RegularTriangulationTraits_3

Definition

Refines

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.

Types

RegularTriangulationTraits_3::Line_3
	The line type.
RegularTriangulationTraits_3::Object_3
	The object type.
RegularTriangulationTraits_3::Plane_3
	The plane type.
RegularTriangulationTraits_3::Ray_3
	The ray type.

We use here the same notation as in Section 22.3. To simplify notation, $$p will often denote in the sequel either the point $$p ³ or the weighted point $$p^(w)=(p,w_p).

RegularTriangulationTraits_3::Weighted_point_3
	The weighted point 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 $$sqrt( wz(p,q,r,s)2 + wt2 ) (which is equivalent to $$(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(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 $$z(p,q,r)(w). Precondition: p, q, r are not collinear and p, q, r, t are coplanar. If all the points have a weight equal to 0, then power_test(p,q,r,s,t) = side_of_oriented_circle(p,q,r,s,t). Oriented_side operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 t), which is the same for collinear points, where $$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(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.

The following predicate is required if a call to nearest_power_vertex or nearest_power_vertex_in_cell is issued:

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.

In addition, only when the dual operations are used, the traits class must provide the following constructor objects:

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);
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.
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.
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.
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.

Operations

The following function gives access to the predicate object:

Power_test_3

traits.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
	traits.construct_weighted_circumcenter_3_object ()
Construct_object_3	traits.construct_object_3_object ()
Construct_perpendicular_line_3
	traits.construct_perpendicular_line_object ()
Construct_plane_3	traits.construct_plane_3_object ()
Construct_ray_3	traits.construct_ray_3_object ()

Has Models

CGAL::Regular_triangulation_euclidean_traits_3
CGAL::Regular_triangulation_filtered_traits_3.