CGAL 5.4 - 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 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.

## ◆ 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
authalic_weight()