Mean Value Weight

#include<CGAL/Weights/mean_value_weights.h>

This weight is computed as \(w = \pm 2 \sqrt{\frac{2 (d_0 d_2 - D)}{(d d_0 + D_0)(d d_2 + D_2)}}\) with notations shown in the figure below and dot products

\(D_0 = (p_0 - q) \cdot (p_1 - q)\), \(D_2 = (p_1 - q) \cdot (p_2 - q)\), and \(D = (p_0 - q) \cdot (p_2 - q)\).

The \(\pm\) sign is a sign of the weight that depends on the configuration.

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 110.1 Notation used for the mean value weight.

Alternative Formulations

• This weight is equal to the Tangent 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

(d * d1 + D1) != 0 && (d * d2 + D2) != 0

Functions
template<typename GeomTraits >
GeomTraits::FT	CGAL::Weights::mean_value_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 mean value weight in 2D at q using the points p0, p1, and p2.
template<typename Kernel >
Kernel::FT	CGAL::Weights::mean_value_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 mean value weight in 2D at q using the points p0, p1, and p2.

Function Documentation

mean_value_weight() [1/2]

template<typename Kernel >

Kernel::FT CGAL::Weights::mean_value_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/mean_value_weights.h>

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

Template Parameters

Kernel

a model of Kernel

mean_value_weight() [2/2]

template<typename GeomTraits >

GeomTraits::FT CGAL::Weights::mean_value_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/mean_value_weights.h>

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

Template Parameters

GeomTraits

a model of AnalyticWeightTraits_2