CGAL 5.0  2D and 3D Linear Geometry Kernel

AdaptableFunctor
(with two arguments)Operations  
Kernel::FT  operator() (const Kernel::Weighted_point_3 &pw, const Kernel::Weighted_point_3 &qw) const 
returns the power product of pw and qw . More...  
Kernel::FT Kernel::ComputePowerProduct_3::operator()  (  const Kernel::Weighted_point_3 &  pw, 
const Kernel::Weighted_point_3 &  qw  
)  const 
returns the power product of pw
and qw
.
Let \( {p}^{(w)}=(p,w_p), p\in\mathbb{R}^3, w_p\in\mathbb{R}\) and \( {q}^{(w)}=(q,w_q), q\in\mathbb{R}^3, w_q\in\mathbb{R}\) be two weighted points.
The power product, also called power distance between \( {p}^{(w)}\) and \( {q}^{(w)}\) is defined as
\[ \Pi({p}^{(w)},{q}^{(w)}) = {\{pq}\^2w_pw_q} \]
where \( \{pq}\\) is the Euclidean distance between \( p\) and \( q\).
The weighted points \( {p}^{(w)}\) and \( {q}^{(w)}\) are said to be orthogonal iff \( \Pi{({p}^{(w)},{q}^{(w)})} = 0\).
Four weighted points have, in 3D, a unique common orthogonal weighted point called the power sphere. The weighted point orthogonal to three weighted points in the plane defined by these three points is called the power circle. The power segment will denote the weighted point orthogonal to two weighted points on the line defined by these two points.