates, 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.