Tangent Weight

#include <CGAL/Weights/tangent_weights.h>

This weight is computed as \(w = 2 \frac{t_1 + t_2}{d}\), where \(t_1 = \frac{2 A_0}{d d_0 + D_0}\) and \(t_2 = \frac{2 A_2}{d d_2 + D_2}\) with notations shown in the figure below and dot products

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

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 tangent weight.

Alternative Formulations

• This weight is equal to the Mean Value 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 && d != 0

Functions
template<typename FT >
FT	CGAL::Weights::half_tangent_weight (const FT tan05, const FT d)
	computes the half value of the tangent weight. More...
template<typename FT >
FT	CGAL::Weights::tangent_half_angle (const FT d, const FT l, const FT A, const FT D)
	computes the tangent of the half angle. More...
template<typename FT >
FT	CGAL::Weights::half_tangent_weight (const FT d, const FT l, const FT A, const FT D)
	computes the half value of the tangent weight. More...
template<typename GeomTraits >
GeomTraits::FT	CGAL::Weights::tangent_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 tangent weight in 2D at q using the points p0, p1, and p2 More...
template<typename Kernel >
Kernel::FT	CGAL::Weights::tangent_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 tangent weight in 2D at q using the points p0, p1, and p2 More...
template<typename GeomTraits >
GeomTraits::FT	CGAL::Weights::tangent_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 tangent weight in 3D at q using the points p0, p1, and p2 More...
template<typename Kernel >
Kernel::FT	CGAL::Weights::tangent_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 tangent weight in 3D at q using the points p0, p1, and p2 More...

Function Documentation

half_tangent_weight() [1/2]

template<typename FT >

FT CGAL::Weights::half_tangent_weight	(	const FT	tan05,
		const FT	d
	)

#include <CGAL/Weights/tangent_weights.h>

computes the half value of the tangent weight.

This function constructs the half of the tangent weight using the precomputed half angle tangent and distance values. The returned value is \(\frac{2\textbf{tan05}}{\textbf{d}}\).

Template Parameters

FT	a model of FieldNumberType

Parameters

tan05	the half angle tangent value
d	the distance value

Precondition

d != 0

See also

tangent_half_angle()

tangent_weight()

half_tangent_weight() [2/2]

template<typename FT >

FT CGAL::Weights::half_tangent_weight	(	const FT	d,
		const FT	l,
		const FT	A,
		const FT	D
	)

#include <CGAL/Weights/tangent_weights.h>

computes the half value of the tangent weight.

This function constructs the half of the tangent weight using the precomputed distance, area, and dot product values. The returned value is \(\frac{2\textbf{t}}{\textbf{d}}\) where \(\textbf{t} = \frac{2\textbf{A}}{\textbf{d}\textbf{l} + \textbf{D}}\).

Template Parameters

FT	a model of FieldNumberType

Parameters

d	the distance value
l	the distance value
A	the area value
D	the dot product value

Precondition

(d * l + D) != 0 && d != 0

See also

tangent_weight()

tangent_half_angle()

template<typename FT >

FT CGAL::Weights::tangent_half_angle	(	const FT	d,
		const FT	l,
		const FT	A,
		const FT	D
	)

#include <CGAL/Weights/tangent_weights.h>

computes the tangent of the half angle.

This function computes the tangent of the half angle using the precomputed distance, area, and dot product values. The returned value is \(\frac{2\textbf{A}}{\textbf{d}\textbf{l} + \textbf{D}}\).

Template Parameters

FT	a model of FieldNumberType

Parameters

d	the distance value
l	the distance value
A	the area value
D	the dot product value

Precondition

(d * l + D) != 0

See also

half_tangent_weight()

tangent_weight() [1/4]

template<typename GeomTraits >

GeomTraits::FT CGAL::Weights::tangent_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/tangent_weights.h>

computes the tangent weight in 2D at q using the points p0, p1, and p2

Template Parameters

GeomTraits

a model of AnalyticWeightTraits_2

Examples:

Weights/weights.cpp.

tangent_weight() [2/4]

template<typename Kernel >

Kernel::FT CGAL::Weights::tangent_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/tangent_weights.h>

computes the tangent weight in 2D at q using the points p0, p1, and p2

Template Parameters

Kernel

a model of Kernel

tangent_weight() [3/4]

template<typename GeomTraits >

GeomTraits::FT CGAL::Weights::tangent_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/tangent_weights.h>

computes the tangent weight in 3D at q using the points p0, p1, and p2

Template Parameters

GeomTraits

a model of AnalyticWeightTraits_3

tangent_weight() [4/4]

template<typename Kernel >

Kernel::FT CGAL::Weights::tangent_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/tangent_weights.h>

computes the tangent weight in 3D at q using the points p0, p1, and p2

Template Parameters

Kernel

a model of Kernel