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.

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.

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.

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.

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.

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.

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 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_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.