CGAL 5.4.4 - Weights
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 104.1 Notation used for the authalic weight.

Alternative Formulations

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. 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. More...

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. More...

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. More...

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. More...

## ◆ authalic_weight() [1/4]

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() [2/4]

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() [3/4]

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

## ◆ authalic_weight() [4/4]

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

## ◆ 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
authalic_weight()