Authalic Weight

#include <CGAL/Weights/authalic_weights.h>

This weight is computed as \(w = 2 \frac{\cot\beta + \cot\gamma}{d^2}\) with notations shown in the figure below.

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

Alternative formulations are explained in Implementation.

Figure 106.1 Notation used for the authalic weight.

Alternative Formulations

• This weight is equal to the Wachspress 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 != 0

Functions
template<typename FT >
FT	CGAL::Weights::half_authalic_weight (const FT cot, const FT sq_d)
	computes the half value of the authalic weight.
template<typename GeomTraits >
GeomTraits::FT	CGAL::Weights::authalic_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 authalic weight in 2D at q using the points p0, p1, and p2.
template<typename Kernel >
Kernel::FT	CGAL::Weights::authalic_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 authalic weight in 2D at q using the points p0, p1, and p2.
template<typename GeomTraits >
GeomTraits::FT	CGAL::Weights::authalic_weight (const typename GeomTraits::Point_3 &p0, const typename GeomTraits::Point_3 &p1, const typename GeomTraits::Point_3 &p2, const typename GeomTraits::Point_3 &q, const GeomTraits &traits)
	computes the authalic weight in 3D at q using the points p0, p1, and p2.
template<typename Kernel >
Kernel::FT	CGAL::Weights::authalic_weight (const CGAL::Point_3< Kernel > &p0, const CGAL::Point_3< Kernel > &p1, const CGAL::Point_3< Kernel > &p2, const CGAL::Point_3< Kernel > &q)
	computes the authalic weight in 3D at q using the points p0, p1, and p2.

Function Documentation

authalic_weight() [1/4]

template<typename Kernel >

Kernel::FT CGAL::Weights::authalic_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/authalic_weights.h>

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

Template Parameters

Kernel

a model of Kernel

authalic_weight() [2/4]

template<typename Kernel >

Kernel::FT CGAL::Weights::authalic_weight	(	const CGAL::Point_3< Kernel > &	p0,
		const CGAL::Point_3< Kernel > &	p1,
		const CGAL::Point_3< Kernel > &	p2,
		const CGAL::Point_3< Kernel > &	q
	)

#include <CGAL/Weights/authalic_weights.h>

computes the authalic weight in 3D at q using the points p0, p1, and p2.

Template Parameters

Kernel

a model of Kernel

authalic_weight() [3/4]

template<typename GeomTraits >

GeomTraits::FT CGAL::Weights::authalic_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/authalic_weights.h>

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

Template Parameters

GeomTraits

a model of AnalyticWeightTraits_2

authalic_weight() [4/4]

template<typename GeomTraits >

GeomTraits::FT CGAL::Weights::authalic_weight	(	const typename GeomTraits::Point_3 &	p0,
		const typename GeomTraits::Point_3 &	p1,
		const typename GeomTraits::Point_3 &	p2,
		const typename GeomTraits::Point_3 &	q,
		const GeomTraits &	traits
	)

#include <CGAL/Weights/authalic_weights.h>

computes the authalic weight in 3D at q using the points p0, p1, and p2.

Template Parameters

GeomTraits

a model of AnalyticWeightTraits_3

half_authalic_weight()

template<typename FT >

FT CGAL::Weights::half_authalic_weight	(	const FT	cot,
		const FT	sq_d
	)

#include <CGAL/Weights/authalic_weights.h>

computes the half value of the authalic weight.

This function computes the half of the authalic weight using the precomputed cotangent and squared distance values. The returned value is \(\frac{2\textbf{cot}}{\textbf{sq_d}}\).

Template Parameters

FT	a model of FieldNumberType

Parameters

cot	the cotangent value
sq_d	the squared distance value

Precondition

sq_d != 0

See also

authalic_weight()