CGAL 5.6 - Weights
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#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

Template Parameters
GeomTraitsa 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
Kernela 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
Kernela 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
GeomTraitsa 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
GeomTraitsa 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
FTa model of FieldNumberType
Parameters
cotthe cotangent value
sq_dthe squared distance value
Precondition
sq_d != 0
See also
authalic_weight()