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\newcommand{\E}{\mathrm{E}} \newcommand{\A}{\mathrm{A}} \newcommand{\R}{\mathrm{R}} \newcommand{\N}{\mathrm{N}} \newcommand{\Q}{\mathrm{Q}} \newcommand{\Z}{\mathrm{Z}} \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }
CGAL 4.12 - 2D and 3D Linear Geometry Kernel
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Kernel::ComputePowerProduct_2 Concept Reference

Definition

Operations

A model of this concept must provide:

Kernel::FT operator() (const Kernel::Weighted_point_2 &pw, const Kernel::Weighted_point_2 &qw) const
 returns the power product of pw and qw. More...
 

Member Function Documentation

◆ operator()()

Kernel::FT Kernel::ComputePowerProduct_2::operator() ( const Kernel::Weighted_point_2 pw,
const Kernel::Weighted_point_2 qw 
) const

returns the power product of pw and qw.

Let {p}^{(w)} = (p,w_p), p\in\mathbb{R}^2, w_p\in\mathbb{R} and {q}^{(w)}=(q,w_q), q\in\mathbb{R}^2, 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)}) = {\|{p-q}\|^2-w_p-w_q}

where \|{p-q}\| 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.

Three weighted points have, in 2D, a unique common orthogonal weighted point 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.