Discrete Harmonic Weight

#include<CGAL/Weights/discrete_harmonic_weights.h>

This weight is computed as \(w = \frac{d_2^2 A_0 - d^2 B + d_0^2 A_2}{A_0 A_2}\) with notations shown in the figure below.

Here, q is a query point and the points p0, p1, and p2 are its neighbors.

This weight supports only planar configurations (see more in section about Coplanarity) while alternative formulations are explained in Implementation.

Figure 109.1 Notation used for the discrete harmonic weight.

Alternative Formulations

• This weight is equal to the Cotangent Weight.
• This weight is a special case of the Three Point Family Weight.

Template Parameters

GeomTraits

a model of AnalyticWeightTraits_2 for 2D points; a model of AnalyticWeightTraits_3 for 3D points

Precondition

A1 != 0 && A2 != 0

Functions
template<typename GeomTraits >
GeomTraits::FT	CGAL::Weights::discrete_harmonic_weight (const typename GeomTraits::Point_2 &p0, const typename GeomTraits::Point_2 &p1, const typename GeomTraits::Point_2 &p2, const typename GeomTraits::Point_2 &q, const GeomTraits &traits)
	computes the discrete harmonic weight in 2D at q using the points p0, p1, and p2.
template<typename Kernel >
Kernel::FT	CGAL::Weights::discrete_harmonic_weight (const CGAL::Point_2< Kernel > &p0, const CGAL::Point_2< Kernel > &p1, const CGAL::Point_2< Kernel > &p2, const CGAL::Point_2< Kernel > &q)
	computes the discrete harmonic weight in 2D at q using the points p0, p1, and p2.

Function Documentation

discrete_harmonic_weight() [1/2]

template<typename Kernel >

Kernel::FT CGAL::Weights::discrete_harmonic_weight	(	const CGAL::Point_2< Kernel > &	p0,
		const CGAL::Point_2< Kernel > &	p1,
		const CGAL::Point_2< Kernel > &	p2,
		const CGAL::Point_2< Kernel > &	q
	)

#include <CGAL/Weights/discrete_harmonic_weights.h>

computes the discrete harmonic weight in 2D at q using the points p0, p1, and p2.

Template Parameters

Kernel

a model of Kernel

discrete_harmonic_weight() [2/2]

template<typename GeomTraits >

GeomTraits::FT CGAL::Weights::discrete_harmonic_weight	(	const typename GeomTraits::Point_2 &	p0,
		const typename GeomTraits::Point_2 &	p1,
		const typename GeomTraits::Point_2 &	p2,
		const typename GeomTraits::Point_2 &	q,
		const GeomTraits &	traits
	)

#include <CGAL/Weights/discrete_harmonic_weights.h>

computes the discrete harmonic weight in 2D at q using the points p0, p1, and p2.

Template Parameters

GeomTraits

a model of AnalyticWeightTraits_2