Three Point Family Weight

#include<CGAL/Weights/three_point_family_weights.h>

This weight is computed as \(w = \frac{d_2^a A_0 - d^a B + d_0^a A_2}{A_0 A_2}\) with notations shown in the figure below and \(a\) any real number being the power parameter.

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 three point family weight.

Specializations

• For \(a = 0\), this weight is equal to the Wachspress Weight and Authalic Weight.
• For \(a = 1\), this weight is equal to the Mean Value Weight and Tangent Weight.
• For \(a = 2\), this weight is equal to the Discrete Harmonic Weight and Cotangent 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::three_point_family_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 typename GeomTraits::FT a, const GeomTraits &traits)
	computes the three-point family weight in 2D at q using the points p0, p1 and p2, and the power parameter a.
template<typename Kernel >
Kernel::FT	CGAL::Weights::three_point_family_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, const typename Kernel::FT a={1})
	computes the three-point family weight in 2D at q using the points p0, p1 and p2, and the power parameter a.

Function Documentation

three_point_family_weight() [1/2]

template<typename Kernel >

Kernel::FT CGAL::Weights::three_point_family_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,
		const typename Kernel::FT	a = {1}
	)

#include <CGAL/Weights/three_point_family_weights.h>

computes the three-point family weight in 2D at q using the points p0, p1 and p2, and the power parameter a.

Template Parameters

Kernel

a model of Kernel

three_point_family_weight() [2/2]

template<typename GeomTraits >

GeomTraits::FT CGAL::Weights::three_point_family_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 typename GeomTraits::FT	a,
		const GeomTraits &	traits
	)

#include <CGAL/Weights/three_point_family_weights.h>

computes the three-point family weight in 2D at q using the points p0, p1 and p2, and the power parameter a.

Template Parameters

GeomTraits

a model of AnalyticWeightTraits_2

Examples

Weights/convergence.cpp.