To be used by Triangulation_3<Traits, Tds>

The geometric traits class Triangulation_Traits_3 of the triangulation class Triangulation_3<Traits, Tds> must define the geometric objects (points, segments, triangles and tetrahedra) forming the triangulation together with a few geometric predicates on these objects: equality, coordinate comparison, orientation in space, orientation in case of coplanar points, and collinearity tests.

Types

Triangulation_Traits_3::Point
	The type must provide a copy constructor and assignment operator.
Triangulation_Traits_3::Segment
	The type must provide a constructor that takes two points as arguments.
Triangulation_Traits_3::Triangle
	The type must provide a constructor that takes three points as arguments.
Triangulation_Traits_3::Tetrahedron
	The type must provide a constructor that takes four points as arguments.

Creation

Only a default constructor is required.

Triangulation_Traits_3 traits;
	A default constructor.

Predicates

To be used by Delaunay_triangulation_3<Traits, Tds>

In addition to the requirements described before, the geometric traits class of a Delaunay triangulation must fulfill the following requirements:

Predicates

Oriented_side	traits.side_of_oriented_sphere ( Point p, Point q, Point r, Point s, Point test)
		Determines on which side of the oriented sphere circumscribing p, q, r, s the point test lies.
Oriented_side	traits.side_of_oriented_circle ( Point p, Point q, Point r, Point test)
		Determines on which side of the oriented circle circumscribing p, q, r the point test lies. Precondition: p, q, r, test are coplanar and p, q, r are not collinear.

To be used by Delaunay_hierarchic_triangulation_3<Traits, Tds>

Not yet implemented

To be used by Regular_triangulation_3<Traits, Tds>

To be used as the traits class for the regular triangulation provided by CGAL, a class must satisfy additional requirements.

We use here the same notation as in Section .

Oriented_side	traits.power_test ( Weighted_point p, Weighted_point q, Weighted_point r, Weighted_point s, Weighted_point t)
		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	traits.power_test ( Weighted_point p, Weighted_point q, Weighted_point r, Weighted_point t)
		Analogous definition as 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	traits.power_test ( Weighted_point p, Weighted_point q, Weighted_point t)
		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 the 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.