CGAL 5.6.3 - Weights

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Functions
Mean Value Weight
Weight Interface Reference » Analytic Weights

##

    #include <CGAL/Weights/mean_value_weights.h>

This weight is computed as \(w = \pm 2 \sqrt{\frac{2 (d_0 d_2 - D)}{(d d_0 + D_0)(d d_2 + D_2)}}\) with notations shown in the figure below and dot products

\(D_0 = (p_0 - q) \cdot (p_1 - q)\), \(D_2 = (p_1 - q) \cdot (p_2 - q)\), and \(D = (p_0 - q) \cdot (p_2 - q)\).

The \(\pm\) sign is a sign of the weight that depends on the configuration.

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

This weight supports only planar configurations (see more in section about Coplanarity) while alternative formulations are explained in Implementation.

Figure 106.1 Notation used for the mean value weight.

Alternative Formulations

- This weight is equal to the Tangent 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 * d1 + D1) != 0 && (d * d2 + D2) != 0

## Functions
template<typename GeomTraits >
GeomTraits::FT | CGAL::Weights::mean_value_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 mean value weight in 2D at `q` using the points `p0`, `p1`, and `p2`.
template<typename Kernel >
Kernel::FT | CGAL::Weights::mean_value_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 mean value weight in 2D at `q` using the points `p0`, `p1`, and `p2`.

## Function Documentation

## ◆ mean_value_weight() [1/2]

template<typename Kernel >

Kernel::FT CGAL::Weights::mean_value_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/mean_value_weights.h>`

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

Template Parameters

  Kernel | a model of `Kernel`

## ◆ mean_value_weight() [2/2]

template<typename GeomTraits >

GeomTraits::FT CGAL::Weights::mean_value_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/mean_value_weights.h>`

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

Template Parameters

  GeomTraits | a model of `AnalyticWeightTraits_2`