#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 105.1 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 d2)
	computes the half value of the authalic weight. More...

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`, given a traits class `traits` with geometric objects, predicates, and constructions.

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`, given a traits class `traits` with geometric objects, predicates, and constructions.

template<typename K >
K::FT	CGAL::Weights::authalic_weight (const CGAL::Point_2< K > &p0, const CGAL::Point_2< K > &p1, const CGAL::Point_2< K > &p2, const CGAL::Point_2< K > &q)
	computes the authalic weight in 2D at `q` using the points `p0`, `p1`, and `p2` which are parameterized by a `Kernel` K.

template<typename K >
K::FT	CGAL::Weights::authalic_weight (const CGAL::Point_3< K > &p0, const CGAL::Point_3< K > &p1, const CGAL::Point_3< K > &p2, const CGAL::Point_3< K > &q)
	computes the authalic weight in 3D at `q` using the points `p0`, `p1`, and `p2` which are parameterized by a `Kernel` K.

Function Documentation

◆ half_authalic_weight()

template<typename FT >

FT CGAL::Weights::half_authalic_weight	(	const FT	cot,
		const FT	d2
	)

#include <CGAL/Weights/authalic_weights.h>

computes the half value of the authalic weight.

This function constructs the half of the authalic weight using the precomputed cotangent and squared distance values. The returned value is \(\frac{2\textbf{cot}}{\textbf{d2}}\).

Template Parameters

FT	a model of `FieldNumberType`

Parameters

cot	the cotangent value
d2	the squared distance value

Precondition: d2 != 0

See also: authalic_weight()

Functions

Function Documentation

◆ half_authalic_weight()